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Page 1: Light Scattering Reviews 3: Light Scattering and Reflection
Page 2: Light Scattering Reviews 3: Light Scattering and Reflection

Light Scattering Reviews 3Light Scattering and Reflection

Page 3: Light Scattering Reviews 3: Light Scattering and Reflection

Published in association with

PPraxisraxis PPublishingublishingChichester, UK

Alexander A. Kokhanovsky (Editor)

Light ScatteringReviews 3Light Scattering and Reflection

Page 4: Light Scattering Reviews 3: Light Scattering and Reflection

EditorDr Alexander A. KokhanovskyInstitute of Environmental PhysicsUniversity of BremenBremenGermany

SPRINGER–PRAXIS BOOKS IN ENVIRONMENTAL SCIENCES (LIGHT SCATTERING SUB-SERIES)SUBJECT ADVISORY EDITOR: John Mason B.Sc., M.Sc., Ph.D.EDITORIAL ADVISORY BOARD MEMBER: Dr Alexander A. Kokhanovsky, Ph.D. Institute of EnvironmentalPhysics, University of Bremen, Bremen, Germany

ISBN 978-3-540-48305- Springer Berlin Heidelberg New York

Springer is part of Springer-Science + Business Media (springer.com)

Bibliographic information published by Die Deutsche Bibliothek

Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data are available from the Internet at http://dnb.ddb.de

Library of Congress Control Number: 2007941067

Apart from any fair dealing for the purposes of research or private study, or criticismor review, as permitted under the Copyright, Designs and Patents Act 1988, thispublication may only be reproduced, stored or transmitted, in any form or by anymeans, with the prior permission in writing of the publishers, or in the case ofreprographic reproduction in accordance with the terms of licences issued by theCopyright Licensing Agency. Enquiries concerning reproduction outside those termsshould be sent to the publishers.

# Praxis Publishing Ltd, Chichester, UK, 2008Printed in Germany

The use of general descriptive names, registered names, trademarks, etc. in thispublication does not imply, even in the absence of a specific statement, that suchnames are exempt from the relevant protective laws and regulations and therefore freefor general use.

Cover design: Jim WilkieProject copy editor: Mike ShardlowAuthor-generated LaTex, processed by EDV-Beratung, Germany

Printed on acid-free paper

2

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Contents

List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI

Notes on the contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XIII

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XXI

Part I Single Light Scattering

1 Observational quantification of the optical properties of cirruscloudTimothy J. Garrett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Measurement of the asymmetry parameter in cirrus . . . . . . . . . . . . . . . 5

1.2.1 Indirect estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Nephelometer measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Reconciling discrepancies between theory and observations . . 11

1.3 Extinction coefficient and effective radius . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Indirect measurement of effective radius . . . . . . . . . . . . . . . . . . 151.3.2 Direct measurement of effective radius . . . . . . . . . . . . . . . . . . . . 161.3.3 Optical evaluation of ice crystal effective radius using halos . . 19

1.4 Summary of outstanding problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Statistical interpretation of light anomalous diffraction bysmall particles and its applications in bio-agent detection andmonitoringMin Xu, A. Katz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Review of recent developments in ADT . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Light anomalous diffraction using geometrical path statisticsof rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 Ray distributions for various shapes . . . . . . . . . . . . . . . . . . . . . . 322.2.3 Gaussian ray approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.4 Performance of Gaussian ray approximation and difference

in optical efficiencies between cylinders and spheroids . . . . . . . 472.2.5 Implications on particle sizing with light scattering techniques 50

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2.3 Applications of light scattering to bacteria monitoring and detection . 522.3.1 Angular scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.2 Bacteria size determined by transmission measurements . . . . . 582.3.3 In vivo monitoring of biological processes in bacteria . . . . . . . 59

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Light scattering by particles with boundary symmetriesMichael Kahnert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2 Symmetries in linear boundary-value problems . . . . . . . . . . . . . . . . . . . . 71

3.2.1 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.2.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.2.3 Boundary symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3 Symmetries in electromagnetic and acoustic scattering problems . . . . 783.3.1 Self-consistent Green’s function formalism . . . . . . . . . . . . . . . . . 783.3.2 Symmetry relations of GΓ+ , G∂Γ+ , and W∂Γ+ . . . . . . . . . . . . . . 823.3.3 Symmetry relations in matrix form . . . . . . . . . . . . . . . . . . . . . . . 853.3.4 Unitary, reducible representations of point-groups . . . . . . . . . . 883.3.5 Explicit symmetry relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.6 Irreducible representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4 Scattering by particles on or near a plane surfaceAdrian Doicu, Roman Schuh and Thomas Wriedt . . . . . . . . . . . . . . . . . . . . . . 1094.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.2 Single particle on or near a plane surface . . . . . . . . . . . . . . . . . . . . . . . . . 1104.3 Single particle on or near a plane surface coated with a film . . . . . . . . 1194.4 System of particles on or near a plane surface . . . . . . . . . . . . . . . . . . . . 1214.5 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Part II Radiative Transfer and Inverse Problems

5 Impact of single- and multi-layered cloudiness on ozonevertical column retrievals using nadir observations ofbackscattered solar radiationV. V. Rozanov and A. A. Kokhanovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.3 Atmospheric and cloud models used for forward simulations . . . . . . . . 1375.4 Forward simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.4.1 Reflection function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.4.2 Weighting function and differential absorption . . . . . . . . . . . . . 142

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5.4.3 Impact of cloud parameters on the integral absorption . . . . . . 1465.4.4 Linear approximation for the reflection function with

respect to the cloud parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.4.5 Scaling approximation and weighting function for ozone

vertical columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.5 Inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.5.1 Retrieval of cloud parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.5.2 Total ozone column retrieval algorithm . . . . . . . . . . . . . . . . . . . 155

5.6 Results of numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.6.1 Single cloud layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.6.2 Two-layered cloud systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.6.3 Three-layered cloud systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

A.1 Gaseous absorber number density WF . . . . . . . . . . . . . . . . . . . . 179A.2 Cloud optical thickness WF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181A.3 Cloud geometrical parameters WFs . . . . . . . . . . . . . . . . . . . . . . 181A.4 LER altitude (surface elevation) WF . . . . . . . . . . . . . . . . . . . . . 182

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6 Remote sensing of clouds using linearly and circularlypolarized laser beams: techniques to compute signal polarizationL. I. Chaikovskaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.2 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.2.1 Matrix describing the light field produced by a normallyincident beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.2.2 Matrices of propagation and near-backward scattering . . . . . . 1996.2.3 Simplified transfer equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

6.3 Polarized light transmission by a cloud . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.3.1 Generalization of the multicomponent technique . . . . . . . . . . . 2126.3.2 Transmission of an infinitely wide beam through water

cloud: computation and discussion . . . . . . . . . . . . . . . . . . . . . . . 2156.4 Polarization of the pulsed lidar return from a cloud . . . . . . . . . . . . . . . 218

6.4.1 Semi-analytical technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.4.2 Backscattering of linearly and circularly polarized pulses

from a water cloud: computation and discussion . . . . . . . . . . . 2206.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7 LIDORT and VLIDORT: Linearized pseudo-spherical scalarand vector discrete ordinate radiative transfer models for use inremote sensing retrieval problemsRobert Spurr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.2 Description of VLIDORT and LIDORT . . . . . . . . . . . . . . . . . . . . . . . . . . 232

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7.2.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2327.2.2 Homogeneous RTE solutions and their linearization . . . . . . . . 2387.2.3 Solar sources: particular integrals and linearization . . . . . . . . . 2427.2.4 Thermal sources: particular integrals and linearization . . . . . . 2467.2.5 Boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2477.2.6 Post processing: source function integration . . . . . . . . . . . . . . . 2487.2.7 Spherical and single-scatter corrections . . . . . . . . . . . . . . . . . . . 2517.2.8 Surface reflectance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

7.3 Performance and benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2617.3.1 Performance considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2617.3.2 VLIDORT validation and benchmarking . . . . . . . . . . . . . . . . . . 266

7.4 Preparation of inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2677.4.1 Example: specification of atmospheric IOP inputs . . . . . . . . . . 2677.4.2 Surface and other atmospheric inputs . . . . . . . . . . . . . . . . . . . . . 269

7.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Part III Bi-directional Reflectance of Light from Natural andArtificial Surfaces

8 Bi-directional reflectance measurements of closely packednatural and prepared particulate surfacesHao Zhang and Kenneth J. Voss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2798.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2798.2 Definitions of bi-directional reflectance and related quantities . . . . . . . 2808.3 BRDF models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

8.3.1 Hapke’s isotropic multiple-scattering approximation (HIMSA)2828.3.2 Hapke’s anisotropic multiple-scattering approximation

(HAMSA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2838.3.3 Lumme–Bowell’s (LB) model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2848.3.4 Mishchenko et al.’s BRF algorithm (MBRF) . . . . . . . . . . . . . . 2848.3.5 The DISORT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2858.3.6 Some remarks on the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

8.4 BRDF instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2868.4.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2868.4.2 An in situ BRDF-meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2878.4.3 A simple goniometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2898.4.4 An example of the calibration measurements . . . . . . . . . . . . . . 290

8.5 Controlled BRDF measurements on prepared packed surfaces andcomparisons with models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2928.5.1 Samples and single-scattering quantities . . . . . . . . . . . . . . . . . . 2928.5.2 Some parameters of packed surfaces and measurement results 2928.5.3 Some discussions on controlled BRDF measurements . . . . . . . 297

8.6 In situ BRDF measurements on benthic sediment floors . . . . . . . . . . . . 3018.6.1 Typical features of benthic sediment BRDF . . . . . . . . . . . . . . . 301

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8.6.2 A simple model for sediment BRDF . . . . . . . . . . . . . . . . . . . . . . 3088.7 Effects of translucent grains and pore liquid complex refractive

index on particulate BRDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3128.7.1 Sample descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3148.7.2 Effects of translucent particle concentrations on wetting . . . . 3148.7.3 Effects of the wetting liquid absorption coefficient . . . . . . . . . . 318

8.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

9 Light scattering from particulate surfaces in geometricaloptics approximationYevgen Grynko and Yuriy G. Shkuratov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3299.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

9.1.1 Practical tasks in optical remote sensing . . . . . . . . . . . . . . . . . . 3299.1.2 Principle and history of the ray tracing method . . . . . . . . . . . . 3339.1.3 Problems of analytical accounting for multiple scattering in

particulate media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3349.1.4 Range of applicability of ray tracing . . . . . . . . . . . . . . . . . . . . . . 336

9.2 Computer modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3379.2.1 Particulate medium generation and description of irregular

shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3379.2.2 Ray tracing algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

9.3 The shadow-hiding effect and multiple scattering in systems ofopaque particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3479.3.1 Ray tracing modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3499.3.2 Results of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

9.4 Single scattering component. Transparent and semitransparentparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3549.4.1 Faceted spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3559.4.2 Binary spheres and faceted ellipsoids . . . . . . . . . . . . . . . . . . . . . 3579.4.3 Perfect and ‘spoiled’ cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3599.4.4 RGF particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

9.5 Incoherent multiple scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3639.5.1 Photometric and polarimetric phase curves . . . . . . . . . . . . . . . . 3639.5.2 Spectrophotometry of particulate surfaces . . . . . . . . . . . . . . . . . 369

9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

10 Laboratory measurements of reflected light intensity andpolarization for selected particulate surfacesYuriy G. Shkuratov, Andrey A. Ovcharenko, Vladimir A. Psarev andSergey Y. Bondarenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38310.2 Laboratory instruments and samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

10.2.1 The wide-phase-angle photometer/polarimeter . . . . . . . . . . . . . 38410.2.2 The small-phase-angle photometer/polarimeter . . . . . . . . . . . . 385

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10.2.3 The laser super-small-phase-angle photometer/polarimeter . . 38710.2.4 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

10.3 Results of measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39110.3.1 Albedo and particle size effects . . . . . . . . . . . . . . . . . . . . . . . . . . 39210.3.2 The contribution of single light scattering . . . . . . . . . . . . . . . . . 39310.3.3 Opposition spikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Page 11: Light Scattering Reviews 3: Light Scattering and Reflection

List of Contributors

Sergey Y. BondarenkoAstronomical Institute of KharkovV. N. Karazin National University35 Sumskaya Str., Kharkov, [email protected]

Ludmila I. ChaikovskayaB. I. Stepanov Institute of PhysicsNational Academy of Sciences of BelarusNezavisimosti Ave. 68, Minsk, [email protected]

Adrian DoicuRemote Sensing Technology InstituteDLR, OberpfaffenhofenD-82234 [email protected]

Timothy GarrettMeteorology DepartmentUniversity of Utah135 S 1460 E, Rm 819Salt Lake City UT [email protected]

Yevgen GrynkoAstronomical Institute of KharkovV. N. Karazin National University35 Sumskaya Str., Kharkov, [email protected]

Michael KahnertSwedish Meteorologicaland Hydrological InstituteFolkborgsvagen 1S-601 76 [email protected]

Alvin KatzInstitute for Ultrafast Spectroscopy andLaserDepartment of PhysicsCity College of New YorkNew York, NY 10031USAemail: [email protected]

Alexander A. KokhanovskyInstitute of Environmental PhysicsUniversity of BremenOtto Hahn Allee 1, D-28334, [email protected]

Andrey A. OvcharenkoAstronomical Institute of KharkovV. N. Karazin National University35 Sumskaya Str., Kharkov, [email protected]

Vladimir A. PsarevAstronomical Institute of KharkovV. N. Karazin National University35 Sumskaya Str., Kharkov, [email protected]

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XII List of Contributors

Vladimir V. RozanovInstitute of Environmental Physics,University of Bremen,Otto Hahn Allee 1, D-28334, [email protected]

Roman SchuhProcess EngineeringUniversity of BremenBadgasteiner Str.3, D-28359 [email protected]

Yuriy G. ShkuratovAstronomical Institute of KharkovV. N. Karazin National University35 Sumskaya Str., Kharkov, [email protected]

Robert SpurrRT SOLUTIONS Inc.9 Channing Street, Cambridge MA [email protected]

Kenneth J. VossPhysics DepartmentUniversity of Miami1320 Campo Sano DriveCoral Gables, FL [email protected]

Thomas WriedtStiftung Institut fur WerkstofftechnikBadgasteiner Str.3, D-28359 [email protected]

Min XuDepartment of PhysicsFairfield University1073 North Benson RoadFairfield, CT [email protected]

Hao ZhangCooperative Institute for Research in theAtmosphereColorado State UniversityandNOAA/NESDIS/STAR5200 Auth Road, Room102Camp Springs, MD [email protected]

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Notes on the contributors

Sergey Y. Bondarenko graduated from Kharkov V. N. Karazin National Universityin 2003. He is currently a research assistant in the Astronomical Institute of KharkovNational University. His research interest is studies of light scattering with laboratoryphotopolarimeters. He has published six papers concerning light scattering measure-ments.

Ludmila I. Chaikovskaya received her Ph.D. degree in physics and mathematicsfrom the B. I. Stepanov Institute of Physics of National Academy of Sciences of Be-larus (NASB) in 1985. Her Ph.D. work was devoted to the problem of polarized lighttransfer in scattering media including media with gyrotropy. Currently, she is a seniorresearcher at the Institute of Physics, NASB. Her research interests involve the de-velopment of approximate methods in the theory of polarized light transfer and theirapplications to actual tasks of passive and active sounding of multiply light scatter-ing media. For example, she derived a number of important theoretical results in thefield of polarized lidar sounding of clouds. She has published over eighty papers on theproblem of polarized light transfer and its applications.

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XIV Notes on the contributors

Adrian Doicu received his Ph.D. in 1996 from the University Politechnica of Buchareston the subject of phase-doppler anemometry. Currently, he is a researcher at the Ger-man Aerospace Center and works in the field of numerical methods in electromagneticscattering and inverse methods for atmospheric remote sensing. He has published twobooks on the application of null-field method in electromagnetic scattering and is theauthor and co-author of about fifty papers in peer-reviewed journals.

Tim Garrett received his Ph.D. in 2000 from the University of Washington on thesubject of Arctic cloud radiative properties, and is now an Assistant Professor in theDepartment of Meteorology at the University of Utah in Salt Lake City, USA. Cur-rent research is devoted to understanding interactions between aerosols, microphysics,radiation and dynamics in atmospheric clouds, using numerical and observational tech-niques. An important component of these studies has been the measurement of ice cloudoptical properties from aircraft in locations ranging from the Arctic to the tropics. DrGarrett has authored or co-authored more than thirty peer-reviewed papers, and hasserved on the editorial boards of the Journal of Atmospheric Science and AtmosphericChemistry and Physics.

Yevgen Grynko graduated from Kharkov National University, Ukraine, and receiveda diploma in physics and astronomy in 2000. He obtained his Ph.D. degree in astro-physics from the Georg-August University of Gottingen, Germany, in 2005. His Ph.D.

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Notes on the contributors XV

work, done at the Max Planck Institute for Solar System Research, was concentrated onthe study of light scattering by cometary dust by means of numerical simulations andthe analysis of the observational data. After defending his thesis he continued researchin the field of reflectance spectroscopy of regolith-like surfaces at MPS, providing the-oretical support for the SIR spectrometer team (ESA SMART-1 lunar mission). He isnow a research scientist at the Institute of Astronomy of Kharkov National University,Department of Remote Sensing of Planets. Yevgen Grynko’s main research interestsinclude theoretical simulations of light scattering in particulate media and scatteringby individual irregular particles, in particular, ray tracing modelling and applicationsof the modelling in the spectroscopy and photopolarimetry of the surfaces of SolarSystem bodies.

Michael Kahnert graduated from the Physics Department of the Free University ofBerlin in 1994 and received his Ph.D. in physics from the University of Alaska Fairbanksin 1998. He is now with the Department for Research and Development of the SwedishMeteorological and Hydrological Institute. His scientific interests include theoreticaland numerical methods in electromagnetic scattering, optical and radiative propertiesof aerosols and ice clouds, and data assimilation of remote sensing observations intoaerosol dynamic and chemical transport models.

Alvin Katz received his B.S. in physics from The City College of New York and hisPh.D. in Physics from The City University of New York. He is currently a member ofthe research staff at the Institute for Ultrafast Lasers and Spectroscopy at The CityCollege of New York. His research interests include investigating the spectroscopicproperties of bacteria and viruses and developing methods to detect bio-agents basedon light scattering and/or fluorescence. He is actively involved in investigating thenative fluorescence properties of tissues for the purpose of developing optical methodsto detect cancer and pre-cancer.

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XVI Notes on the contributors

Alexander A. Kokhanovsky graduated from the Physical Department of the Be-larusian State University, Minsk, Belarus, in 1983. He received his Ph.D. in opticalsciences from the Institute of Physics, National Academy of Sciences of Belarus, Minsk,Belarus, in 1991. His Ph.D. work was devoted to modelling the light scattering prop-erties of aerosol media and foams. Alexander Kokhanovsky is currently a memberthe SCIAMACHY/ENVISAT algorithm development team (Institute of Environmen-tal Physics, University of Bremen). His research interests are directed towards mod-elling light propagation and scattering in the terrestrial atmosphere. Dr Kokhanovskyis the author of the books Light Scattering Media Optics: Problems and Solutions(Chichester: Springer–Praxis, 1999, 2001, 2004), Polarization Optics of Random Me-dia (Berlin: Springer–Praxis, 2003), Cloud Optics (Berlin: Springer, 2006), and AerosolOptics (Springer–Praxis, 2008). He has published more than one hundred papers in thefield of environmental optics, radiative transfer, and light scattering. Dr Kokhanovskyis a member of the European and American Geophysical Unions and also he is a mem-ber of the Belarusian Physical Society.

Andrey Ovcharenko graduated from Kharkov State University in 1995. He receivedhis Ph.D. in optics and laser physics from Kharkov State University in 1999. He iscurrently a senior researcher at the Remote Sensing Department of the AstronomicalInstitute of Kharkov National University. His research interests are directed to experi-mental investigations of light scattering by surfaces with complicated structure at smalland extremely small phase angles. He has published more than fifty papers in the fieldof light scattering experiments.

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Notes on the contributors XVII

Vladimir A. Psarev graduated from Kharkov State University, Ukraine, in 1970. Hereceived his Ph.D. in physics and mathematics in 1982 from the Main AstronomicalObservatory of the Ukrainian National Academy of Sciences, Kyiv, Ukraine. He iscurrently a deputy director of the Astronomical Institute of Kharkov V. N. KarazinNational University. His research interests are directed to studies of light scattering,photometric and polarimetric research of the Moon, Mars, and small bodies of theSolar System. He has published about thirty papers in this field.

Vladimir V. Rozanov graduated from the University of St Petersburg, Russia, in1973. He received his Ph.D. degree in physics and mathematics from the University ofSt Petersburg, Russia, in 1977. From 1973 until 1991 he was a research scientist at theDepartment of Atmospheric Physics of the University of St Petersburg. In 1990–1991he worked at the Max-Planck Institute of Chemistry, Mainz, Germany. In July 1992he joined the Institute of Remote Sensing at the University of Bremen, Germany. Themain directions of his research are atmospheric radiative transfer and remote sensing ofatmospheric parameters (including aerosols, clouds, and trace gases) from space-bornespectrometers and radiometers. He is the author and co-author of about a hundredpapers in peer-reviewed journals.

Roman Schuh graduated from the Physics Department of the University of Old-enburg, Germany, 1998. He is now with the Department of Process Engineering ofthe University of Bremen. His current research is mainly focused on optical particlecharacterization and light scattering simulations.

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XVIII Notes on the contributors

Yuriy G. Shkuratov graduated from the Physics Department of Kharkov State Uni-versity in 1975. He received his Ph.D. in mathematical physics and his Doctor of Sci-ences degree in physics and mathematics from Moscow and Kharkov State Universitiesin 1980 and 1993, respectively. His works are devoted to photopolarimetric laboratorystudies of particulate surfaces, theoretical investigations of shadow-hiding effects forrandomly rough surfaces, and planetary physics. He is Professor and Director of theAstronomical Institute of Kharkov V. N. Karazin National University. He has publishedover five hundred papers.

Robert Spurr is director of RT Solutions, Inc., a newly formed company for con-sultation in radiative transfer and remote sensing applications. Dr Spurr received hisfirst degree from Cambridge University, England, in the 1970s, and was a professionalmeteorologist in the 1980s. He joined the remote sensing community in 1991 and has16 years’ experience with radiative transfer in the Earth’s atmosphere and ocean, andwith the retrieval of ozone and other atmospheric constituents. He obtained his Ph.D.on linearized radiative transfer modelling in 2001. He spent 10 years working at theHarvard-Smithsonian Center for Astrophysics before leaving in January 2005 to set upRT Solutions. Dr Spurr is the main author of the LIDORT family of discrete ordinateradiative transfer codes. He is a member of the American Geophysical Union and aFellow of the Royal Meteorological Society.

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Notes on the contributors XIX

Kenneth Voss is a Professor in the Physics Department at the University of Mi-ami. His specialty is experimental environmental optics, in particular, ocean and at-mospheric optics. He received his Ph.D. in physics at Texas A&M University (1984),where he built an instrument to measure the polarized light scattering in seawater. Hispost-doctoral experience was at Scripps Institution of Oceanography, where he workedwith Ros Austin at the Visibility Laboratory. Here he worked at developing instru-mentation to measure different aspects of the in-water light field. He has been at theUniversity of Miami since 1989. Since arriving at Miami he has been involved withremote sensing, through the SeaWiFS and MODIS projects, along with in-water opticsand instrumentation. In 2003 he was elected fellow of the Optical Society of America.

Thomas Wriedt has been the head of the Department of Particle Technology andParticle Characterization at the Institut fur Werkstofftechnik, Bremen, Germany, since1989. He studied Electrical Engineering at the University of Applied Science, Kiel, andat the University of Bremen and obtained his Dr.-Ing. degree on numerical design ofmicrowave antennas and components in 1986. From 1986 to 1989 he carried out researchon signal processing with Phase Doppler Anemometry at the University of Bremen.His current research is mainly focused on optical particle characterization and lightscattering theory.

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XX Notes on the contributors

Min Xu received his B.S. and M.S. in Physics from Fudan University, China, in 1992and 1995, respectively, and his Ph.D. in physics from The City University of New Yorkin 2001. Dr Xu is currently an Assistant Professor in the Department of Physics atFairfield University, Connecticut. His research interests include wave scattering andpropagation in random media, radiative transfer of polarized light, random processesand Monte Carlo methods, biomedical optics, and inverse problems in applied physicsand engineering. Dr. Xu’s recent work in biomedical optics has been on modellinglight scattering by cells and human tissues and developing optical spectroscopic andtomographic methods for cancer detection. He has published over thirty peer-reviewedpapers and is co-author of the book Random Processes in Physics and Finance (OxfordUniversity Press, 2006).

Hao Zhang took his graduate study at the University of Miami, where he receivedhis Ph.D. in physics, under the supervision of Kenneth Voss, in 2004. His graduateresearch was concentrated on the measurement of the bi-directional reflectance distri-bution function of closely packed particulate layers such as benthic sediment floors.After graduation he stayed on as a postdoctoral fellow until 2006. He then became apostdoctoral fellow affiliated with Colorado State University’s Cooperative Institutefor Research in the Atmosphere, working on ocean colour remote sensing at NationalOceanic and Atmospheric Administration in Camp Springs, Maryland. His researchinterests include Raman and infrared spectroscopy, BRDF of natural and artificialsurfaces, and satellite remote sensing.

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Preface

This volume of Light Scattering Reviews describes some recent advances in abroad area of light scattering media optics. It is composed of three parts. Thefirst part is concerned with single light scattering by small nonspherical par-ticles such as crystals in clouds or suspensions of various cells. Most naturalmedia are characterized by a variety of shapes and often a particular selectionof particle shape is not representative of the ensemble of shapes encounteredduring measurements. This being the case, theoretical calculations are only oflimited value and comprehensive experimental studies must be performed to elu-cidate the question of optical response for a collection of particles having diverseshapes. The volume opens with the paper of Timothy Garrett, which describesa number of recent in situ airborne experiments related to the quantificationof optical properties of cirrus clouds. Min Xu and Alvin Katz discuss a novelstatistical approach to deal with the light extinction and absorption propertiesof small nonspherical particles having a refractive index close to that of the hostmedium. In this case, van de Hulst’s anomalous diffraction theory can be used.The new approach is applied to studies of bio-agent detection and monitoring.Michael Kahnert presents a comprehensive theoretical treatment of symmetriesin linear boundary value problems. The use of symmetries simplifies electromag-netic scattering problems to a great extent. In the final section of the first partAdrian Doicu, Roman Schuh and Thomas Wriedt address the treatment of lightscattering by particles situated on or near a plane surface. This problem is ofgreat importance for a number of technological applications including the de-velopment of surface scanners for wafer inspection, laser cleaning, and scanningnear-field optical spectroscopy.

The second part of the book is devoted to selected topics in remote sensingand radiative transfer. Vladimir Rozanov and Alexander Kokhanovsky describea novel approach to the simultaneous retrieval of total ozone concentration andcloud parameters (e.g., cloud top height and optical thickness) using exact ra-diative transfer theory for a number of cloud models including single and multi-layered clouds having various thermodynamic states and microphysical charac-teristics. Errors related to the Lambertian cloud model are studied in depth.Ludmila Chaikovskaya presents a review of modern semi-analytical methods tostudy the propagation and backscattering of polarized laser beams in cloudymedia. The state of the art of using the polarization of light to investigate cloudproperties is outlined. In the final paper of this part, Robert Spurr describes thewell known LIDORT and VLIDORT linearized radiative transfer models. These

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XXII Preface

recently developed models have been used in a broad range of atmospheric andoceanic remote sensing problems.

The volume concludes with three papers on the reflective properties of variousparticulate media, including several natural (e.g., dry, wet and submerged sand)and artificial (e.g., composed of boron carbide, alumina, polystyrene, and sili-con glass particles) surfaces. Hao Zhang and Kenneth Voss describe instrumen-tation for the measurement of bi-directional reflectance distribution functions(BRDFs). They present results of BRDF measurements and their interpretationusing various approximate and exact radiative transfer models. One of the aimsis to quantify the wetting-induced darkening effect; the authors also study theinfluence of close-packed effects on BRDFs. Yevgen Grynko and Yuriy Shkura-tov describe their theoretical model for the description of BRDFs of particu-late surfaces composed of particles with sizes much larger than the wavelengthof incident light. The geometrical optics approximation (ray tracing) is used togenerate reflective properties of various turbid media. Both the intensity and po-larization of reflected light are calculated for various types of polydispersed lightscattering media composed of irregularly shaped particles. The book finisheswith the experimental study of Yuriy Shkuratov, Andrey Ovcharenko, VladimirPsarev, and Sergey Bondarenko. The aim of this study is the quantification ofthe reflective properties of planetary regoliths. In order to measure reflected lightproperties at very large scattering angles (up to 0.008 degrees from the backscat-tered direction), the laser super-small-phase-angle photometer/polarimeter hasbeen constructed. Measurements and their interpretations are given for varioustypes of surfaces.

In summary, this volume provides a detailed coverage of modern theoreticaltrends and new results in light scattering media optics. The material presentedhere is of interest not only to those working in the field of optical physics but alsoto geophysicists, chemists, astronomers, and biophysicists dealing with variouslight scattering problems in their practical work.

Bremen, Germany Alexander A. KokhanovskyOctober, 2007

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Part I

Single Light Scattering

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1 Observational quantification of the opticalproperties of cirrus cloud

Timothy J. Garrett

1.1 Introduction

By blocking sunlight and trapping heat, cirrus clouds play a central role inthe climate of Earth (Manabe and Strickler, 1964; Cox 1971; Ramaswamy andRamanathan, 1989). Temperature contrasts are created by the consequent redis-tribution of radiative energy, which in turn plays a role in driving atmosphericmotions and regional climate. Fundamentally, it is ice crystals that are respon-sible for radiative interactions, and their crystal shape and size distributions areconsequently a major research focus. This chapter focuses in particular on lightscattering by ice crystals in terrestrial cirrus.

When an incident beam of electromagnetic radiation sinusoidally acceleratesan electron field around an atomic nucleus, the radiation is scattered. The os-cillating electron field creates its own ‘dipole’ or Rayleigh radiation field. Theangular distribution of the scattered intensity is symmetric about the poles, andthe mean cosine of the scattering angle μ is zero.

Within particles of condensate, however, individual molecular dipoles areclose together, and this induces interactions between radiating fields. A deviationfrom Rayleigh scattering occurs that becomes increasingly pronounced with in-creasing particle size. Forward scattering dominates, and for particles large withrespect to the wavelength of light, forward diffraction assumes approximatelyhalf of all incident energy independent of wavelength. With respect to visiblelight, terrestrial tropospheric clouds are normally composed of particles that fallinto this large-particle ‘geometric optics’ regime.

For the purpose of the radiative description of a cloud layer in climate models,a group of dimensionless ‘single-scattering’ parameters is normally employed.These characteristic values – those associated with a single encounters of incidentphotons – are the single-scattering albedo (ωo), the phase function (p), and theextinction optical depth (τ).

For the special case of conservative scattering, there is no light absorptionand ω0 = 1. In this case, the probability distribution of angular scattering isrepresented fully by p (μ). For radiative flux calculations in climate simulations,however, p (μ) is normally simplified to the asymmetry parameter g – ultimately

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4 Timothy J. Garrett

only two directions are of interest in climate: up and down. From a mechanicsperspective, g represents the portion of forward momentum maintained by anincident light beam:

g =12

∫ 1

−1p (μ)μdμ (1.1)

Here, μ is the cosine of the scattering angle θ.Optical extinction, the amount of scattered energy removed from the direc-

tion of the incident beam, is represented by cloud extinction optical depth τ . Inpractice, τ is often expressed as the vertical integral with respect to height, z,of the extinction coefficient, βext:

τ =∫ h

0βext (z) dz (1.2)

where βext may be inferred from cloud microphysical properties in the form ofa size distribution of crystal area projected normal to the beam n (P ):

βext =∫ ∞

0Qext (P )

dn (P )dP

P dP (1.3)

For cloud particles at visible wavelengths, Qext is approximately equal to 2.The amount of solar radiation reflected to outer space by clouds is propor-

tional to the cloud albedo (α). In the limiting case of thin clouds, visible cloudalbedo can be roughly approximated by:

α � (1 − g) τ

2μ(1.4)

where, μ is a two-stream quadrature angle, set variously to√

3 and 1/2 (Liou,2002). Equation (1.4) shows how cloud albedo is proportional to cloud opticaldensity and depth, but also that small values of g can correspond to high cloudalbedo and thus lower surface solar irradiance and warming.

Normally, climate models employ some functional dependence of cirrus cloudg and βext on an ice crystal ‘effective radius’ re. For example, at visible wave-lengths (e.g. Foot, 1988),

βext =3W2ρre

(1.5)

where, ρ is the bulk density of ice, and W is its density in air. With respect tog, (e.g. Fu, 1996)

g =∑

i

airbie

where, a and b are constants.Unfortunately, re itself must be parameterized, and it remains unclear

whether widely used parameterizations are appropriate for real clouds. Second,while existing parameterizations of g (re) for ice crystals are theoretically well-grounded, they are usually based on some assumed ice crystal habit. Thus the

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1 Observational quantification of the optical properties of cirrus cloud 5

scattering models are correct only to the extent that idealized shapes adequatelyrepresent environmental crystal shapes.

This chapter shows that there are considerable discrepancies between single-scattering parameterizations developed from measurements and models. The dif-ferences are potentially important. Over the range of the mismatch, cirrus cloudmass may ultimately resist or amplify forecast greenhouse warming (Stephens etal., 1990). Also, full global climate model simulations reproduce significant cli-mate differences over regional spatial and temporal scales (Kristjansson, 2000).The current absence of consensus about how to represent cirrus cloud light scat-tering impedes accurate representation of Earth’s climate system.

1.2 Measurement of the asymmetry parameter in cirrus

1.2.1 Indirect estimates

Until relatively recently, two indirect techniques were available for deriving avalue of the asymmetry parameter appropriate for implementation in cirrus cloudradiative transfer models. In principle, the most straightforward of these involvedcalculating how a beam of radiation interacts with idealized model ice crystals.The most commonly used approach has been ray-tracing, sometimes with correc-tions for wave effects that are applied to smaller size parameter crystals (Iaquintaet al., 1995; Macke et al., 1996, 1998; Yang and Liou, 1998). The approach hasthe virtue of precision, but also has a notable drawback: by definition, modelingscattering by ice crystals requires idealization of crystal geometry.

The most idealized shapes afford relative computational facility: values canbe readily derived for ice hexagonal prisms of arbitrary size and aspect ratio.Table 1.1 shows that idealized columns have values of g ranging from 0.77 to0.86. Bullet rosettes have values that are similar, independent of the number ofbranches. Plates have values that are somewhat higher. In general, low values ofg are associated with small, isometric ice crystals (Macke et al., 1996).

Table 1.1. Theoretically derived values of the asymmetry parameter for pristine icecrystal shapes

g Crystal habit Reference

0.79–0.88 bullet rosettes Iaquinta et al. (1995)0.80–0.92 plates (Macke et al. (1998)0.77–0.86 columns (Macke et al. (1998)

The first attempts to test these idealized ice crystal models were airborneand indirect. Values of g were inferred with radiative transfer models from mea-sured cirrus microphysics and shortwave radiative flux profiles (Table 1.2). Thesepointed, with exceptions, to values that generally lay between 0.7 and 0.8, sug-gesting high ice crystal backscattering (Stephens et al., 1990; Wielicki et al.,

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6 Timothy J. Garrett

Table 1.2. Values of the asymmetry parameter for cirrus clouds inferred from ra-diometer observations

g Reference

0.7 Stephens et al. (1990)0.7 Stackhouse and Stephens (1991)0.7 Wielicki et al. (1990)0.75 Shiobara and Asano (1994)0.8 Francis et al. (1994)

>0.84 Mitchell et al. (1996)0.6–0.7 Spinhirne et al. (1996)

1990; Stackhouse and Stephens, 1991; Francis et al., 1994; Shobar and Asano,1994; Mitchell et al. 1996; Spinhirne et al., 1996). The values should be consid-ered with some caution, however. The microphysics measurements had consider-able uncertainties and, inherently, the approach requires potentially importantassumptions about the nature of radiation transport within the cirrus clouds,in particular that the ice clouds are physically homogeneous and plane-parallel.Nonetheless, these approximate techniques pointed to values of g that were aconsiderable deviation from those for idealized ice crystals (Table 1.1).

1.2.2 Nephelometer measurements

More recently, instruments have been developed for the direct measurement of gin situ that employ a nephelometer principle: that is they measure the intensityof light scattered from a collimated beam by a cloud of particles.

A nephelometer does not measure a normalized phase function p (μ), butrather the angular intensity distribution of laser light scattered by particles I (μ).In an idealized nephelometer, the asymmetry parameter may then be calculatedfrom

g =

∫ 1−1 I (μ)μdμ∫ 1−1 I (μ) dμ

(1.6)

As a practical matter, sensing the full angular distribution of scattering wouldrequire an infinitely long sensor. Obviously this is not possible. And becausesuch a large fraction of visible light is diffracted by cloud particles, this is not atrivial omission. For example, the fraction of energy diffracted by 10 μm spheresinto the forward 1 degree (μ = 0.9999) approaches 10%.

Gerber et al. (2000) found a solution to this problem in the developmentof the Cloud Integrating Nephelometer (or CIN), an airborne probe for themeasurement of the extinction coefficient and asymmetry parameter in clouds.While the precise angular distribution of diffraction is sensitive to both thesize and shape of the effective aperture, diffraction approaches one half of totalscattered energy in the geometric optics regime, independent of particle sizeand shape (see Fig. 1.1). Gerber et al. (2000) showed how this consequence of‘Babinet’s Principle’ can be exploited. At some angle from the forward direction

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1 Observational quantification of the optical properties of cirrus cloud 7

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f(θ′

)

θ′ (degrees)

Hollowed−ColumnsRough−AggregatesPlatesColumns θ′ for CIN

Fig. 1.1. Fraction of energy f scattered into angles smaller than θ′ for various icecrystal habits with an equivalent area diameter of 20 μm. The value of θ′ chosen forexperimental nephelometer (CIN) measurements is shown by the vertical dashed line.

θ′ = arccosμ′, the value of dp (μ) /dμ declines. At this point, the value of f (θ′),the integrated forward scattered energy, becomes insensitive to the precise choiceof θ′.

Accordingly, Gerber et al. designed the CIN to measure I as an integratedquantity, portioned into forward and backscattering regimes such that

F =∫ μ′

0I (μ)μdμ (1.7)

B =∫ 0

μ′′I (μ)μdμ (1.8)

F =∫ μ′

0I (μ) dμ (1.9)

B =∫ 0

μ′′I (μ) dμ (1.10)

Because Eqs. (1.7) to (1.10) are truncated in μ, Eq. (1.6) is modified to Gerber(2000):

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8 Timothy J. Garrett

g = f + (1 − f)F − B

F + B(1.11)

where

f =∫ 1

μ′I (μ) dμ (1.12)

Gerber et al. (2000) showed that a suitable choice for μ′ is θ′ = 10◦. In thiscase, over a wide range of ice crystal habits and particle sizes, the fraction ofenergy f scattered into μ > μ′ is 0.57 ± 0.02 (Fig. 1.2). While the uncertaintyin the calculation of f may seem implausibly small, it is more realistic whenconsidered in the light of it representing a zeroth order quantity (diffraction �0.5) plus a first-order correction from the refraction of light into the forward 10◦

(0.07±0.02). Equation (1.11) is merely a modified form of the more fundamentalrelationship (van de Hulst, 1981)

g =12

+12ggeom (1.13)

where, ggeom represents the portion of light scattering that can treated withray-tracing.

We note that the utility of f is diminished for idealized plate crystals(Fig. 1.2), because there is significant delta-function forward transmission through

100

101

102

103

0.2

0.3

0.4

0.5

0.6

0.7

0.8

f

Deq

(μm)

Hollowed−ColumnsRough−AggregatesPlatesColumns

Fig. 1.2. Value of f assuming that θ′ = 10◦, for various ice crystal habits and equiv-alent area diameters Deq.

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1 Observational quantification of the optical properties of cirrus cloud 9

Fig. 1.3. Schematic of the Cloud Integrating Nephelometer. Laser light passes througha cloud of particles, and is scattered to four sensors, two that measure forward-scattering (F and F ) and two that measure backscattering (B and B). Cosine masksweight the scattered light by the cosine of the scattering angle (after Gerber et al.,2000).

their basal facets if they are smooth, parallel and opposite. As will be discussed,such perfection is unlikely in nature.

Truncation occurs in the rear scattering regime also, for which Gerber et al.chose a truncation angle of cos 175◦ = μ′′. The derived value of g is not highlysensitive to this value because the scattered momentum is relatively small inthe rear few degrees. A schematic of the CIN design is shown in Fig. 1.3. Laserlight is scattered by particles into four Lambertian sensors, two forward and twoback, and then amplified by photomultipliers. For measurement of F and B,the scattered light is cosine-weighting by a quarter-circle mask. Baffles excludescattering from the forward 10◦ and the rear 5◦.

Another instrument with a similar design to the CIN is the Polar Neph-elometer (PN) (Gayet et al. 1997). Whereas the CIN distinguishes only betweenforward and back scattering, the PN uses a circular array of 33 photodiodes tomeasure the spectrum of scattered intensity in 33 intervals between 3.49◦ and169◦ (Fig. 1.4). The primary functional advantage of the PN over the CIN isthat it conveys information about particle shape through partial measurementof p (μ). For example, Auriol et al. (2001) described the detection by the PNof the 22◦ and 46◦ halos in cirrus. Because these did not correlate well withobserved habits of large (>100 μm across) ice crystals, it was inferred that cloudscattering had been dominated by ice crystals that were smaller.

Like the CIN, the PN has been used to measure g through application ofequation 1.6. The PN measures angular scattering, but also misses about onehalf of scattered energy (Fig. 1.1) and additionally it suffers from light pollutionat forward angles. To calculate g, Auriol et al. (2001) used Eq. (1.11), andestimated the amount of energy scattered at angles less than 15◦ by using valuesof f estimated by Gerber et al. (2000) for energy scattered at angles less than10◦; the difference in f associated with this angular discrepancy turns out to besmall (Fig. 1.1).

A summary of measurements by the CIN and PN of g in cirrus is shown inTable 1.3. In general, values are generally in the vicinity of 0.75 and are nearlyindependent of location and temperature. For example, values of g obtained inanvil cirrus stemming from Florida deep convection are effectively identical to

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10 Timothy J. Garrett

Fig. 1.4. Optical scheme of the polar nephelometer (after Gayet et al., 1997).

Table 1.3. Typical values of cirrus cloud asymmetry parameter g measured in situ

g Location Reference

0.76 mid-latitude NH Auriol et al. (2001)0.74 Arctic Gerber et al. (2000); Garrett et al. (2001)0.75 Florida anvil Garrett et al. (2003)0.77 mid-latitude NH and SH Gayet et al. (2004)0.74 Antarctic Baran et al. (2005)

those measured within synoptic cirrus in the Arctic and at mid-latitudes, andice cloud measured on the ground in the Antarctic.

Measurements also show a dependence of g on particle effective radius re thatis remarkably weak compared to theoretical estimates that have been derived byassuming hexagonal prism shapes (Fig. 1.5). Fu (1996) developed parameter-izations for the single-scattering properties of cirrus clouds for use in climatemodels. In the visible, non-absorbing range of wavelengths, the theoretically cal-culated value for g increases by 0.06 (25% in momentum removal) as re increasesfrom 5 to 70 μm. By comparison, values of g increase by just 0.01 to 0.02 (∼6%in momentum removal) over the same size range (Garrett et al., 2003; Gayet etal., 2004). Measurements and models only agree for the smallest sizes.

Thus, a fundamental inconsistency exists between values of g that are theo-retically derived from idealized representations of ice crystal shapes, and valuesmeasured within cirrus cloud. The differences are important, for they imply up

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1 Observational quantification of the optical properties of cirrus cloud 11

Fig. 1.5. Parameterized dependence of the asymmetry parameter g on ice crystaleffective radius re (Fu, 1996), compared with in situ measurements (Garrett et al.,2003; Gayet et al., 2004).

to 50% greater backscattered solar energy for a given cirrus optical depth (Eq.(1.4)).

1.2.3 Reconciling discrepancies between theory and observations

Why should such discrepancies in observed and calculated g exist? An obviouspossible explanation is that the idealized shapes assumed in scattering calcula-tions lack some basic but important natural feature.

In general, the factors that contribute to g might be seen as coming from aseries of components. The first, as described, is diffraction. For particles largewith respect to the wavelength of light, diffraction contributes one half to gindependent of particle shape (Mischchenko and Macke, 1997). This componentcan then be added to a second contribution from refraction. The two componentscan be combined to obtain a total.

If a particle has many components in close proximity, however, as for exam-ple within aggregate ice crystals, there are additional contributions to g frominterference and interactions between component fields (Videen et al., 1998). In-terference adds high-frequency structure to the scattering phase function thattends to add to g. Interaction is associated with enhanced multiple scatteringbetween component structures, increasing backscattering and lowering g. Botheffects are maximum for small component separation distances. Unfortunately,

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12 Timothy J. Garrett

quantifying the relative magnitudes of the interaction and interference contribu-tions for anything other than the most simple shapes (e.g. two adjacent spheres)has proved difficult (Holler et al., 2000).

It is possible to obtain a qualitative perspective of the influence of morecomplex particle geometry on g. Consider that individual column ice crystalshave similar scattering characteristics to radial assemblages of columns or bulletrosettes (Table 1.1). Unless the branches are very numerous, the electromag-netic interactions between branches are weak (Macke, 1993; Iaquinta, 1995). If,however, an assemblage of columns is not radially oriented, but rather clustered,the interactions are stronger and backscattering more pronounced. Values of gderived for such aggregated crystal forms are about 0.75 (Yang and Liou, 1998).

Although a value of 0.75 for g for aggregates would appear to be encouragingbecause it agrees well with values obtained in situ, the explanation is unlikelyto be so simple. Large aggregated crystals in cirrus are normally rare (Kajikawaand Heymsfield, 1989), even when in situ measurements of g are low (Garrettet al., 2001). Also, many of the in situ measurements by the CIN and PN inmid- and low-latitude cirrus indicate that optical scattering is dominated by verysmall crystals, less than 50 μm across (Garrett et al., 2003, 2007; Gayet et al.,2002, 2004). Such small crystals are unlikely to have more than a singular grosscomponent.

Another possible explanation for observed values of g is that the gross phys-ical shape of an ice crystal may play only a contributing role to the scatteringprofile. Rather, features at sub-crystal scales control g, and the reason g is lowis due to electromagnetic interactions between these features.

Obviously, a hexagonal prism represents the ice crystal lattice structure ofice at most terrestrial temperatures below freezing. However, this does not meanthat the structure is maintained uniformly as the ice nucleus grows from molec-ular to super-micrometer scales – a change of about four orders of magnitude.

For example, depositional growth is not uniform on a crystal surface butrather occurs in steps or spirals that are up to 0.2 μm high, sufficiently large toscatter visible light. These can bunch together to form steps with larger featuresthat extend into the geometric optics regime (Pruppacher and Klett, 1997). Also,under supersaturated conditions growth is favored along corners and edges inregions of higher surface area density. Sub-saturated conditions can cause anice crystal to develop a fibrous or pitted appearance (Cross, 1969; Davy andBranton, 1970). As a cloud evolves, an ice crystal may easily be exposed toa wide range of saturation conditions, and these should be reflected as a morecomplex surface structure. Attempts have been made to incorporate some versionof roughness on ice crystal surfaces using either idealized fractal geometry (Mackeet al., 1996) or a surface roughness model (Yang and Liou, 1998; Shcherbakov etal., 2006). Complexity at the ice crystal surface is usually associated with lowervalues of g, closer to 0.75, and hence substantially higher backscattering.

A second mechanism that is sometimes considered for explaining low valuesof g is the presence of bubbles within ice crystal interiors. The density of bub-bles need not be extreme to have a significant impact. Introducing bubbles thathave an effective radius of 1.0 μm and a mean free path length between them of

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1 Observational quantification of the optical properties of cirrus cloud 13

20 μm lowers the asymmetry parameter of a hexagonal ice crystal from 0.81 to0.69 (Macke et al., 1996). Mischchenko and Macke (1997) showed that spheres,if they are heavily included, can have values of g as low as 0.65. Model crystalsfilled with bubbles have scattering phase functions that have been observed toreproduce PN measurements well (Labonnote et al., 2000; Shcherbakov et al.,2006). Unfortunately, there is scant evidence that bubbles are in fact a normalcomponent of ice crystals. A few examples can be seen in photographs from theSouth Pole (Tape, 1994), but even there, ice crystal bubbles are largely absent.Bubbles may be missed simply due to insufficient photographic resolution. How-ever, even in this case it is not clear that a good physical mechanism to explaintheir formation exists. Macke et al. (1996) proposed that aerosol particles be-come trapped in an ice crystal lattice without going into solution, but did notexplain how this might happen in significant numbers. In any case, it is now un-derstood that cirrus ice crystals form primarily from the homogeneous freezingof solution aerosol (Koop et al., 2000), in which case the aerosol freezes.

Table 1.4. Values of the asymmetry parameter for cirrus clouds estimated from someprevious theoretical studies

g Crystal habit Reference

0.75 aggregates of columns Yang and Liou (1998)0.74 randomized Koch fractals Macke et al. (1996)0.69 bubbles in a column Macke et al. (1996)

Thus, it remains unanswered why exactly values of ice crystal g are low. Thebest candidate may be that ice crystal surfaces are roughened. It would seem auseful avenue of theoretical research to pursue efforts to quantify these small-scale effects (e.g. Videen et al., 1998; Holler et al., 2000). However, regardlessof what specific physical ice crystal property dominates scattering, the observedabsence of a strong proportionality between g and re, as shown in Fig. 1.5,suggests that sub-crystal-scale features play an important role. Current climatemodel parameterizations for g do not represent this physics, favoring insteadmore idealized representations.

Of course, it is also possible that measurements by the CIN and PN are inerror. A problem that may be associated with airborne cloud probes in generalis the shattering of snow particles on cloud probe instrument inlets. Because thiswould artificially inflate concentrations of small particles, measured scatteringwould not be precisely representative of true cloud. The potential importance ofthis issue is discussed further in the following section.

1.3 Extinction coefficient and effective radius

The extinction coefficient spans a large range in terrestrial cirrus. At its upperend, values well in excess of 100 km−1 are found in the outflow of deep con-vective cloud (Fig. 1.6). Convection rapidly transports warm moist air to cold

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14 Timothy J. Garrett

W (g m−3)

β ext (

km−

1 )

WB−57FCitation

0.001 0.01 0.1 1

1

10

100

Fig. 1.6. Probability distribution contours of βext and W in anvil cirrus measured inFlorida by two aircraft during the 2002 CRYSTAL-FACE mission. The WB-57F andCitation sampled cloudy air up to 15.2 and 12.6 km, respectively (details in Garrett etal., 2003).

temperatures, forcing near-complete condensation of available water vapor. Atthe lower end, a continuum of extinction exists along a line of cloud evaporation,giving cirrus cloud its distinctively fuzzy edges. Sustained thin cloud also exists.Values of βext in ‘ultra-thin tropopause cirrus’ may be as low as 10−3 km−1

over a horizontal extent of 100 km2 (Luo et al., 2003). Thus, in such context ofat least five orders of magnitude of variability, accurate prescription of βext inmodel simulations would appear to be of first-order importance.

The problem is considerably simplified by introducing to models a particleeffective radius re (Eq. (1.5)). Because the effective radius represents an ensembleof hydrometeors through the ratio of their physical volume to their radiativeextinction cross-section at visible wavelengths, it is therefore an optical lengthscale, and not a morphological one. Attempts have been made to treat ice crystalre strictly on the basis of geometric considerations (McFarquhar and Heymsfield,1998), but only because this approach has been necessitated by the types ofmeasurements made.

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1 Observational quantification of the optical properties of cirrus cloud 15

The utility of using an effective radius comes from the fact that climatemodels predict densities of condensate mass, but not the radiative cross-sectionsrequired for radiative transfer calculations. The effective radius provides a linkbetween the two quantities. Also, despite covering orders of magnitude, βext andW strongly covary (e.g. Fig. 1.6). Therefore, the variability in re is typicallysmall and more easily treated.

1.3.1 Indirect measurement of effective radius

Climate model parameterizations of re are generally based on in situ airbornemeasurements. Derived values have traditionally relied on in situ measurementsof ice crystal size and shape distributions n(P ).

Deriving re from n(P ) is difficult for several important reasons. First, therelationship between P and ice crystal mass must be assumed where it is infact individual to each ice crystal. For simple shapes, these assumptions may bereasonable. But for the case of an ensemble of ice crystals like those shown inFig. 1.7, the relationship between P and mass is nearly impossible to infer withany accuracy. Second, there exists potential for compounding errors in n and Pwhen summing projected areas. For example, particle imaging probes often havecoarse size resolution, small and ambiguous sample volumes, include shatteringartifacts in the data, and have reduced sensitivity to particles with sizes below∼100 μm diameter (Korolev et al., 1998; Korolev and Isaac, 2005).

Fig. 1.7. Image of an ice crystal sampled within Florida cumulonimbus outflow. Thedimensions of the image are approximately 400 μm.

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16 Timothy J. Garrett

A consequence of these uncertainties is that there is currently considerabledisagreement within the cloud physics community as to which values of re shouldbe applied to represent mass-specific scattering by cirrus cloud ice crystals. To il-lustrate, values from measurements of n(P ) usually fall either in a range between5 and 30 μm (Heymsfield and Platt, 1984; Knollenberg et al., 1993; Boudala etal., 2002; Field et al., 2003; Gayet et al., 2004), or between 30 and 220 μm (Dowl-ing and Radke 1990; Heymsfield and McFarquhar, 1996; Wyser, 1998; Baum etal., 2005; Field et al., 2005). Within either range, variability can often be at-tributed to natural variations in temperature T and ice water content W (Mc-Farquhar et al., 2003). However, the discrepancy between the two ranges arisesfrom subjective interpretations of how in situ cloud size distribution probes per-form in ice cloud. In particular, questions have been raised about the reliabilityof in situ measurements of small ice crystals <50 μm across. It has been arguedthat larger ice crystals can shatter on the inlets of cloud probes. Shattering wouldproduce artificially high concentrations of small ice crystals and correspondinglylarge scattering densities (Gardiner and Hallett, 1985; Gayet et al., 1996; Fieldet al., 2003; Heymsfield et al., 2006). Where W is conserved, βext is inflated andre correspondingly reduced.

It remains to be shown unambiguously the extent to which high concen-trations of small ice crystals can be attributed to shattering (e.g. Field et al.,2003). In the meantime, many studies have chosen to be conservative, and basedice crystal scattering calculations solely on measurements of ice crystals largerthan 50 to 100 μm across (e.g. Baum et al., 2005a,b), choosing to omit or pa-rameterize more contentious measurements of the concentrations of ice crystalswith smaller sizes. Whether or not this approach is valid remains a subject ofextensive debate. One perspective on this problem is presented next.

1.3.2 Direct measurement of effective radius

A promising approach for deriving re, one that does not require measurement ofn(P ), is to compare direct airborne measurements of βext and W . The advantageof the ‘direct’ approach is that by measuring the integral scattering of an icecrystal ensemble, it obviates errors and assumptions involved associated withintegration. It does not implicitly remove the potential for hydrometer shatteringon inlets, and this could remain a concern for the measurement of βext. It doesnot obviously affect measurement of W , as mass is conserved under breakup.Therefore, the potential is for underestimation of re.

Several techniques exist for the measurement of W directly. These derivemass density from the absorption by laser light by evaporated condensate, orfrom the fluorescence of photodissociated OH fragments (e.g. Weinstock et al.,1994; Twohy et al., 1997). The instruments have been extensively validated andcompare well in cirrus (Davis et al., 2007).

Measurements of βext can be obtained directly using either the CIN or PNbecaused ice particle extinction and scattering are equivalent in the visible. Aswith measurement of g, the fraction f of light scattered into the forward 10◦ mustbe assumed, and used to scale the fraction of scattered light that is measured.

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1 Observational quantification of the optical properties of cirrus cloud 17

Thus, the extinction coefficient measured by the CIN follows from Eqs. (1.9) and(1.10)

βext =F + B

(1 − f)(1.14)

Attempts have also been made to measure cloud βext using a Nevzorov airbornetransmissometer (Korolev et al., 1999). The transmissometer approach has theadvantage of being first principles and ostensibly more accurate than the CIN be-cause it measures extinction directly, rather than inferring it from some angularfraction of scattered light. A light source is reflected by a mirror 2.33 m distantto a smaller sensor. In the single-scattering regime, the loss of light intensityat the sensor is directly proportional to the extinction coefficient. Because theinstrument is open path and in the free airstream with no inlet, the instrumenthas no possible sensitivity to hydrometeor shattering.

The Nevzorov probe has not been flown extensively in cirrus. Nonetheless,the transmissometer probe has been useful for the validation of the CIN in wa-ter clouds (Garrett, 2007). Observed discrepancy between Nevzorov and CINmeasurements of βext is small – about 13% – and well within the range of uncer-tainty in both instruments (Fig. 1.8). Complementary conclusions were reached

0 50 100 150 200 2500

50

100

150

200

250

Transmissometer βext

(km−1)

CIN

β ext (

km−

1 )

CIN = 1.13*Transmissometer, r2 = 0.89

>1 per liter >100 μm diameter

Fig. 1.8. Comparison of the extinction coefficient βext measured with the CIN and theNevzorov transmissometer in warm cumulus. Circles indicate time periods when theconcentration of particles >100 μm diameter was greater than 1 per liter (maximummeasured was 26 per liter).

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18 Timothy J. Garrett

by Gerber (2007), who compared the CIN to a ground-based transmissometer,and found on average differences in measurements of βext that were within 5%.In the case of the airborne data, it is notable that the level of agreement showedno sensitivity to the presence of precipitation-sized particles that might havebroken up on the CIN inlet. This is important; it implies that to the extent thatparticles did shatter on the CIN, they contributed only a negligible amount tothe CIN measurements.

In the meantime, measurements of re derived from bulk measurements ofβext and W remain contentious (Heymsfield et al., 2006). This is because theypoint to values that are generally smaller than those implied by size distributionprobes. For example, Garrett et al. (2003, 2007) found values of re ranging fromless than 10 μm at very cold temperatures below −50◦C to about 30 μm closeto freezing (Fig. 1.9). The result held in both Florida anvil cirrus, where theice particles were nucleated within deep convection, and within mid-latitudesynoptic cirrus. The values in synoptic cirrus were slightly larger, by about 50%

−70 −60 −50 −40 −30 −200

5

10

15

20

25

30

35

40

T °C

r e (μm

)

MidCiX

C−F

re(MidCiX) = 6e[(T + 75)/35]

re(C−F) = 5e[(T + 75)/39]

10 cm/s50 cm/s100 cm/s

Fig. 1.9. Thirty-second mean values of ice crystal effective radius re obtained usingbulk βext and IWC data in mid-latitude synoptic cirrus (Garrett et al., 2007). Datapoints explicitly noted by the WB-57F back-seater as being associated with 22◦ and46◦ halos, are marked with single and double circles, respectively. The black dot corre-sponds to the photograph in Fig. 1.11. The gray box represents the region below which,theoretically, it should be expected that diffraction effects would begin to smooth outa distinct halo. A fit to the data is shown by the dashed line. A fit (solid line) todata obtained in Florida anvil cirrus (C-F) (Garrett et al., 2003) is shown for compar-ison. Filled symbols represent theoretical results derived by (Karcher and Lohmann,2002) for homogeneous nucleation of haze aerosol at specified updraft velocities andtemperatures.

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1 Observational quantification of the optical properties of cirrus cloud 19

at any given temperature, but a general exponential temperature dependence inre was observed in both cases. It may be added that Kokhanovsky and Nauss(2005) observed similar particle effective sizes and temperature dependencies inmulti-spectral satellite retrievals of the tops of Hurricane Jeanne, just prior toits 2004 landfall in Florida.

Combined, these results suggest a general link between the scattering char-acteristics of cirrus clouds and their thermodynamic and dynamic properties.Karcher and Lohmann (2002) showed that ice crystal size is determined to firstorder by temperature: from the Clausius–Clapeyron relation the amount of vaporavailable for condensation increases exponentially with temperature. In addition,fewer haze aerosol homogeneously freeze at warmer temperatures. Ice crystalnumber is thus lower, and particle size larger. The rate of cooling also matters.Ice crystals nucleated in more vigorous clouds nucleate more, and consequentlysmaller ice crystals.

Thus, physical considerations indicate that re depends primarily on tempera-ture, but that variability in updraft velocity plays a role also. Ice crystals formedunder weaker updraft conditions, as in synoptic cirrus, tend to be larger for agiven temperature. While the proportionality of the exponential temperature de-pendence may vary, these general characteristics are consistent between theoryand observations.

1.3.3 Optical evaluation of ice crystal effective radius using halos

Ice crystal size can also be inferred from observation of cirrus optical effects.Most commonly, these effects are light that is focused at 22◦ and 46◦ from thesun, either in patches of light or in continuous circular halos. These optical ef-fects form when light is diffracted through two prism faces in the case of 22◦

effects, and a prism and a basal face in the case of 46◦ effects (Fig. 1.10). Usingthese effects to infer ice crystal size is approximate, but it can be first princi-ples, and it is not associated with instrument artifacts from shattering. To testwhether shattering is important to cloud-probe data, comparisons can be madebetween simultaneous halo observations and ice crystal measurements made insitu.

Fig. 1.10. Ray paths through a hexagonal prism ice crystal associated with a 46◦ halo(left) and 22◦ halo (right).

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20 Timothy J. Garrett

Diffraction smooths out ray-optics effects when ice crystals are small by creat-ing an angular intensity spread about the refractive angle of minimum deviation.Fraser (1979) developed an analysis of this problem and showed how, for non-absorbing wavelengths, the angular width from a central intensity peak to itshalf-power point can be approximated by

θ1/2 � λ

2re(1.15)

The approximate spread about the angle of minimum deviation is thus 1◦ for a20 μm radius crystal, and 4◦ for a 5 μm crystal. Fraser (1979) estimated that10 μm radius is approximately the minimum crystal size required for ice crystalsto produce distinct ray-optics effects. Similar conclusions have been reachedusing T-matrix calculations (Mishchenko and Macke, 1999) and from laboratorystudies (Sassen and Liou, 1979).

This lower limit provided by diffraction for the formation of ice crystal opticaleffects can be used as a test of the accuracy of in situ measurements of icecrystal size. If these effects were observed when in situ probes showed valuesof re substantially less than 10 μm, the probes were likely in error. During theMidCiX field program of April and May, 2004 (Garrett et al., 2007), airbornemeasurements were obtained over Texas and Louisiana within moist synopticcirrus outflow from the Gulf of Mexico. Concurrently, cirrus optics was notedand photographed within the cirrus itself. Fig. 1.9 shows all re measurementsduring MidCiX, with the time periods highlighted for the subset when 46◦ and22◦ halos were explicitly photographed (they were casually noted much moreoften). There is some uncertainty about whether the 46◦ halo cases photographedwere in fact circumhorizon arcs caused by oriented plate crystals (Tape, 1994),but the crystal ray paths are identical regardless. Note that no 46◦ halos werephotographed in synoptic cirrus when bulk probe measurements indicated thatice crystal re was smaller than about 15 μm. The fact that 46◦ halos werenot observed at smaller sizes suggests that diffraction effects had smoothed outthe halo peak. 22◦ halos were occasionally noted when the bulk in situ probesindicated values of re as small as 8 μm. However, even for these times, closerexamination of halo photographs showed that that despite the photographs beingtaken near cloud top – and therefore in the single-scattering regime – the 22◦

halos appeared ‘washed-out’, and had no distinct boundaries, color separationor accompanying 46◦ halo (Fig. 1.11).

Therefore, the nature of cirrus optics can provide an independent check ofin situ measurements of re. Very small ice crystal effective sizes were measured,and these were consistent with the optical characteristics of the 46◦ and 22◦

halos observed. Although it cannot be excluded, there is no indication fromthis independent test that measured values of re were artificially small due to ashattering bias.

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1 Observational quantification of the optical properties of cirrus cloud 21

Fig. 1.11. Photograph of a 22◦ halo taken at 12.1 km altitude, less than 100 m belowlocal cloud top, and corresponding to the black point in Fig. 1.9. In the inset is showna profile along the dashed line of the camera’s red-colored pixel intensity (linear scale).

1.4 Summary of outstanding problems

Looking at the Earth from outer space, cloud is obviously of first-order im-portance to the planetary radiation balance. At any given cloudy location, theamount of solar radiation reflected to outer space depends on the composition ofthe cloud, even at sub-micrometer scales. Cirrus cloud is particularly widespread.However, there remain substantial gaps in our understanding of the relationshipsbetween cloud microstructures and cirrus radiative properties, even after severaldecades of study.

The two primary single-scattering properties of relevance here are the asym-metry parameter g and the effective radius re. In the case of the asymmetryparameter, in situ measurements of ice crystals show values of g that are gener-ally lower and show less size sensitivity than values implied from computationalcalculations for idealized ice crystal shapes. If the measurements are accurate towithin stated uncertainties, there are several implications. First, cirrus cloud ismore highly reflective per unit optical depth than implied by most current cirrusmodels, by as much as a factor of 2. Second, the gross morphology of ice crystalsmay play a secondary role to internal inclusions or surface roughness when itcomes to determining the scattering profile of atmospheric ice crystals.

A second concern is whether cirrus cloud radiative properties are dominatedby ice crystals with characteristically small optical sizes. It has been argued thatmeasurements that suggest small ice crystal predominate are contaminated byshattering on in situ instrument inlets. If, however, ice crystal effective radii

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22 Timothy J. Garrett

are indeed small, the optical scattering density of cirrus per unit mass is cor-respondingly high, and for thin cirrus at least, so is the cloud albedo. Currentpublications point to a very wide range of values of re, even when obtainedwithin the same cloud, depending on the measurement technique used. Becauseclimate model mass-specific optical density is represented by measurement-basedparameterizations of re, greater consensus will be required. Parameterizationsof re are most easily derived from cloud probes that measure cloud mass andextinction directly, rather than by inferring re from size and shape distributionsof individual ice crystals. The ‘direct’ technique provides values of re that arephysically defensible and optically justified, despite being on the small end ofthose normally considered.

Today, few remote sensing techniques or climate models employ parameteri-zations consistent with the smaller values of g and re described here. Any switchhas the potential to imply important differences in models and measurements ofregional climate and atmospheric heating.

Acknowledgments

The author is grateful for support from NASA, collaboration with Hermann Ger-ber, and contributions from Ping Yang, Paul Lawson, Alexander Kokhanovsky,and Jean-Francois Gayet.

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Garrett, T. J., M. B. Kimball, G. G. Mace, and D. G. Baumgardner, 2007: Observingcirrus halos to constrain in situ measurements of ice crystal size. Atmos. Chem.Phys. Disc., 7, 1295–1325.

Gayet, J.-F., G. Febvre, and H. Larsen, 1996: The reliability of the PMS FSSP in thepresence of small ice crystals. J. Atmos. Oceanic. Technol., 13, 1300–1310.

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Gayet, J.-F., F. Auriol, A. Minikin, J. Strom, M. Seifert, R. Krejci, A. Petzold, G. Feb-vre, and U. Schumann, 2002: Quantitative measurement of the microphysical andoptical properties of cirrus clouds with four different in situ probes: Evidence ofsmall ice crystals. Geophys. Res. Lett., 29, doi:10.1029/2001GL014342.

Gayet, J.-F., J. Ovarlez, V. Shcherbakov, J. Strom, U. Schumann, A. Minikin, F. Auriol,A. Petzold, and M. Monier, 2004: Cirrus cloud microphysical and optical propertiesat southern and northern midlatitudes during the INCA experiment. J. Geophys.Res., 109, doi:10.1029/2004JD004803.

Gerber, H., 2007: Comment on ‘Effective radius of ice cloud particle populations derivedfrom aircraft probes’ by Heymsfield et al. J. Atmos. Oceanic. Technol., in press.

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24 Timothy J. Garrett

Heymsfield, A. J. and G. M. McFarquhar, 1996: High albedos of cirrus in the tropicalPacific warm pool: Microphysical interpretations from CEPEX and from Kwajelein,Marshall Islands. J. Atmos. Sci., 53, 2424–2451.

Heymsfield, A. J. and C. M. R. Platt, 1984: A parameterization of the particle sizespectrum of ice clouds in terms of the ambient temperature and the ice watercontent. J. Atmos. Sci., 41, 846–855.

Heymsfield, A. J., C. Schmitt, A. Bansemer, G.-J. van Zadelhoff, M. J. McGill,C. Twohy, and D. Baumgardner, 2006: Effective radius of ice cloud particle popu-lations derived from aircraft probes. J. Atmos. Oceanic. Technol., 23, 361–380.

Holler, S., J.-C. Auger, B. Stout, Y. Pan, J. R. Bottiger, R. K. Chang, and G. Videen,2000: Observations and calculations of light scattering from clusters of spheres.Appl. Opt., 39, 6873–6887.

Iaquinta, J., H. Isaka, and P. Personne, 1995: Scattering phase function of bullet rosetteice crystals. J. Atmos. Sci., 52, 1401–1413.

Kajikawa, M. and A. J. Heymsfield, 1989: Aggregation of ice crystals in cirrus. J.Atmos. Sci., 46, 3108–3121.

Karcher, B. and U. Lohmann, 2002: A parameterization of cirrus cloud forma-tion: Homogeneous freezing of supercooled aerosols. J. Geophys. Res., 107, 4–1,doi:10.1029/2001JD000470.

Knollenberg, R. G., K. Kelly, and J. C. Wilson, 1993: Measurements of high numberdensities of ice crystals in the tops of tropical cumulonimbus. J. Geophys. Res., 98,8639–8664.

Kokhanovsky, A. A. and T. Nauss, 2005: Satellite-based retrieval of icecloud properties using a semianalytical algorithm. J. Geophys. Res., 110,doi:10.1029/2004JD005744.

Koop, T., B. Luo, A. Tsias, and T. Peter, 2000: Water activity as the determinant forhomogeneous ice nucleation in aqueous solutions. Nature, 406, 611–614.

Korolev, A. and G. A. Isaac, 2005: Shattering during sampling by OAPs and HVPS.Part I: Snow particles. J. Atmos. Oceanic. Technol., 22, 528–542.

Korolev, A. V., J. W. Strapp, and G. A. Isaac, 1998: Evaluation of the accuracy ofPMS optical array probes. J. Atmos. Oceanic. Techol., 15, 708–720.

Korolev, A. V., G. A. Isaac, J. W. Strapp, and A. N. Nevzorov, 1999: In situ measure-ments of effective diameter and effective droplet number concentration. J. Geophys.Res., 104, 3993–4004, doi:10.1029/1998JD200071.

Kristjansson, J. E., J. M. Edwards, and D. L. Mitchell, 2000: Impact of a new schemefor optical properties of ice crystals on climates of two GCMs. J. Geophys. Res.,105, 10063–10080, doi:10.1029/2000JD900015.

Labonnote, L. C., G. Brogniez, M. Doutriaux-Boucher, J.-C. Buriez, J.-F. Gayet, andH. Chepfer, 2000: Modeling of light scattering in cirrus clouds with inhomogeneoushexagonal monocrystals. Comparison with in situ and ADEOS-POLDER measure-ments. Geophys. Res. Lett., 27, 113–116, doi:10.1029/1999GL010839.

Liou, K., 2002: An Introduction to Atmospheric Radiation. International GeophysicsSeries, Academic Press.

Luo, B. P., T. Peter, S. Fueglistaler, H. Wernli, M. Wirth, C. Kiemle, H. Flentje, V. A.Yushkov, V. Khattatov, V. Rudakov, A. Thomas, S. Bormann, G. Toci, P. Mazz-inghi, J. Beuermann, C. Schiller, F. Cairo, G. Di Donfrancesco, A. Adriani, C. M.Volk, J. Strom, K. Noone, V. Mitev, R. A. Mackenzie, K. S. Carslaw, T. Trautmann,V. Santacesaria, and L. Stefanutti, 2003: Dehydration potential of ultrathin cloudsat the tropical tropopause. Geophys. Res. Lett., 30, doi:10.1029/2002GL016737.

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Macke, A., 1993: Scattering of light by polyhedral ice crystals. Appl. Opt., 32, 2780–2788.

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Macke, A., J. Mueller, and E. Raschke, 1996b: Single scattering properties of atmo-spheric ice crystals. J. Atmos. Sci., 53, 2813–2825.

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Mischchenko, M. I. and A. Macke, 1997: Asymmetry parameters of the phase functionfor isolated and densely packed spherical particles with multiple internal inclusionsin the geometric optics limit. J. Quant. Spectrosc. Radiat. Transfer , 57, 767–794.

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26 Timothy J. Garrett

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Wielicki, B. A., J. T. Suttles, A. J. Heymsfield, R. M. Welch, J. D. Spinhirne, M.-L. Wu,D. C. O’Starr, L. Parker, and R. F. Arduini, 1990: The 27-28 October 1986 FIREIFO cirrus case study: Comparison of radiative transfer theory with observationsby satellite and aircraft. Mon. Wea. Rev., 118, 2356–2376.

Wyser, K., 1998: The effective radius in ice clouds. J. Climate, 11, 1793–1802.Yang, P. and K.-N. Liou, 1998: Single scattering properties of complex ice crystals in

terrestrial atmosphere. Contr. Atmos. Phys., 71, 223–248.

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2 Statistical interpretation of light anomalousdiffraction by small particles and itsapplications in bio-agent detection andmonitoring

Min Xu and A. Katz

2.1 Introduction

Light scattering by small particles is one of the most powerful techniques forprobing the properties of particulate systems and has numerous applications inparticle characterization and remote sensing of, for example, clouds and aerosols,interplanetary dust, marine environment, bacteria, biological cells and tissues.This subject, governed by Maxwell’s electromagnetic theory of light, developedin the later nineteenth century, was first summarized in van de Hulst’s clas-sic 1957 work [1], since Lorentz [2], Mie [3], Rayleigh [4] and Tyndall [5] laid thefoundations of light scattering. The field is yet vigorous and ever expanding,documented by the current interest and the increasing number of publications.Light scattering by small particles is actively being pursued, especially for non-spherical particles (see, for example, the review volume edited by Mishchenko,Hovenier and Travis [6]). Alongside the availability of computational capabilityand the advance of numerical methods based on an exact theory, approximatetheories of light scattering are still attractive in providing both simpler alterna-tives and much more direct physical interpretations. Approximation theories areappealing in inverse problems such as remote sensing where the error introducedby the approximate theory can be negligible compared to that introduced bya priori assumptions. Approximation theories are sometimes also mandatory incases (for example, computation of the optical efficiencies of particles of large sizeparameters and aspect ratios) where the exact numerical methods such as theT-matrix method [7] fail due to the limitation of current computational resourcesand floating point accuracy.

Many approximate theories for light scattering by small particles havebeen proposed, including the Rayleigh–Gans–Debye (RGD) approximation, theWentzel–Kramers–Brillouin (WKB) approximation, Fraunhofer diffraction ap-proximation, anomalous diffraction theory (ADT), and geometrical optics ap-proximation [8]. Each approximation is valid in a particular domain of the sizeparameter and the relative refractive index of the scatterer. The anomalousdiffraction theory introduced by van de Hulst [1] may be one of the most usefuland intuitive approximate theories for light scattering by optically soft particles

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28 Min Xu and A. Katz

and has been applied to diverse fields such as remote sensing of cirrus cloudsand climate research, biophysical and biomedical research, and other applica-tions [6, 9–15].

This chapter will be limited to a review of some recent development inanomalous diffraction theory of light and its applications in bio-agent detectionand monitoring. Readers are directed to notable books and reviews by van deHulst [1], Kerker [8], Bayvel and Jones [16], Bohren and Huffman [17], Gouesbetand Grehan [9], Shifrin and Tonna [18], Stephens [19], Jones [11], Kokhanovsky[20], Xu [21], Mishchenko, Travis and Lacis [22], and Sharma and Sommerford [23]for broader discussions.

In section 2.2, a statistical interpretation of light anomalous diffraction willbe presented after a brief review of other recent developments in ADT. Theoptical efficiencies are shown to be determined solely by the probability distri-bution function of the geometrical paths of the rays inside the particles, takinginto account particle shape, orientation and polydispersity simultaneously. Thisstatistical view provides a systemic approach to find optical efficiencies for parti-cles of arbitrary shape and at any orientation. The analytical expressions for theoptical efficiencies for spheroids and finite circular cylinders are then presentedfollowed by a discussion of the origin of their difference. This statistical view ofADT reveals what matters for the optical efficiencies of optically soft particlesis the mean and mean-squared-root geometrical paths of the rays. The Gauss-ian ray approximation (GRA) in ADT based on this observation yields simpleanalytical expressions for optical efficiencies of a system of randomly orientedand/or polydisperse scatterers. Section 2.2 ends with discussions on implicationson particle sizing with light scattering techniques.

In section 2.3, experimental results will be presented. Light scattering andextinction measurements were performed to obtain the particle size distributionof three species of bacteria of distinctive shapes (Staphylococcus aureus, Bacillussubtilis, and Pseudomonas aeruginosa) from first a rescaled spectrum of angularlight intensities and then light transmission measurements from 0.35 to 0.60μm.The particle sizes obtained by a simple analysis of light transmission using theGRA to ADT agree well with those obtained by both electron microscopy andangular scattering measurements. Light transmission from 0.40 to 1.0μm wasthen employed to measure changes in the refractive index and the size of B. sub-tilis spores as a method to in situ monitor the germination process of spores.

2.2 Review of recent developments in ADT

Anomalous diffraction theory (ADT) introduced by van de Hulst [1] for lightextinction and scattering is one of the simplest and most powerful approxima-tions of electromagnetic radiation interaction with spherical and nonsphericalsoft particles. The anomalous diffraction theory is based on the premise thatthe extinction of light by a particle is primarily a result of the interference be-tween the rays that pass through the particle with those that do not [24]. Thisapproximation is most applicable to so called ‘soft particles’ with the complex

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2 Statistical Iinterpretation of ADT and its applications 29

relative refractive index m near one and with a characteristic dimension of sizer exceeding the wavelength λ of the incident radiation in the host medium toachieve a high degree of accuracy [24–27]. The first condition

|m − 1| � 1 (2.1)

ensures that the reflection and refraction of the incident light by the particlecan be ignored and the incident ray does not bend when it enters and leaves theparticle. The second condition on the size parameter

x = 2πr/λ � 1 (2.2)

ensures the ray picture is applicable [1, 24].The validity of the anomalous diffraction theory has been investigated by

many authors. [24,25,27–29] Farone and Robinson [25] compared the ADT resultto that from the exact Mie theory for a sphere. Maslowska et al. [29] studied therange of validity of ADT for a cube. Liu et al. [27] compared the ADT result tothat from an exact T-matrix calculation for a finite circular cylinder and foundthat ADT solutions approach the rigorous T-matrix results when the refrac-tive indices approach unity and differences in extinctions between ADT and ex-act solutions generally decrease with nonsphericity. Ackerman and Stephens [24]showed the validity of ADT improves for a polydisperse medium. Sharma sug-gested that the softness condition may be relaxed to |m − 1|2 � |m + 1|2 asopposed to Eq. (2.1) [30]. Sharma and Sommerford [23, 31] recently comparedvarious approximations including ADT for computation of light scattering andabsorption characteristics of optically soft particles.

To improve the accuracy of ADT, in particular, in applications of remotesensing of clouds, the edge effect has been investigated extensively by, for ex-ample, Jones [32, 33], Ackerman and Stephens [24], and Nussenzveig and Wis-combe [34, 35]. The edge effect comes from tunneling [36] where photons whichdo not pass through the particle still interact with the particle. Mitchell [37]significantly improves the accuracy of ADT for light absorption and extinc-tion by spherical particles by parameterizing the missing physics (internal re-flection/refraction, photon tunneling and edge diffraction). Zhao and Hu [38]recently proposed a bridging technique to include the edge contribution in theADT computation. Yang et al. [39] introduced tuning parameters in ADT to bestfit the ADT solution to the Mie solution for spherical particles. The anomalousdiffraction theory has also been used to help understand why Mie theory over-predicted absorption and hence underpredicted the sizes of ice crystals in cirrusclouds [10]. The edge effect is not as important for nonspherical particles as forspherical ones since the ability of a particle to support tunneling and surfaceeffects that arise from tunneling depends on its surface geometry [36].

Another recent development is a simple statistical interpretation of theanomalous diffraction theory by Xu et al. [40] The optical efficiencies in ADTare determined solely by the probability distribution of the geometrical pathsof the rays inside the particles. This ray distribution succinctly summarizes thedependence of the optical efficiencies on the nonsphericity, orientation and poly-dispersity of the scatterers.

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30 Min Xu and A. Katz

Exact analytical results in ADT have only been obtained for particles of a fewshapes, including spheres, spheroids [26,41] and infinite cylinder [42] at arbitraryorientations, and cubes [29], finite cylinders and columns at some special orien-tations [43] due to the difficulty of evaluating the optical efficiencies for complexgeometries. The statistical view provides a straightforward recipe to evaluateoptical efficiencies of particles of arbitrary shape and at any orientation in theframework of ADT. A closed-form analytical formula of optical efficiencies fora finite circular cylinder at an arbitrary orientation has been derived using thismethod [44] for the first time.

The main feature of the optical efficiencies is characterized by just two pa-rameters: the mean and mean-squared-root geometrical paths of the rays. TheGaussian ray approximation (GRA) for ADT [40] that assumes the probabilitydistribution of the geometrical paths of the rays can be approximated by a Gaus-sian function produces simple analytical expressions for the optical efficienciesof a system of randomly oriented and/or polydisperse scatterers [40, 44]. GRAreduces to the exact ADT in the intermediate case [45] of light scattering foran arbitrary soft particle. It has been successfully applied to determine bacte-ria sizes [46] and monitor in situ changes in the refractive index and the size ofspores [12] (see section 2.3).

The following subsections will review this statistical interpretation of ADT(section 2.2.1), ray distributions for various shapes (section 2.2.2), and Gaussianray approximation (section 2.2.3). The performance of GRA and the influence ofthe shape differences between cylinders and spheroids on their optical efficienciesis presented in section 2.2.4. The implication on probing the size and shape ofparticles from light extinction is discussed in section 2.2.5.

2.2.1 Light anomalous diffraction using geometrical path statisticsof rays

The anomalous diffraction theory of light extinction by soft particles has beenshown to be determined by a probability distribution function of the geometricalpaths of individual rays inside the particles [40]. The light extinction by particlesmeasures this probability distribution function of a collection of scatterers ratherthan the size or shape of the individual particles.

In the framework of ADT [1], the extinction, absorption and scattering effi-ciencies of a particle are given by:

Qext =2P

Re∫ ∫

P

[1 − exp(−ikl(mr − 1)) exp(−klmi)] dP, (2.3)

Qabs =1P

∫ ∫P

[1 − exp(−2klmi)] dP,

Qsca = Qext − Qabs,

where Re represents the real part, the wave number is k = 2π/λ for wavelengthλ, the complex relative refractive index m = mr−imi, l is the geometrical path ofan individual ray inside the particle, and P is the projected area of the particle

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2 Statistical Iinterpretation of ADT and its applications 31

in the plane perpendicular to the incident light over which the integration isperformed. The optical efficiencies for a system of randomly oriented and/orpolydisperse particles are averaged over all the sizes and orientations of particlesweighted by their projection areas, i.e.,

Q =∑

PQ∑P

. (2.4)

The integration in Eq. (2.3) over the projected area for a single particle at afixed orientation or the averaging in Eq. (2.4) over the combined projected areafrom all sizes and orientations of particles can be reinterpreted as an averagingover a distribution of the geometrical path l of rays. By dividing the (combined)projection area into equal-area elements and counting the resulting geometricalpaths corresponding to each projection area element according to their lengths,a probability function p(l) dl can be found which describes the probability thatgeometrical path l of a ray is within [l, l+dl), i.e., l ≤ geometrical path < l+dl.The probability function is normalized to

∫p(l) dl = 1. By this interpretation,

we can rewrite the optical efficiencies in Eq. (2.3) as expected values under theprobability distribution p(l) of the geometrical paths of rays. The extinction andabsorption efficiencies in Eq. (2.3) can be expressed as:

Qext = 2Re∫

[1 − exp(−ikl(mr − 1)) exp(−klmi)] p(l) dl, (2.5)

Qabs =∫

[1 − exp(−2klmi)] p(l) dl.

The probability distribution function of the geometrical paths of rays (inshort, ray distribution) p(l) unifies the dependence of the optical efficiencies onthe nonsphericity, orientation and polydispersity of the particle. In the absenceof absorption, Eq. (2.5) can be rewritten as

1 − 12Qext(k) =

∫p(l) cos[kl(mr − 1)] dl (2.6)

and its conjugate

p(l) =π

2|mr − 1|

∫ [1 − 1

2Qext(k)

]cos[kl(mr − 1)] dk. (2.7)

One can further introduce the phase delay distribution p′(l′) ≡ |mr − 1| p( l′|mr−1| )

where l′ = (mr − 1)l is the phase delay and the distribution p′(l′) satisfies∫p′(l′) dl′ = 1, and rewrite Eqs (2.6) and (2.7) as

1 − 12Qext(k) =

∫p′(l′) cos(kl′) dl′, (2.8)

p′(l′) =π

2

∫ [1 − 1

2Qext(k)

]cos(kl′) dk.

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32 Min Xu and A. Katz

Thus, the ray distribution p(l), or more precisely, the phase delay distributionp′(l′) ≡ |mr − 1| p(l′/ |mr − 1|), is the Fourier cosine transform of the spectrumof the extinction efficiency [1 − Qext(k)/2] if light absorption by the particles isnegligible.

Assume the ray distribution for one particle with a unit size is p0(l), the raydistribution for a particle with the same shape, orientation and a different size Lis given by p(l) = 1

Lp0( lL ) from scaling of length. A system of such particles of a

common shape whose size distributes according to a probability density functionn(x) has a ray distribution function

ppol(l) =

∫ 1xp0( l

x )n(x)x2 dx∫n(x)x2 dx

, (2.9)

weighted by the projection area of individual particles which is proportional tox2. The ray distribution for a randomly oriented particle of size L is given by aweighted average over the full 4π solid angle:

prn(l) =∫

p(l)Σ(Ω) dΩ∫Σ(Ω) dΩ

(2.10)

where Σ(Ω) is the projection area of the particle at the orientation Ω. Thesubscript ‘pol’ or ‘rn’ is used to denote a polydisperse particle or a randomlyoriented particle.

2.2.2 Ray distributions for various shapes

By the statistical interpretation of light anomalous diffraction by small parti-cles, the computation of optical efficiencies are reduced to the evaluation of theprobability distribution function p(l) (the ray distribution) of the geometricalpaths of rays inside particles. The ray distribution for particles of even complexshapes can be computed using an approach similar to ray tracing.

2.2.2.1 Spheroids

Let’s first consider a spheroid with a semiaxis b of revolution and an axial ratio ε.The semiaxis of the spheroid in the direction perpendicular to the axis of revolu-tion is a = εb. The incident light forms an angle χ between the revolution axis ofthe spheroid and the propagation direction of the incident beam (see Fig. 2.1).The geometrical length of a ray and the projection area for such a spheroidhas been calculated [26]. We will use the ray tracing approach to compute thegeometrical length of a ray inside the spheroid here.

Inside the coordinate system x′yz′ whose z′ axis coincides with the incidencedirection cosχz + sinχx (0 ≤ χ < π/2) of light (see Fig. 2.1), the spheroid isbounded by

(x′ cosχ + z′ sinχ)2

a2 +y2

a2 +(−x′ sinχ + z′ cosχ)2

b2 = 1. (2.11)

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2 Statistical Iinterpretation of ADT and its applications 33

z

y

x

z’

x’

χ

b

a

Fig. 2.1. A spheroid whose axis of revolution makes an angle of χ with the incidentbeam.

The geometrical path of a ray passing through (x′, y, 0) on the z′ = 0 planeis given by:

l =2ab√

a2 cos2 χ + b2 sin2 χ

√1 −

(a2 cos2 χ + b2 sin2 χ

)−1x′2 − a−2y2. (2.12)

The projected area of the spheroid is the area formed by the points oftangency (l = 0) on the z′ = 0 plane, yielding an ellipse with semiaxes√

a2 cos2 χ + b2 sin2 χ and a. The projection area is given by

Σ = πa

√a2 cos2 χ + b2 sin2 χ = πεb2

√ε2 cos2 χ + sin2 χ. (2.13)

Rewrite x′ =√

a2 cos2 χ + b2 sin2 χρ cos θ and y = aρ sin θ where 0 ≤ ρ ≤ 1 and0 ≤ θ ≤ 2π, the geometrical path (2.12) can be written as

l =2ab√

a2 cos2 χ + b2 sin2 χ

(1 − ρ2)1/2

. (2.14)

The area inside the projection area Σ which results in a geometrical path within[l, l + dl) is then

Page 55: Light Scattering Reviews 3: Light Scattering and Reflection

34 Min Xu and A. Katz

dΣ = a

√a2 cos2 χ + b2 sin2 χ2πρ dρ =

π(ε2 cos2 χ + sin2 χ

)3/2

2εl dl. (2.15)

Thus, the geometrical path distribution of the rays is given by

psph(l) =dΣ

Σ

=1

2ε2b2 (ε2 cos2 χ + sin2 χ)l H(2εb√

ε2 cos2 χ + sin2 χ− l),

l ≥ 0 (2.16)

where H(x) is a Heaviside function. This reduces to

psph(l) =1

2b2 l H(2b − l), ε = 1 (2.17)

for a sphere (ε = 1).The ray distribution for randomly oriented spheroids is obtained by using

Eq. (2.10) to be:

psphrn (l) =

38ε2b2 l

{[1 +

23

ε3

ε + (ε2 − 1)−1/2 ln(ε +

√ε2 − 1

)]H(2b − l) H(l)

+23αγ3 + αγ +

(ε2 − 1

)−1/2 ln(γ + α

√ε2 − 1

)ε + (ε2 − 1)−1/2 ln

(ε +

√ε2 − 1

) H(l − 2b) H(2εb − l)

}(2.18)

when ε > 1, and

psphrn (l) =

38ε2b2 l

{[1 +

23

ε3

ε + (1 − ε2)−1/2 arcsin√

1 − ε2

]H(2εb − l) H(l)

+23ε

3 + ε − αγ(1 + 2

3γ2)

+(1 − ε2

)−1/2 [arcsin√

1 − ε2 − arcsin(α√

1 − ε2)]

ε + (1 − ε2)−1/2 arcsin√

1 − ε2

× H(l − 2εb) H(2b − l)} (2.19)

when ε ≤ 1. The parameters α =

√4ε2b2 − l2

(ε2 − 1)l2and γ = 2εb/l.

The ray distribution for a system of such spheroids at a fixed orientation χwith a lognormal size distribution [47]

n(x) =1

(2π)1/2σr−1 exp

[− ln2(r/am)

2σ2

](2.20)

for the semiaxis r of revolution is given by

psphpol (l) =

(ε−2 sin2 χ + cos2 χ)l4

erfc(1√2σ

ln(ε−2 sin2 χ + cos2 χ)1/2l

2am)

a2m exp(2σ2)

(2.21)

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2 Statistical Iinterpretation of ADT and its applications 35

utilizing Eq. (2.9) where erfc(x) is the complementary error function. The raydistribution becomes

psphpol,rn(l) =

∫ 10 psph

pol (l)πε2√

ε−2 sin2 χ + cos2 χd cosχ∫ 10 πε2

√ε−2 sin2 χ + cos2 χd cosχ

(2.22)

for such polydisperse spheroids when they are randomly oriented.It is worth noting here that the ray distribution for a single spheroid at a fixed

orientation given by Eq. (2.16) is a triangular regardless of the axial ratio of thespheroid. This fundamental geometrical characteristics enables a simple radiusrescaling to calculate the optical efficiencies from a sphere for a spheroid [26].

2.2.2.2 Finite cylinders

Now consider a finite cylinder with radius a and height L. The cylinder isbounded by side I: z = L/2, side II: z = −L/2 and side III: x2/a2 + y2/a2 = 1.The incident light is in the direction of cosχz +sinχx (0 ≤ χ < π/2). Let us ro-tate the coordinate system xyz along the y axis for an angle χ to x′yz′ such thatz′ axis coincides with the incident direction of light, the boundary of cylinder isthen given by

−x′ sinχ + z′ cosχ = L/2, side I (2.23)−x′ sinχ + z′ cosχ = −L/2, side II

(x′ cosχ + z′ sinχ)2

a2 +y2

a2 = 1, side III

in the new coordinate system x′yz′ (see Fig. 2.2).The incident beam passing through (x′, y, 0) on the z′ = 0 plane intersects

the boundaries of the cylinder at points (x′, y, z′1,2) = a(ξ cosχ, η, ζ1,2) according

to

ζ2 =

⎧⎪⎪⎨⎪⎪⎩e + ξ sinχ cosχ

cosχif |ξ + β| ≤

√1 − η2√

1 − η2 − ξ cos2 χ

sinχotherwise

(2.24)

ζ1 =

⎧⎪⎪⎨⎪⎪⎩−e + ξ sinχ cosχ

cosχif |ξ − β| ≤

√1 − η2

−√

1 − η2 − ξ cos2 χ

sinχotherwise

in the coordinate system x′yz′ where β ≡ L tanχ/2a = e tanχ and e ≡ L/2a. ξand η form a set of unitless bases for the projected area of the cylinder.

Page 57: Light Scattering Reviews 3: Light Scattering and Reflection

36 Min Xu and A. Katz

z

y

x

z’

x’

a

χ

L

I

II

III

Fig. 2.2. A finite circular cylinder is bounded by sides I, II and III. The axis ofrevolution of the cylinder makes an angle of χ with the incident beam.

Hence, we obtain the geometrical path l inside the cylinder

l/a = ζ2 − ζ1 =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

ξ + β +√

1 − η2

sinχif |ξ + β| ≤

√1 − η2

−ξ + β +√

1 − η2

sinχif |ξ − β| ≤

√1 − η2

2√

1 − η2

sinχotherwise

(2.25)

if β ≥√

1 − η2, and

l/a = ζ2 − ζ1 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ξ + β +√

1 − η2

sinχif −

√1 − η2 − β ≤ ξ ≤ −

√1 − η2 + β

2e/ cosχ if −√

1 − η2 + β ≤ ξ ≤√

1 − η2 − β

−ξ + β +√

1 − η2

sinχif√

1 − η2 − β ≤ ξ ≤√

1 − η2 + β

2√

1 − η2

sinχotherwise

(2.26)if β <

√1 − η2.

Page 58: Light Scattering Reviews 3: Light Scattering and Reflection

2 Statistical Iinterpretation of ADT and its applications 37

The projected area of the cylinder is the area formed by the points of tangency(l = 0) on the z′ = 0 plane. This area is enclosed inside η = ±1 and (ξ±β)2+η2 =1 by equating Eqs (2.25) and (2.26) to zero and hence is given by (see Fig. 2.3):

Σ = 2aL sinχ + πa2 cosχ = 4a2 cosχ(β +π

4). (2.27)

The geometrical path l in Eqs (2.25) and (2.26) can be rewritten as

t ≡ l sinχ

a= (ζ2 − ζ1) sinχ =

⎧⎪⎪⎨⎪⎪⎩ξ + β +

√1 − η2 in I

−ξ + β +√

1 − η2 in II2β in III

2√

1 − η2 in IV

(2.28)

η

ξ

I IIIII

IV

IV

CO

A

B

D

E F

G

β−β

(a)

η

ξ

I IIIV

IV

C

O

A D

E F

G

β−β

(b)

Fig. 2.3. The projected area of a cylinder whose revolution axis makes an angle of χwith the incident light. (a) β < 1 and (b) β ≥ 1.

Page 59: Light Scattering Reviews 3: Light Scattering and Reflection

38 Min Xu and A. Katz

inside different regions I, II, III and IV of the projected area as specified inFig. (2.3) where the region III is absent for β ≥ 1. The quantity t defined in(2.28) will be called the scaled geometrical path later.

Due to the symmetry presented here, we only need to consider the firstquadrant in Fig. 2.3 in the calculation of the geometrical path distribution of raysfor the cylinder. The distribution function p(l) of the geometrical paths of rays isproportional to the total area density q(t) inside the first quadrant. Here q(t) dtdescribes the area inside the first quadrant through which the scaled geometricalpath of rays is within [t, t+dt). After a straightforward computation [44], we find:

q(t)=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

[√4 − t2

2+

t2 (β − t

2 )√4 − t2

]H(2β − t)+

arccosβ − β√

1 − β2

2δ(t − 2β) β < 1[√

4 − t2

2+

t2 (β − t

2 )√4 − t2

]H(2 − t) β ≥ 1

(2.29)where H(x) is the Heaviside function and δ(x) is the Dirac delta function.Eq. (2.29) satisfies ∫ +∞

0q(t) dt = β +

π

4, (2.30)

yielding the projected area inside the first quadrant on the ξη plane as expected.The geometrical path distribution of rays for a finite cylinder whose revolutionaxis forms an angle of χ with the incident light is given by:

pcyl(l) =sinχ

a

q( l

asinχ

)β +

π

4

. (2.31)

The geometrical path distribution for a randomly oriented cylinder is ob-tained by

pcylrn (l) =

∫ π/2

0dχ

1a

sin2 χ cosχq( l

asinχ

)∫ π/2

0dχ sinχ cosχ

(e tanχ +

π

4

) (2.32)

weighted by the projection area given in Eq. (2.27) of the particle oriented at anangle of χ with respect to the incident light. The geometrical path distributionfor a randomly oriented cylinder is found to be,

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2 Statistical Iinterpretation of ADT and its applications 39

pcylrn (l) =

1π8 (1 + 2e)a

{H(e − x)D(arctan e−1, x) (2.33)

+H(x − e) H(√

e2 + 1 − x)[D(arctan e−1, x) − D

(arccos

e

x, x)]

+H(1 − x)[D(π

2, x)

− D(arctan e−1, x)]

+H(x − 1) H(√

e2 + 1 − x)[D(arcsinx−1, x) − D(arctan e−1, x)

]+H(x − e) H(

√e2 + 1 − x)

· e2

4x3

[arccos

√x2 − e2 −

√(x2 − e2)(1 + e2 − x2)

]}in which x ≡ l/2a and the function D(χ, x) is defined as:

D(χ, x) =∫

dχ sin2 χ cosχ

[Δ +

x sinχ(e tanχ − x sinχ)2Δ

](2.34)

=6x2 sin2 χ + 1

16x2 sinχΔ − 116x3 arcsin(x sinχ)

+e sinχ cosχΔ

6x+

e(2 + x2)6x3 F (χ, x) − e(1 + x2)

3x3 E(χ, x)

where Δ ≡√

1 − x2 sin2 χ and F (χ, x) and E(χ, x) are elliptic integral of thefirst and second kind:

F (χ, x) =∫ χ

0

da√1 − x2 sin2 a

, E(χ, x) =∫ χ

0

√1 − x2 sin2 a da. (2.35)

The ray distribution for polydisperse cylinders can be obtained similarly.Table 2.1 summaries the ray distribution for spheres, spheroids and finite cylin-ders.

Table 2.1. The ray distribution for spheres of radius b, spheroids with a semiaxis ofrevolution b and aspect ratio ε = a/b, and cylinders with radius a and height L andaspect ratio ε = e−1 = 2a/L. Spheroids and cylinders are either oriented at an angleχ or randomly oriented

Ray distribution p(l)

Sphere 12b2

l H(l)H(2b − l)

Spheroid1

2ε2b2 (ε2 cos2 χ+sin2 χ)l H(l)H(

2εb√ε2 cos2 χ+sin2 χ

−l

)Randomly oriented spheroid Eqs (2.18) and (2.19)

Cylinder Eq. (2.31)

Randomly oriented cylinder Eq. (2.33)

Page 61: Light Scattering Reviews 3: Light Scattering and Reflection

40 Min Xu and A. Katz

2.2.2.3 Comparison of ray distributions for spheroids and cylinders

The ray distributions from a finite circular cylinder, a randomly oriented cylin-der, a system of polydisperse cylinders at a fixed orientation and a system ofrandomly oriented polydisperse cylinders are plotted in Fig. 2.4. Figure 2.4(a)and Fig. 2.4(b) show the ray distribution for a cylinder of an axial ratio ε = 0.5and ε = 2, respectively. The ray distribution for a monosized cylinder at afixed orientation (solid lines in Fig. 2.4) is close to flat except for a peak at thelargest value of the ray path. The peak is a delta-function for a cylinder satisfy-ing β = ε−1 tanχ < 1, originating from the second term in Eq. (2.29) where therays intersect with sides I and II of the cylinder [Fig. 2.4(b)]. Two peaks occur atthe values of the ray path of the diameter 2a and the height L of the cylinder forthe ray distribution of randomly oriented monosized cylinders (long dash lines inFig. 2.4). The ray distribution for a polydisperse finite circular cylinder, eitherat a fixed orientation (dashed lines in Fig. 2.4) or randomly oriented (short dashlines in Fig. 2.4), pertains a smooth bell shape whose left wing is pushed up dueto the nonzero probability density of zero geometrical paths originated from thesharp edges of cylinders [44]. This probability density of zero geometrical pathsapproaches zero with the decrease of the axial ratio ε (like a disk rather than aneedle) and the increase of the dispersion σ of the particle size distribution ofthe cylinder.

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3

Pro

babi

lity

Den

sity

Ray Path

FXRN

POL FXPOL RN

0

1

2

0 2 4

n(x)

x

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7 8

Pro

babi

lity

Den

sity

Ray Path

Pδ(l−2√2) FXRN

POL FXPOL RN

0

1

2

0 2 4

n(x)

x

(b)

Fig. 2.4. The ray distributions for a finite circular cylinder at a fixed orientationχ = π/4 (FX), randomly oriented (RN), polydisperse at a fixed orientation χ = π/4(POL FX), and randomly oriented polydisperse (POL RN). The axial ratio of the cylin-der is (a) ε = 0.5, (b) ε = 2. The height of the cylinder L = 2 for the monosized cylin-der. The lognormal size distribution n(x) of the half height (L/2) of the polydispersecylinder with am = 1 and σ = 0.2 is also plotted as an inset. The height of the delta-function peak in (b) for the monosized cylinder at a fixed orientation is P = (4π−3

√3)/

6(π + 2) � 0.239.

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2 Statistical Iinterpretation of ADT and its applications 41

The ray distributions for spheroids with axial ratios ε = 0.5 and ε = 2 areplotted in Fig. 2.5. The ray distribution for a spheroid at a fixed orientation (solidline in Fig. 2.5) is triangular. One peak occurs at the value of the ray path of theshorter diameter 2a of the spheroid for the ray distribution of randomly orientedmonosized spheroids (long dash lines in Fig. 2.5). The ray distribution for apolydisperse spheroid, either at a fixed orientation (dashed lines in Fig. 2.5) orrandomly oriented (short dash lines in Fig. 2.5), has a smooth bell shape. Onesignificant feature of the ray distribution of spheroids is the zero probabilitydensity of zero ray path.

The different characteristics of the shapes of cylinders and spheroids produceunique features in the geometrical path distribution. One significant differencebetween a cylinder and a spheroid is the presence of the appreciable nonzeroprobability density of zero ray paths for the cylinder due to its sharp edgeswhile the probability density of zero ray paths for spheroids is always zero.

The ray distribution for a randomly oriented cylinder has two peaks locatedat its diameter 2a and height L. By contrast, only one peak appears in theray distribution for a randomly oriented spheroid at the length of its shorteraxis. The second peak in the ray distribution of the cylinder diminishes with thedecrease of the aspect ratio, yielding a ray distribution more similar to that ofthe spheroid.

0

0.5

1

1.5

2

0 1 2 3

Pro

babi

lity

Den

sity

Ray Path

FXRN

POL FXPOL RN

0

1

2

0 2 4

n(x)

x

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7 8

Pro

babi

lity

Den

sity

Ray Path

FXRN

POL FXPOL RN

0

1

2

0 2 4

n(x)

x

(b)

Fig. 2.5. The ray distributions for a spheroid at a fixed orientation χ = π/4 (FX),randomly oriented (RN), polydisperse at a fixed orientation (POL FX), and randomlyoriented polydisperse (POL RN). The axial ratio of the spheroid is (a) ε = 0.5 and(b) ε = 2. The semiaxis of revolution of the monosized spheroid is one. The lognormalsize distribution n(x) with am = 1 and σ = 0.2 for the semiaxis of revolution of thespheroid is also plotted as an inset.

Page 63: Light Scattering Reviews 3: Light Scattering and Reflection

42 Min Xu and A. Katz

2.2.2.4 Mean and mean-squared-root geometrical paths for spheroidsand cylinders

The mean and mean-squared geometrical path of rays can be easily obtainedfrom the geometrical path distribution of rays. The mean geometrical path andmean-squared geometrical path are given by:

〈l〉sph =43

εb√ε2 cos2 χ + sin2 χ

(2.36)

⟨l2⟩sph

=2ε2b2

ε2 cos2 χ + sin2 χ

for a spheroid oriented at an angle χ with respect to the incident light, respec-tively. The corresponding paths become:

〈l〉sphrn =

⎧⎪⎪⎨⎪⎪⎩8ε/3

ε + (1 − ε2)−1/2 arcsin√

1 − ε2b ε ≤ 1

8ε/3

ε + (ε2 − 1)−1/2 ln(ε +

√ε2 − 1

)b ε > 1(2.37)

⟨l2⟩sphrn =

⎧⎪⎪⎪⎨⎪⎪⎪⎩4 arcsin

√1 − ε2

ε√

1 − ε2 + arcsin√

1 − ε2ε2b2 ε ≤ 1

4 ln(ε +

√ε2 − 1

)ε√

ε2 − 1 + ln(ε +

√ε2 − 1

)ε2b2 ε > 1

if the spheroid is randomly oriented.The mean and mean-squared geometrical paths for a cylinder oriented at an

angle χ are given by

〈l〉cyl =a

sinχ

πβ

2(β + π4 )

(2.38)

and

⟨l2⟩cyl

=

⎧⎪⎪⎨⎪⎪⎩a2

sin2 χ

1β+ π

4

[83β− π

4+

12(1 + 4β2) arccosβ− 1

6β(2β2+13)

√1−β2

]β<1

a2

sin2 χ

83β − π

4

β + π4

β≥1

(2.39)from Eq. (2.31), respectively.

The mean and mean-squared geometrical path for a randomly oriented fi-nite cylinder are an average over all the orientation angles weighted by theirprojection area:

〈l〉cylrn =

∫ π/2

0dχ sinχ 〈l〉 cosχ

(β +

π

4

)∫ π/2

0dχ sinχ cosχ

(β +

π

4

) =4e

1 + 2ea (2.40)

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2 Statistical Iinterpretation of ADT and its applications 43

and

⟨l2⟩cylrn =

∫ π/2

0dχ sinχ

⟨l2⟩cosχ

(β +

π

4

)∫ π/2

0dχ sinχ cosχ

(β +

π

4

) (2.41)

=8a2e2

2e + 1

{43e

+e

6(e +√

e2 + 1)− 13

√e2 + 112e

+ 1

+ lne +

√e2 + 1

2e− 1

4e2 ln(e +√

e2 + 1)

}

=8a2e2

2e + 1

⎧⎪⎨⎪⎩43e−1 − 1

4

(ln 2e +

54

)e−2 − 1

96e−4 +

1768

e−6 + O(e−7) e→∞

ln1e

+ 1 − ln 2 +23e − 1

6e2 +

130

e3 − 1420

e5 + O(e6) e→0

respectively. Note e ≡ ε−1. The mean geometrical path 〈l〉rn = V/ 〈Σ〉rn whereV is the volume of the particle and 〈Σ〉 is the mean geometrical projection areaof the particle, and 〈Σ〉rn = S/4 for a randomly oriented convex particle whosesurface area is S.

The mean and mean-squared ray paths of a polydisperse particle relate simplyto that of a monosized particle by

〈l〉pol = 〈l〉0 exp(5σ2/2) (2.42)

and ⟨l2⟩pol =

⟨l2⟩0 exp(6σ2) (2.43)

where the polydisperse particle has a lognormal particle size distribution givenby Eq. (2.20) characterized by am and σ, and 〈〉0 is the corresponding value ofa particle with the radius am.

Table 2.2 lists the average geometrical cross-sections, the mean and the mean-squared-root geometrical paths for a sphere of radius a, a randomly orientedspheroid with a semiaxis b of revolution and aspect ratio ε = a/b, and a randomlyoriented finite circular cylinder with height L and radius a.

Fig. 2.6 plots the mean and mean-squared-root geometrical paths for a ran-domly oriented cylinder and spheroid. The cylinder and the spheroid are of acommon aspect ratio ε and of a common surface area equal to that of a sphere ofradius as. The mean and mean-squared-root ray paths of the spheroid are largerthan those of the cylinder when the aspect ratio is near one. The situation isreversed in both limits of small and large aspect ratios. In the limit of a smallaspect ratio ε � 1, both the spheroid and the cylinder approaches a needle withthe mean and mean-squared-root ray paths given by

〈l〉sph =32

3π3/2 ε1/2as � 1.92ε1/2as,

√〈l2〉sph = 4√

πε1/2as � 2.26ε1/2as

〈l〉cyl = 2ε1/2as,

√〈l2〉cyl = 4√

3ε1/2as � 2.31ε1/2as

Page 65: Light Scattering Reviews 3: Light Scattering and Reflection

44 Min Xu and A. Katz

Table 2.2. Average geometrical cross sections, mean, and mean-squared-root geomet-rical paths of spheres, randomly oriented spheroids and finite circular cylinders. Theradius of the sphere is a. The spheroid has a semiaxis of revolution b with an aspectratio ε = a/b. The finite circular cylinder has a height of L and radius a with an aspectratio ε = e−1 = 2a/L.

Properties sphere spheroid Cylinder

〈Σ〉rn πa2

π

2ε2b2+

π

2εb2(1−ε2)−1/2

arcsin√

1−ε2 ε ≤ 1

π

2ε2b2+

π

2εb2 (

ε2−1)−1/2

ln(ε +

√ε2−1

)ε > 1

π2 a2(2e + 1)

〈l〉rn 43a

8ε/3ε + (1 − ε2)−1/2 arcsin

√1 − ε2

b ε ≤ 1

8ε/3ε + (ε2 − 1)−1/2 ln

(ε +

√ε2 − 1

) b ε > 1

2a

1 + a/L

√〈l2〉rn√

2a

√arcsin

√1 − ε2

ε√

1 − ε2 + arcsin√

1 − ε22εb ε ≤ 1

√ln

(ε +

√ε2 − 1

)ε√

ε2 − 1 + ln(ε +

√ε2 − 1

)2εb ε > 1

√Eq. (2.41)

respectively. In the limit of a large aspect ratio ε � 1, both the spheroid andthe cylinder approaches a disk with the mean and mean-squared-root ray pathsgiven by

〈l〉sph =83

√2ε−1as � 3.77ε−1as,

√〈l2〉sph = 2

√2

√ln εε as � 2.83

√ln εε as

〈l〉cyl = 4√

2ε−1as � 5.66ε−1as,

√〈l2〉cyl = 4

√ln εε as

respectively. The difference in the mean and mean-squared-root ray paths of thecylinder and the spheroid tends to be negligible for small aspect ratios (needle-like) but is significant for large aspect ratios (disk-like). This is consistent withthe observation that the second peak in the ray distribution of a randomlyoriented cylinder diminishes with the decrease of the aspect ratio and becomesmore similar to that of the spheroid.

Fig. 2.7 plots the ratio of the ray path dispersion (√

〈l2〉 − 〈l〉2/ 〈l〉) overthe mean geometrical path for cylinders and spheroids. Near the region of unityaspect ratio, the ratio for spheroids is much less than that of cylinders, as thespheroid has a smaller dispersion in the geometrical ray paths than the cylin-der.

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2 Statistical Iinterpretation of ADT and its applications 45

0

0.5

1

1.5

2

0.01 0.1 1 10 100

Leng

th/a

s

ε

Spheroid √<l2><l>

Cylinder √<l2><l>

Fig. 2.6. The mean and mean-squared-root geometrical path for a randomly orientedcylinder and spheroid with a common aspect ratio ε and a common surface areas of asphere of radius as.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.01 0.1 1 10 100

√<l2 >

−<l>

2 /<

l>

ε

SpheroidCylinder

Fig. 2.7. The ratio of the ray path dispersion over the mean geometrical path forcylinders and spheroids.

Page 67: Light Scattering Reviews 3: Light Scattering and Reflection

46 Min Xu and A. Katz

2.2.3 Gaussian ray approximation

The ray distributions from a single spheroid, a single randomly oriented spheroid,a system of polydisperse spheroids at a fixed orientation and a system of ran-domly oriented polydisperse spheroids are plotted in Fig. 2.5. It is clear from thefigure that the shape characteristics of an individual particle are washed out bythe averaging over the polydispersity and the orientations of the particle. Theshape characteristics of an individual particle are expected to be further washedout if particles of different shapes are involved. Thus, the ray distribution p(l)for a system of particles such as a bacteria suspension, biological cells or cirrusclouds where particles are polydisperse, randomly oriented and/or of multipleshapes approaches a probability density function p(l) characterized essentiallyby the mean geometrical path 〈l〉 =

∫lp(l)dl and mean-squared geometrical

path⟨l2⟩

=∫

l2p(l)dl of rays inside the particles. One natural choice here is theGaussian probability distribution function following the same spirit of the wellknown central limit theorem [48]. We should point out that this choice does notsatisfy p(l < 0) = 0 but the contribution from near the l = 0 region in the raydistribution is much smaller compared to that from other regions and hence canbe ignored.

Let’s now assume the ray distribution is given by a Gaussian distribution

p(x) =1√2πν

exp(

− (x − μ)2

2ν2

), (2.44)

the extinction and scattering efficiencies are then given by

Qext = 2 − 2 cos[k(mr − 1)(μ − kν2mi)

]exp

[−kμmi − k2ν2((mr − 1)2 − m2

i )2

]Qabs = 1 − exp

[−2kmi(μ − kmiν

2)]

(2.45)

from Eq. (2.5) after a straightforward integration. The optical efficiencies Eq.(2.45), in the intermediate case limit (k(mr − 1)l � 1 and kmil � 1 where l isthe geometrical path) [45], reduce to

Qext = 2kmi 〈l〉 + k2 [(mr − 1)2 − m2i] ⟨

l2⟩

(2.46)

Qabs = 2kmi 〈l〉 − 2k2m2i⟨l2⟩

Qsca = k2 |m − 1|2⟨l2⟩

where the mean and mean-squared geometrical paths are given by 〈l〉 = μand

⟨l2⟩

= μ2 + ν2 respectively. These results agree exactly with those of theintermediate region over which the Rayleigh–Gans–Debye approximation andthe anomalous diffraction theory of light scattering from small particles over-lap [1, 45]. This means that Eq. (2.45) from our Gaussian ray approximationreduces to the exact ADT in the intermediate case.

For nonabsorbing scatterers, the scattering efficiency in the GRA reduces to

Qsca = 2 − 2 cos [k(m − 1)μ] exp[−k2ν2(m − 1)2

2

]. (2.47)

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2 Statistical Iinterpretation of ADT and its applications 47

If the soft scatterer is not too large (k |m − 1|μ < 1), Eq. (2.47) can be expandedand simplified to

Qsca = k2(m − 1)2(μ2 + ν2) − k4

12(m − 1)4(μ4 + 6μ2ν2 + 3ν4), (2.48)

by retaining only the first two leading terms.GRA provides simple analytical expressions (2.47) and (2.48) to analyze the

optical density of a system of soft particles. The spectrum of the optical densityOD or the light extinction K of the system of soft particles is proportionalto the scattering cross-section Csca = GQsca where G is the mean geometricalcross-section. From Eq. (2.48), Csca (and OD, K) can be written in a form ofC2λ

−2 + C4λ−4. The two parameters C2 and C4 can then be easily fitted from

the spectrum and provide a simple means to find the changes in the refractiveindex and the size of soft particles noting that (m − 1) ∝ C4/C

3/22 and the

characteristic size a ∝ C2/C1/24 . GRA and, in particular, the above method, has

been used successfully in characterization of bacteria sizes and in situ monitoringof spore germination (see section 2.3).

2.2.4 Performance of Gaussian ray approximation and difference inoptical efficiencies between cylinders and spheroids

Both random orientation and polydispersity of particles tend to smear the char-acteristic features of the ray distribution of the particle. The ray distributionsof the polydisperse cylinder and spheroid, either randomly oriented or not, ap-proach a bell shape. The characteristic features of the ray distribution of aparticle are gradually washed out (see Figs 2.4 and 2.5). The main feature ofthe ray distribution is captured by its mean and mean-squared-root geometricalpaths.

Thus the Gaussian ray approximation, dependent only on the mean andmean-squared-root geometrical paths, becomes a good approximation for anoma-lous light diffraction for polydisperse and/or randomly oriented particles. Figs 2.8–2.11 plot the optical efficiencies of cylinders and spheroids with a common surfacearea and with aspect ratios ε = 0.5 and ε = 2 for cases: with a fixed orientation,randomly oriented, polydisperse and with a fixed orientation, and polydisperseand randomly oriented, respectively. The relative refractive index of cylindersand spheroids is m = 1.05 − i0.0005.

The Gaussian ray approximation reduces to the exact ADT in both limitsof small and large size parameters of the particle. The maximum relative errorof the absorption efficiency of the Gaussian ray approximation compared withthe exact ADT is less than 0.3% for the cylinders in Figs 2.8–2.11, except forthe cylinder with an aspect ratio ε = 0.5 and fixed in an orientation of χ = π/4whose maximum relative error equals 3.1% (see Fig. 2.8). The maximum relativeerror of the absorption efficiency is less than 0.07% for the spheroids as shownin Figs 2.8–2.10 and about 0.25% as shown in Fig. 2.11 for the polydisperse andrandomly oriented spheroids.

Page 69: Light Scattering Reviews 3: Light Scattering and Reflection

48 Min Xu and A. Katz

0

0.5

1

1.5

2

2.5

3

3.5

0 50 100 150 200 250 300 350

Opt

ical

Effi

cien

cy

Equivalence Size Parameter

Qext CYL ADTGaus

SPH ADTGaus

Qabs CYL ADTGaus

SPH ADTGaus

(a)

0

0.5

1

1.5

2

2.5

3

3.5

0 50 100 150 200 250 300 350

Opt

ical

Effi

cien

cy

Equivalence Size Parameter

Qext CYL ADTGaus

SPH ADTGaus

Qabs CYL ADTGaus

SPH ADTGaus

(b)

Fig. 2.8. The extinction and absorption efficiencies of cylinders and spheroids withaspect ratios (a) ε = 0.5 and (b) ε = 2. The equivalence size parameter is the sizeparameter of the sphere whose surface area is the same as the cylinder and the spheroid.Both the cylinder and the spheroid are oriented at a fixed orientation χ = π/4. Therelative refractive index of both cylinders and spheroids is m = 1.05 − i0.0005.

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350

Opt

ical

Effi

cien

cy

Equivalence Size Parameter

Qext CYL ADTGaus

SPH ADTGaus

Qabs CYL ADTGaus

SPH ADTGaus

(a)

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350

Opt

ical

Effi

cien

cy

Equivalence Size Parameter

Qext CYL ADTGaus

SPH ADTGaus

Qabs CYL ADTGaus

SPH ADTGaus

(b)

Fig. 2.9. The extinction and absorption efficiencies of cylinders and spheroids withaspect ratios (a) ε = 0.5 and (b) ε = 2. Both the cylinder and the spheroid are randomlyoriented.

The maximum relative error of the extinction efficiency is about 35% for themonosized cylinder fixed in the orientation χ = π/4 as shown in Fig. 2.8 andprogressively reduces to less than 8% for the polydisperse and random orientedcylinders as shown in Fig. 2.11. The maximum relative error of the extinctionefficiency of the spheroid is about 25% when the spheroid is fixed in an orien-

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2 Statistical Iinterpretation of ADT and its applications 49

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350

Opt

ical

Effi

cien

cy

Equivalence Size Parameter

Qext CYL ADTGaus

SPH ADTGaus

Qabs CYL ADTGaus

SPH ADTGaus

(a)

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350

Opt

ical

Effi

cien

cyEquivalence Size Parameter

Qext CYL ADTGaus

SPH ADTGaus

Qabs CYL ADTGaus

SPH ADTGaus

(b)

Fig. 2.10. The extinction and absorption efficiencies of cylinders and spheroids withaspect ratios (a) ε = 0.5 and (b) ε = 2. The equivalence size parameter is the sizeparameter of the sphere of an equivalent surface area of the respective particles of sizeam. The dispersion of the lognormal size distribution of the cylinder and the spheroidis σ = 0.2. Both the cylinder and the spheroid are polydisperse and oriented at a fixedorientation χ = π/4.

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350

Opt

ical

Effi

cien

cy

Equivalence Size Parameter

Qext CYL ADTGaus

SPH ADTGaus

Qabs CYL ADTGaus

SPH ADTGaus

(a)

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350

Opt

ical

Effi

cien

cy

Equivalence Size Parameter

Qext CYL ADTGaus

SPH ADTGaus

Qabs CYL ADTGaus

SPH ADTGaus

(b)

Fig. 2.11. The extinction and absorption efficiencies of cylinders and spheroids withaspect ratios (a) ε = 0.5 and (b) ε = 2. Both the cylinder and the spheroid arepolydisperse and randomly oriented.

tation of χ = π/4. The maximum relative error reduces to be less than 4% forthe monosized and randomly oriented spheroid. This value becomes about 8%for the polydisperse and randomly oriented spheroids. The abnormal increaseof error in the Gaussian ray approximation for a polydisperse and randomly

Page 71: Light Scattering Reviews 3: Light Scattering and Reflection

50 Min Xu and A. Katz

oriented spheroid compared to that for a monosized randomly oriented spheroidreflects that the ray distribution of the latter is closer to a Gaussian distribution.This means excessive polydispersity may occasionally degrade the accuracy ofthe Gaussian ray approximation of anomalous light diffraction.

Some comparisons between the optical efficiencies of cylinders and spheroidsare in order. The Gaussian approximation works extremely well for the absorp-tion efficiency of both cylinders and spheroids. For the extinction efficiency, theGaussian ray approximation works better for spheroids than for cylinders. Thiscan be attributed to the appreciable nonzero probability density of zero raypaths for the cylinder (the left wing of the bell shape of the ray distribution ispushed up). For the aspect ratios ε = 0.5 and ε = 2 plotted in Figs 2.8–2.11, theabsorption efficiency of the spheroids is larger than that of the cylinders, mainlydue to a larger mean geometrical path of rays for spheroids in that region ofaspect ratios (see Fig. 2.6). This same fact also explains why the first peak ofthe extinction curve of the spheroid is higher than that of the cylinder.

The periodic structure of extinction curves links closely to the peaks pre-sented in the ray distribution of the particles. One sharp peak in the ray dis-tribution produces a train of exponentially decaying sinusoidal peaks in theextinction curve whose spacing is inversely proportional to the positioning ofthe peak in the ray distribution. This is most evident in Fig. 2.9(b) where theextinction curve exhibits the composite of two trains of exponentially decayingsinusoidal peaks originated from the two peaks presented in its ray distributionof the randomly oriented cylinder of an aspect ratio ε = 2.

2.2.5 Implications on particle sizing with light scattering techniques

The statistical interpretation of ADT opens a new way to compute and appre-ciate optical efficiencies of soft particles of different shapes using the probabilitydistribution of the geometrical paths of individual rays inside particles. Lightextinction is determined by the probability distribution of the geometrical pathsof individual rays inside the particles rather than the size or shape of an individ-ual particle. Thus the optical efficiency equivalence [49] can be easily achievedfrom different-shaped particles or particles of different size distribution as longas they share a common geometrical path distribution of rays.

The geometrical path distribution of rays can be approximated by a Gaussianprobability distribution function for a system of particles in which particles arerandomly oriented, polydisperse, and/or multiple-shaped. For such a system ofparticles, the light extinction measurements essentially determine the mean andmean-squared-root geometrical paths of rays from all particles in the system.The shape and size of an individual particle can only be deduced with a prioriinformation of the shape and/or size distribution of the particles involved.

For particles of internal structures and with a nonuniform refractive indexdistribution, the geometrical length of a ray can be replaced by the effective geo-metrical length l = (1/(m − 1))

∑(mi−1)li of the ray where li is the geometrical

length of the segment of the ray within the region of a refractive index mi andm is the volume average of the refractive index of the particle. An effective ray

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2 Statistical Iinterpretation of ADT and its applications 51

distribution p(l) can then be obtained following the same procedure discussedin section 2.2.1. Eq. (2.5) now should read

Qext = 2Re∫ [

1 − exp(−ikl(mr − 1)) exp(−klmi)]p(l) dl, (2.49)

Qabs =∫ [

1 − exp(−2klmi)]p(l) dl.

This variation is used in section 2.3.3 in obtaining changes in the refractiveindex and the size from light extinction Eqs (2.55), (2.56) and (2.57) for sporesmodeled as coated spheres using GRA.

The pursuit of the mean and the mean-squared-root path from fitting GRAto experimental data, or the general geometrical path distribution of rays p(l)of particles from solving the inverse problem in Eq. (2.5) or (2.49) provides anintuitive viewpoint and approach of particle sizing and shape determination [40].The first approach is detailed in section 2.3.

In the latter approach, one usually cannot simply cosine transform the mea-sured spectrum of the optical extinction efficiency to obtain the ray distributionusing Eqs (2.6–2.8) due to either the lack of sufficiently wide spectrum or thepresence of light absorption by scatterers at some wavelengths. One can, how-ever, discretize p(l) into a set of probabilities

pj =∫ (j+1)h

jh

p(l) dl ≥ 0

of the geometrical length of the ray within jh ≤ l < (j+1)h for j = 0, 1, . . . , J−1where

∑pj = 1, h > 0 is the step size, J is the total number of the segments, and

(J −1)h should be at least the maximum geometrical length of the ray inside theparticle. Under the assumption that light absorption by the particles is negligible,the ray distribution pj now relates to the total scattering cross-section Csca ofthe system (easily accessible by light transmission measurements) via a matrixequation:

2G − 2G∑

j

pj cos [kjh(m − 1)] = Csca(k), (2.50)

where G is the mean geometrical cross-section of the scatterers. This inverseproblem given by (2.50) is ill-posed and requires regularization to stabilize itssolution [50, 51]. By employing an appropriate regularization taking account ofa priori information on p(l), the ray distribution can then be recovered fromlight transmission measurements by inverting Eq. (2.50) [18, 52, 53]. The latterapproach was used by Li et al. [54] for inverse light scattering problems of ir-regular shaped particles after this idea was first proposed by Xu et al. [40] Thesize and shape of the scatterers may further be deduced from the recovered raydistribution.

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52 Min Xu and A. Katz

2.3 Applications of light scattering to bacteria monitoringand detection

The formalism described in section 2.2 provides the theoretical foundation for acomputational efficient and straightforward approach to analyze light scatteringand extinction by small particles, in particular, systems of randomly orientedand/or polydisperse soft particles. The Gaussian ray approximation capturesthe main feature of light extinction by such systems. Light extinction is deter-mined by three parameters: the mean and mean-squared-root of the geometricalpath of rays inside the scatterers and the relative refractive index of the scat-terers, rather than their detailed orientations, their shapes and the particle sizedistribution. This approach is most effective in probing bacteria suspensions orbacteria colonies using visible and near-infrared light (400–1000 nm). Light ex-tinction takes a simple quadratic form in λ−2 (see section 2.2.3) in the GRA. Byplotting the measured spectrum of light extinction versus λ−2, one can visuallyappreciate the refractive index and the size of the scatterers from the slope andcurvature. The values of the refractive index and the size of the scatterers canbe readily fitted by comparing the measured spectrum of light extinction to thequadratic expression.

The physical properties of bacteria make them well suited for investigationby light scattering in the 400 to 1000 nm wavelength range. In this spectralregion, light absorption by bacteria is weak and optical extinction is dominatedby scattering. In the UV, proteins and nucleic acid in bacteria absorb strongly,while water absorption becomes substantial at longer wavelengths. Most bacte-rial cells range in size from 0.5 to 2 micrometers [55] and have a refractive index of∼1.39 [56,57]. Therefore, when suspended in aqueous media, light scattering bymost bacteria cells satisfy the requirements for the ADT (|m − 1| � 1 and2πr/λ � 1). Prokaryotic cells, which include bacteria and archaea, lack thecomplex organelles and nuclear organization found in eukaryotic cells; there-fore their scattering properties are dominated by their size, shape and refractiveindex.

The application of light scattering for particle characterization has a long his-tory. In 1968, Wyatt [58] discussed the feasibility of light scattering for bacteriaidentification: ‘the characteristic of each distinct microorganism that scatters isan essentially unique scattering pattern.’ It has since been extensively applied indiverse areas: characterization of cell populations [59], biological dispersive sys-tems [53], and many atmospheric, oceanic and biological systems [9, 18]. In par-ticular, light scattering has the potential to provide a real-time, in situ means foridentification and classification of bacteria based on size and shape, therefore al-lowing a rapid response to contamination. Light scattering techniques can be animportant improvement over the current method for identifying bacteria, whichis time-consuming and labor-intensive. The current method consists of isolatinga specimen, growing the bacteria in culture media for 24 to 48 h, in order to pro-duce a sufficient population, followed by staining and microscopic examination.In contrast, light scattering techniques do not necessarily need a large sample

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2 Statistical Iinterpretation of ADT and its applications 53

population. In point of fact, angular scattering measurements require that thepopulation be sufficiently dilute to avoid multiple scattering events.

Several different elastic scattering techniques have been applied to the prob-lem of bacteria identification. Three light scattering techniques, turbidimetry,differential light scattering and quasi-elastic light scattering, applied to micro-biology have been review by Harding [60]. Differential light scattering was em-ployed by Wyatt to identify bacteria [58, 61]. It was later used to measure theeffects of heat treatment of S. epidermidis [62]. Quasi-elastic light scatteringhas been used to determine charge distributions in bacteria [63], motility in Es-cherichia coli [64,65], motility in both prokaryotic and eukaryotic cells [66], andto study cultures of marine bacteria [67]. Laser diffractometry was used to mea-sure water content of B. sphaericus spores [68]. Polarized light scattering wasused to investigate bacterial cell structure and to measure the size distribu-tion of bacteria colonies [69–74]. Recently, light extinction has been applied todetermine the size of different-shaped bacterial cells [75]. Light scattering hasalso been proposed to classify bacterial based on the structure of the bacterialcolonies based on Zernike moments of scatter pattern [76].

Due to its non-invasive nature, light scattering is also an ideal choice forin situ monitoring of biological processes. The growth behavior of E. coli wasinvestigated using UV/visible light transmission [77, 78]. The effects of wigglingmotion of bacteria on the light scattering spectrum has been investigated [79].Light transmission has been analyzed in the context of the ADT to monitorthe germination process in B. subtilis spores through changes in cell size andrefractive index [57].

In the following subsections, experimental results of light angular scatteringand extinction measurements on three species of bacteria of distinctive shapes(S. aureus, B. subtilis, and P. aeruginosa) and the retrieval of their particle sizedistributions are presented in sections 2.3.1 and 2.3.2, respectively. The particlesizes obtained by both methods agree well with those obtained by electron mi-croscopy. The monitoring of the changes of the refractive index and size of B.subtilis spores during germination by in situ light transmission is then presentedin section 2.3.3.

2.3.1 Angular scattering

One can apply the ADT either to analyze the spectra of scattering light, that ismeasure the angular distribution of scattered light, or to analyze transmissionmeasurements, that is the light, in the absence of absorption, not scattered. Ifthe concentration of particles is small, mostly single scattering events occur andangular scattering measurements are the superior method for determining par-ticle size distribution. However, if the concentration of scatters is large, multiplescattering events will occur which complicates the analysis of angular scattering.In this situation, transmission measurements are better suited for determiningparticle size.

A typical experimental setup for measuring angular scattering is shown inFig. 2.12. The output of a broadband light source is collimated and directed

Page 75: Light Scattering Reviews 3: Light Scattering and Reflection

54 Min Xu and A. Katz

Lamp

Collimator

Sample

CollectionFiberAperture

Spectrometer

PhotoDetector

Fig. 2.12. Experimental setup for angular scattering measurements.

through a cuvette, which contains the target sample suspended in a medium ofknown refractive index. For biological samples, the medium should be aqueousand contain the necessary nutrients and minerals so that the bacteria populationremains stable over the course of the experiment. The scattered light is collectedby an optical fiber mounted on a rotating arm. The spectra are collected atseveral different angles relative to the incident beam. Since for soft particles,most of the light is forward scattered, the scattered intensity will be greater atsmaller angles. The divergence of the incident beam determines the minimumangle at which the scattered light spectrum can be acquired without leakagefrom the transmitted light. In the single scattering regime, most of the light willbe transmitted, not scattered. The diameter of the aperture mounted in front ofthe collection fiber defines the angular resolution of the system. The scatteringangle, in the cuvette, is calculated from the position of the fiber aperture andthe refractive index of the medium. The output of the collection fiber is directedto a spectrometer and photodetector array. The photodetector signal is digitizedand transferred to a computer for analysis. The scattered light spectrum at eachangle θ must be normalized by the incident intensity prior to additional analysis.

For a collection of randomly oriented particles, with size r and a particle sizedistribution (PSD) given by n(r), the scattered light spectrum at angle θ in theRayleigh–Gans–Debye approximation, is given by [1]:

ISC(k, θ) ∝ k4(1 + cos2 θ)∫

r6R2(2k sin θ

2

)n(r) dr (2.51)

where R(θ) is the form factor [1]. The spectra acquired at different angles canthen be combined to form one rescaled spectrum of ISC(θ)/

[I0k

4(1 + cos2 θ)]

vsk sin(θ/2), from which the PSD can be determined.

Katz et al. [75] applied such a rescaled spectrum technique to analyze theangular scattering from three species of bacteria: S. aureus, B. subtilis, and P.aeruginosa at concentrations ranging from 107 to 109 bacteria/cm3. The three

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2 Statistical Iinterpretation of ADT and its applications 55

species were chosen for their differences in size and shape. S. aureus are spherical-shaped bacteria of radius ∼0.4μm. B. subtilis are rod-shaped bacteria of length∼2.5μm and diameter ∼1μm. P. aeruginosa are short rod-shaped bacteria oflength ∼1.5μm and diameter ∼0.6μm.

For angular scattering, it is necessary to confirm that one is in the singlescattering regime. In single scattering, the form factor is independent of particleconcentration. Thus the shape of the angular distribution of scattered light isindependent of particle concentration. This condition is satisfied when shape ofISC(λ, θ)/ITS(λ) is independent of concentration, where ISC(λ, θ) is the scat-tered light spectrum at a given angle and ITS(λ) is the spectrum of the totalscattered light. In the absence of absorption, ITS(λ) = I0(λ) − IT (λ), whereI0(λ) and IT (λ) are the incident and transmitted light intensity, respectively.Fig. 2.13 plots ISC(λ, θ)/ITS(λ) at a scattering angle of 5o (in air) for the threebacteria species at five different concentrations for each species. For all threespecies, the two most dilute concentrations, ISC(λ, θ)/ITS(λ) are close to beingequal, indicating that for these concentrations mostly single scattering is occur-ring. For the more concentrated suspensions, ISC(λ, θ)/ITS(λ) is concentrationdependent signaling that multiple scattering is occurring.

The rescaled scattering spectra for the three bacteria samples are plotted inFig. 2.14. The spectra from different scattering angles are seen to align with eachother, indicating the applicability of Eq. (2.51).

A least square fitting algorithm was used to determine the radius of the S.aureus, and the radius and aspect ratio for the B. subtilis and P. aeruginosa. Foran assumed Gamma Function PSD, the PSDs were calculated and are plottedin Fig. 2.15.

The particle size as determined by electron microscopy and angular scatteringmeasurements are summarized in Table 2.3.

Table 2.3. Comparison of bacteria cell sizes obtained by electron microscopy, angularscattering and transmission measurements

Electron microscopy Angular scattering TransmissionRadius Length Radius Length Radius Length(μm) (μm) (μm) (μm) (μm) (μm)

P. aeruginosa 0.3 1.5 0.28 1.4 0.30 1.5S. aureus 0.4 – 0.39 – 0.43 –B. subtilis 0.5 3 0.51 2.6 0.49 2.4

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56 Min Xu and A. Katz

0.35 0.40 0.45 0.50 0.55 0.600

2

4

6

8

10

12

14

0

2

4

6

8

10

12

14

0

2

4

6

8

10

12

14

B. subtilis N = 5.5 × 107 /ml N = 3.0 × 107 /ml N = 2.0 × 107 /ml N = 1.46 × 107 /ml N = 1.0 × 107 /ml

I SC/I TS

x10

3

Wavelength ( m)

P. aeruginosa N = 3.40 × 108 /ml N = 1.65 × 108 /ml N = 8.0 × 107 /ml N = 4.1 × 107 /ml N = 2.4 × 107 /ml

I SC/I TS

x10

3I SC

/I TS x

103

S. aureus N = 4.88 × 108 /ml N = 2.18 × 108 /ml N = 9.5 × 107 /ml N = 4.2 × 107 /ml N = 2.1 × 107 /ml

Fig. 2.13. Plot of ISC(λ, θ)/ITS(λ) for different bacteria concentrations. For each ofthe species, the two most dilute concentrations are close in shape, indicating singlescattering.

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2 Statistical Iinterpretation of ADT and its applications 57

I SC(k

,)/(

I ok4 (1 +

cos

2)) (a) Pseudomonas aeruginosa

(c) Bacillus subtilis

I SC(k

,)/(

I ok4 (1 +

cos

2))

2k sin( /2)

I SC(k

,)/(

I ok4 (1 +

cos

2)) (b) Staphylococcus aureus

Fig. 2.14. Rescaled angular scattering spectra.

Page 79: Light Scattering Reviews 3: Light Scattering and Reflection

58 Min Xu and A. Katz

0.0 0.5 1.0 1.50.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

P. aeruginosaS. aureusB. subtilis

Radius ( m)

N(r

)

Fig. 2.15. Particle size distribution for three different-shaped bacteria as determinedby angular scattering.

2.3.2 Bacteria size determined by transmission measurements

The Gaussian ray approximation in ADT provides a simple analytical expressionfor the scattering cross-section from Eq. (2.48):

Csca = G

[4n2π2

λ2 (m − 1)2(μ2 + ν2)− 4n4π4

3λ4 (m − 1)4(μ4 + 6μ2ν2 + 3ν4)]

(2.52)in the absence of absorption where G is the geometric cross-section, μ = 〈l〉 is themean light path through the scatterer, and σ is the dispersion of the geometricalpaths given by ν2 = 〈l2〉 − 〈l〉2. The optical density is given by:

OD = 0.4343 (Csc(λ)NL) , (2.53)

where N is the concentration and L is the optical path length.The mean geometric path and mean-squared-root geometrical ray path can

be determined by transmission measurements by fitting experimental OD toEq. (2.53) with the help of Eq. (2.52). From the mean geometric path and mean-squared-root geometric ray path, nominal particle size can be deduced, providedthe particle shape is known.

For intermediate-sized particles, the second term on the right-hand side ofEq. (2.52) is negligible and particle size can only be determined when indepen-dent knowledge of particle shape and concentration is available. In this case, theOD plotted as a function of 1/λ2 is linear and particle size can be calculatedfrom the slope of OD vs. λ-2. For intermediate-sized particles the total scatteringcross-sections for spheres and cylinders are given by:

Csphsca =

8n2π3

λ2 |m − 1|2 r4

Ccylsca =

32n2π3

3λ2 |m − 1|2 r3l, (random orientation, l � r)(2.54)

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2 Statistical Iinterpretation of ADT and its applications 59

in GRA where r is the radius of the sphere or cylinder, l is the length of thecylinder, n is the index of the surrounding media, and λ is the wavelength invacuum. The size of the intermediate-sized particles can be calculated from theOD if the particle concentration, refractive index and, for cylinders, the aspectratio are known.

The determination of bacteria cell size from transmission measurements wasinvestigated [46] for S. aureus, B. subtilis and P. aeruginosa. Plots of OD vs. 1/λ2

are shown in Fig. 2.16(a–c) for five different bacteria concentrations. The plotsare close to linear for the smaller-sized bacteria (S. aureus and P. aeruginosa).The deviation from linearity may be due to absorption at shorter wavelengths.P. aeruginosa contain siderophores (iron-chelating molecules) which absorb inthe blue region. For the larger sized B. subtilis, the second term in Eq. (2.52)cannot be ignored, and the spectra plotted in Fig. 2.16 are quadratic in 1/λ2.From the slope of the graphs in Fig. 2.16, it is determined that the radius ofthe S. aureus is 0.43μm; the radius and length of the P. aeruginosa are 0.30μmand 1.5 μm, respectively. For the B. subtilis, the radius and length are calculatedfrom the coefficient of the λ−2 term and is determined to be 0.49μm and 2.4 μm,respectively. The radii and lengths of the three bacteria species as determinedby transmission measurements are in good agreement with those obtained bymicroscopy and angular scattering measurements (see Table 2.3).

2.3.3 In vivo monitoring of biological processes in bacteria

The real-time, non-invasive nature of light scattering measurements providesan in situ method to monitor biological process in cells. Light scattering canbe employed to measure changes in concentration, cell size or refractive index.Refractive index changes can reflect structural changes occurring in the cells.

Light transmission was used to measure size and refractive index changes inB. subtilis spores during germination [57, 80]. Two genera of bacteria, Bacillusand Clostridium, form spores in response to hostile environments. When condi-tions become more receptive, spores revert to vegetative cells by the process ofgermination. B. anthraces spores are potential bio-warfare agents. C. botulinumis a common causative agent in food poisoning. The structural changes occurringduring germination are also important to biologists.

Spores consist of a dehydrated, high-density core surrounded by a lower-density spore coat composed of cross-linked polypeptides. Spores have a higherrefractive index than vegetative cells. The process of germination begins withactivation during which the spore coat is shed and the core re-hydrates. Thisprocess reduces the spore density and refractive index. During germination thecell grows in size. In nature, germination occurs randomly, but heat shock canstimulate activation, causing an entire population to initiate germination [81].The shock activation is accomplished by subjecting spores to a 30-min water bathat 80oC. Fig. 2.17 plots the optical extinction of B. subtilis spores at differenttimes following heat shock activation. The wavelength range in Fig. 2.17 is0.4 to 1.0μm, a region for which spores have minimal absorption. Immediatelyafter heat shock, the optical extinction (scattering) has not changed significantly,

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60 Min Xu and A. Katz

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6 7 80.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

2.0

(b) Staphylococcus aureus OD Linear Fit

Opt

ical

Den

sity

-2 ( m-2)

(c) Bacillus subtilis OD Quadratic Fit

Opt

ical

Den

sity

(a) Pseudomonas aeruginosa OD Linear Fit

Opt

ical

Den

sity

Fig. 2.16. Optical density plotted as a function of 1/λ2 for (a) P. aeruginosa, (b)S.aureus and (c) B. subtilis.

indicating that while heat shock triggers activation, no major structural changesoccur during the bath. However, within 30 min of activation, there is a reductionin the amount of scattering. Since spore size is not decreasing, the decrease inscattering is a result of a decrease in the refractive index. The change in refractiveindex is a direct result of the spores rehydrating and decreasing in density.

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2 Statistical Iinterpretation of ADT and its applications 61

1 2 3 4 5 60.0

0.1

0.2

0.3

0.4

0.5

0.6

Opt

ical

Ext

inct

ion

-2 ( m-2)

Unshocked SporesT = 1 minT = 30 minT = 1 hrT = 2 hrT = 6 hrT = 12 hr

Fig. 2.17. Optical extinction of B. subtilis at different times after heat shock activa-tion. Dotted lines are the second-order polynomial fit to the data.

From Eq. (2.52), the light extinction, in the absence of absorption, can bewritten as:

K = CscaNL = C2λ−2 + C4λ

−4. (2.55)

A least squares fit of the form given in Eq. (2.55) is also plotted in Fig. 2.17 foreach spectrum. The coefficients C2 and C4, are plotted in Fig. 2.18 for a timeperiod up to 20 h after activation. Most of the changes in scattering occur in thefirst 2.5 h while the changes observed in the scattering after 3 h are minimal.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 6 12 18-0.008

-0.006

-0.004

-0.002

0.00

0.04

0.08

0.12

Time (hours)

C4 ( m4)

-2 a

nd

-4 C

oeffi

cien

ts

C2 ( m2)

unshocked spores

Fig. 2.18. Fitting coefficients, C2 and C4, for optical extinction of spores, followingheat shock activation.

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62 Min Xu and A. Katz

Modeling the spore as a spherical spore core surrounded by a spore coat‘shell’, the two coefficients in the optical extinction Eq. (2.55) are given by:

C2 = 4αn2π3r4NL, C4 = −43βn4π4r6NL, (2.56)

after applying GRA, respectively, with α and β given by:

α = 2(m1 − 1)2 − 8(m1 − 1)(m1 − m0)ε + . . . ,

β =46081

(m1 − 1)4 −[54427

(m1 − m0)(m1 − 1)3 − 2(m1 − m0)3]ε + . . .

(2.57)

where m0 and m1 are the refractive indices of the spore coat and core, respec-tively, n is the refractive index of the host medium, and ε is the ratio of the coatthickness to the spore radius.

The spore coat index, m0, is assumed to be 1.39 (equal to the refractive indexof vegetative cells). The initial spore radius is estimated to be 400 nm and thespore coat thickness is ∼70 nm [82]; therefore, the initially value of ε is 0.175. Thespore coat thickness is assumed to decrease linearly to zero in time period of 0.5to 3 h after activation [57]. The spore radius and refractive index are calculatedusing Eqs (2.56) and (2.57) and the results of these calculations are plotted inFig. 2.19. As can be seen in Fig. 2.19, the spore radius increased from 0.38 μmto 0.5 μm within 3 h after heat shock. The refractive index of the spores priorto heat shock is 1.51, and at 2.5 h after heat activation, the refractive indexdecreases to that of vegetative cells (n = 1.39), indicating that the spores haverehydrated. The spores did not complete the germination process and attain thesize of vegetative cells because the spore medium was prepared lacking specificnutrients needed to complete germination.

0.0 0.5 1.0 1.5 2.0 2.5 6 12 180.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.5 1.0 1.5 2.0 2.5 6 12 181.30

1.35

1.40

1.45

1.50

1.55

1.60

Rad

ius

(m

)

Time (hours)

Ref

ract

ive

Inde

x

Time (hours)

Fig. 2.19. Spore radius and refractive index following heat shock activation.

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2 Statistical Iinterpretation of ADT and its applications 63

2.4 Conclusion

We have reviewed here the statistical interpretation of light anomalous diffrac-tion and its applications in bio-agent detection and monitoring. This statisticalview shows that the optical efficiencies in ADT is determined solely by theprobability distribution of the geometrical paths of the rays inside the particles.Optical efficiency equivalence can be easily achieved from different-shaped par-ticles or particles of different size distribution as long as they share a commongeometrical path distribution of rays. The main feature of optical efficiencies ofsoft particles is characterized by the mean and mean-squared-root geometricalpaths of the rays. The Gaussian ray approximation based on this observation forrandomly oriented and/or polydisperse soft scatterers can be used for a quick insitu particle characterization and monitoring.

The applications of light scattering and extinction measurements to bacteriacharacterization and in situ spore germination monitoring presented here demon-strate light scattering is a valuable tool to aid in the detection, classification andmonitoring of bacteria and other bio-agents. Although not species-specific, theability of light scattering to determine size and shape in real-time and in situmakes it a valuable tool in the detection and classification of bio-agents andbacterial contamination. Real-time. in situ monitoring of the changes of cellsize, shape, and structure, and, of its refractive index, by simple light extinctionmeasurements may prove to be potentially useful for microbiology where lightscattering can be used to monitor population growth, spore germination andother metabolic process occurring in bacteria colonies.

Acknowledgement

This work has been supported in part by the Department of Army (Grant#DAMD17-02-1-05 16), the Department of Defense (Grant# DAMD01-0084 andDAMD98-8147), a NASA URC – The Center of Optical Sensing and Imagingat The City College of New York (Grant #NCC-1-03009), and grants from NewYork State Office of Science, Technology and Academic Research (NYSTAR).M.X. also acknowledges support by Fairfield University.

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3 Light scattering by particles with boundarysymmetries

Michael Kahnert

3.1 Introduction

Symmetries are frequently exploited in physics to simplify the mathematicaldescription of nature. For instance, the idealised concept of a point source inNewton’s law of gravity or in Coulomb’s law yields a simple, radially symmetricforce field ∼ 1/r2. The isotropy of space in such a force field entails a dynamicsymmetry, namely conservation of angular momentum. The point-source con-cept constitutes a drastic simplification compared to the source distributionsencountered in nature. In spite of that, it has been highly successful for threereasons. First, the anisotropic field from a general source distribution can berepresented by superposition of the fields of individual point sources, which canbe expressed as a multipole expansion. Second, viewed from a large distancefrom the source distribution, the field is approximated with sufficient accuracyby the field of a point source, since the higher-order multipole terms can beneglected and only the monopole term survives. Third, nearly-spherical sourcedistributions, for which the higher-order multipole terms are small even in thenear field, are frequently encountered in nature, e.g. in planetary sciences.

In quantum mechanics one often makes symmetry assumptions to facilitatethe computation of the eigenstates of the Hamiltonian. The symmetries implythat the eigenstates are degenerate. Higher-order effects that break the symme-tries and lead to a splitting of the energy levels, such as spin–orbit interaction inan atom, can be treated by perturbation theory as corrections to the symmetricsolution.

These examples illustrate that symmetries are often not inherent in naturebut enter into physics through idealising assumptions. However, those idealisa-tions often lead to a description of nature that is sufficiently accurate, or thatonly requires small corrections, or that allows us to construct a general solutionto the problem from a simpler solution based on symmetry assumptions. In elec-tromagnetic scattering theory the situation is quite similar. In some cases we aredealing with light scattering particles or surfaces for which we can assume a sym-metric geometry, such as spherical cloud droplets, axisymmetric rain drops, orpristine ice crystals. In other cases, however, the particles are completely irregu-

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70 Michael Kahnert

lar, such as mineral aerosols. In spite of that one can often model the scatteringand absorption properties of such particles by assuming idealised shapes. Thisis because in the process of taking an ensemble-average of the optical propertiesover sizes, shapes, chemical compositions, and orientations of the particles oneloses information about the physical and chemical properties of the particles inthe ensemble. As a consequence, one can often find a suitable ensemble of simple,symmetric model particles that reproduces the aerosol optical properties withsufficiently high accuracy [1–6].

The earliest solution to the electromagnetic scattering problem based on sym-metry assumptions is Mie’s solution for the sphere [7]. For spherical symmetryone obtains an analytic solution. For any lower degree of symmetry the scatteringproblem needs to be solved numerically. For axisymmetric particles this has beendone by various methods, see, for example, [8,9]. For spheres and axisymmetricparticles the usual choice of spherical or spheroidal wave functions as a basis ofthe function space automatically accounts for the particle’s symmetries. For non-axisymmetric particles, exploiting symmetries in the calculations is less trivial.

Borghese et al. [10] applied group theory to scattering by a cluster of spheresto model scattering by molecular systems. Tarasov [11] and Zakharov et al.[12, 13] took a general approach to symmetries in boundary value problems.Applications were presented for electrostatics [14] and for a boundary-integralequation solution to Laplace’s equation [15]. Zagorodnov and Tarasov appliedgroup theory to a Green’s function formulation of the scattering problem [16]to a boundary-integral equation formulation [17], and presented applicationsto various polyhedral geometries [16–19]. Weiland and Zagorodnov [20] inves-tigated the exploitation of symmetries in Maxwell’s equations, thus exploitingspatial symmetries in the solution of an initial-value problem instead of a spatialboundary-value problem.

A systematic application of group theory to the T -matrix description ofthe electromagnetic scattering problem was presented in [21]. Applications ofthe method have been reported for general polyhedral prisms [2, 3, 5, 6, 22] andhexagonal prisms [23]. From a practical point of view, the approach of exploitingsymmetries within the T -matrix formalism is rather general, since the T -matrixcan be computed with a variety of different methods, such as boundary-integralequation methods [24], the separation of variables method [25], the generalisedpoint-matching method [26], volume-integral equation methods [27], and super-position methods for clusters of particles [28]. The use of symmetries in theT -matrix formulation was originally limited to exploiting only reducible rep-resentations of groups [21, 22]. However, it has been demonstrated [22] thatthis relatively simple approach is sufficient to reduce CPU-time requirementsin boundary-integral equation computations of the T -matrix by a factor M2

o ,where Mo denotes the order of the symmetry group. For instance, for hexag-onal prisms this amounts to a reduction of computation time on the order of242 ≈ 600. Interestingly enough, the more sophisticated method by Zagorodnov[17] based on using irreducible representations of groups resulted in exactly thesame reduction of CPU-time by a factor M2

o .

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3 Light scattering by particles with boundary symmetries 71

The exploitation of particle symmetries in the T -matrix formulation waslater extended to incorporate irreducible representations [29]. It could then beclarified that the main advantage of irreducible representations is to increasethe numerical stability of electromagnetic scattering computations by improvingthe conditioning of the notorious matrix inversion problem. For instance, forhexagonal prisms the range of accessible size parameters could be increased by50% thanks to the use of irreducible representations [29]. Although CPU-timerequirements could also be further reduced by the use of irreducible represen-tations, this was a rather minor effect. For example, for hexagonal prisms theuse of irreducible representations resulted in an extra reduction of computationtime by a factor of 3.6, so in total the computation time is reduced by a factorof 242 × 3.6 ≈ 2000. Thus, if computational speed is the main concern, thena simple approach based on using reducible representations of groups is oftensufficient. However, if numerical stability is the main problem, then it is worththe extra effort to exploit irreducible representations.

An investigation of the surface Green’s function in the presence of boundarysymmetries revealed a close formal analogy to the corresponding treatment ofsymmetries within the T -matrix formalism [30]. Recently, a more general treat-ment of symmetries in boundary-value problems was presented and applied toelectromagnetic and acoustic scattering theory [31]. This formulation is basedon Rother’s self-consistent Green’s function formulation [32–35] and allows us totreat symmetries in boundary-integral equation methods, volume-integral equa-tion methods, and in the T -matrix formulation under a common theoreticalframework.

The present review of boundary symmetries in acoustic and light scatteringtheory is meant to be reasonably self-contained. The main goal is to make thesubject accessible to a broad readership. Thus, at the risk of sacrificing somemathematical elegance and conciseness, it is not assumed that the reader hasextensive previous knowledge of group theory. At the same time, this reviewaims at treating symmetries from a general perspective rather than focusing ona specific method for solving the scattering problem, which requires at least acertain degree of mathematical rigour. Thus this article constitutes a compro-mise between a more intuitive and a more formal treatment of the subject. Westart with a general treatment of symmetries in boundary-value problems. Weproceed by applying the general concepts to the Helmholtz equation within theself-consistent Green’s function formalism. From this general, coordinate-freetreatment, we derive the symmetry relations of the T -matrix in the particular,reducible basis of spherical wave functions. Finally, we present a method forfinding for any arbitrary symmetry group the irreducible basis and discuss theadvantages of such an approach.

3.2 Symmetries in linear boundary-value problems

We start by deriving symmetry relations of the Green’s function for generalboundary-value problems. Thus this section is not limited to the Helmholtz

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72 Michael Kahnert

equation. Section 3.2.1 provides a brief review based on [36] of some basic Green’sfunction concepts, whereas the rest of section 3.2 mainly follows [31].

3.2.1 Green’s functions

Let ψ denote some physical field, x = (x1, . . . , xN ) the variables on which ψdepends, and A a linear operator that describes the physical conditions governingψ. A may be, for example, a differential operator, an integral operator, or anintegro-differential operator. The variables x can include spatial coordinates,time, momentum, energy, etc.

We consider the homogeneous equation

Aψ = 0, (3.1)

and the corresponding inhomogeneous equation

Aψ = −ρ, (3.2)

where the source field ρ depends, in general, on x.The Green’s function G belonging to operator A is a solution to the inhomo-

geneous problem with a delta-source term, i.e.

AG(x;x0) = −δ(x − x0). (3.3)

Thus G(x;x0) represents the field at x caused by a unit point source at x0. Wewish to find the solution to Eq. (3.2) for an arbitrary source distribution ρ.

Let Γ ⊂ RN denote the physical space of interest, and let ∂Γ denote theboundary of Γ . We shall consider Dirichlet boundary conditions on ∂Γ . A generalsource distribution ρ can be represented by a linear superposition of unit pointsources. If ψ satisfies homogeneous Dirichlet boundary conditions on ∂Γ , i.e.ψ(xs) = 0 ∀xs ∈ ∂Γ , then, owing to the linearity of A, a general solution to Eq.(3.2) must be given by a linear superposition of solutions G of Eq. (3.3) weightedwith the source distribution, i.e.

ψ(x0) =∫

Γ

G(x0;x)ρ(x) dΓ, (3.4)

where dΓ = dx1 · · ·dxN denotes the N -dimensional volume-element.Now suppose ψ satisfies inhomogeneous boundary conditions on ∂Γ . Then we

can obtain a solution to Eq. (3.2) by use of a generalisation of Green’s theorem[36]

uAv − vAu = ∇x · P[u, v] (3.5)

where P is known as the bilinear concomitant [36]. It needs to be determinedfrom Eq. (3.5). A denotes the adjoint operator of A. The corresponding adjointGreen’s function is a solution to the equation

AG(x;x0) = −δ(x − x0). (3.6)

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3 Light scattering by particles with boundary symmetries 73

It can be shown thatG(x;x0) = G(x0;x). (3.7)

Example 1: Consider either the Laplace operator AL=∇23=∂2

x +∂2y +∂2

z , or theHelmholtz operator AH=∇2

3 + k2. It is straightforward to show that

uAL,Hv − vAL,Hu

= ∂x [u∂xv − v∂xu] + ∂y [u∂yv − v∂yu] + ∂z [u∂zv − v∂zu]= ∇3 · [u∇3v − v∇3u] . (3.8)

Similarly, we obtain for the wave operator AW =∇24=∇2

3 + ∂2t /c

2

uAW v − vAW u

= ∇4 · [u∇4v − v∇4u] . (3.9)

Comparison with Eq. (3.5) shows that AL, AH , and AW are self-adjoint, andthat

P[u, v] = u∇v − v∇u, (3.10)

where ∇=∇3 for the Laplace and Helmholtz operators, and ∇=∇4=(∇3, ∂t/c)for the wave operator.

Example 2: Consider the diffusion operator Ad =∇23 − a2∂t. We derive

u[∇23 − a2∂t]v − v[∇2

3 + a2∂t]u= ∂x [u∂xv − v∂xu] + ∂y [u∂yv − v∂yu] + ∂z [u∂zv − v∂zu] − ∂t

[a2uv

]= (∇3, ∂t) ·

(u∇3v − v∇3u,−a2uv

). (3.11)

By comparison with (3.5), one can see that A =∇23 + a2∂t, and

P[u, v] =(u∇3v − v∇3u,−a2uv

). (3.12)

Example 3: Consider a linear integral operator Ai=1 −∫ xmax

xminK(x,x0) · · ·dx0.

One obtains

uAiv − vAiu

= − ddx

∫ x

x1

dx′∫ xmax

xmin

dx0K(x′, x0)[u(x′)v(x0) − v(x′)u(x0)], (3.13)

thus A=A, and

P[u, v] = −∫ x

x1

dx′∫ xmax

xmin

dx0K(x′, x0)[u(x′)v(x0) − v(x′)u(x0)]. (3.14)

Returning to the solution of Eq. (3.2) subject to inhomogeneous Dirichletboundary conditions on ∂Γ , we substitute in Eq. (3.5) u = G and v = ψ, and

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74 Michael Kahnert

integrate over Γ . This results in

ψ(x0) =∫

Γ

ρ(x)G(x0;x) dΓ +∫

∂Γ

n · P[G(x0;xs), ψ(xs)] dS(xs), (3.15)

x0 ∈ Γ,

where we have applied Gauss’s theorem to the right-hand side of the equation,as well as Eqs (3.2), (3.6), and (3.7). dS(xs) denotes a surface element, and ndenotes an outside-pointing normal unit vector on the boundary ∂Γ . Equation(3.15) is the general solution to the inhomogeneous problem given in Eq. (3.2)subject to inhomogeneous Dirichlet boundary conditions ψ = ψ(xs) for xs ∈ ∂Γ .

Comparison of Eqs (3.4) and (3.15) shows that in the case of homogeneousboundary conditions we have∫

∂Γ

n · P[G(x;xs), ψ(xs)] dS(xs) = 0. (3.16)

This allows us to determine the boundary conditions that have to be satisfiedby the Green’s function. Consider as an example the Helmholtz equation. Sub-stitution of Eq. (3.10) into Eq. (3.16) yields∫

∂Γ

[G(x;xs)

∂ψ(xs)∂n

− ψ(xs)∂G(x;xs)

∂n

]= 0, (3.17)

where we introduced the notation

n · ∇f(x)|x=xs∈∂Γ =∂f(xs)

∂n. (3.18)

Equation (3.17) makes it obvious that one cannot simultaneously require homo-geneous Dirichlet and homogeneous Neumann conditions for ψ on ∂Γ , becausein such case the surface integral trivially vanishes, and the boundary condi-tions that need to be satisfied by the Green’s function remain undetermined. Bycontrast, if we require either homogeneous Dirichlet or homogeneous Neumannconditions for ψ on ∂Γ , then Eq. (3.17) shows that the Green’s function has tosatisfy the same conditions as ψ on ∂Γ in order for the surface integral to vanish.

From Eq. (3.15) we also obtain the solution to the homogeneous problem(ρ = 0) subject to inhomogeneous boundary conditions:

ψ(x0) =∫

∂Γ

n · P[G(x0;xs), ψ(xs)] dS(xs). (3.19)

3.2.2 Groups

Let G denote a set and •: G × G → G with (g1, g2) �→ g1 • g2 denote a binaryoperation. (G, •) is called a group, if

∃E ∈ G; g • E = E • g = g ∀g ∈ G. (3.20)∀g ∈ G ∃g−1 ∈ G; g • g−1 = g−1 • g = E. (3.21)g1 • (g2 • g3) = (g1 • g2) • g3 ∀g1, g2, g3 ∈ G. (3.22)

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3 Light scattering by particles with boundary symmetries 75

Per definition, a binary operation satisfies closure, i.e.

g1 • g2 ∈ G ∀g1, g2 ∈ G. (3.23)

To apply the abstract operations represented by the group elements to theelements of a specific vector space, one needs to map the group elements ontolinear transformations acting on that vector space. In most general terms, alinear representation is a homomorphic map D : G → L, g �→ D(g) = Dg, withthe properties

D(g1 • g2) = D(g1) · D(g2) ∀g1, g2 ∈ G (3.24)D(E) = EL (3.25)

The vector space L is called the representation space, and EL denotes the unitelement in L. For our purposes, we shall define a representation of a groupas a linear map D : G → GL(N), where g �→ D(g) = Dg maps each groupelement g ∈ G onto an automorphism Dg. GL(N) denotes the general lineargroup, i.e. the equivalence class of automorphisms in N dimensions, where theequivalence relation is given by group isomorphy. The important properties ofautomorphisms are

(i) Injectivity. f : V1 → V2 is called injective, if [f(x) = f(y)] ⇒ [x = y]∀x, y ∈ V1.

(i) Surjectivity. f : V1 → V2 is called surjective, if ∀Y ∈ V2 ∃x ∈ V1; f(x) = Y .

A map that is both injective and surjective is called bijective. Bijective mapsare invertible. Thus our definition of a representation of a group ensures thatrepresentations preserve group property (3.21). Note that the definition of anautomorphism further requires that V1=V2. This ensures that Dg1 ◦ Dg2 (cor-responding to g1 • g2) is well defined and an element of the same group ofautomorphisms, thus ensuring closure.

Given a representation D : G → GL(N), where g �→ D(g) = Dg, each vectorx ∈ Γ transforms under the action of a group element g ∈ G according to

xg�→ x = Dg(x). (3.26)

Similarly, a scalar field ψ : Γ → R, with x �→ ψ(x) transforms under the actionof g according to

ψ(x)g�→ ψ(x) = ψ(D−1

g (x)). (3.27)

This is illustrated in Fig. 3.1 for the example of Γ = R3 and g representing areflection in the x1x2 plane.

3.2.3 Boundary symmetries

Suppose now that we transform the source field ρ in Eq. (3.2) according to

ρ(x)g�→ ρ(x) = ρ(D−1

g (x)). (3.28)

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76 Michael Kahnert

ψ( )=ψ(^

ψ

x

x D ( ))g−1 x

x

x

x

1

2

3

Fig. 3.1. Transformation of a scalar function ψ under a reflection in the x1x2 plane.

x2

x3

x1

x2

x3

x1

ψ( )^ x0

x0x

ρ( )^ x

ρ(D ( ))g−1x

ψ g−1 x0(D ( ))

gD

∂Γ

∂Γ

Fig. 3.2. Transformation of the source field ρ under a reflection in the x1x2 plane,and its effect on the solution ψ in Γ if ∂Γ is invariant under the transformation.

This is illustrated in Fig. 3.2 for a three-dimensional case. Further, for the caseof inhomogeneous boundary conditions, we transform the field at the boundaryaccording to

ψ(xs)g�→ ψ(xs) = ψ(D−1

g (xs)). (3.29)

In general, the transformed source field ρ will give rise to a different solution ψin Γ . However, if the boundary ∂Γ is invariant under the transformation, i.e.if g corresponds to a symmetry element of the boundary, then the old and new

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3 Light scattering by particles with boundary symmetries 77

solutions ψ and ψ will be related according to

ψ(x0) = ψ(D−1g (x0)) ∀x0 ∈ Γ. (3.30)

Substitution of Eq. (3.15) into Eq. (3.30) and application of Eqs (3.28) and (3.29)yields∫

Γ

ρ(D−1g (x))G(x0;x) dΓ +

∫∂Γ

n · P[G(x0;xs), ψ(D−1g (xs))] dS(xs)

=∫

Γ

ρ(x)G(D−1g (x0);x) dΓ +

∫∂Γ

n · P[G(D−1g (x0);xs), ψ(xs)] dS(xs).

(3.31)

This equation describes the symmetry of the solution to the inhomogeneous prob-lem with inhomogeneous boundary conditions if the boundary has g-symmetry.

From Eq. (3.31) we obtain a corresponding symmetry relation for the ho-mogeneous problem with inhomogeneous boundary conditions by setting ρ = 0.This yields ∫

∂Γ

n · P[G(x0;xs), ψ(D−1g (xs))] dS(xs)

=∫

∂Γ

n · P[G(D−1g (x0);xs), ψ(xs)] dS(xs).

(3.32)

This is an implicit symmetry relation for the Green’s function that can bebrought into explicit form once we have determined P for a given operator A.

For the inhomogeneous problem (ρ �= 0) with homogeneous Dirichlet bound-ary conditions (ψ(xs) = 0 ∀xs ∈ ∂Γ ) Eq. (3.31) yields for all g′ ∈ G∫

Γ

ρ(D−1g′ (x))G(x0;x) dΓ =

∫Γ

ρ(x)G(D−1g′ (x0);x) dΓ (3.33)

This equation can also be derived directly from Eqs (3.4), (3.28), (3.29), and(3.30). Equation (3.33) describes the symmetry properties of the Green’s functionand has to be valid for any source field ρ. We can therefore choose ρ(x)= δ(x −D−1

g′ (x1)). Thus∫Γ

G(x0;x)δ(D−1g′ (x) − D−1

g′ (x1)) dΓ = G(D−1g′ (x0);D−1

g′ (x1)). (3.34)

D−1g′ is injective, so δ(D−1

g′ (x) − D−1g′ (x1)) can be replaced by δ(x − x1) [see

remark after Eq. (3.64)]. Consequently,

G(x0;x1) = G(D−1g′ (x0);D−1

g′ (x1)). (3.35)

Finally, since G is a group, g′−1 =: g ∈ G. Thus we can replace D−1g′ by Dg and

obtain an explicit symmetry relation for the Green’s function:

G(x0;x1) = G(Dg(x0);Dg(x1)) (3.36)∀g ∈ G, ∀x0,x1 ∈ Γ.

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78 Michael Kahnert

3.3 Symmetries in electromagnetic and acoustic scatteringproblems

Throughout this section, we shall limit ourselves to the scalar Helmholtz equa-tion. Note that an electromagnetic (vector) scattering problem can be treatedby use of the Debye potentials as two scalar scattering problems, one for theTE- and one for the TM-mode. Also, we shall limit ourselves to the exteriorscattering problem for impenetrable or metallic scatterers. An extension to pen-etrable or dielectric scatterers will involve an additional Green’s function forthe interior problem, namely Rother’s generalisation of the Green’s function ofthe third kind [35]. Further, we shall only consider the case that the solutionto Helmholtz equation satisfies inhomogeneous Dirichlet boundary conditions.Finally, our discussions focus on scattering by objects of finite extent. An ap-plication of the concepts to scattering by surfaces will involve an extension ofpoint-group to space-group symmetries.

The short review of the self-consistent Green’s function formalism given insection 3.3.1 follows a series of papers by Rother [32–35]. The treatment ofirreducible representations in section 3.3.6 is mainly based on [29]. The remainderof section 3.3 follows, unless otherwise stated, [31].

3.3.1 Self-consistent Green’s function formalism

The self-consistent Green’s function formalism developed by Rother [32–35] al-lows us to treat volume-integral equation methods, boundary-integral equationmethods, and methods based on the T -matrix under a common theoretical frame-work. It therefore provides an ideal basis for discussing boundary symmetries inconjunction with the Helmholtz equation in most general terms. A thoroughdiscussion of the formalism is given in [32–35]. In the following, only a briefsummary of the main results is given.

We consider the Helmholtz operator AH=∇2 + k20, where k0 denotes the

wavenumber in free space. The free-space Green’s function G0 is a solution to

(∇2 + k20)G0(x;x′) = −δ(x − x′), (3.37)

subject only to the radiation condition

lim|x|→∞

(x|x| · ∇x − ik0

)G0(x;x′) = 0. (3.38)

Now we consider a scattering object occupying a finite region Γ− in three-dimensional space with boundary ∂Γ . Let Γ+=R3\Γ− denote the surroundingmedium, which is assumed to be non-absorbing. The volume Green’s functionGΓ+ is a solution to Eq. (3.37) subject to the radiation condition (3.38) andsubject to homogeneous Dirichlet boundary conditions on ∂Γ , i.e.

GΓ+(xs;x′) = 0 ∀xs ∈ ∂Γ, ∀x′ ∈ Γ+. (3.39)

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3 Light scattering by particles with boundary symmetries 79

Thus, a solution to the homogeneous scattering problem

(∇2 + k20)ψ(k0,x) = 0, (3.40)

subject to the radiation condition

lim|x|→∞

(x|x| · ∇x − ik0

)ψ(k0,x) = 0, (3.41)

and subject to inhomogeneous boundary conditions

ψ(k0,xs) = −f(xs) ∀xs ∈ ∂Γ (3.42)

can be obtained by use of GΓ+ in Eq. (3.19) and substitution of Eq. (3.10):

ψ(k0,x) =∮

∂Γ

∂GΓ+(x;xs)∂n−

f(xs) dS(xs), (3.43)

where we have used the notation introduced in Eq. (3.18), and where we haveexploited Eq. (3.39). n− denotes the unit normal vector on ∂Γ that points outof the region Γ+. The quantity

G∂Γ+(xs;x) =∂GΓ+(x;xs)

∂n−, (3.44)

which was introduced by Morse and Feshbach [36], is called the surface Green’sfunction. Note that the negative sign in Eq. (3.42) ensures that the inhomogene-ity f exactly cancels the external field ψ. Thus the total field on the boundaryof the metallic or impenetrable scatterer is identically zero, as it should.

GΓ+ can be expressed in terms of G∂Γ+ , as was first shown by Rother [32] byuse of Green’s theorem. We can also start, even more generally, from Morse andFeshbach’s [36] generalisation of Green’s theorem given in Eq. (3.5). Substitutingu = G0(x1;x0) and v = G(x;x1) in Eq. (3.5), integrating over Γ , and using (3.3),(3.6), and G0(x1;x0)=G0(x1;x0), we arrive at

G(x;x0) = G0(x;x0) +∫

∂Γ

n · P[G0(x0;xs), G(x;xs)] dS(xs). (3.45)

x,x0 ∈ Γ.

Setting G = GΓ+ , we obtain for the Helmholtz problem with Eqs (3.10) and(3.44)

GΓ+(x;x0) = G0(x;x0) +∮

∂Γ

G0(x0;xs)G∂Γ+(xs;x) dS(xs) (3.46)

x,x0 ∈ Γ+.

The third essential quantity in the self-consistent Green’s function formalism,besides GΓ+ and G∂Γ+ , is the interaction operator W∂Γ+ . It is defined by theequation [32]

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80 Michael Kahnert

GΓ+(x;x0) = G0(x;x0)

+∮

∂Γ

∮∂Γ

G0(x;xs)W∂Γ+(xs;x′s)G0(x′

s;x0) dS(xs) dS(x′s). (3.47)

By comparison with Eq. (3.46), it can be related to G∂Γ+ according to [32]

G∂Γ+(xs;x) =∮

∂Γ

G0(x;x′s)W∂Γ+(x′

s;xs) dS(x′s). (3.48)

Note that the interaction operator is only defined on the boundary surface. Bycontrast, the so-called transition operator introduced by Tsang et al. [37] relatesthe total to the incident electric field via a volume integral expression. However,for dielectric scatterers, Rother has generalised the surface interaction operatorto a volume interaction operator [35], which is essentially identical with thetransition operator.

The volume Green’s function can be related to volume-integral equationmethods, the surface Green’s function to boundary-integral equation methods,and the interaction operator to the T -matrix. The details of these relations aregiven in [35]. We shall limit the discussion to briefly reviewing the relation be-tween W∂Γ+ and the T -matrix.

Following [32] we formally define the matrix elements of the interaction op-erator according to

Tn,m;n′,m′

= −(ik0)∮

∂Γ

∮∂Γ

ψn,m(k0,xs)W∂Γ+(xs;x′s)ψn′,m′(k0,x′

s) dS(xs) dS(x′s),

(3.49)

where

ψn,m(k0,x) =

√2n + 1

4π(n − m)!(n + m)!

jn(k0r)P (m)n (cos θ) exp(imφ) (3.50)

denote regular spherical wave functions, the P(m)n denote associated Legendre

functions, and the jn denote spherical Bessel functions. The tilde notation isdefined according to

ψn,m(k0,x) = (−1)mψn,−m(k0,x). (3.51)

To bring the formal definition given in Eq. (3.49) into an explicit, numericallyapplicable form, we consider the well-known expansion of the free-space Green’sfunction [38]

G0(x,x′) = ik0

∞∑n=0

n∑m=−n

{ϕn,m(k0,x)ψn,m(k0,x′), | x |>| x′ |ψn,m(k0,x)ϕn,m(k0,x′) | x |<| x′ |

(3.52)

where

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3 Light scattering by particles with boundary symmetries 81

ϕn,m(k0,x) =

√2n + 1

4π(n − m)!(n + m)!

h(1)n (k0r)P (m)

n (cos θ) exp(imφ) (3.53)

denote outgoing radiating spherical wave functions, and h(1)n denote spherical

Hankel functions of the first kind. Substitution of (3.52) into (3.47) yields with(3.49)

GΓ+(x;x0) = G0(x;x0)

+ ik0

Ncut∑n,n′=0

n∑m=−n

n′∑m′=−n′

Tn,m;n′,m′ϕn,m(k0,x)ϕn′,m′(k0,x0), (3.54)

where we have approximated the, in principle, infinite series by a finite seriesextending up to a cut-off parameter Ncut. If x = xs∈ ∂Γ , then the homogeneousboundary conditions (3.39) require that this expression vanishes on the boundarysurface. Applying once more expansion (3.52) this yields

ik0

Ncut∑n,n′=0

n∑m=−n

n′∑m′=−n′

Tn,m;n′,m′ϕn,m(k0,xs)ϕn′,m′(k0,x0)

= −ik0

Ncut∑n′=0

n′∑m′=−n′

ψn′,m′(k0,xs)ϕn′,m′(k0,x0) (3.55)

Let t denote the transformation matrix that transforms the set of functions ψn,m

into ϕn,m, i.e. t is defined by

ψn′,m′(k0,xs) =Ncut∑n,=0

n∑m=−n

ϕn,m(k0,xs)tn,m;n′,m′ . (3.56)

Substitution of Eq. (3.56) into (3.55) yields, by comparison,

Tn,m;n′,m′ = −tn,m;n′,m′ . (3.57)

Now we can obtain an explicit expression for the matrix elements of the inter-action operator. Let {χn,m} be a set of not yet specified weighting functions.Multiplication of Eq. (3.56) with χ∗

n′′,m′′ and integration over the boundary sur-face yields

Rn′′,m′′;n′,m′ =Ncut∑n,=0

n∑m=−n

Qn′′,m′′;n,mtn,m;n′,m′ , (3.58)

where

Rn′′,m′′;n′,m′ =∮

∂Γ

χ∗n′′,m′′(xs)ψn′,m′(k0,xs) dS(xs) (3.59)

Qn′′,m′′;n,m =∮

∂Γ

χ∗n′′,m′′(xs)ϕn,m(k0,xs) dS(xs). (3.60)

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82 Michael Kahnert

By use of Eq. (3.57), we obtain in compact matrix notation

T = −Q−1 · R. (3.61)

Note that the use of Eq. (3.52) in this derivation has introduced restrictions onthe convergence of the solution to the scattering problem related to the casedistinction in Eq. (3.52)

The special choice of weighting functions

χn,m(x) = n− · ∇ψn,m(k0,x) (3.62)

in Eqs (3.59) and (3.60) yields with Eq. (3.61) a matrix representation of theinteraction operator that is identical with Waterman’s T -matrix [24] for theexterior scalar scattering problem, as was first noted by Rother [32].

3.3.2 Symmetry relations of GΓ+ , G∂Γ+ , and W∂Γ+

The boundary symmetries considered in section 3.2.3 where quite general. Forinstance, the N -dimensional space Γ can contain a time-coordinate. A symmetryof the physical system with respect to, for example, time reversal t → −t is aspecific boundary symmetry with respect to the boundary t = 0. For the spe-cial case of the Helmholtz equation, the physical space of interest is R3, wherethe three dimensions are the three spatial coordinates. Thus we are now dealingwith spatial symmetries, i.e. with invariances of the boundary surface ∂Γ underspatial coordinate transformations, such as rotations, reflections, inversion ofthe spatial coordinates, and rotation-reflections. Note that for an object of finiteextent Γ− cannot have translations as symmetry elements. Thus the symmetryoperations of the boundary surface and any combinations thereof must leaveat least one point in space unchanged. Therefore, the corresponding symmetrygroups are called point-groups. They play an important role in theoretical chem-istry, especially in electronic structure calculations and in the classification ofmolecular vibrations [39]. When dealing with scattering on infinite surfaces withboundary symmetries, one also needs to include translations and combinations oftranslations with point-group operations. The corresponding symmetry groupsare called space groups, or, in two dimensions, ‘wallpaper’ groups. They havebeen extensively applied in solid state physics. Note that periodic, symmetricstructures of infinite extent can only be built up of unit cells that have zero-,one-, two-, three-, four-, or six-fold rotational symmetry, where N -fold rotationalsymmetry refers to a rotation by an angle 2π/N . This so-called crystallographicconstraint limits the number of point-groups that play a role in space-groups to32. However, when dealing with finite objects, the crystallographic constraintdoes not apply. Thus there exist infinitely many point-groups for finite objects.

In this section we derive symmetry relations for the volume and surfaceGreen’s functions and for the interaction operator for the case that the closedboundary ∂Γ of the finite domain Γ− has point-group symmetries. Most im-portantly, we shall prove that the symmetry relations of GΓ+ , G∂Γ+ , and W∂Γ+

are equivalent. Before proceeding, we shall list and reiterate a few importantproperties that will be used repeatedly throughout the derivations.

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3 Light scattering by particles with boundary symmetries 83

– Since free space is homogeneous and isotropic, the free-space Green’s functionG0 is invariant under all point- (and space-) group operations, i.e.

G0 (Dg(x0);Dg(x)) = G0(x0;x) ∀x0;x ∈ R3, ∀g ∈ G, (3.63)

where G is an arbitrary point- or space-group.– In general,

δ (Dg(x) − Dg(x′)) = δ (Dg(x − x′)) =1

| detJ(Dg) |δ (x − x′) , (3.64)

where we have used the linearity of Dg. The Jacobi matrix J(Dg) of a lineartransformation is a constant matrix. Since Dg is an automorphism, its Jacobi-determinant is nonzero. For point-groups Dg represents a rotation, reflection,inversion, rotation-reflection, or identity operation, which are represented inR3 by orthogonal transformations with |det J(Dg)| = 1. Thus it follows that

δ (Dg(x) − Dg(x′)) = δ (x − x′) . (3.65)

Since this holds for all g ∈ G, it is, of course, also valid for D−1g .

– Since Dg is an automorphism, and since we assume the boundary ∂Γ to haveg-symmetry, Dg it is surjective on ∂Γ , i.e.

∀xs ∈ ∂Γ ∃x′s ∈ ∂Γ ; xs = Dg(x′

s). (3.66)

– Since the boundary ∂Γ is assumed to have g-symmetry, i.e. the boundarysurface is invariant under the corresponding coordinate transformation Dg,so is the surface element dS(xs). Thus

dS(Dg(xs)) = dS(xs) ∀xs ∈ ∂Γ, ∀g ∈ G. (3.67)

For a general homogeneous linear equation subject to inhomogeneous bound-ary conditions, where the boundary surface has g-symmetry, we obtained theimplicit symmetry relation given in Eq. (3.32). Thus, for the homogeneousHelmholtz problem (3.40) subject to inhomogeneous boundary conditions (3.42),an explicit symmetry relation is obtained by substituting Eq. (3.10) into Eq.(3.32) (with G = GΓ+), and using Eqs (3.39), (3.42), and (3.44). This yields∮

∂Γ

f(D−1

g′ (xs))

G∂Γ (xs;x0) dS(xs)

=∮

∂Γ

f(xs)G∂Γ

(xs;D−1

g′ (x0))

dS(xs), (3.68)

Since this relation holds for any inhomogeneity f , we can choose f(xs)=δ(xs − D−1

g′ (x′s)). Exploiting Eq. (3.65) for D−1

g′ , we obtain

G∂Γ (x′s;x0) = G∂Γ

(D−1

g′ (x′s);D

−1g′ (x0)

). (3.69)

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84 Michael Kahnert

Since this holds for all g′ ∈ G, we can replace g′ by g−1, thus D−1g′ by Dg, and

obtain

G∂Γ (xs;x0) = G∂Γ (Dg(xs);Dg(x0)) (3.70)∀x0 ∈ Γ+, ∀xs ∈ ∂Γ, ∀g ∈ G.

Thus, as a first important result, we have derived from the general implicitsymmetry relation (3.32) an explicit symmetry relation for the surface Green’sfunction of the Helmholtz equation.

Next we derive the symmetry properties of the volume Green’s function.From Eq. (3.46) we obtain

GΓ+ (Dg(x);Dg(x0)) = G0 (Dg(x);Dg(x0))

+∮

∂Γ

G0(xs;Dg(x0))G∂Γ (xs;Dg(x)) dS(xs)

= G0 (Dg(x);Dg(x0))

+∮

∂Γ

G0 (Dg(x′s);Dg(x0))G∂Γ (Dg(x′

s);Dg(x)) dS (Dg(x′s))

= G0 (x;x0) +∮

∂Γ

G0(x′s;x0)G∂Γ (x′

s;x) dS(x′s) (3.71)

Here we have first used the surjectivity (3.66), and then we used the symmetryproperties (3.63), (3.67), and (3.70) of the free-space Green’s function, the surfaceelement, and the surface Green’s function, respectively. Thus, by comparing Eqs(3.46) and (3.71) we arrive at

GΓ+ (Dg(x);Dg(x0)) = GΓ+(x;x0) (3.72)∀x,x0 ∈ Γ+, ∀g ∈ G.

This is of course nothing else but the symmetry relation of the Green’s functiongiven in Eq. (3.36), which we have already derived for a general linear operatorA and for homogeneous boundary conditions. What we have achieved in thepreceding derivation is to show that the symmetry relation of GΓ+ can be derivedfrom that of G∂Γ , i.e. Eq. (3.70) ⇒ Eq. (3.72).

Next, we derive the symmetry relations of the interaction operator. Substitu-tion of Eq. (3.47) into Eq. (3.72) and use of the symmetry relation of G0 (3.63)yields ∮

∂Γ

∮∂Γ

G0(Dg(x);xs)W∂Γ (xs;x′s)G0(x′

s;Dg(x0))dS(xs) dS(x′s)

=∮

∂Γ

∮∂Γ

G0(x;xs)W∂Γ (xs;x′s)G0(x′

s;x0)dS(xs) dS(x′s). (3.73)

Using the surjectivity (3.66) of Dg and the symmetry properties of G0 (3.63), wecan write G0(Dg(x);xs) = G0(Dg(x);Dg(x′

s)) = G0(x;x′s) = G0(x;D−1

g (xs)),and similarly for the second G0-term on the left hand side. Thus we obtain

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3 Light scattering by particles with boundary symmetries 85∮∂Γ

∮∂Γ

G0(x;D−1g (xs))W∂Γ (xs;x′

s)G0(D−1g (x′

s);x0) dS(xs) dS(x′s)

=∮

∂Γ

∮∂Γ

G0(x;xs)W∂Γ (xs;x′s)G0(x′

s;x0) dS(xs) dS(x′s). (3.74)

To get rid of the integrals, we apply (∇2x + k2)(∇2

x0+ k2) to both sides of the

equation and exploit Eq. (3.37). Since W∂Γ is only defined on the boundary ∂Γ ,we have to set x = xs ∈ ∂Γ , x0 = xs ∈ ∂Γ . This yields

W∂Γ (Dg(xs);Dg(xs)) = W∂Γ (xs; xs) (3.75)∀xs, xs ∈ ∂Γ, ∀g ∈ G.

We have once more used Eqs (3.65) and (3.66), which allow us to writeδ(xs − D−1

g (xs)) = δ(D−1g (x′

s) − D−1g (xs)) = δ(x′

s − xs) = δ(Dg(xs) − xs),and similarly δ(D−1

g (x′s) − xs) = δ(x′

s − Dg(xs)).Thus, as a third important result, we have derived the symmetry relations of

the interaction operator, and we have so far shown that Eq. (3.70) ⇒ Eq. (3.72)⇒ Eq. (3.75). As a final step we show the equivalence of the three symmetryrelations by proving the implication Eq. (3.75) ⇒ Eq. (3.70). From Eq. (3.48) itfollows that

G∂Γ (Dg(xs);Dg(x)) =∮

∂Γ

G0(Dg(x);x′s)W∂Γ (x′

s;Dg(xs)) dS(x′s). (3.76)

Using Eqs (3.63) and (3.66) (with Dg(x′′s ) = x′

s), this becomes

G∂Γ (Dg(xs);Dg(x)) =∮

∂Γ

G0(x;x′′s )W∂Γ (Dg(x′′

s );Dg(xs)) dS(Dg(x′′s )).

(3.77)Now we use the symmetry relations of the interaction operator (3.75) and thesurface element (3.67) and obtain, by comparison with Eq. (3.48),

G∂Γ (Dg(xs);Dg(x)) =∮

∂Γ

G0(x;x′′s )W∂Γ (x′′

s ;xs) dS(x′′s )

= G∂Γ (xs;x), (3.78)

which is identical with the symmetry relation (3.70). Thus we have derived Eq.(3.70) from Eq. (3.75).

In summary, we have shown that Eq. (3.70) ⇒ Eq. (3.72) ⇒ Eq. (3.75) ⇒Eq. (3.70). Thus the symmetry relations for the surface Green’s function [Eq.(3.70)], the volume Green’s function [Eq. (3.72)], and the interaction operator[Eq. (3.75)] have been shown to be equivalent to each other.

3.3.3 Symmetry relations in matrix form

The symmetry relations derived in the previous section shall now be broughtinto more explicit form. As mentioned earlier, the elements of the T -matrix are

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86 Michael Kahnert

the matrix elements of the interaction operator [see Eq. (3.49)]. Thus, we shallderive explicit symmetry relations of the T -matrix from the symmetry relationsof the interaction operator given in Eq. (3.75).

Let g �→ U(g) denote a representation of the symmetry group G in the basis{ψn,m}. Then the spherical wave functions transform under each g ∈ G accordingto

ψn,m(k0,x)g�→ ψn,m(k0,x) = ψn,m(k0, D

−1g (x))

=Ncut∑n′=0

n′∑m′=−n′

ψn′,m′(k0,x)Un′,m′;n,m(g). (3.79)

Again, we have approximated the infinite series by a finite series extending up toa cut-off parameter Ncut. The T -matrix is truncated accordingly. U can, withoutloss of generality, be assumed to be unitary. Thus

ψ∗n,m(k0,x)

g�→ ψ∗n,m(k0,x) = ψ∗

n,m(k0, D−1g (x))

=Ncut∑n′=0

n′∑m′=−n′

ψ∗n′,m′(k0,x)

[U†(g)

]n,m;n′,m′

=Ncut∑n′=0

n′∑m′=−n′

[U−1(g)

]n,m;n′,m′ ψ

∗n′,m′(k0,x). (3.80)

Note that for the non-absorbing surrounding medium k0 is real. In this case,ψn,m is identical with ψ∗

n,m. Using Eqs (3.49), (3.79), and (3.80), we derive forthe transformation of the T -matrix

Ncut∑n1,n2=0

n1∑m1=−n1

n2∑m2=−n2

[U−1(g)

]n,m;n1,m1

Tn1,m1;n2,m2Un2,m2;n′,m′(g)

= −(ik0)Ncut∑

n1,n2=0

n1∑m1=−n1

n2∑m2=−n2

∮∂Γ

∮∂Γ

[U−1]

n,m;n1,m1(g)ψ∗

n1,m1(k0,xs)

×W∂Γ+(xs;x′s)ψ(k0,x′

s)n2,m2Un2,m2;n′,m′(g) dS(xs) dS(x′s)

= −(ik0)∮

∂Γ

∮∂Γ

ψ∗n,m(k0, D

−1g (xs))W∂Γ+(xs;x′

s)

×ψ(k0, D−1g (x′

s))n′,m′ dS(xs) dS(x′s)

= −(ik0)∮

∂Γ

∮∂Γ

ψ∗n,m(k0, D

−1g (xs))W∂Γ+(D−1

g (xs);D−1g (x′

s))

×ψ(k0, D−1g (x′

s))n′,m′ dS(D−1g (xs)) dS(D−1

g (x′s)). (3.81)

In the last step we have used the symmetry properties of W∂Γ+ given in Eq.(3.75), as well as the symmetry properties of the surface element (3.67). Due tothe surjectivity of D−1

g on ∂Γ , we can set D−1g (xs) = x′′

s and D−1g (x′

s) = x′′′s .

Thus, by comparison with Eq. (3.49),

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3 Light scattering by particles with boundary symmetries 87

Ncut∑n1,n2=0

n1∑m1=−n1

n2∑m2=−n2

[U−1]

n,m;n1,m1Tn1,m1;n2,m2Un2,m2;n′,m′

= −(ik0)∮

∂Γ

∮∂Γ

ψ∗n,m(k0,x′′

s )W∂Γ+(x′′s ;x′′′

s )ψn′,m′(k0,x′′′s ) dS(x′′

s ) dS(x′′′s )

= Tn,m;n′,m′ , (3.82)

or, in compact matrix notation

U−1(g) · T · U(g) = T. (3.83)

Alternatively, this can be written as

[T,U(g)] = 0 ∀g ∈ G (3.84)

where the brackets denote the commutator of the two arguments.Note the analogy with dynamic symmetries in quantum mechanics, which

are expressed as commutator relations [H, Q] = 0, where the Hamiltonian Hdescribes the dynamics of the system, and Q represents the generating observ-able of some dynamic symmetry group. For instance, the momentum operatorP is the generating observable of the translation group, the angular momentumoperator J is the generating observable of the rotation group. Thus [H, P ] = 0expresses conservation of momentum in dynamic processes, [H, J ] = 0 expressesconservation of angular momentum. In the T -matrix formalism, we are consid-ering the spatial pattern of the scattered field. The T -matrix characterises thescattering and absorption properties of the scattering object at a given wave-length. Thus Eq. (3.84) expresses the invariance of the object’s optical propertiesunder the spatial coordinate transformation represented by U(g).

The surface Green’s function can be expanded according to [30,32]

G∂Γ (xs;x) =Ncut∑

n,n′=0

n∑m=−n

n′∑m′=−n′

[A−1

∂Γ

]n,m;n′,m′ ϕn,m(k0,xs)ϕ∗

n′,m′(k0,x).

(3.85)Just as the T -matrix is a matrix representation of the interaction operator, sois A−1

∂Γ a matrix representation of the surface Green’s function. Note that thematrices A−1

∂Γ and T depend, in general, on the truncation parameter Ncut.Substitution of Eq. (3.85) into Eq. (3.70) yields, together with Eqs (3.79)

and (3.80)[31], [A−1

∂Γ ,U(g)]

= 0 ∀g ∈ G. (3.86)

This is completely analogous to the symmetry relations of the T -matrix given inEq. (3.84). Note that the representations U(g) in the bases ψn,m and ϕn,m areidentical. This is because the basis functions ψn,m and ϕn,m only differ in theradial dependence. However, since point-groups do not involve translations, theradial functions have to remain unaffected by point-group operations, providedthat one chooses the origin of the coordinate system to coincide with the invariantpoint of the point-group.

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88 Michael Kahnert

3.3.4 Unitary, reducible representations of point-groups

To go any further with the commutator relations (3.84) or (3.86), one needsto explicitly derive the unitary representations U(g). We consider elements ofso-called non-cubic finite point-groups.

3.3.4.1 Unit element

The simplest group element is the unit element g = E. Since this correspondsto the identity operation, it trivially follows with Eq. (3.79)

Un,m;n′,m′(E) = δn,n′ δm,m′ . (3.87)

3.3.4.2 Rotations

Let g = CN denote a rotation about the z-axis by an angle 2π/N . This isillustrated in Fig. 3.3 for the example of a four-fold rotational symmetry, i.e.g = C4. If g = CN is a member of the group, then so is g = CN •CN = C2

N , and,more generally, also g = Cj

N , where j = 1, . . . , N − 1. (Note that CNN = E.)

C

C’(0)

(0)C’2

(1)

(1)

C’’2

C’’2

4

2

Fig. 3.3. Rotation axes C4, C′(0)2 , C

′(1)2 , C

′′(0)2 , and C

′′(1)2 .

The transformation behaviour of the spherical wave functions under rotationshas been studied extensively within the quantum theory of angular momentum[40] (since the spherical harmonics are the eigenfunctions of J2 and Jz, whereJ and Jz denote the angular momentum operator and its z-component, respec-tively). Let (α, β, γ) denote three Euler angles, such that one first rotates aboutthe z-axis by an angle γ, then about the y-axis by an angle β, and finally againabout the z-axis by an angle α. (This is equivalent to first rotating about thez-axis by an angle α, then about the y′-axis in the rotated coordinate systemby an angle β, and finally about the z′′-axis in the double-rotated coordinatesystem by an angle γ – see discussion in [41].) A representation of this mostgeneral rotation of the coordinate system in the basis of spherical harmonics isgiven by the Wigner d-functions

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3 Light scattering by particles with boundary symmetries 89

Un,m;n′,m′(α, β, γ)

= δn,n′D(n)m,m′(α, β, γ) = δn,n′ exp(−imα)d(n)

n,n′(β) exp(−im′γ), (3.88)

where the d(n)m,m′ are called the Wigner d-functions. They can be expressed in

terms of Jacobi polynomials [42].From Eq. (3.88) we immediately obtain a representation of g = Cj

N by settingα = 2πj/N and β = γ = 0, and by using d

(n)n,n′(0) = δn,n′ . Thus

Un,m;n′,m′(CjN ) = δn,n′δm,m′ exp

(−i

2πjm

N

). (3.89)

The so-called dihedral point-groups contain, in addition to the main rota-tional symmetry element CN , N two-fold rotational symmetry axes perpendic-ular to the main axis (see Fig. 3.3). These symmetry elements can all be in thesame class of the group, in which case they are denoted by C

′(0)2 , . . . , C

′(N−1)2 ,

or they can be contained in two different classes, in which case they are denotedby C

′(0)2 , . . . , C

′(N/2−1)2 and C

′′(0)2 , . . . , C

′′(N/2−1)2 , respectively.

From Eq. (3.88) we see that a rotation C(y)2 about π around the y-axis is

represented by

Un,m;n′,m′(0, π, 0) = δn,n′δm,−m′(−1)n+m, (3.90)

where we have used the property d(n)m,m′(π) = δm,−m′(−1)n+m. The x-axis shall

be aligned with the C′(0)2 axis. Thus C

′(0)2 = C−1

4 •C(y)2 •C4, and C

′(j)2 = (Cj

N )−1•C

′(0)2 • Cj

N . With Eqs (3.89) and (3.90) it follows

Un,m;n′,m′(C ′(j)2 ) = δn,n′δm,−m′(−1)n exp

(−i

4πjm

N

). (3.91)

The class of C ′′2 operations is obtained by rotating a C ′

2 axis by an angle π/N

(see Fig. 3.3). Thus C′′(0)2 = C−1

2N •C′(0)2 •C2N , and C

′′(j)2 = (Cj

N )−1 •C′′(0)2 •Cj

N ,which yields

Un,m;n′,m′(C ′′(j)2 ) = δn,n′δm,−m′(−1)n exp

(−i

4π(j + 1/2)mN

). (3.92)

3.3.4.3 Reflections

Consider a reflection plane coinciding with the xy-plane, as shown in Fig. 3.4.The corresponding group element is denoted by g = σh. The correspondingcoordinate transformation z → −z, or, in spherical coordinates, θ → π − θ, onlyaffects the associated Legendre functions in the spherical wave functions givenin Eqs (3.50) and (3.53). Since

P (m)n (cos(π − θ)) = (−1)n+mP (m)

n (cos θ), (3.93)

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90 Michael Kahnert

σ

σσ

h

vd(0)(0)

Fig. 3.4. Reflection planes σh, σ(0)v , and σ

(0)d .

we findUn,m;n′,m′(σh) = δn,n′δm,m′(−1)n+m. (3.94)

Let g = σ(0)v denote a group element corresponding to a reflection plane

coinciding with the xz-plane, as illustrated in Fig. 3.4. Then, as can be seen inFig. 3.5, σ

(0)v = C

′(0)2 · σh.

Let further σ(j)v denote a group element corresponding to a reflection plane

that contains the z-axis and that makes an angle 2πj/N with the xz-plane. Thenσ

(j)v = (Cj

N )−1 · σ(0)v · Cj

N . Using Eqs (3.89), (3.91), and (3.94) yields

Un,m;n′,m′(σ(j)v ) = δn,n′δm,−m′(−1)m exp

(−i

4πjm

N

). (3.95)

In some point-groups, j = 0, . . . , N −1. In other point-groups j = 0, . . . , N/2−1.The latter contain another class of reflection elements denoted by σ

(j)d , j =

0, . . . , N/2 − 1. These make an angle π/N with the nearest σ(j)v reflection plane

a bcd

e f

gh

a b

cd

e f

gh

a

b c

d

e

f g

h

σ

2C’(0)

hvσ(0)

x

Fig. 3.5. Illustration of the equality σ(0)v =C

′(0)2 · σh.

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3 Light scattering by particles with boundary symmetries 91

(see Fig. 3.4). Thus σ(0)d = (C2N )−1 · σ(0)

v · C2N and σ(j)d = (Cj

N )−1 · σ(0)d · Cj

N ,which yields

Un,m;n′,m′(σ(j)d ) = δn,n′δm,−m′(−1)m exp

(−i

4π(j + 1/2)mN

). (3.96)

3.3.4.4 Rotation-reflections

Combinations of rotations and reflections yield new symmetry elements denotedby g = SN = CN • σh. There exist symmetry groups that contain rotation-reflection elements SN , but not the corresponding rotation CN or reflection σh.An example is given by the polyhedron in Fig. 3.6, which belongs to the pointgroup D4d. The particle has S8-symmetry, but not C8- or σh-symmetry. On theother hand, the rectangular parallelepiped shown in Fig. 3.3 has both S4- andC4- and σh-symmetry. The polyhedron in Fig. 3.6 also has the symmetry elementS

(3)8 = C3

8 • σh. In general, we denote S(j)N = Cj

N • σh. The cardinal number jlies between 0 and N − 1, but in some point-groups it only takes on odd values.Use of Eqs (3.89) and (3.94) yields a representation for the rotation-reflectionelements

Un,m;n′,m′(S(j)N ) = δn,n′δm,m′(−1)n+m exp

(−i

2πjm

N

). (3.97)

S8

Fig. 3.6. Example of a polyhedron with S8-symmetry.

3.3.4.5 Inversion

An inversion g = I of all spatial coordinates (x, y, z)→ (−x,−y,−z) is equivalentto a rotation-reflection S

(N/2)N with even N . Thus we immediately obtain from

Eq. (3.97)Un,m;n′,m′(I) = δn,n′δm,m′(−1)n. (3.98)

The representations of symmetry operations derived here comprise all possi-ble symmetry elements occurring in finite, non-cubic point-groups. (The so-calledcubic point-groups are special groups that contain more than one CN elementwith N ≥ 3.) A complete list of all non-cubic, finite point-groups can be foundin [29]. The special cases of infinite point-groups will be discussed briefly insection 3.3.5.

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92 Michael Kahnert

3.3.5 Explicit symmetry relations

In the previous sections we have derived explicit representations U(g) for varioussymmetry elements given in Eqs (3.87), (3.89), (3.91), (3.92), and (3.94)–(3.98).Now we can return to the problem of bringing the symmetry relations of thematrix quantities given in Eqs (3.84) and (3.86) into explicit form. Before doingso we note the following. Suppose that g1, g2, g3 ∈ G and g3 = g1 • g2. Thus thecorresponding representations satisfy

U(g3) = U(g1) · U(g2). (3.99)

The symmetry relations for, for example, the T -matrix (3.84) are

T = U−1(g1) · T · U(g1), (3.100)T = U−1(g2) · T · U(g2), (3.101)T = U−1(g3) · T · U(g3). (3.102)

Multiplication of Eq. (3.100) from the left by U−1(g2) and from the right byU(g2) yields

U−1(g2) · T · U(g2) = U−1(g2) · U−1(g1) · T · U(g1) · U(g2). (3.103)

Using Eqs (3.99) and (3.101), this is seen to be equivalent to Eq. (3.102). Thusthe T -matrix symmetry relation for g3 can be derived from the correspondingrelations for g1 and g2. It follows that once we have identified for a given pointgroup a set of generating group elements from which all other group elements canbe obtained by combination, then only this subset of group elements provides uswith independent symmetry relations for the matrix elements of the interactionoperator or of the surface Green’s function.

Example: Consider the point group C3v. an example geometry belonging tothis group is given in Fig. 3.7. The group is of order 6, i.e. it contains six elements,namely, E, C3, C2

3 , σ(0)v , σ

(1)v , and σ

(2)v , where the three reflection planes each

contain the C3 axis and intercept one of the corners of the triangular bottomfacet.

Table 3.1 shows the multiplication table for this group. Note that this groupis non-Abelian, so in general g1 • g2 �= g2 • g1. The table is to be interpreted

C3

Fig. 3.7. Example of a polyhedron belonging to the point group C3v.

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3 Light scattering by particles with boundary symmetries 93

Table 3.1. Multiplication table for the point group C3v

E C3 C23 σ

(0)v σ

(1)v σ

(2)v

E C3 C23 σ

(0)v σ

(1)v σ

(2)v

C3 C3 C23 E σ

(2)v σ

(0)v σ

(1)v

C23 C2

3 E C3 σ(1)v σ

(2)v σ

(0)v

σ(0)v σ

(0)v σ

(1)v σ

(2)v E C3 C2

3

σ(1)v σ

(1)v σ

(2)v σ

(0)v C2

3 E C3

σ(2)v σ

(2)v σ

(0)v σ

(1)v C3 C2

3 E

such that the column element is to be applied first. For example C3 •σ(0)v = σ

(2)v ,

where σ(0)v is applied first. From the table we can see by inspection that all

group elements can be generated by combination of the generators C3 and σ(0)v .

Thus from Eq. (3.84) we obtain with Eqs (3.89) and (3.95) two independentcommutator relations for the T -matrix in explicit form, namely

Tn,m;n′,m′ = exp(

i2π(m − m′)

3

)Tn,m;n′,m′ (3.104)

Tn,m;n′,m′ = (−1)m+m′Tn,−m;n′,−m′ . (3.105)

Equation (3.104) can also be expressed in the form

Tn,m;n′,m′ = 0 unless |m − m′| = 0, 3, 6, . . . . (3.106)

The symmetry relations (3.105) and (3.106) reduce the number of non-zero,independent matrix elements that need to be computed by a factor of six, whichis equal to the order Mo of the group. Note that, for example, in boundary-integral equation methods the area over which the boundary integrals need tobe evaluated is also reduced by a factor equal to Mo, thus resulting in a totalreduction in computation time by a factor of M2

o . This is discussed in detail in[22].

Apart from the finite point-groups, there are three infinite point-groups. Par-ticles with axial symmetry belong either to the group C∞v, such as circularcones and Chebyshev particles of odd order, or to the point group D∞h, suchas spheroids, finite circular cylinders, linear clusters of spheres, and Chebyshevparticles of even order.

The generalisation of Eq. (3.106) for CN -symmetry is

Tn,m;n′,m′ = 0 unless |m − m′| = 0, N, 2N, . . . . (3.107)

By taking the limit N → ∞, we obtain for axial symmetry

Tn,m;n′,m′ = δm,m′Tn,m;n′,m, (3.108)

which agrees with the result derived in [9].

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94 Michael Kahnert

The group C∞v contains infinitely many ‘vertical’ reflection planes contain-ing the rotational symmetry axis. The symmetry relation corresponding to σv-symmetry was given in Eq. (3.105). Equations (3.108) and (3.105) are the onlyindependent symmetry relations for the C∞v group.

For the group D∞h we obtain, in addition to the commutator relation (3.108),independent symmetry relations from σh and C

′(0)2 -symmetry (‘D’ stands for

dihedral symmetry). Thus, by substituting the corresponding representations(3.91) and (3.94) into the general commutator relation (3.84) we obtain

Tn,m;n′,m′ = (−1)n+n′Tn,−m;n′,−m′ . (3.109)

Tn,m;n′,m′ = (−1)n+m+n′+m′Tn,m;n′,m′ , (3.110)

where the last relation can be written as

Tn,m;n′,m′ = 0 unless (n + m + n′ + m′) even. (3.111)

Eqs (3.108), (3.109), and (3.111) are the independent symmetry relations for theD∞h-group. Note that we can, for instance, derive Eq. (3.105) from Eqs (3.109)and (3.111). This is not surprising, since the group D∞h also contains ‘vertical’reflection planes. In fact, C∞v is a subgroup of D∞h.

Finally, we mention the highly special case of spherical symmetry, which isonly fulfilled for spherical particles. The corresponding symmetry group is de-noted by K. The T -matrix is invariant under an arbitrary rotation characterisedby the Euler angles (α,β,γ). Since we already have derived the symmetry rela-tion for axial symmetry – see Eq. (3.108) – we can limit ourselves to rotationsabout an arbitrary Euler angle β around the y axis, which are represented bythe Wigner d-functions. Thus, using Eq. (3.88) for (α, β, γ) = (0, β, 0), the com-mutator relation of the T -matrix for a rotation about an angle β around they-axis reads

Tn,m;n′,m′ = δm,m′

n∑m1=−n

d(n)m,m1

(−β)Tn,m1;n′,m1 d(n′)m1,m(β), (3.112)

where we have exploited the symmetry relation (3.108). Since this relation hasto hold for any β, we can integrate over all β. Using d

(n)m,m1(−β) = d

(n)m1,m(β) and

the orthogonality properties of the Wigner d-functions [40] , the integration ofEq. (3.112) over β yields

Tn,m;n′,m′ =1

2n + 1δm,m′δn,n′ Tn, (3.113)

where we have defined∑n

m1=−n Tn,m1;n,m1 = Tn. Thus the T -matrix is inde-pendent of m and diagonal in all indices.

We can see from the examples given here that the explicit symmetry relationsof the T -matrix are considerably less appealing to our intuition than the corre-sponding coordinate-free symmetry relations of the surface and volume Green’sfunction and of the interaction operator. However, they can be readily appliedin numerical computations.

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3 Light scattering by particles with boundary symmetries 95

3.3.6 Irreducible representations

In practice, computation of the T -matrix often involves computation of two ma-trices R and Q, of which the latter needs to be inverted – see Eq. (3.61). Thismatrix inversion often causes ill-conditioning problems in numerical computa-tions. The method described in this section aims at reducing these problems bypre-conditioning the matrix quantities in an efficient and numerically inexpensiveway.

We assume that the matrices R and Q satisfy the same symmetry relationsas the T -matrix, i.e.

[R,U(g)] = 0 ∀g ∈ G (3.114)[Q,U(g)] = 0 ∀g ∈ G. (3.115)

For a T -matrix given by Eqs (3.59)–(3.61), this can be shown by investigating thetransformation of the integral expressions (3.59) and (3.60), and by making useof the transformation properties of the functions appearing under the boundaryintegrals. Such an approach has been taken in [30].

As shown in the previous section, the symmetry relations (3.114) and (3.115)reduce the number of matrix elements that need to be computed numerically,thus expediting numerical computations. We shall now go one step further andexploit boundary symmetries for constructing a similarity transformation thatbrings the Q-matrix into block-diagonal form, thus achieving the aforementionedpre-conditioning of the matrix. This will be done in two steps. First we bringthe representations U(g) into block-diagonal form. Second we show that thecorresponding similarity transformation also block-diagonalises the Q-matrix.

As an example for what we are aiming at, consider the rotation group andits representations in the basis of spherical harmonics given in Eq. (3.88). Thefactor δn,n′ occurring in this equation is quite remarkable. It implies that thespherical harmonics Yn,m transform under a general rotation (α, β, γ) accordingto

Yn,m −→∞∑

n′=0

n′∑m′=−n′

δn,n′D(n′)m,m′(α, β, γ)Yn′,m′

=n∑

m′=−n

D(n)m,m′(α, β, γ)Yn,m′ . (3.116)

Thus, a rotation of Yn,m results in a linear combination of functions Yn,m′ be-longing to the same order n. More generally, a rotation of any element of thesubspace Tn spanned by the functions {Yn,−n, Yn,−n+1, . . . , Yn,n} yields againan element of Tn. This subspace is said to be invariant under rotations. We canwrite Eq. (3.116) in a more suggestive way:⎛⎜⎜⎜⎝

Y0Y1Y2...

⎞⎟⎟⎟⎠ −→

⎛⎜⎜⎜⎝D(0) 0 0 · · ·0 D(1) 0 · · ·0 0 D(2) · · ·...

......

. . .

⎞⎟⎟⎟⎠ ·

⎛⎜⎜⎜⎝Y0Y1Y2...

⎞⎟⎟⎟⎠ , (3.117)

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96 Michael Kahnert

where Yn is a vector with elements Yn,m, m = −n, . . . , n, and D(n) is a matrixwith elements D

(n)m,m′ . Thus the matrix representation D of the rotation group is

built up of block matrices D(n) along the matrix diagonal. We write symbolically

D =∞⊕

n=0

D(n). (3.118)

The matrix D(n) is a representation of the rotation group in the invariant sub-space Tn.

If we take away any of the basis vectors Yn,−n, Yn,−n+1, . . . , Yn,n then thesubspace spanned by the reduced set of basis functions is no longer invariantunder rotations. The subspaces Tn, n = 0, 1, 2, . . . are therefore the smallestinvariant subspaces (apart from the improper subspace that only contains thenull-vector). The Tn are therefore called irreducible invariant subspaces. Conse-quently, the matrix representation D cannot be decomposed into smaller blockmatrices. The matrices D(n) are therefore called irreducible representations ofthe rotation group.

More generally, given an N -dimensional vector space VN and a representationD : G → GL(N) with g �→ Dg (where the automorphisms in GL(N) operateon the elements of VN ), a subspace T ⊂ V is called an invariant subspace, ifDg(x) ∈ T ∀g ∈ G ∀x ∈ T . If VN contains a proper invariant subspace T (i.e.0 < dimT < dimVN = N), then D is called a reducible, otherwise an irreduciblerepresentation of G.

Unlike in the case of the representation of the rotation group in the basis ofspherical harmonics given in Eq. (3.118), an irreducible representation can occurmore than one time in a reducible representation, i.e. in general

U =r⊕

μ=1

αμU(μ) (3.119)

where the U(μ) denote the irreducible representations, and the cardinal numbersαμ indicate how many times each of the block matrices U(μ) occurs on thediagonal of the matrix U. In the case of finite point-groups, the number r ofirreducible representations is finite.

The irreducible, block-diagonal structure of the representations of the rota-tion group given in Eq. (3.117) is not a coincidence, but a result of the choice ofbasis functions. The spherical harmonics are eigenfunctions of the operator J2

with corresponding quantum number n, where J denotes the angular momen-tum operator. However, the operator J2 is an invariant operator of the rotationgroup (a so-called Casimir operator). This is why the subspace characterised bythe order (quantum number) n is invariant under rotations. Thus, if we had cho-sen a set of basis functions that are not eigenfunctions of the Casimir operatorthen the representation of the group elements would not have been irreduciblyblock-diagonal. For instance, for spheroidal wave functions, the reducible repre-sentations of the rotation group have been derived in [25].

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3 Light scattering by particles with boundary symmetries 97

The representations U of the point-groups derived in section 3.3.4 are notirreducible. One therefore needs to find a transformation into a new basis inwhich the representations U become block-diagonal. This new choice of basisfunctions will depend on the point group in question. Since there exist infinitelymany point-groups, we need to have an automatised procedure for constructingsuch a basis transformation for an arbitrary group.

The strategy is to find the invariant subspaces. Suppose that we truncate,as before, the function space to a finite dimension n. Suppose further that wehave a set of basis functions B0 = {f1, . . . , fn} spanning the function space, inwhich each g ∈ G is represented by a reducible representation Ured(g). Thus thefunctions fj transform under the group action according to

fjg−→ fj =

n∑i=1

U redi,j (g)fi. (3.120)

Suppose also that there are r (still unknown) irreducible representationsU(μ), and that the μ-th irreducible representation occurs αμ times in the re-ducible representation Ured. Let us further assume that we have a set of basisfunctions Bμ,l= {h(μ,l)

1 , . . . , h(μ,l)nμ } in each invariant subspace Tμ,l, i.e.

Tμ,l = 〈〈h(μ,l)1 , . . . , h(μ,l)

nμ〉〉 (3.121)

μ = 1, . . . r, l = 1, . . . , αμ,

where 〈〈. . .〉〉 denotes the set of all linear combinations of the basis functions. Theset of basis functions Bμ =

⋃αμ

l=1 Bμ,l span a subspace Tμ that is also invariant,but not irreducible, i.e.

Tμ = 〈〈h(μ,1)1 , . . . , h(μ,αμ)

nμ〉〉 (3.122)

μ = 1, . . . r.

The union of all basis vectors B =⋃r

μ=1 Bμ is a basis of the entire function

space. The subspace Tμ,l spanned by the basis Bμ,l is invariant, so the h(μ,l)j

transform under g ∈ G according to

h(μ,l)j

g−→ h(μ,l)j =

nμ∑i=1

U(μ)i,j (g)h

(μ,l)i (3.123)

μ = 1, . . . , r, l = 1, . . . , αμ.

Each fi can be expanded in the basis B, i.e.

fi =r∑

μ=1

αμ∑l=1

nμ∑j=1

a(μ,l)i,j h

(μ,l)j , i = 1, . . . , n. (3.124)

Likewise, each h(μ,l)i can be expanded in the basis B0, i.e.

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98 Michael Kahnert

h(μ,l)i =

n∑j=1

b(μ,l)i,j fj , μ = 1, . . . , r, l = 1, . . . , αμ, i = 1, . . . , nμ. (3.125)

By substituting Eq. (3.124) into Eq. (3.125) we obtainn∑

j=1

b(μ,l)i,j a

(ν,m)j,p = δi,pδμ,νδl,m. (3.126)

Further, we can derive a relation between the reducible and irreducible represen-tations by re-investigating the transformation of the fi under the group action:

fi =r∑

μ=1

αμ∑l=1

nμ∑j=1

a(μ,l)i,j h

(μ,l)j

=r∑

μ=1

αμ∑l=1

nμ∑j=1

nμ∑p=1

a(μ,l)i,j U

(μ)p,j (g)h(μ,l)

p

=r∑

μ=1

αμ∑l=1

nμ∑j=1

nμ∑p=1

n∑q=1

a(μ,l)i,j U

(μ)p,j (g) b(μ,l)

p,q fq. (3.127)

In steps 1–3 above we have used Eqs (3.124), (3.123), and (3.125), respectively.Since the fq are linearly independent, comparison with Eq. (3.120) yields

U redj,i (g) =

r∑μ=1

αμ∑l=1

nμ∑p,q=1

a(μ,l)i,p U (μ)

q,p (g)b(μ,l)q,j . (3.128)

We only know that the a- and b-matrices in this relation exist and that theysatisfy Eq. (3.126). Thus Eq. (3.128) is only a formal relation between the re-ducible and irreducible representations. To get any further we need to make useof additional information from group theory about the irreducible representa-tions. According to the so-called great orthogonality theorem, the irreduciblerepresentations satisfy the orthogonality property∑

g∈GU

(μ)i,l (g)U (ν)∗

j,m(g) =Mo

nμδμ,νδi,jδl,m, (3.129)

where Mo denotes the order of the group. A proof of this theorem can be foundin the standard literature [39,43,44]. We exploit this theorem by multiplying Eq.(3.128) by U (ν)∗

k,k(g), and by summing over k and over all g ∈ G. This yields

P(ν)j,i

def=∑g∈G

nν∑k=1

U (ν)∗k,k(g)U red

j,i (g)

=r∑

μ=1

αμ∑l=1

nμ∑p,q=1

nν∑k=1

a(μ,l)i,p b

(μ,l)q,j

∑g∈G

U (ν)∗k,k(g)U (μ)

q,p (g)

=Mo

nν∑k=1

αν∑l=1

a(ν,l)i,k b

(ν,l)k,j . (3.130)

In the last step we have used Eq. (3.129).

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3 Light scattering by particles with boundary symmetries 99

Application of the operator P(ν) to the elements fj of the reducible basis B0yields

n∑j=1

P(ν)j,i fj =

Mo

n∑j=1

nν∑k=1

αν∑l=1

a(ν,l)i,k b

(ν,l)k,j fj

=Mo

r∑μ=1

αμ∑m=1

nμ∑p=1

nν∑k=1

αν∑l=1

a(ν,l)i,k

n∑j=1

b(ν,l)k,j a

(μ,m)j,p h(μ,m)

p

=Mo

nν∑k=1

αν∑l=1

a(ν,l)i,k h

(ν,l)k

∈ Tν . (3.131)

In the above steps, we used, respectively, Eqs (3.130), (3.124), (3.126), and(3.122). Thus the operator P(ν) can be interpreted as a projection operatorthat projects any arbitrary vector into the ν-th invariant subspace.

The matrix traces

χ(ν)(g) def=nν∑

k=1

U (ν)k,k(g), g ∈ G (3.132)

are called the characters of the ν-th irreducible representation. Thus the projec-tors P(ν) introduced in Eq. (3.130) can be written as

P(ν)i,j =

∑g∈G

χ(ν)∗(g)U red

i,j (g). (3.133)

The crucial point is that the characters can be computed with standard grouptheoretical methods [45, 46] without a priori knowledge of the irreducible sub-spaces or the irreducible representations. For many frequently used point-groupsthe characters can even be found tabulated in the literature, see, for example,[39, 47]. Thus Eq. (3.133) provides us with a numerically useful method to con-struct the desired similarity transformation from a reducible basis to a basis inwhich the vector space is divided up into invariant subspaces Tν .

A basis in each of these subspaces is given by Bν = {h(ν,l)k | k = 1, . . . , nν ,

l = 1, . . . , αν}. Consequently, the n × n projection matrix P(ν) will containβν

def= nν · αν linearly independent column and row vectors. Let P′(ν) denote aβν ×n matrix containing βν linearly independent row vectors of the matrix P(ν).Then the matrix

P =

⎛⎜⎝ P′(1)...P′(r)

⎞⎟⎠ (3.134)

is an n × n matrix that performs the desired transformation into a new basis inwhich, according to our earlier remarks, the reducible representations Ured(g)become block-diagonal, i.e.

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100 Michael Kahnert

U′(g) = P−1 · Ured(g) · P (3.135)

=

⎛⎜⎜⎜⎝U′(1)(g)

U′(2)(g). . .

U′(r)(g)

⎞⎟⎟⎟⎠ (3.136)

=r⊕

μ=1

U′(μ)(g). (3.137)

The block-matrices U′(μ)(g) are βμ × βμ matrices.A comparison of Eqs (3.119) and (3.137) reveals that our method is not ca-

pable of fully reducing the reducible representations. In other words, the trans-formation given by Eqs (3.134) only allows us to find the invariant subspacesTμ [see Eq. (3.122)], but not the irreducible invariant subspaces Tμ,l [see Eq.(3.121)]. We shall see shortly that this is of no practical concern, since our ulti-mate interest is to block-diagonalise not Ured(g) but the Q-matrix. It remainsto be shown that the transformation P reduces the Q-matrix to block-diagonalform.

The starting point is the commutator relations for the Q-matrix in Eq.(3.115). Multiplication with P−1 from the left and P from the right yields

P−1 · Ured(g) · P · P−1 · Q · P = P−1 · Q · P · P−1 · Ured(g) · P. (3.138)

Defining P−1 · Q · P = Q′ and using (3.135), we obtain

U′(g) · Q′ = Q′ · U′(g) ∀g ∈ G. (3.139)

Thus the commutator relations are form-invariant under the transformation P.The reason is that a similarity transformations is a conformal mapping, whichdoes not change the symmetry properties of the boundary surface.

To exploit the block-diagonal structure of U′ given in Eq. (3.136), we splitup the matrix Q′ according to

Q′ =

⎛⎜⎝Q′(1,1) · · · Q′(1,r)

......

Q′(r,1) · · · Q′(r,r)

⎞⎟⎠ , (3.140)

where the submatrix Q′(μ,ν) has βμ rows and βν columns. With (3.136) and(3.140) we can bring Eq. (3.139) into the form

U′(μ)(g) · Q′(μ,ν) = Q′(μ,ν) · U′(ν)(g) ∀g ∈ G, μ, ν = 1, . . . , r. (3.141)

The goal is to show that Q′ is block diagonal, i.e. Q′(μ,ν) = 0 for μ �= ν. Thestrategy is to apply the great orthogonality theorem to Eq. (3.141). However,this theorem involves the irreducible representations U(μ)(g). Thus we first haveto express U′(μ)(g) in terms of U(μ)(g). To this end, we write Eq. (3.119) in theform

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3 Light scattering by particles with boundary symmetries 101

Uirr(g) =r⊕

μ=1

U(μ)(g), (3.142)

where each of the matrices

U(μ)(g) =

⎛⎜⎝U(μ)(g). . .

U(μ)(g)

⎞⎟⎠ μ = 1, . . . , r, (3.143)

is a βμ × βμ matrix that contains the irreducible matrix U(μ)(g) αμ times alongits diagonal. Thus both U′(μ)(g) and U(μ)(g) act on the βμ-dimensional invariantsubspace Tμ. Therefore, there must exist a similarity transformation S so that

U′(μ)(g) = S(μ) · U(μ)(g) · S−1(μ). (3.144)

To exploit in Eqs (3.141) and (3.144) the block-diagonal structure of U(μ)(g)given in Eq. (3.143), we split up the matrices U′(μ)(g), S, and Q′(μ,ν) into sub-matrices according to

U′(μ)(g) =

⎛⎜⎝U′(μ;1,1) · · · U′(μ;1,αμ)

......

U′(μ;αμ,1) · · · U′(μ;αμ,αμ)

⎞⎟⎠ , (3.145)

S(μ) =

⎛⎜⎝ S(μ;1,1) · · · S(μ;1,αμ)

......

S(μ;αμ,1) · · · S(μ;αμ,αμ)

⎞⎟⎠ , (3.146)

and

Q′(μ,ν) =

⎛⎜⎝Q′(μ,ν;1,1) · · · Q′(μ,ν;1,αν)

......

Q′(μ,ν;αμ,1) · · · Q′(μ,ν;αμ,αν)

⎞⎟⎠ . (3.147)

Each of the submatrices S(μ;l,m) and U′(μ;l,m) is an nμ × nμ matrix, whereaseach of the submatrices Q′(μ,ν;l,m) is an nμ × nν matrix.

Equations (3.141) and (3.144) can now be written asαμ∑

l′=1

U′(μ;l,l′)(g) · Q′(μ,ν;l′,m) =αν∑

l′′=1

Q′(μ,ν;l,l′′) · U′(ν;l′′,m)(g) (3.148)

∀g ∈ G, μ, ν = 1, . . . , r,l = 1, . . . , αμ, m = 1, . . . , αν ,

and

U′(μ;l,m)(g) =αμ∑

l′=1

S(μ;l,l′) · U(μ)(g) · S−1(μ;l′,m), (3.149)

∀g ∈ G, μ,= 1, . . . , r,l,m = 1, . . . , αμ.

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102 Michael Kahnert

Substitution of (3.149) into (3.148) yields, written out in components,nμ∑

p,p′,p′′=1

αμ∑l′,l′′=1

S(μ;l,l′)i,p U

(μ)p,p′(g)[S−1](μ;l′,l′′)

p′,p′′ Q′(μ,ν;l′′,m)p′′,j

=nν∑

q,q′,q′′=1

αν∑m′,m′′=1

Q′(μ,ν;l,m′)i,q S

(ν;m′,m′′)q,q′ U

(ν)q′,q′′(g)[S−1](ν;m′′,m)

q′′,j ,

∀g ∈ G, μ, ν = 1, . . . , r, (3.150)l = 1, . . . , αμ, m = 1, . . . , αν ,

i = 1, . . . , nμ, j = 1, . . . , nν .

We have succeeded, as planned, to express the commutator relation (3.141) interms of the irreducible representations U(μ)(g), and thus prepared the equationfor application of the great orthogonality theorem given in Eq.(3.129) Thus,multiplication of both sides of Eq. (3.150) with χ(μ)∗(g) =

∑nμ

k=1 U(μ)∗k,k (g) and

summation over all g ∈ G yields, together with (3.129),

Q′(μ,ν;l,m)i,j = δμ,νQ

′(μ,μ;l,m)i,j (3.151)

μ, ν = 1, . . . , r,i = 1, . . . , nμ, j = 1, . . . , nν ,

l = 1, . . . , αμ, m = 1, . . . , αν .

(3.152)

Together with Eq. (3.140) we obtain

Q′ = P−1 · Q · P =

⎛⎜⎜⎜⎝Q′(1,1) 0 · · · 00 Q′(2,2) · · · 0...

.... . .

...0 0 · · · Q′(r,r)

⎞⎟⎟⎟⎠ . (3.153)

Thus, the similarity transformation P reduces the Q-matrix to block-diagonalform, Q.E.D. This result has been obtained by exploiting the symmetry relationsof the Q-matrix given by the commutator relations (3.115) [or, equivalently, Eq.(3.139)], and the orthogonality property of the irreducible representations givenin (3.129).

Suppose we knew the similarity transformation S which, according to Eq.(3.144), fully reduces the representations of the group. Can we say anythingabout the transformation of the matrix Q′ under S? As we saw before in Eq.(3.139), the symmetry relations of the Q-matrix are form-invariant under similar-ity transformations. Thus, the commutator relations will also be form-invariantunder the transformation S. Defining Q′′(μ;l,m) =

∑αμ

l′,l′′=1 S(μ;l,l′) · Q′(μ,μ;l′,l′′) ·[S−1](μ;l′,m), the commutator relations given in (3.148) transform under S to

nμ∑p=1

U(μ)i,p (g)Q′′(μ;l,n)

p,j =nμ∑q=1

Q′′(μ;l,n)i,q U

(μ)q,j (g). (3.154)

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3 Light scattering by particles with boundary symmetries 103

We try to apply the same method as before in order to obtain information aboutthe structure of Q′′. Thus we multiply by χ(μ)∗(g)=

∑nμ

k=1 U(μ)∗k,k (g), sum over

all g ∈ G, and apply (3.129). However, this only results in the trivial relation

Q′′(μ;l,n)i,j = Q

′′(μ;l,n)i,j . (3.155)

We can conclude that the commutator relations of the Q-matrix do not yield anyadditional information about the structure of the Q-matrix when transformedunder S into the irreducible basis. Apparently, it is in general not possible toreduce the matrix Q′ any further. Thus the similarity transformation P that weconstructed in this section is the desired similarity transformation that bringsthe Q-matrix into irreducible, block-diagonal form.

The similarity transformation P has been constructed via Eq. (3.133) by useof the reducible representations Ured(g). Inspection of the reducible representa-tions derived in section 3.3.4 reveals that these are highly sparse matrices. Asa consequence, P is highly sparse as well. For numerical applications this hasthree crucial consequences. First, the matrix can be stored in compact form andrequires hardly any memory. Second, The CPU-time requirement for the trans-formation of the Q-matrix into block-diagonal form according to Eq. (3.153) issmall. Third, the inversion of P in Eq. (3.153) is very stable and can be done witha suitable sparse-matrix inversion algorithm. Subsequently, the block-diagonalmatrix Q′ can be numerically inverted by inverting each block matrix separately.As confirmed by numerical experiments [29], this improves the conditioning ofthe matrix inversion problem significantly. Further, since all the zero elements inthe off-diagonal block matrices no longer enter the matrix inversion algorithm,the CPU-time requirements for the numerical matrix inversion is reduced. Thisreduction outweighs the extra CPU-time required for the construction and ap-plication of the similarity transformation P [29]. This is a remarkable resultthat clearly demonstrates the power of group theory. Usually, attempts to in-crease the stability of the numerical matrix inversion, such as use of extendedprecision variables [48, 49], result in significantly increased computation time.By contrast, the pre-conditioning method based on irreducible representationsof point-groups increases the conditioning of the matrix inversion and, at thesame time, increases the overall computational speed.

Incidentally, the reason why the T -matrix approach has been numerically sosuccessful for axisymmetric particles [50] is related to the symmetry structureof the matrices. As we saw in Eq. (3.108), the Q-matrix becomes block-diagonalin the basis of spherical wave functions (due to the factor δm,m′), thus reducingthe problem of inverting the Q-matrix to the task of inverting each block-matrixseparately, which substantially improves the conditioning of the matrix inver-sion problem. Thus in the particular case of axial symmetry the spherical wavefunctions are already a basis in which the matrix is reduced to block-diagonalform.

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104 Michael Kahnert

3.4 Concluding remarks

The present review started out from a rather general treatment of symmetriesin boundary-value problems and proceeded to more and more specific formula-tions of the problem. First, we investigated boundary symmetries for arbitrarylinear differential or integral equations. Based on this general treatment of theproblem, we considered the more specific case of the Helmholtz equation inthe presence of spatial boundary symmetries. This was done within the self-consistent Green’s function formalism, which encompasses boundary-integralequation methods, volume-integral equation methods, and the T -matrix for-malism. The corresponding ‘coordinate-free’ quantities are the surface-Green’sfunction, the volume-Green’s function, and the interaction operator, respectively.We derived for each of these quantities the symmetry relations and showed thatthey are equivalent. Subsequently, we proceeded to a more explicit formulationby choosing a specific basis of the function space, thus deriving commutator rela-tions for the matrix representations of the Green’s function and of the interactionoperator. The latter is identical with the T -matrix. The commutator relations re-duce computational requirements in T -matrix computations by at least a factorof Mo, in boundary-integral equation computations even by a factor of M2

o , whereMo denotes the order of the symmetry group. Finally, we exploited irreduciblerepresentations of groups for deriving a similarity transformation that bringsall matrix quantities into block-diagonal form, thus achieving a pre-conditioningthat significantly alleviates the notorious ill-conditioning problems in T -matrixcomputations.

The scalar treatment in this paper can be readily generalised to a dyadic for-mulation. Representations of point-group elements in the basis of vector sphericalwave functions are given in [29]. One can also use a basis other than the sphericalwave functions. In [25], the commutator relations of the T -matrix for axisym-metric particles are derived in the basis of vector spheroidal wave functions.Further, it is possible to generalise the treatment from metallic (or, in acousticscattering, impenetrable) scatterers to dielectric (or penetrable) scatterers. Inthe simplest case of a homogeneous scatterer, the symmetries of the solution tothe problem are completely determined by the symmetries of the boundary sur-face. However, for inhomogeneous particles one needs to consider the symmetryproperties of both the interior domain inside the particle and of the boundarysurface. For example, an eccentrically coated sphere only has C∞v-symmetry andnot K-symmetry, even though the interior particle and the coating both havespherical boundaries.

References

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3 Light scattering by particles with boundary symmetries 105

2. F. M. Kahnert, J. J. Stamnes, and K. Stamnes. Can simple particle shapes be usedto model scalar optical properties of an ensemble of wavelength-sized particles withcomplex shapes? J. Opt. Soc. Am. A, 19:521–531, 2002.

3. F. M. Kahnert, J. J. Stamnes, and K. Stamnes. Using simple particle shapes tomodel the Stokes scattering matrix of ensembles of wavelength-sized particles withcomplex shapes: possibilities and limitations. J. Quant. Spectrosc. Radiat. Transfer,74:167–182, 2002.

4. T. Nousiainen and K. Vermeulen. Comparison of measured single-scattering ma-trix of feldspar particles with T-matrix simulations using spheroids. J. Quant.Spectrosc. Radiat. Transfer, 79-80:1031–1042, 2003.

5. F. M. Kahnert. Reproducing the optical properties of fine desert dust aerosolsusing ensembles of simple model particles. J. Quant. Spectrosc. Radiat. Transfer,85:231–249, 2004.

6. T. Nousiainen, M. Kahnert, and B. Veihelmann. Light scattering modeling of smallfeldspar aerosol particles using polyhedral prisms and spheroids. J. Quant. Spec-trosc. Radiat. Transfer, 101:471–487, 2006.

7. G. Mie. Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen. Ann.Phys., 25:377–445, 1908.

8. S. Asano and G. Yamamoto. Light scattering by a spheroidal particle. Appl. Opt.,14:29–49, 1975.

9. M. I. Mishchenko. Light scattering by randomly oriented axially symmetric parti-cles. J. Opt. Soc. Am. A., 8:871–882, 1991.

10. F. Borghese, P. Denti, R. Saija, G. Toscano, and O. I. Sindoni. Use of group theoryfor the description of electromagnetic scattering from molecular systems. J. Opt.Soc. Am. A, 1:183–191, 1984.

11. R. P. Tarasov. Harmonic analysis on finite groups and methods for the numeri-cal solution of boundary equations in boundary-value problems with non-Abeliansymmetry group. Comp. Maths. Math. Phys., 32:1367–1369, 1992.

12. E. V. Zakharov, S. I. Safronov, and R. P. Tarasov. Finite-order Abelian groupsin the numerical analysis of linear boundary-value problems of potential theory.Comput. Maths. Math. Phys., 32:34–50, 1992.

13. Ye. V. Zakharov, S. I. Safronov, and R. P. Tarasov. Reduction to a boundary-valueproblem with finite non-Abelian symmetry group using the interlacing operator.Comput. Maths. Math. Phys., 35:1275–1282, 1995.

14. E. V. Zakharov, S. I. Safronov, and R. P. Tarasov. Finite group algebras in iter-ational methods of solving boundary-value problems of potential theory. Comput.Maths. Math. Phys., 33:907–917, 1993.

15. R. P. Tarasov. Numerical solution of convolution-type equations on finite non-commutative groups. Comp. Maths. Math. Phys., 33:1589–1597, 1993.

16. I. A. Zagorodnov and R. P. Tarasov. The problem of scattering from bodies witha noncommutative finite group of symmetries and its numerical solution. Comput.Maths. Math. Phys., 37:1206–1222, 1997.

17. I. A. Zagorodnov and R. P. Tarasov. Finite groups in numerical solution of elec-tromagnetic scattering problems on non-spherical particles. In Light scattering bynonspherical particles: Halifax contributions, pages 99–102. Army Research Labo-ratory, Adelphi, MD, 2000.

18. I. A. Zagorodnov and R. P. Tarasov. Numerical solution of the problems of scat-tering by platonic bodies in the classes of functions invariant under symmetrytransformations. Comput. Maths. Math. Phys., 38:1247–1259, 1998.

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19. I. A. Zagorodnov and R. P. Tarasov. Numerical solution of the problems of scat-tering by platonic bodies in the classes of functions invariant under symmetrytransformations. Comput. Maths. Math. Phys., 40:1456–1478, 2000.

20. T. Weiland and I. Zagorodnov. Maxwell’s equations for structures with symmetries.J. Comput. Phys., 180:297–312, 2002.

21. F. M. Schulz, K. Stamnes, and J. J. Stamnes. Point group symmetries in electro-magnetic scattering. J. Opt. Soc. Am. A, 16:853–865, 1999.

22. F. M. Kahnert, J. J. Stamnes, and K. Stamnes. Application of the extended bound-ary condition method to homogeneous particles with point group symmetries. Appl.Opt., 40:3110–3123, 2001.

23. S. Havemann and A. J. Baran. Extension of T -matrix to scattering of electromag-netic plane waves by non-axisymmetric dielectric particles: application to hexago-nal ice cylinders. J. Quant. Spectrosc. Radiat. Transfer, 70:139–158, 2001.

24. P. C. Waterman. Matrix formulation of electromagnetic scattering. Proc. IEEE,53:805–812, 1965.

25. F. M. Schulz, K. Stamnes, and J. J. Stamnes. Scattering of electromagnetic wavesby spheroidal particles: A novel approach exploiting the T-matrix computed inspheroidal coordinates. Appl. Opt., 37:7875–7896, 1998.

26. T. A. Niemen, H. Rubinsztein-Dunlop, and N. R. Heckenberg. Calculation of theT -matrix: general considerations and application of the point-matching method. J.Quant. Spectrosc. Radiat. Transfer, 79-80:1019–1029, 2003.

27. D. W. Mackowski. Discrete dipole moment method for calculation of the T matrixfor nonspherical particles. J. Opt. Soc. Am. A, 19:881–893, 2002.

28. D. W. Mackowski and M. I. Mishchenko. Calculation of the T matrix and thescattering matrix for ensembles of spheres. J. Opt. Soc. Am. A, 13:2266–2278,1996.

29. M. Kahnert. Irreducible representations of finite groups in the T matrix formula-tion of the electromagnetic scattering problem. J. Opt. Soc. Am. A, 22:1187–1199,2005.

30. T. Rother, M. Kahnert, A. Doicu, and J. Wauer. Surface Green’s function of theHelmholtz equation in spherical coordinates. In J. A. Kong, editor, Progress inelectromagnetic research (PIER), volume 38, pages 47–95. EMW Publishing, Cam-bridge, MA, 2002.

31. M. Kahnert. Boundary symmetries in linear differential and integral equation prob-lems applied to the self-consistent Green’s function formalism of acoustic and elec-tromagnetic scattering. Opt. Commun., 265:383–393, 2006.

32. T. Rother. Self-consistent Green’s function formalism for acoustic and light scat-tering, Part 1: Scalar notation. Opt. Commun., 251:254–269, 2005.

33. T. Rother. Self-consistent Green’s function formalism for acoustic and light scat-tering, Part 2: Dyadic notation. Opt. Commun., 251:270–285, 2005.

34. T. Rother. Self-consistent Green’s function formalism for acoustic and light scat-tering, Part 3: Unitarity and symmetry. Opt. Commun., 266:380–389, 2006.

35. T. Rother. Scalar Green’s function for penetrable or dielectric scatterers. Opt.Commun., 274:15–22, 2007.

36. P. M. Morse and H. Feshbach. Methods of theoretical physics. McGraw-Hill, NewYork, 1953.

37. L. Tsang, J. A. Kong, and R. T. Shin. Radiative transfer theory for active remotesensing of a layer of nonspherical particles. Radio Sci., 19:629–642, 1984.

38. A. Sommerfeld. Partial differential equations. Academic Press, New York, 1949.39. D. M. Bishop. Group theory and chemistry. Dover Publications, Mineola, 1993.

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40. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii. Quantum theory ofangular momentum. World Scientific, Singapore, 1988.

41. G. Baym. Lectures on quantum mechanics. Addison-Wesley Publishing, Reading,1993.

42. M. Abramowitz and I. A. Stegun. Handbook of mathematical functions. DoverPublications, New York, 1972.

43. M. Hamermesh. Group theory and its application to physical problems. Dover Pub-lications, New York, 1989.

44. T. Inui, Y. Tanabe, and Y. Onodera. Group theory and its applications in physics.Springer, Berlin, 1996.

45. J. D. Dixon. High speed computation of group characters. Numerische Mathematik,10:446–450, 1965.

46. J. J. Cannon. Computers in group theory: a survey. Commun. ACM, 12:3–11, 1969.47. D. C. Harris and M. D. Bertolucci. Symmetry and spectroscopy. Oxford University

Press, New York, 1978.48. M. I. Mishchenko and L. D. Travis. T-matrix computations of light scattering by

large spheroidal particles. Opt. Commun., 109:16–21, 1994.49. S. Havemann and A. J. Baran. Calculation of the phase matrix elements of elon-

gated hexagonal ice columns using the T-matrix method. In T. Wriedt, editor,Electromagnetic and light scattering – Theory and applications VII, pages 107–110.Universitat Bremen, Bremen, 2003.

50. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski. T-matrix computations oflight scattering by nonspherical particles: A review. J. Quant. Spectrosc. Radiat.Transfer, 55:535–575, 1996.

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4 Scattering by particles on or near a planesurface

Adrian Doicu, Roman Schuh and Thomas Wriedt

4.1 Introduction

Computation of light scattering from particles deposited upon a surface is ofgreat interest in the simulation, development and calibration of surface scan-ners for wafer inspection [1]. More recent applications include laser cleaning [2],scanning near-field optical microscopy (SNOM) [3] and plasmon resonances ef-fects in surface-enhanced Raman spectroscopy (SERS) [4]. Several studies haveaddressed this scattering problem using different methods. Simplified theoret-ical models have been developed on the basis of Lorenz–Mie theory and Fres-nel surface reflection [5–8]. A coupled-dipole algorithm has been employed byTaubenblatt and Tran [9] and Nebeker et al. [10] using a three-dimensional ar-ray of dipoles to model a feature shape and the Sommerfeld integrals to describethe interaction between a dipole and a surface. The theoretical aspects of thecoupled-dipole model has been fully outlined by R. Schmehl [11]. A model basedon the discrete source method has been given by Eremin and Orlov [12, 13],whereas the transmission conditions at the interface are satisfied analyticallyand the fields of discrete sources are derived by using the Green tensor for aplane surface. More details on computational methods and experimental resultscan be found in a book edited by Moreno and Gonzales [14].

Similar scattering problems have been solved by Kristensson and Strom [15],and Hackmann and Sammelmann [16] in the framework of the null-field method.Acoustic scattering from a buried inhomogeneity has been considered by Kris-tensson and Strom on the assumption that the free-field T-matrix of the parti-cle modifies the free-field T-operator of the arbitrary surface. By projecting thefree-field T-operator of the surface onto a spherical basis, an infinite system oflinear equations for the free-field T-matrix of the particle has been derived. Incontrast, Hackmann and Sammelmann assumed that the free-field T-operatorof the surface modifies the free-field T-matrix of the particle, and in this case,the free-field T-matrix of the particle has been projected onto a rectangular ba-sis and an integral equation for the spectral amplitudes of the fields has beenobtained.

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110 Adrian Doicu, Roman Schuh and Thomas Wriedt

More recently Mackowski [24] extended the T-matrix approach to multiplespheres on a plane interface.

In this contribution we use the null-field method to analyze the scattering byparticles on a plane surface. Our intention is to treat several scattering geometrieswhich occur in practice. These include the scattering by single particles on aplane surface or on a plane surface coated with a film, and the scattering bysystems of particles.

4.2 Single particle on or near a plane surface

In this section we extend the results of Bobbert and Vlieger [5] to the caseof axisymmetric particles situated on or near a plane surface. The scatteringproblem is a multiple particles problem and the solution method is the separationof variables technique. To model the scattering problem in the framework of theseparation of variables technique one must address how the radiation interactswith the particle. The incident field strikes the particle either directly or afterinteracting with the surface, while the fields emanating from the particle mayalso reflect off the surface and interact with the particle again. The transitionmatrix relating the incident and scattered field coefficients is computed in theframework of the null-field method and the reflection matrix characterizing thereflection of the scattered field by the surface is computed by using the integralrepresentation for the vector spherical wave functions.

The geometry of the scattering problem is shown in Fig. 4.1. An axisymmetricparticle is situated in the neighborhood of a plane surface Σ, so that its axis ofsymmetry is normal to the plane surface. The z-axis of the particle coordinatesystem Oxyz is directed along the axis of symmetry and the origin O is situatedat the distance z0 below the plane surface. The incident radiation is a linearlypolarized vector plane wave propagating in the ambient medium (the mediumbelow the surface Σ)

O x

z

ke

0z0

mrs

mr

Fig. 4.1. Geometry of an axisymmetric particle situated near a plane surface. Theexternal excitation is a vector plane wave propagating in the ambient medium.

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4 Scattering by particles on or near a plane surface 111

Ee(r) = (Ee0,βeβ + Ee0,αeα) ejke·r, (4.1)

and the wave vector ke, ke = ksek, is assumed to be in the xz-plane and toenclose the angle β0 with the z-axis.

The incident wave strikes the particle either directly or after interacting withthe surface. The direct and the reflected incident fields are expanded in termsof regular vector spherical wave functions

Ee(r) =∞∑

n1=1

n1∑m=−n1

a0mn1

M1mn1

(ksr) + b0mn1

N1mn1

(ksr) (4.2)

and

ERe (r) =

∞∑n1=1

n1∑m=−n1

aRmn1

M1mn1

(ksr) + bRmn1

N1mn1

(ksr), (4.3)

respectively, and we define the total expansion coefficients amn1 and bmn1 by therelations

amn1 = a0mn1

+ aRmn1

,

bmn1 = b0mn1

+ bRmn1

.

The coefficients a0mn1

and b0mn1

are the expansion coefficients of a vector planewave travelling in the (β0, 0) direction,

a0mn1

= − 4jn1√2n1 (n1 + 1)

[jmπ|m|

n1(β0)Ee0,β + τ |m|

n1(β0)Ee0,α

],

b0mn1

= − 4jn1+1√2n1 (n1 + 1)

[τ |m|n1

(β0)Ee0,β − jmπ|m|n1

(β0)Ee0,α

],

while the coefficients aRmn1

and bRmn1

are the expansion coefficients of a vectorplane wave travelling in the (π − β0, 0) direction. The part of the incident fieldthat reflects off the surface will undergo a Fresnel reflection and it will be outof phase by an amount of exp(2jksz0 cosβ0). This phase factor arises from thephase difference between the plane wave and its reflected wave in O. The resultingexpressions for aR

mn1and bR

mn1are

aRmn1

= − 4jn1√2n1 (n1 + 1)

[jmπ|m|

n1(π − β0)ER

e0,β + τ |m|n1

(π − β0)ERe0,α

],

bRmn1

= − 4jn1+1√2n1 (n1 + 1)

[τ |m|n1

(π − β0)ERe0,β − jmπ|m|

n1(π − β0)ER

e0,α

],

where

ERe0,β = r‖(β0) e2jksz0 cos β0Ee0,β ,

ERe0,α = r⊥(β0) e2jksz0 cos β0Ee0,α,

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112 Adrian Doicu, Roman Schuh and Thomas Wriedt

O x

z

ke

0 0 keR

keT

Ee ERe

ETe

mrs

z0

Fig. 4.2. Reflection and refraction of a vector plane wave (propagating in the ambientmedium) at the interface Σ.

with r‖(β0) and r⊥(β0) being the Fresnel reflection coefficients for parallel andperpendicular polarizations, respectively. The Fresnel reflection coefficients aregiven by

r‖(β0) =mrs cosβ0 − cosβ

mrs cosβ0 + cosβ,

r⊥(β0) =cosβ0 − mrs cosβ

cosβ0 + mrs cosβ,

where mrs is the relative refractive index of the substrate with respect to the am-bient medium and β is the angle of refraction (Fig. 4.2). The angles of incidenceand refraction are related to each other by Snell’s law,

sinβ =1

mrssinβ0,

cosβ = ±√

1 − sin2 β,

and for real values of mrs, the sign of the square root is plus, while for complexvalues of mrs, the sign of the square root is chosen such that Im{mrs cosβ} > 0.This choice guarantees that the amplitude of the refracted wave propagating inthe positive direction of the z-axis would tend to zero with increasing distancez.

The scattered field is expanded in terms of radiating vector spherical wavefunctions

Es(r) =∞∑

n=1

n∑m=−n

fmnM3mn(ksr) + gmnN3

mn(ksr), (4.4)

and the rest of our analysis concerns the calculation of the expansion coefficientsfmn and gmn. In addition to the fields described by (4.2)–(4.4), a fourth fieldexists in the ambient medium. This field is a result of the scattered field reflectingoff the surface and striking the particle. It can be expressed as

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4 Scattering by particles on or near a plane surface 113

ERs (r) =

∞∑n=1

n∑m=−n

fmnM3,Rmn(ksr) + gmnN3,R

mn(ksr), (4.5)

where M3,Rmn(ksr) and N3,R

mn(ksr) are the radiating vector spherical wave func-tions reflected by the surface. Accordingly to Videen [6–8], the field ER

s will bedesignated as the interacting field. For r inside a sphere enclosed in the parti-cle and a given azimuthal mode m, we anticipate an expansion of the reflectedvector spherical wave functions of the form(

M3,Rmn(ksr)

N3,Rmn(ksr)

)=

∞∑n1=1

(αmnn1

γmnn1

)M1

mn1(ksr) +

(βmnn1

δmnn1

)N1

mn1(ksr). (4.6)

Inserting (4.6) into (4.5), we derive a series representation for the interactingfield in terms of regular vector spherical wave functions,

ERs (r) =

∞∑n1=1

n1∑m=−n1

fRmn1

M1mn1

(ksr) + gRmn1

N1mn1

(ksr)], (4.7)

where (fR

mn1

gRmn1

)=

∞∑n=1

(αmnn1

βmnn1

)fmn +

(γmnn1

δmnn1

)gmn. (4.8)

In the null-field method, the scattered field coefficients are related to theexpansion coefficients of the fields striking the particle by the transition matrixT. For an axisymmetric particle, the equations become uncoupled, permittinga separate solution for each azimuthal mode. Thus, for a fixed azimuthal modem, we truncate the expansions given by (4.2)–(4.4) and (4.7), and derive thefollowing matrix equation:[

fmn

gmn

]= [Tmn,mn1 ]

([amn1

bmn1

]+[

fRmn1

gRmn1

]), (4.9)

where n and n1 ranges from 1 to Nrank, and m ranges from −Mrank to Mrank,with Nrank and Mrank being the maximum expansion and azimuthal orders,respectively. The expansion coefficients of the interacting field are related to thescattered field coefficients by a so-called reflection matrix:[

fRmn1

gRmn1

]= [Amn1n]

[fmn

gmn

], (4.10)

where, in view of (4.8),

[Amn1n] =[

αmnn1 γmnn1

βmnn1 δmnn1

].

Now it is apparent that the scattered field coefficients fmn and gmn can beobtained by combining the matrix equations (4.9) and (4.10), and the result is[17]

Page 133: Light Scattering Reviews 3: Light Scattering and Reflection

114 Adrian Doicu, Roman Schuh and Thomas Wriedt

(I − [Tmn,mn1 ] [Amn1n])[

fmn

gmn

]= [Tmn,mn1 ]

[amn1

bmn1

]. (4.11)

To derive the expression of the reflection matrix we use the integral repre-sentations for the radiating vector spherical wave functions,

(M3

mn(ksr)N3

mn(ksr)

)= − 1

2πjn+11√

2n(n + 1)

2π∫0

π/2−j∞∫0

[(mπ

|m|n (β)

τ|m|n (β)

)eβ

+j

|m|n (β)

mπ|m|n (β)

)eα

]ejmα ejk(β,α)·r sinβ dβ dα, (4.12)

where (ks, β, α) are the spherical coordinates of the wave vector k, and (ek, eβ , eα)are the spherical unit vectors of k. Each reflected plane wave in (4.12) will con-tain a Fresnel reflection term and a phase term equivalent to exp(2jksz0 cosβ).The reflected vector spherical wave functions can be expressed as

(M3,R

mn(ksr)N3,R

mn(ksr)

)= − 1

2πjn+11√

2n(n + 1)

2π∫0

π/2−j∞∫0

[(mπ

|m|n (β)

τ|m|n (β)

)r‖(β)eβR

+ j

|m|n (β)

mπ|m|n (β)

)r⊥(β)eαR

]ejmα e2jksz0 cos β ejkR(βR,αR)·r

× sinβ dβ dα,

where βR = π − β, αR = α, (ks, βR, αR) are the spherical coordinates of thereflected wave vector kR, and (ekR, eβR, eαR) are the spherical unit vectors ofkR. For r inside a sphere enclosed in the particle, we expand each plane wave interms of regular vector spherical wave functions(

eβR

eαR

)ejkR·r = −

∞∑n1=1

n1∑m1=−n1

4jn1√2n1 (n1 + 1)

[(jm1π

|m1|n1 (π − β)

τ|m1|n1 (π − β)

)

× M1m1n1

(ksr) +

(jτ |m1|

n1 (π − β)

m1π|m1|n1 (π − β)

)N1

m1n1(ksr)

]e−jm1α,

and obtain the following expressions for the elements of the reflection matrix:

αmnn1 =2jn1−n√

nn1(n + 1)(n1 + 1)

π/2−j∞∫0

[m2π|m|

n (β)π|m|n1

(π − β)r‖(β)

+ τ |m|n (β)τ |m|

n1(π − β)r⊥(β)

]e2jksz0 cos β sinβ dβ, (4.13)

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4 Scattering by particles on or near a plane surface 115

βmnn1 =2jn1−n√

nn1(n + 1)(n1 + 1)

π/2−j∞∫0

m[π|m|

n (β)τ |m|n1

(π − β)r‖(β)

+ τ |m|n (β)π|m|

n1(π − β)r⊥(β)

]e2jksz0 cos β sinβ dβ, (4.14)

γmnn1 =2jn1−n√

nn1(n + 1)(n1 + 1)

π/2−j∞∫0

m[τ |m|n (β)π|m|

n1(π − β)r‖(β)

+ π|m|n (β)τ |m|

n1(π − β)r⊥(β)

]e2jksz0 cos β sinβ dβ, (4.15)

δmnn1 =2jn1−n√

nn1(n + 1)(n1 + 1)

π/2−j∞∫0

[τ |m|n (β)τ |m|

n1(π − β)r‖(β)

+ m2π|m|n (β)π|m|

n1(π − β)r⊥(β)

]e2jksz0 cos β sinβ dβ, (4.16)

An approximate expression for the reflection matrix can be derived if weassume that the interacting radiation strikes the surface at normal incidence.Assuming r(0) = r⊥(β) = −r‖(β), changing the variable from β to βR = π −β, and using the relations

π|m|n (π − βR) = (−1)n−|m|π|m|

n (βR),

τ |m|n (π − βR) = (−1)n−|m|+1τ |m|

n (βR),

yields the following simplified integral representations for the reflected vectorspherical wave functions:(

M3,Rmn(ksr)

N3,Rmn(ksr)

)= − (−1)n−|m|r(0)

2πjn+11√

2n(n + 1)

×2π∫0

π∫π/2+j∞

[(−mπ

|m|n (βR)

τ|m|n (βR)

)eβR

+ j

(−τ

|m|n (βR)

mπ|m|n (βR)

)eαR

]

× ejmαR e−2jksz0 cos βR ejkR(βR,αR)·r sinβR dβR dαR.

To compute M3,Rmn and N3,R

mn we introduce the image coordinate system O′x′y′z′

by shifting the original coordinate system a distance 2z0 along the positive z-axis. The geometry of the image coordinate system is shown in Fig. 4.3. Takinginto account that kR · r′ = kR · r−2ksz0 cosβR, where r′ = (x′, y′, z′),we identifyin the resulting equation the integral representations for the radiating vector

Page 135: Light Scattering Reviews 3: Light Scattering and Reflection

116 Adrian Doicu, Roman Schuh and Thomas Wriedt

Ox

z’

z0

x’

z

O’

z0r’

rP

Fig. 4.3. Image coordinate system.

spherical wave functions in the half-space z < 0:(M3,R

mn(ksr)N3,R

mn(ksr)

)= (−1)n−|m|r(0)

(−M3mn(ksr′)

N3mn(ksr′)

).

In this case the interacting field is the image of the scattered field and the expan-sion (4.6) can be derived by using the addition theorem for vector spherical wavefunctions. The elements of the reflection matrix are the translation coefficients,and as a result, the amount of computer time required to solve the scatteringproblem is significantly reduced. In this regard it should be mentioned that theformalism using the approximate expression for the reflection matrix has beenemployed by Videen [6–8].

In most practical situations we are interested in the analysis of the scatteredfield in the far-field region and below the plane surface, i.e., for θ > π/2. Inthis region we have two contributions to the scattered field: the direct electricfar-field pattern Es∞(θ, ϕ),

Es∞(θ, ϕ) =1ks

∞∑n=1

n∑m=−n

(−j)n+1 [fmnmmn(θ, ϕ) + jgmnnmn(θ, ϕ)] (4.17)

and the interacting electric far-field pattern ERs∞(θ, ϕ),

ERs∞(θ, ϕ) =

1ks

∞∑n=1

n∑m=−n

(−j)n+1 [fmnmR

mn(θ, ϕ) + jgmnnRmn(θ, ϕ)

], (4.18)

where mmn and nmn are the vector spherical harmonics, and mRmn and nR

mn arethe reflected vector spherical harmonics,

mRmn(θ, ϕ) =

1√2n(n + 1)

e−2jksz0 cos θ

×[jmπ|m|

n (π − θ)r‖(π − θ)eθ − τ |m|n (π − θ)r⊥(π − θ)eϕ

]ejmϕ,

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4 Scattering by particles on or near a plane surface 117

nRmn(θ, ϕ) =

1√2n(n + 1)

e−2jksz0 cos θ

×[τ |m|n (π − θ)r‖(π − θ)eθ + jmπ|m|

n (π − θ) r⊥(π − θ)eϕ

]ejmϕ.

Thus, the solution of the scattering problem in the framework of the separa-tion of variables method involves the following steps:

1. calculation of the T-matrix relating the expansion coefficients of the fieldsstriking the particle to the scattered field coefficients;

2. calculation of the reflection matrix A characterizing the reflection of vectorspherical wave functions by the surface;

3. computation of an approximate solution by solving the matrix equation(4.11);

4. computation of the far-field pattern by using (4.17) and (4.18).

In practice, we must compute the integrals in (4.13)–(4.16), which are of theform

I =

π/2−j∞∫0

f (cosβ) e2jq cos β sinβ dβ.

Changing variables from β to x = −2jq (cosβ − 1) , we have

I =e2jq

2jq

∞∫0

f

(1 − x

2jq

)e−x dx,

and integrals of this type can be computed efficiently by using the Laguerrepolynomials [5].

Scanning near-field optical microscopy [18,19] requires a rigorous analysis ofthe evanescent scattering by small particles near the surface of a dielectric prism[20–23]. The scattering of evanescent waves can be analyzed by extending ourformalism to the case of an incident plane wave propagating in the substrate(Fig. 4.4).

For the incident vector plane wave given by (4.1), the transmitted (or therefracted) vector plane wave is

EeT(r) =(ET

e0,βeβT + ETe0,αeαT

)ejkeT ·r,

where

ETe0,β = t‖(β0) ejksz0(cos β−mrs cos β0)Ee0,β ,

ETe0,α = t⊥(β0) ejksz0(cos β−mrs cos β0)Ee0,α,

β0 is the incident angle and (ekT, eβT, eαT) are the spherical unit vectors of thetransmitted wave vector keT. The Fresnel transmission coefficients are given by

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118 Adrian Doicu, Roman Schuh and Thomas Wriedt

O x

z

ke

0

z0

mrs

mr

keT

Fig. 4.4. Geometry of an axisymmetric particle situated near a plane surface. Theexternal excitation is a vector plane wave propagating in the substrate.

t‖(β0) =2mrs cosβ0

cosβ0 + mrs cosβ,

t⊥(β0) =2mrs cosβ0

mrs cosβ0 + cosβ,

while the angle of refraction is computed by using Snell’s law:

sinβ = mrs sinβ0,

cosβ = ±√

1 − sin2 β.

Evanescent waves appear for real mrs and incident angles β0 > β0c, where β0c =arcsin (1/mrs). In this case, sinβ > 1 and cosβ is purely imaginary. For negativevalues of z, we have

exp(jkeT · r) = exp(−jksz cosβ + jksx sinβ) = exp(jks |z| cosβ + jksx sinβ) ,

and we choose the sign of the square root such that Im{cosβ} > 0. This choiceguarantees that the amplitude of the refracted wave propagating in the negativedirection of the z-axis decreases with increasing the distance |z|. The expansioncoefficients of the transmitted wave are

aTmn1

= − 4jn1√2n1 (n1 + 1)

[jmπ|m|

n1(π − β)ET

e0,β + τ |m|n1

(π − β)ETe0,α

],

bTmn1

= − 4jn1+1√2n1 (n1 + 1)

[τ |m|n1

(π − β)ETe0,β − jmπ|m|

n1(π − β)ET

e0,α

],

and we see that our previous analysis remains unchanged if we replace the totalexpansion coefficients amn1 and bmn1 by the expansion coefficients of the trans-mitted wave aT

mn1and bT

mn1.

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4 Scattering by particles on or near a plane surface 119

4.3 Single particle on or near a plane surface coated with afilm

The scattering by a particle situated on a plane surface coated with a film canbe also treated with the above formalism. The major changes concern with thecalculation of the Fresnel reflection coefficients which enter in the expressionof the reflection matrix and the reflected incident field. To compute the Fres-nel reflection coefficients we consider the scattering of a plane wave by a layeredplane-parallel structure. The scattering geometry is shown in Fig. 4.5. The thick-ness of the film is d, while the relative refractive indices of the film and of thesubstrate are mrf and mrs, respectively. The electric fields in the three regions(ambient medium, film and substrate) are given by

E(r) = (Eβeβ + Eαeα) ejks·r + (EβReβR + EαReαR) ejkR·r,

Ef(r) = (EβTeβT + EαTeαT) ejkT·r + (EβfReβfR + EαfReαfR) ejkfR·r,

Es(r) = (EβfTeβfT + EαfTeαfT) ejkfT·r,

while the magnetic fields read as

H(r) = (−Eαeβ + Eβeα) ejks·r + (−EαReβR + EβReαR) ejkR·r,

Hf(r) =√

εrf (−EαTeβT + EβTeαT) ejkT·r

+√

εrf (−EαfReβfR + EβfReαfR) ejkfR·r,

Hs(r) =√

εrs (−EαfTeβfT + EβfTeαfT) ejkfT·r.

The incident wave vector encloses the angle β0 with the z-axis, and as a result,the wave vectors in the different regions can be expressed as.

O

x

z

ks

00 kRe

mrs

z0

kT

kfT

kfR

e R

e T e fR

e fT

d mrf

film

Fig. 4.5. Scattering geometry of a particle on a plane surface coated with a film.

Page 139: Light Scattering Reviews 3: Light Scattering and Reflection

120 Adrian Doicu, Roman Schuh and Thomas Wriedt

ks = ks (sinβ0ex + cosβ0ey) ,

kR = ks (sinβ0ex − cosβ0ey) ,

kT = mrfks (sinβ1ex + cosβ1ey) ,

kfR = mrfks (sinβ1ex − cosβ1ey) ,

kfT = mrsks (sinβ2ex + cosβ2ey) .

The angles of incidence and refraction are related to each other by Snell’s law:

sinβ0 = mrf sinβ1 = mrs sinβ2,

and the cosine of the refraction angles are computed accordingly to the relations

cosβ1 = ±√

1 − sin2 β1, Im (mrf cosβ1) > 0,

cosβ2 = ±√

1 − sin2 β2, Im (mrs cosβ2) > 0.

Imposing the boundary conditions at the interfaces between the regions

ez × E(r) = ez × Ef(r), ez × H(r) = ez × Hf(r), r = z0ez

and

ez × Ef(r) = ez × Es(r), ez × Hf(r) = ez × Hs(r), r = (z0 + d) ez

yields the desired relations

EβR = r⊥(β0)Eβ , EαR = r‖(β0)Eα,

where the Fresnel reflection coefficients are now given by

r‖(β0) =r01‖ (β0) + r12

‖ (β0) e2jmrfksd cos β1

1 + r01‖ (β0)r12

‖ (β0) e2jmrfksd cos β1e2jksz0 cos β ,

r⊥(β0) =r01⊥ (β0) + r12

⊥ (β0) e2jmrfksd cos β1

1 + r01⊥ (β0)r12

⊥ (β0) e2jmrfksd cos β1e2jksz0 cos β ,

with

r01‖ (β0) =

mrf cosβ0 − cosβ1

mrf cosβ0 + cosβ1,

r01⊥ (β0) =

cosβ0 − mrf cosβ1

cosβ0 + mrf cosβ1,

r12‖ (β0) =

mrs cosβ1 − mrf cosβ2

mrs cosβ1 + mrf cosβ2,

r12⊥ (β0) =

mrf cosβ1 − mrs cosβ2

mrf cosβ1 + mrs cosβ2.

Page 140: Light Scattering Reviews 3: Light Scattering and Reflection

4 Scattering by particles on or near a plane surface 121

It is apparent that when d → 0, then

r01‖ (β0) + r12

‖ (β0) e2jmrfksd cos β1

1 + r01‖ (β0)r12

‖ (β0) e2jmrfksd cos β1→ mrs cosβ0 − cosβ2

mrs cosβ0 + cosβ2= r02

‖ (β0)

and similarly,

r01⊥ (β0) + r12

⊥ (β0) e2jmrfksd cos β1

1 + r01⊥ (β0)r12

⊥ (β0) e2jmrfksd cos β1→ cosβ0 − mrs cosβ1

cosβ0 + mrs cosβ1= r02

⊥ (β0).

In this case, the solution corresponds to a particle situated on the plane surface.For mrf = mrs, the identities r12

‖ (β0) = r12⊥ (β0) = 0, imply that

r‖(β0) = r01‖ (β0) e2jksz0 cos β , r⊥(β0) = r01

⊥ (β0) e2jksz0 cos β ,

and we obtain the solution corresponding to a particle situated on the film.When the film is absorbing and d → ∞, we see that

e2jmrfksd cos β1 → 0,

and as before, we obtain the solution corresponding to a particle situated on thefilm.

4.4 System of particles on or near a plane surface

To compute the scattering characteristics of a system of particles on a planesurface we have to account for the surface interaction among the particles. Inthe following analysis we follow the formulation presented by Mackowski [24] forsphere clusters on a plane interface.

The situation under examination is illustrated in Fig. 4.6. The system consistsof N particles each characterized by a position vector r0i, while the plane surfaceis placed at the distance z0 with respect to the origin of a global coordinatesystem.

The field exciting the particle i consists of the direct and the reflected inci-dent field and the contribution from the individual particles. This contributionincludes the direct and the reflected components of the scattered field due to theparticle j, and we have the representation

Eexc,i (ri) = Ee (ri) + ERe (ri) + ER

s,i (ri) +N∑

j �=i

Es,j (rj) + ERs,j (rj) .

The incident field is expressed in the global coordinate system

Ee (r) + ERe (r) =

∞∑m1n1

am1n1M1m1n1

(ksr) + bm1n1N1m1n1

(ksr),

Page 141: Light Scattering Reviews 3: Light Scattering and Reflection

122 Adrian Doicu, Roman Schuh and Thomas Wriedt

ON

Oi

O

O1

Oj

r0jx

z

i, is, s

Rr0i

rji

z0

mrs

Fig. 4.6. Scattering geometry of a collection of particles on a plane surface.

whence using the addition theorem for regular spherical vector wave functions[M1

m1n1(ksr)

N1m1n1

(ksr)

]=[T 11

m1n1,mn (ksr0i)] [M1

mn (ksri)N1

mn (ksri)

],

yields a representation centered about the origin of the ith particle

Ee (ri) + ERe (ri) =

∞∑mn

ai,mnM1mn(ksri) + bi,mnN1

mn(ksri)

with [ai,mn

bi,mn

]=[T 11

m1n1,mn (ksr0i)] [ am1n1

bm1n1

].

For the field scattered by the jth particle, we consider the series representa-tion

Es,j (rj) =∑

m1n1

fj,m1n1M3m1n1

(ksrj) + gj,m1n1N3m1n1

(ksrj) ,

and use the addition theorem[M3

m1n1(ksrj)

N3m1n1

(ksrj)

]=[T 31

m1n1,mn (ksrji)] [M1

mn (ksri)N1

mn (ksrj)

],

which is valid for ri < rji, to derive

Es,j (ri) =∑mn

fij,mnM1mn (ksri) + gij,mnN1

mn (ksri) ,

with [fij,mn

gij,mn

]=[T 31

m1n1,mn (ksrji)] [ fj,m1n1

gj,m1n1

].

Page 142: Light Scattering Reviews 3: Light Scattering and Reflection

4 Scattering by particles on or near a plane surface 123

The reflected field scattered by the jth particle,

ERs,j (rj) =

∑m1n1

fj,m1n1M3,Rm1n1

(ksrj) + gj,m1n1N3,Rm1n1

(ksrj) ,

is first expressed in terms of regular spherical vector wave functions

ERs,j (rj) =

∑m2n2

fRj,m2n2

M1m2n2

(ksrj) + gRj,m2n2

N1m2n2

(ksrj) ,

where [fR

j,m2n2

gRj,m2n2

]= [Am1n1,m2n2 ]

[fj,m1n1

gj,m1n1

],

and A is the reflection matrix. Further using the transformation[M1

m2n2(ksrj)

N1m2n2

(ksrj)

]=[T 11

m2n2,mn (ksrji)] [M1

mn (ksri)N1

mn (ksrj)

],

we obtain a series representation centered about the origin of the ith particle,that is,

ERs,j (ri) =

∑mn

fRij,mnM1

mn (ksri) + gRij,mnN1

mn (ksri) ,

with [fR

ij,mn

gRij,mn

]=[T 11

m2n2,mn (ksrji)][Am1n1,m2n2 ]

[fj,m1n1

gj,m1n1

].

Thus, the field exciting the ith particle can be expressed in terms of regularvector spherical wave functions centered at the origin Oi:

Eexc,i (ri) =∑mn

ai,mnM1mn (ksri) + bi,mnN1

mn (ksri) ,

with the expansion coefficients being given by[ai,mn

bi,mn

]=[T 11

m1n1,mn (ksr0i)] [ am1n1

bm1n1

]+ [Am1n1,mn]

[fi,m1n1

gi,m1n1

]+

N∑j �=i

([T 31

m1n1,mn (ksrji)]+[T 11

m2n2,mn (ksrji)][Am1n1,m2n2 ]

)×[

fj,m1n1

gj,m1n1

],

Using the T-matrix equation[fi,m′n′

gi,m′n′

]= [Tm′n′,mn]

[ai,mn

bi,mn

]

Page 143: Light Scattering Reviews 3: Light Scattering and Reflection

124 Adrian Doicu, Roman Schuh and Thomas Wriedt

we obtain the interaction equations as

(I − [Tm′n′,mn] [Am′n′,mn])[

fi,m′n′

gi,m′n′

]−

N∑j �=i

[Tm′n′,mn]([

T 31m1n1,mn (ksrji)

]+[T 11

m2n2,mn (ksrji)][Am1n1,m2n2 ]

)×[

fj,m1n1

gj,m1n1

]= [Tm′n′,mn]

[T 11

m1n1,mn (ksr0i)] [ am1n1

bm1n1

].

Ensembling the interaction equations for all particles into a global system ofequations, and using a direct or an iterative solution method, yield the expressionof the scattered field coefficients.

The scattered field will be the sum of the direct and the reflected scatteredfields of all particles. In practice, we use the far-field representation of the fieldscattered by the ith particle in the direction er (θ, ϕ),

Es,i(r) =ejksri

ri

{Es∞,i(er) + O

(1ri

)}and the approximation

ejksri

ri=

ejksre−jkser·r0i

r

[1 + O

(1r

)],

to define the angular-dependent vector of scattering coefficients[fmn (er)gmn (er)

]=

N∑l=1

e−jkser·r0l

[fi,mn

gi,mn

].

To account of multiple scattering effects, we then consider the expressions of thedirect and the interacting electric far-field patterns Es∞(θ, ϕ) and ER

s∞(θ, ϕ) asgiven by (4.17) and (4.18) respectively, but with the angular-dependent scatter-ing coefficients fmn (er) and gmn (er), in place of the scattering coefficients fmn

and gmn.

4.5 Numerical simulation

In this section we present scattering results for an axisymmetric particle situatedon or near a plane surface. As reference we use a computer program based onthe discrete sources method [12,13].

Figs 4.7, 4.8 and 4.9 show the differential scattering cross-sections forFe-, Si- and SiO-spheroids with semi-axes a = 0.05 μm and b = 0.025 μm.The relative refractive indices are: mr = 1.35+1.97j for Fe, mr = 4.37+0.08j forSi, and and mr = 1.67 for SiO. The particles are situated on a silicon substrate,the wavelength of the incident radiation is λ = 0.488 μm, and the incident angle

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4 Scattering by particles on or near a plane surface 125

10-6

10-5

10-4

10-3

10-2

10-1

100

101

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

TPARTSUB - parallelTPARTSUB - perpendicularDSM - parallelDSM - perpendicular

Fig. 4.7. Normalized differential scattering cross-sections of a Fe-spheroid computedwith the TPARTSUB routine and the discrete sources method (DSM).

10-6

10-5

10-4

10-3

10-2

10-1

100

101

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

TPARTSUB - parallelTPARTSUB - perpendicularDSM - parallelDSM - perpendicular

Fig. 4.8. Normalized differential scattering cross-sections of a Si-spheroid computedwith the TPARTSUB routine and the discrete sources method (DSM).

is β0 = 45◦. The plotted data show that the T-matrix method leads to accurateresults.

In the next example we investigate scattering of evanescent waves by particlessituated on a glass prism. We note that evanescent wave scattering is importantin various sensor applications such as the total internal reflection microscopyTIRM [25]. Choosing the wavelength of the external excitation as λ = 0.488 μmand taking into account that the glass prism has a refractive index of mrs =

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126 Adrian Doicu, Roman Schuh and Thomas Wriedt

10-6

10-5

10-4

10-3

10-2

10-1

100

101

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

TPARTSUB - parallelTPARTSUB - perpendicularDSM - parallelDSM - perpendicular

Fig. 4.9. Normalized differential scattering cross-sections of a SiO-spheroid computedwith the TPARTSUB routine and the discrete sources method (DSM).

1.5, we deduce that the evanescent waves appear for incident angles exceeding41.8◦. In Figs. 4.10, 4.11 and 4.12 we plot the differential scattering cross-sectionfor Ag-, diamond and Si-spheres with a diameter of d = 0.2 μm. The relativerefractive indices of Ag- and diamond particles are mr = 0.25 + 3.14j and mr =2.43, respectively. The scattering plane coincides with the incident plane and theangle of incidence is β0 = 60◦. The plotted data show a good agreement betweenthe discrete sources and the T-matrix solutions.

10-5

10-4

10-3

10-2

10-1

100

101

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

TPARTSUB - parallelTPARTSUB - perpendicularDSM - parallelDSM - perpendicular

Fig. 4.10. Normalized differential scattering cross-sections of a metallic Ag-spherecomputed with the TPARTSUB routine and the discrete sources method (DSM).

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4 Scattering by particles on or near a plane surface 127

10-5

10-4

10-3

10-2

10-1

100

101

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

TPARTSUB - parallelTPARTSUB - perpendicularDSM - parallelDSM - perpendicular

Fig. 4.11. Normalized differential scattering cross-sections of a Diamond-sphere com-puted with the TPARTSUB routine and the discrete sources method (DSM).

10-5

10-4

10-3

10-2

10-1

100

101

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

TPARTSUB - parallelTPARTSUB - perpendicularDSM - parallelDSM - perpendicular

Fig. 4.12. Normalized differential scattering cross-sections of a Si-sphere computedwith the TPARTSUB routine and the discrete sources method (DSM).

In Figs 4.13 and 4.14 we plot the differential scattering cross-sections for aspherical particle with radius a = 0.05 μm situated on a plane surface coatedwith a film. The relative refractive indices are mr = 1.67, mrf = 1.46 + 0.1j andmrs = 1.5. The wavelength of the incident radiation is λ = 0.488 μm, and theincident angle is β0 = 45◦. When the thickness d of the film is very small or

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128 Adrian Doicu, Roman Schuh and Thomas Wriedt

10-8

10-7

10-6

10-5

10-4

10-3

10-2

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

d = 5 micronsd = 5.e-2 micronsd = 5.e-4 microns

Fig. 4.13. Differential scattering cross-sections for parallel polarization of a sphericalparticle situated on a plane surface coated with a film.

10-8

10-7

10-6

10-5

10-4

10-3

10-2

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

d = 5 micronsd = 5.e-2 micronsd = 5.e-4 microns

Fig. 4.14. Differential scattering cross-sections for perpendicular polarization of aspherical particle situated on a plane surface coated with a film.

very large, the differential scattering cross-sections correspond to the extremesituations of a particle situated on a plane surface with the refractive indicesmrs and mrf, respectively.

The differential scattering cross-sections of two prolate spheroids with semi-axes a = 0.1 μm and b = 0.05 μm is shown in Fig. 4.15. Both particles aresituated on the plane surface and the distance between their centers is 0.3 μm.The relative refractive indices are mr = 1.5 and mrs = 1.5, while the wavelength

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4 Scattering by particles on or near a plane surface 129

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

DS

CS

90 120 150 180 210 240 270Scattering Angle (deg)

parallelperpendicular

Fig. 4.15. Differential scattering cross-sections of two prolate spheroids situated on aplane surface

of the incident radiation and the incident angle are is λ = 0.628 μm and β0 = 45◦,respectively

4.6 Conclusions

Relations for calculating the light-scattering from particles on or near a surfaceare provided. The formalism is based on the null-field method and the integralrepresentation of vector spherical wave functions over plane waves. An approx-imate model is obtained as a special case by assuming that the scattered fieldreflecting off the surface and interacting with the particle is incident upon thesurface at near-normal incidence. The formalism is of general use and can alsobe applied to the scattering of particles on a plane surface coated with a filmand of a system of particles. The intention of this work has not been the com-prehensively examine of the scattering features of particles on plane surfaces.Rather, the objective has been to develop a formulation and a code which maketractable the exact calculations of such features.

Acknowledgement

We would like to acknowledge support of this research by DFG (DeutscheForschungsgemeinschaft). We are especially grateful to Daniel Mackowski forproviding the submitted version of his paper.

References

1. J. C. Stover: Optical Scattering: Measurement and Analysis, 2nd edn (SPIE Press,Bellingham, WA 1995).

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130 Adrian Doicu, Roman Schuh and Thomas Wriedt

2. B. Luk’yanchuk: Laser Cleaning (World Scientific, River Edge, NJ 2002).3. S. Kawata, M. Ohtsu, M. Irie: Near-Field Optics and Surface Plasmon Polaritons

(Springer, Berlin Heidelberg New York 2001).4. A. Campion, P. Kambhampati: Surface-enhanced Raman scattering. Chemical So-

ciety Reviews 27, 241 (1998).5. P. A. Bobbert, J. Vlieger: Light scattering by a sphere on a substrate. Physica

137, 209 (1986).6. G. Videen: Light scattering from a sphere on or near a surface. J. Opt. Soc. Am.

A 8, 483 (1991).7. G. Videen: Light scattering from a sphere behind a surface. J. Opt. Soc. Am. A

10, 110 (1993).8. G. Videen: Scattering from a small sphere near a surface. J. Opt. Soc. Am. A 10,

118 (1993).9. M. A. Taubenblatt, T. K. Tran: Calculation of light scattering from particles and

structures on a surface by the coupled-dipole method. J. Opt. Soc. Am. A 10, 912(1993).

10. B. M. Nebeker, G. W. Starr, E. D. Hirleman: Light scattering from patternedsurfaces and particles on surfaces. In Optical Characterization Techniques for highPerformance Microelectronic Device Manufacturing II, ed. by J. K. Lowell, R. T.Chen, J. P. Mathur (Proc. SPIE 2638, 1995), pp. 274–284.

11. R. Schmehl: The coupled-dipole method for light scattering from particles on planesurfaces. Diplomarbeit, Universitat Karlsruhe (TH), Karlsruhe 1994.

12. Y. Eremin, N. Orlov: Simulation of light scattering from a particle upon a wafersurface. Appl. Opt. 35, 6599 (1996).

13. Y. A. Eremin, N. V. Orlov: Analysis of light scattering by microparticles on thesurface of a silicon wafer. Optics and Spectroscopy 82, 434 (1997).

14. F. Moreno, F. Gonzalez: Light Scattering from Microstructures (Springer, Berlin2000).

15. G. Kristensson, S. Strom: Scattering from buried inhomogeneities – a general three-dimensional formalism. J. Acoust. Soc. Am. 64, 917 (1978).

16. R. H. Hackman, G. S. Sammelmann: Acoustic scattering in an homogeneous waveg-uide: Theory. J. Acoust. Soc. Am. 80, 1447 (1986).

17. T. Wriedt, A. Doicu: Light scattering from a particle on or a near surface. Opt.Commun. 152, 376 (1998).

18. R. C. Reddick, R. J. Warmack, T. L. Ferrell: New form of scanning optical mi-croscopy. Phys. Rev. 39, 767 (1989).

19. R. C. Reddick, R. J. Warmack, D. W. Chilcott, S. L. Sharp, T. L. Ferrell: Photonscanning tunneling microscopy. Rev. Sci. Instrum. 61, 3669 (1990).

20. P. C. Chaumet, A. Rahmani, F. Fornel, J.-P. Dufour: Evanescent light scattering:The validity of the dipole approximation. Phys. Rev. 58, 2310 (1998).

21. C. Liu, T. Kaiser, S. Lange, G. Schweiger: Structural resonances in a dielectricsphere illuminated by an evanescent wave. Opt. Commun. 117, 521 (1995).

22. M. Quinten, A. Pack, R. Wannemacher: Scattering and extinction of evanescentwaves by small particles. Appl. Phys. 68, 87 (1999).

23. R. Wannemacher, A. Pack, M. Quinten: Resonant absorption and scattering inevanescent fields. Appl. Phys. 68, 225 (1999).

24. D. Mackowski: Exact solution for the scattering and absorption properties of sphereclusters on a plane surface. J. Quant. Spec. Rad. Transf. 109, 770 (2007).

25. D. C. Prieve: Measurement of colloidal forces with TIRM. Advances in Colloid andInterface Science 82, 93 (1999).

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Part II

Radiative Transfer and Inverse Problems

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5 Impact of single- and multi-layered cloudinesson ozone vertical column retrievals using nadirobservations of backscattered solar radiation

V. V. Rozanov and A. A. Kokhanovsky

5.1 Introduction

The ozone shields the biosphere of the Earth from the harmful UV radiation.Therefore, the monitoring of ozone concentration is of primary importance.Quantitatively the concentration of ozone in the atmosphere can be describedby the vertical profile of its number density. The integral of the number den-sity over entire atmosphere is usually referred to as a vertical column (VC). Thelong-time information about ozone number density and its VC at the global scaleare obtained using the satellite measurements of the reflected solar radiation. Inparticular, series of TOMS (Total Ozone Mapping Spectrometer) (Ahmad et al.,2004; Herman and Celarier, 1997; Niu et al., 1992), SAGE (Stratospheric Aerosoland Gas Experiment) (McDermid et al., 1990), and SBUV (Solar BackscatteredUltraviolet) (Klenk et al., 1982; Mateer et al., 1971; McPeters et al., 1994) pro-vided a wealth of useful information on the ozone vertical column distributionsaround the globe. These measurements were complemented by retrievals fromGlobal Ozone Monitoring Experiment (GOME) (Burrows et al., 1999; Weber etal., 2005), Scanning Imaging Absorption Spectrometer for Atmospheric Char-tographY (SCIAMACHY) (Bovensmann et al., 1999) and Ozone MonitoringInstrument (OMI) (Levelt et al., 2005).

The accuracy of the ozone vertical column retrieval depends on numerousfactors. Among other the presence of clouds in the field-of-view of a satelliteinstrument can significantly limit the accuracy of retrievals. The influence ofclouds on ozone vertical column retrievals using, for example, TOMS has beenstudied by Liu (2002); Liu et al. (2004); Newchurch et al. (2001), and Ahmad etal. (2004) among others. The corresponding investigations for the GOME mea-surements were performed by Koelemeijer and Stammes (1999) and Koelemeijer(2001).

From the theoretical point of view clouds can be accounted for in the ra-diative transfer model used for the retrieval of vertical column. However, thisrequires information on numerous cloud parameters such as cloud top and bot-tom heights, the vertical distribution of liquid water and ice crystals in the cloud,etc., as well as a distance between cloud layers in the case of multi-layered cloud

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134 V. V. Rozanov and A. A. Kokhanovsky

systems. Unfortunately complete information about cloud parameters is usuallyunavailable. Therefore, simplified cloud models are used to take clouds into ac-count in the retrieval of ozone vertical columns using radiative transfer inversionalgorithms. The main requirement to the involved cloud model is that all neededcloud parameters are estimated using the measurement of the same instrumentas used for the vertical column retrieval or other instruments placed on the samespace platform.

The Lambert Equivalent Reflector (LER) model is widely used in the ozonevertical column retrieval algorithms (Burrows et al., 1999; Koelemeijer, 2001; Liuet al., 2004). This model contains only two parameters, i.e., the cloud sphericalalbedo and the position of the cloud approximated as LER above the surface.Both parameters can be estimated using the measurements of the reflected ra-diation at the top of atmosphere (TOA). The spherical albedo can be obtainedusing the measurement in the spectral range where gaseous absorption is weak,whereas the position of LER can be estimated using the measurement of thereflected radiation within absorption bands of gaseous absorbers with knownconcentration such as, for example, O2. The main feature of this model is thatthe radiative transfer processes between the ground surface and the position ofLER are fully ignored. As a consequence, the application of the LER model tothe retrieval of the cloud top height (CTH) leads to the systematic underesti-mation of the cloud upper boundary altitude. Moreover, the application of LERto the retrieval of vertical column allows to estimate the ozone vertical columnbetween the TOA and the position of the lambertian reflector only. The luckof information about the vertical column between LER and surface is usuallycompensated by the usage of a climatological data.

The vertically homogeneous cloud (VHC) model has been introduced bySaiedy et al. (1965) to improve the cloud top height retrieval accuracy. TheVHC model is described by the following set of parameters: the cloud opticalthickness (COT), the cloud top height (CTH), and the cloud bottom height(CBH). Although this model describes properly the radiative transfer processesin the entire atmosphere, this is an approximation not only for the multi-layeredcloud systems but also for a single-layer cloud due to the possible influence ofthe vertical cloud inhomogeneity. Nevertheless, it was shown in the frameworkof TOMS retrieval algorithm (Liu, 2002; Liu et al., 2004) that the employing ofthe VHC model to the determination of the ozone vertical column in the cloudyatmosphere leads to the improved accuracy of retrievals as compared to the LERmodel.

In addition to the VHC and LER models we propose here the verticallyinhomogeneous cloud (VIC) model. This model can be used in a solution ofsuch inverse problems as the retrieval of vertical columns of gaseous absorbersusing the satellite measurements of the backscattered solar radiation. The VICmodel is described by the same parameters as the VHC model, i.e., COT, CTHand CBH. However, it has in contrast to the VHC model very specific verticaldistribution (see below) of such physical cloud parameters as effective radii ofcloud droplets and ice crystals as well as the liquid water and ice water content.

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5 Impact of single and multi-layered cloudiness 135

This model can be of advantage for the retrieval of gaseous absorber verticalcolumns in the case of multi-layered cloud systems.

The main goal of this chapter is to investigate the impact of cloudiness onthe ozone vertical column retrieval accuracy using hyperspectral measurementsof the backscattered solar radiation like GOME and SCIAMACHY, employingthe advanced differential optical absorption spectroscopy (DOAS) retrieval tech-nique. As compared to the previous publications cited above our investigation ischaracterized by several new features.

– First of all, taking into account that GOME and SCIAMACHY data containnot only O3 but also O2-A absorption band spectral range, we employ in ourinvestigation the simultaneous solution of the cloud parameters and ozoneVC inverse problems. This allows us to obtain the realistic estimation of thecloud parameters which should be used in the ozone VC retrieval process.

– The cloud scenarios used for the simulation of experimental data contain notonly a single water cloud with the fixed geometrical thickness (1 km) as used,for example, by Koelemeijer (2001), but also an vertically inhomogeneous icecloud as well as water and ice/water two- and three-layered cloud systems.

– To mitigate the influence of several limitations which are usually used in theoperational cloud parameters and ozone VC retrieval algorithms we have em-ployed the general weighting function approach and exact radiative transfercalculations for the solution of corresponding forward and inverse problems.This allows us to investigate the ozone VC retrieval errors caused by theemploying in the retrieval process different cloud models such as VHC, VIC,and LER.

The layout of this chapter is as follows. Section 5.2 describes the methodo-logy of our investigation. The atmospheric and cloud scenarios used for forwardsimulations are described in section 5.3. The selected results of the forwardsimulations are presented and discussed in section 5.4. Section 5.5 contains thebrief overview of the retrieval algorithms used for the determination of cloudparameters and ozone vertical columns. Results of numerical experiments arepresented in section 5.6. The analytical expressions for the weighting function ofall relevant parameters are given in Appendix A.

5.2 Methodology

The investigation of the impact of different cloud models on the retrieval accu-racy of the ozone VC using the satellite data is a very complicated task. The mainreason is the lack of detailed information on the cloud properties especially inthe case of multi-layered cloud systems. Due to this fact the investigations basedon the experimental data are usually more qualitative than quantitative. The ex-amples of such investigations are presented by Wagner et al. (2004). Therefore,to obtain quantitative estimations of the impact of different cloud parameterson the accuracy of a gaseous absorber vertical column retrieval, we have chosenhere the end-to-end numerical experiment technique. The main difference of our

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136 V. V. Rozanov and A. A. Kokhanovsky

approach from many others used up to date is that both cloud parameters andozone vertical column are retrieved simultaneously using exact radiative transfercalculations. The conceptual flow of our investigations is as follows:

– we formulate the main scenario for the clear atmosphere including the verticalprofiles of pressure, temperature, number densities of gaseous components(O3, NO2, and O2), aerosol properties, and surface albedo;

– we formulate the different cloudiness scenarios including the vertically inho-mogeneous water and/or ice single and multi-layered cloud systems;

– using the radiative transfer code SCIATRAN (Rozanov et al., 2005), wecalculate for these scenarios the spectra of the reflected radiation at TOAfor different solar zenith angles in the spectral ranges 325–336 nm and 758–770 nm containing the absorption bands of ozone and oxygen, respectively;

– the simulated spectra are referred to as synthetic data;– the cloud parameters i.e., COT, CTH and CBH for VHC and VIC models

and the position of LER are retrieved using the synthetic data in 758–770 nmspectral range;

– VHC, VIC and LER models described by the corresponding estimated pa-rameters are employed in the retrieval of the ozone vertical column using thesynthetic data in the spectral range 325–336 nm;

– the retrieval errors of cloud parameters and the ozone vertical columns ob-tained using different cloud models are analyzed.

The described end-to-end approach requires the usage of certain algorithms toretrieve parameters of the cloud models and the ozone vertical column in cloudyatmosphere. Although there are different algorithms to retrieve cloud parame-ters, they have certain disadvantages. Thus, for example, the algorithms for CTHand CBH as well as for the altitude of LER retrievals suggested by Rozanov andKokhanovsky (2004) and by Koelemeijer et al. (2001), respectively, use certainapproximations of the reflection function which can introduce an additional er-ror. Therefore, we have employed here the exact radiative transfer model to cal-culate the reflected intensity and the general weighting function approach for theretrieval of parameters of VHC, VIC and LER models. The weighting functiondifferential optical absorption spectroscopy (WFDOAS) technique (Buchwitz etal., 2000; Weber et al., 2005) is used to obtain the ozone vertical column. Thesmall modifications needed to include VHC and VIC models in the WFDOASretrieval algorithm are described below.

The accuracy of the ozone VC retrieval depends not only on an instrument’sspectral range but also on the assumed spectral sampling and on the spectralresolution. The investigation of the influence of these spectral parameters on theaccuracy of the vertical column retrieval is beyond the scope of this chapter.Hence, we have used for our simulations spectral parameters close to those ofthe instruments GOME (Burrows et al., 1999) and SCIAMACHY (Bovensmannet al., 1999). Moreover, we restrict this study to the error estimation of cloudparameters and ozone vertical column retrieval in the case of fully cloudy con-ditions. The influence of broken cloudiness on cloud top height retrievals using

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5 Impact of single and multi-layered cloudiness 137

nadir observations of backscattered solar radiation in the oxygen A-band is con-sidered by Kokhanovsky et al. (2007a) and an error analysis of ozone verticalcolumn retrieval under broken cloud conditions is given by Kokhanovsky et al.(2007b).

5.3 Atmospheric and cloud models used for forwardsimulations

For the simulation of the reflected radiation all relevant atmospheric processeswere considered: Rayleigh scattering, aerosol scattering and absorption, molec-ular absorption, and multiple scattering of light in clouds. A surface albedo of0.05 was assumed for all considered scenarios, which is a good choice over darksurfaces such as oceans in the near-IR. The vertical profiles of the pressure, tem-perature, and vertical density profiles of O3 and NO2 were used according tothe MPI model (Bruhl and Crutzen, 1993) (northern hemisphere at 45◦N, June15th). The ozone vertical column is equal for this atmospheric model to 302 DU(1 DU = 2.6867×1016 molecules/cm2). The vertical profile of O2 concentrationwas used according to the US-standard atmospheric model (NASA, 1976). Theaerosol properties were attributed as specified by Kneizys et al. (1996) in fouratmospheric layers positioned in the ranges 0–2 km, 2–10 km, 10–30 km, and 30–60 km. In the boundary layer (0–2 km) a maritime aerosol model with a humidityof 80% and a visibility of 23 km was used. The same humidity and visibility wasassumed for the tropospheric aerosol model. The stratospheric and mesosphericaerosol were set to the background and normal mesosphere, respectively. Thevertically integrated Rayleigh and aerosol optical thicknesses are equal to 0.85and 0.39 at 325 nm and to 0.026 and 0.26 at 760 nm, respectively.

Water and ice clouds with different geometrical and optical thicknesses werepositioned at various levels in the atmosphere. Water droplet and ice crystalphase functions were assumed to be constant within a cloud. For water cloudsC1 droplet size distribution was used (Deirmendjian, 1969). The phase func-tion was calculated using Mie theory. For ice clouds, the fractal particle modelwas assumed (Macke et al., 1996) and the calculations were performed usingthe Monte Carlo geometrical optics code. In contrast to the phase functionsthe extinction and absorption coefficients of water and ice clouds were assumedto be dependent on the altitude within a cloud. The extinction coefficients ofwater droplets, Ke1(z), and ice crystals, Ke2(z), were calculated employing thefollowing analytical approximations (Kokhanovsky, 2007):

Ke1(z) =32

l1(z)r1(z)ρ1

{1 + Ax

−2/31 (z) − B

[1 − eCx

−2/31 (z)

]}, (5.1)

Ke2(z) =32

l2(z)r2(z)ρ2

, (5.2)

where subscripts ‘1’ and ‘2’ correspond to water droplets and ice crystals, re-spectively; l1(z) and l2(z) are the liquid water content (LWC) and ice water

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138 V. V. Rozanov and A. A. Kokhanovsky

content (IWC), respectively; r1(z) and r2(z) are the effective radii of particles(rk(z) = 3Vk(z)/Gk(z) , k = 1, 2, where Vk(z) and Gk(z) are the average volumeand the average surface area of particles); xk(z) = 2πrk(z)/λ; λ is the wave-length; ρ1 and ρ2 are the densities of the water and ice, respectively. ConstantsA, B, and C are calculated employing the Mie theory: A = 1.1, B = 1.7 × 10−6

and C = 56.3 (Kokhanovsky, 2007).The absorption coefficients of water droplets, Ka1(z), and ice crystals,

Ka2(z), were calculated as follows (Kokhanovsky, 2007):

Ka1(z) =l1(z)ρ1

4πχ1

λA1

[1 − A2x1(z)χ1

][1 + A3

(1 − e−A4λ/r1(z)

)], (5.3)

Ka2(z) =32

l2(z)r2(z)ρ2

D[1 − e−2ηx2(z)χ2

]. (5.4)

Here, χk represents the imaginary part of the refractive index mk = nk − iχk

of water (k = 1) and ice (k = 2), A1 = 1.23, A2 = 2.6, A3 = 0.34, A4 = 8,D = 0.47, and η depends on the assumed shape of ice crystals. It is equal to 3.6for fractal particles used in this work. The accuracy of these simple equationswas studied by Kokhanovsky (2007).

Taking into account that a cloud can be located at different altitudes and itcan have different values of the geometrical thickness, we employ a dimensionlessvariable to describe vertical profiles of cloud parameters. As such a variable weintroduce, following Feigelson (1981):

x =ht − z

ht − hb, (5.5)

where ht and hb are the cloud top height and cloud bottom height, respectively.The variable x is dimensionless and ranges from 0 at the vertical coordinateequal to ht to 1 at the cloud bottom. The corresponding vertical profiles of theeffective radii of water droplets and ice crystals used in the following forwardsimulations are shown in the left panel of Fig. 5.1.

If the vertical profiles of LWC and IWC are defined, equations (5.1)–(5.4)can be directly used to calculate extinction and absorption coefficients withinthe cloud. However, it is more convenient in the framework of our investigation tocharacterize optical properties of the cloud by an integral parameter such as, forexample, the optical thickness of a cloud. Therefore, we introduce instead of theabsolute values of LWC and IWC profiles used in Eqs (5.1)–(5.4), the functionss1(z) and s2(z), which describe the shape of the corresponding vertical profiles.These shape functions are shown in the right panel of Fig. 5.1. The form of theLWC profile is close to that obtained for stratus clouds (Feigelson, 1981) andthe IWC shape profile is constructed using the experimental and modeling datareported by Seo and Liu (2006). In this case the vertical profile of the cloudextinction coefficient was calculated to match a given cloud optical thickness asfollows:

Kek(z) = CkKsk

(z) , k = 1, 2 . (5.6)

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5 Impact of single and multi-layered cloudiness 139

Fig. 5.1. Vertical profiles of the effective radii (left panel), and ice/water content shapefunctions (right panel) used in the forward simulations.

Here, the scaling factor Ck is defined as

Ck = τk

[∫ ht

hb

Ksk(z) dz

]−1

, (5.7)

where τk is a given optical thickness of water (k = 1) or ice cloud (k = 2), Ksk(z)

is calculated employing Eq. (5.1) or Eq. (5.2) for a given profile of effective radiusand LWC/IWC shape profiles (s1(z)/s2(z), shown in the right panel of Fig. 5.1).

Thus, according to the introduced approach the single-layer cloud used inthe forward simulations is described by the following set of parameters:

pf = {hb, ht, τ, ts} , (5.8)

where the cloud bottom height, hb, and cloud top height, ht, will be given in km,τ presents the optical thickness, and thermodynamic state of a cloud, ts, isdescribed by ts = 1 and ts = 2 corresponding to the water or to ice cloud,respectively. Table 5.1 contains the description of the cloud scenarios which wereused in the forward simulations. We have also considered within each scenariofive additional cases (i = 1, 2, . . . , 5 in Table 1) describing the different positionsand optical thicknesses of cloud layers. Thus, for example, the scenario 1-1 givenin Table 5.1 as {i, 1 + i, 20, 1}, i = 1, 2, . . . , 5 denotes that the cloud bottomheight and the cloud top height of a water cloud having optical thickness equalto 20 increase from 1 km to 5 km and from 2 km to 6 km, respectively, with thestep 1 km.

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140 V. V. Rozanov and A. A. Kokhanovsky

Table 5.1. Cloud scenarios used in the forward simulations

Scenario Lower layer Middle layer Upper layer

number hb ht τ ts hb ht τ ts hb ht τ ts

1-1 i 1+i 20 11-2 1 1+i 30×i 11-3 4+i 6+i 5 2

2-1 1 2 20 1 2+i 3+i 10 12-2 1 2 10 1 2+i 3+i 5 12-3 1 2 20 1 2+i 3+i 2 2

3-1 1 2 20 1 2+i 3+i 10 1 9 9.5 2 23-2 1 2 10 1 2+i 3+i 5 1 9 9.5 2 2

5.4 Forward simulations

5.4.1 Reflection function

The calculations of the reflected light intensity at TOA were performed for thenadir viewing geometry and the solar zenith angles equal to 10◦, 30◦ and 60◦

using the radiative transfer code SCIATRAN 2.1 (Rozanov et al., 2005). Theresults are presented in terms of the reflection function or reflectivity defined asfollows:

R(λ) =πI(λ)μ0F (λ)

, (5.9)

where I(λ) is the intensity of reflected radiation, F (λ) is the incident solar flux,and μ0 is the cosine of the solar zenith angle (SZA) at the TOA. If the inci-dent solar flux, F (λ), is assumed to be equal π it follows that R(λ) = I(λ)/μ0.The examples of the reflection function, R(λ), calculated in the ozone and O2-Aabsorption spectral bands for two positions of the water cloud having opticalthickness 20 and geometrical thickness equal to 1 km are shown in Fig. 5.2. Tobetter demonstrate the impact of a cloudiness on the reflection function, thereflection functions for the cloud-free atmosphere are given in Fig. 5.2 as well.It can clearly be seen that the cloudiness increases significantly the reflectionfunction in both considered spectral regions. This is a well known effect relatedto the enhancement of multiple scattering processes within the cloud. The re-flection function of the atmosphere containing a cloud of a given optical andgeometrical thicknesses depends on the cloud position as well. Fig. 5.2 showsthat the shift of the cloud upward results in the increasing of the reflectionfunction (compare the solid and dashed lines without symbols in this figure).This can be explained by the fact that the extinction of the direct solar and thereflected cloud radiation by the aerosol particles and molecules above a cloudis smaller for the cloud positioned at a higher level as compared to the lowerone. An another effect related to the cloud position in the atmosphere is the

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5 Impact of single and multi-layered cloudiness 141

Fig. 5.2. Reflection function at the TOA for clear and cloudy atmosphere. Leftpanel: 1,3, reflection functions corresponding to the cloud parameters {hb, ht, τ, ts} ={1, 2, 20, 1} (see Eq. (5.8)) with and without ozone absorption in the atmosphere, re-spectively; 2,4, the same but for cloud parameters {5, 6, 20, 1}; 5, reflection function forthe cloud-free atmosphere. Right panel: the same as in the left panel but in the O2-Aabsorption band. The calculations are performed for the solar zenith angle 10◦.

decreasing of the radiation absorption by the moving of the cloud upward. Thiseffect can clearly be seen in the right panel of Fig. 5.2 comparing the depth ofthe O2-A absorption band (the difference between maximal and minimal valuesof the reflection function) for two different cloud positions (solid and dashedlines without symbols, respectively). This dependence is widely used to retrievethe cloud top height from measurements of the backscattered solar radiation inO2-A absorption band (see, for example, Rozanov and Kokhanovsky (2004) andreferences therein).

Clearly, the absorption of the radiation in the ozone absorption band dependson the cloud position as well. However, this is not as obvious as in the case ofO2-A absorption band. To demonstrate this let us introduce as a measure of thegaseous absorption for a given wavelength, λ, the relative difference between thereflection functions as follows:

A(λ) =R0(λ) − R(λ)

R0(λ), (5.10)

where R0(λ) and R(λ) are reflection functions calculated excluding and includ-ing gaseous absorption, respectively. The reflection functions without gaseousabsorption, R0(λ), corresponding to the cloud position {1, 2} km and {5, 6} kmare shown in Fig. 5.2 by solid and dashed lines with symbols, respectively. Theintroduced according to Eq. (5.10) quantity A(λ) will be referred to as the inte-gral absorption.

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142 V. V. Rozanov and A. A. Kokhanovsky

The integral absorptions, A(λ), were calculated for the cloud positioned be-tween 1 and 2 km, A12(λ), and between 5 and 6 km, A56(λ), as well as forthe clear sky condition, Acl(λ). Fig. 5.3 shows the difference of the integral ab-sorptions calculated according to Eq. (5.10) between the cloudy and clear skyconditions for SZA of 10◦. In particular, the well known effect of the enhancementof the radiation absorption in the presence of cloud is clearly seen. It follows thatthe differences A12(λ) − Acl(λ) and A56(λ) − Acl(λ) (solid and dashed lines inFig. 5.3) are positive. The enhancement of absorption is caused by the increasingof the photon path length due to multiple scattering in clouds. This effect wasconsidered for the reflected radiation by Kurosu et al. (1997) and in the caseof the transmitted radiation was investigated by Mayer et al. (1998). Moreover,we can see that the enhancement of the absorption in the cloudy atmosphere ascompared to the clear one depends significantly on the cloud position (comparesolid and dashed lines in Fig. 5.3). In the case under consideration the absorp-tion of the radiation in the ozone band is smaller for the higher cloud positionas compared to the lower one (dotted line in Fig. 5.3) analogously to the case ofO2-A absorption band.

The introduced according to Eq. (5.10) measure of the integral radiationabsorption allows us to obtain the integral effect only. However, it does not showhow strong is the absorption of radiation in the different atmospheric layerslocated above, within or beneath of a cloud. In the following subsection wedemonstrate that the differential absorption, i.e., the absorption of the radiationin an infinitesimal layer located at the altitude z, is related to the variationalderivative of the reflection function with respect to the absorber number densityprofile.

5.4.2 Weighting function and differential absorption

To better demonstrate the impact of the cloudiness on the absorption of radiationin the different atmospheric layers we consider further the weighting function(WF) for the vertical profile of the ozone number density. The WF providesthe linear relationship between the variation of the reflection function and thevariation of the atmospheric parameters. In the case under consideration thislinear relationship can be obtained considering reflection function as a functionalof the absorber number density and expanding it in the functional Taylor seriesas follows:

R′(λ) = R(λ) +

H∫0

W (λ, z)δn(z)n(z)

dz + εl(λ) , (5.11)

where R′(λ) and R(λ) are reflection functions corresponding to the perturbed,n′(z), and unperturbed, n(z), ozone number density profiles, respectively,W (λ, z) is the variational derivative of the reflection function with respect tothe relative variation of the ozone number density (referred to as the weight-ing function here), δn(z) = n′(z) − n(z) is the absolute variation of the ozone

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5 Impact of single and multi-layered cloudiness 143

concentration, the integration is carried out over entire atmosphere, H is theTOA altitude, and εl(λ) is the linearization error containing contributions ofhigher-order terms of the Taylor series expansion with respect to δn(z)/n(z).A complete discussion of the mathematical aspects related to functionals andvariational derivatives is presented by Volterra (1959) among others.

The weighting function for the relative variation of a gaseous absorber num-ber density introduced according to Eq. (5.11) is closely related to the differentialabsorption of the radiation in the atmosphere at a given altitude, z. To demon-strate this let us assume that the perturbed value of the ozone concentration,n′(z), is equal to zero. In this case R′(λ) given by Eq. (5.11) describes the re-flection function without ozone absorption. Substituting the relative variationof the ozone number density δn(z)/n(z) = −1 corresponding to n′(z) = 0 intoEq. (5.11) and neglecting the linearization error, εl(λ), the linear approximationof the reflection function without ozone absorption, R0(λ), is obtained:

R0(λ) = R(λ) −H∫

0

W (λ, z) dz . (5.12)

Substituting the obtained expression for the reflection function without absorp-tion, R0(λ), instead of R0(λ) into Eq. (5.10), we have

A(λ) = − 1R0(λ)

H∫0

W (λ, z) dz . (5.13)

Introducing the function a(λ, z) as follows:

a(λ, z) = − W (λ, z)R0(λ)

, (5.14)

we rewrite Eq. (5.13) in the following equivalent form:

A(λ) =

H∫0

a(λ, z) dz . (5.15)

Thus, we can see that the introduced function, a(λ, z), represents the differentialabsorption of the radiation at a given altitude, z, and it is proportional to theweighting function for the relative variation of the absorber number density.

To calculate WFs for the atmospheric and surface parameters one can use sev-eral approaches (see, for example, Rozanov and Rozanov (2007) and referencestherein). Here, the WFs are calculated according to the analytical expressionsobtained using the forward-adjoint approach which has been implemented in ourradiative transfer code SCIATRAN. The analytical expression for the absorbernumber density WF is given in Appendix A. The differential absorptions of theradiation at the wavelength 325 nm for clear and cloudy conditions calculated

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144 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.3. Enhancement of the absorption in the cloudy atmosphere; 1, A12(λ)−Acl(λ);2, A56(λ) − Acl(λ); 3, A56(λ) − A12(λ); 4, the same as 3 but calculated in the linearapproximation employing Eq. (5.17).

according to Eq. (5.14) are shown in Fig. 5.4. It follows that the differentialabsorption for cloudy conditions are greater than that for the clear atmosphereabove clouds and smaller below them. Comparing a(λ, z) corresponding to thecloud positions {1, 2} km (solid line) and {5, 6} km (dashed line) presented inFig. 5.4, one can see that the absorption for the upper cloud is slightly greaterabove the cloud and significantly smaller below it. This comparison shows thatthe shift of a cloud upward results in the increasing of the absorption in theatmosphere above the cloud and in the decreasing of absorption below a cloud.The integral effect is a sum of these opposite effects. It can result in the decreas-ing or increasing of the absorption in the entire atmosphere depending on cloudposition, cloud optical and geometrical parameters, geometry of observation, etc.In the considered case of the cloud having geometrical and optical thicknessesequal to 1 km and to 20, respectively, the integral effect results in the decreasingof the absorption for the cloud positioned at a higher altitude. This is shown inFig. 5.3 (dotted line) using the calculated reflection functions with and withoutozone absorption for two cloud positions.

The integral effect of the impact of the cloud position on the absorption ofradiation in the entire atmosphere can be estimated not only calculating thereflection functions with and without ozone absorption according to Eq. (5.10)but also using Eq. (5.13) containing the WF. Indeed, introducing the integratedweighting function as

Wc(λ) =

H∫0

W (λ, z) dz , (5.16)

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5 Impact of single and multi-layered cloudiness 145

Fig. 5.4. The vertical profiles of differential absorption at 325 nm for cloudy andclear sky conditions; 1, cloud is characterized by {hb, ht} = {1, 2} km; 2, cloud ischaracterized by {hb, ht} = {5, 6} km; 3, clear sky. Right panel is the same but foraltitude range 0–8 km

the integral absorption in the entire atmosphere given by Eq. (5.13) can berewritten in the linear approximation as follows:

A(λ) = −Wc(λ)R0(λ)

. (5.17)

Employing Eq. (5.17), we have calculated the linear estimation of the absorp-tion enhancement corresponding to the different cloud positions, i.e., ΔA(λ) =A56(λ) − A12(λ), where A12(λ) and A56(λ) are integral absorptions correspond-ing to the cloud positions {1, 2} km and {5, 6} km, respectively. The result isshown in Fig. 5.3 by symbols. Comparing the obtained result to the exact cal-culation of ΔA(λ) given in Fig. 5.3 by dotted line, one can see that the linearapproximation given by Eq. (5.17) works better for smaller values of light absorp-tance by ozone. The difference between ΔA(λ) and ΔA(λ) presented in Fig. 5.3by the dotted line and symbols, respectively, demonstrates the impact of thelinearization error.

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146 V. V. Rozanov and A. A. Kokhanovsky

5.4.3 Impact of cloud parameters on the integral absorption

In the previous subsection we have shown the dependence of the integral absorp-tion of radiation on the position of a cloud having a given optical (τ = 20) andgeometrical (1 km) thickness. Here, we give more attention to demonstrating thedependence of the radiation absorption in the ozone spectral band on the cloudparameters and on the position of LER. For this purpose we have calculatedA(λ) according to Eq. (5.17) for different cloud optical and geometrical thick-nesses, different cloud top heights and different positions of LER. The obtainedresults for the solar zenith angles 10◦ and 60◦ are shown in the left and rightpanels of Fig. 5.5, respectively. It follows that for a cloud with a given geomet-rical thickness (1 km) the absorption decreases with increasing of the cloud topheight. This dependence is similar for clouds with optical thicknesses 5, 20 and100 (see lines 1, 2 and 3 in Fig. 5.5) as well as for LER (line 4 in Fig. 5.5). Thegradient of the absorption with respect to the cloud top height depends on thecloud optical thickness and it increases with increasing the optical thickness.

In the considered examples the increase in the cloud optical thickness iscaused by the increase the cloud scattering coefficient because the geometricalthickness of cloud was fixed. However, in the case of a vertically extended cloud(such as, for example, a deep convection cloud) the increasing of the opticalthickness is caused by the increasing of the geometrical thickness of the cloud

Fig. 5.5. The integral absorption of radiation for different cloud models at the wave-length 330 nm as a function of the cloud top height for SZA 10◦ (left panel) and 60◦

(right panel); 1, τ = 5, geometrical thickness 1 km; 2, τ = 20, geometrical thickness1 km; 3, τ = 100, geometrical thickness 1 km; 4, LER model; 5, cloud parameters aregiven by Eq. (5.18)

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5 Impact of single and multi-layered cloudiness 147

rather than being due to the increase of its scattering coefficient. To demonstratethe dependence of the absorption on the cloud top height in this case we haveconsidered an extended cloud characterizing by the following set of parameters:

pf = {hb, ht, τ, ts} = {1, 1 + i, 30 × i, 1}, i = 1, 2, . . . , 7 , (5.18)

i.e., a water cloud having the constant CBH (1 km) and constant scatteringcoefficient (∼30 km−1), which geometrical and optical thickness increase withthe step 1 km and 30, respectively. The dependence of the absorption on CTHin this case is represented by dashed-double-dotted line (line 5) in Fig. 5.5. Itfollows that this dependence is not monotonous as compared to one obtainedfor the cloud with fixed optical and geometrical thicknesses. In particular, it isclearly seen that the absorption increases for the increasing of CTH between2 km and 3 km. This example shows that the absorption of radiation in theatmosphere containing a cloud with a given top height depends not only on itsoptical thickness but also on its geometrical thickness as well. Indeed, it is easyto estimate that for the scattering coefficient equal to ∼30 km−1 the opticalthickness of the extended cloud reaches ∼100 at the geometrical thickness equalto ∼3.33 km. Taking into account that the extended cloud has the constantCBH equal to 1 km, we obtain that the optical thickness 100 corresponds toCTH of extended cloud at ∼4.33 km. Comparing for this CTH the absorptioncorresponding to the cloud with τ = 100 and geometrical thickness 1 km (dashed-dotted line in Fig. 5.5) to one of the extended cloud corresponding to the sameτ but for the geometrical thickness 3.33 km (dashed-double-dotted line) we canclearly see that the absorption of radiation in the cloudy atmosphere dependsnot only on CTH and optical thickness of cloud but on its geometrical thicknessas well.

5.4.4 Linear approximation for the reflection function with respectto the cloud parameters

The linear approximation can be employed to estimate the variation of the re-flection function caused not only by the variation of ozone number density asgiven by Eq. (5.11) but also by the variation of the cloud parameters such as thecloud top and bottom heights as well as the cloud optical thickness. Consideringthe reflection function, R(λ), as a function of CTH and CBH, and expanding itin the Taylor series in a small range around their values h′

t and h′b, we obtain in

the linear approximation:

R′(λ) = R(λ) + Wt(λ)(h′t − ht) + Wb(λ)(h′

b − hb) . (5.19)

Here, the reflection functions R′(λ) and R(λ) correspond to the cloud parame-ters {h′

b, h′t, τ, ts} and {hb, ht, τ, ts}, respectively, Wt(λ) and Wb(λ) are the par-

tial derivatives of the reflection function with respect to the cloud top and cloudbottom heights, respectively, which will be referred to as weighting functions.This linear representation is of great importance for the solution of inverse prob-lems because it allows us to reduce an initial nonlinear inverse problem to the

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148 V. V. Rozanov and A. A. Kokhanovsky

linear one. Assuming that the cloud is shifted upwards without the changing itsgeometrical thickness i.e., h′

t − ht = h′b − hb, we obtain

R′(λ) = R(λ) +[Wt(λ) + Wb(λ)

](h′

t − ht) . (5.20)

Thus, the variation of the reflection function caused by the moving of a cloudwithout changing of its geometrical thickness can be described in the linearapproximation by WF which is the sum of CTH and CBH weighting functions.

The variation of reflection function caused by the variation of the cloud opti-cal thickness can be written in a linear approximation analogously to Eq. (5.20)as follows:

R′(λ) = R(λ) + Wτ (λ)(τ ′ − τ) , (5.21)

where R′(λ) is the reflection function corresponding to the cloud parameters{hb, ht, τ

′, ts} and Wτ (λ) is the weighting function for optical thickness.The weighting functions for the different cloud parameters have been de-

rived by Rozanov et al. (2007) and implemented in the radiative transfer codeSCIATRAN 2.1. The analytical expressions for CTH, CBH, and cloud opticalthickness WFs are given in Appendix A. Fig. 5.6 illustrates the application of

Fig. 5.6. Linear approximation of the reflection function caused by the variationof cloud position (left panel) and by the variation of cloud optical thickness (rightpanel). Left panel: 1, initial reflection function corresponding to the cloud parame-ters {1, 2, 20, 1}; 2, exact reflection function corresponding to the cloud parameters{5, 6, 20, 1}; 3, linear approximation of the reflection function obtained employingEq. (5.20) for h′

t − ht = 4 km. Right panel: 1, initial reflection function correspond-ing to the cloud parameters {1, 2, 20, 1}; 2, exact reflection function corresponding tothe cloud parameters {1, 2, 22, 1}; 3, linear approximation of the reflection functionobtained employing Eq. (5.21) for τ ′ − τ = 2.

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5 Impact of single and multi-layered cloudiness 149

Eqs (5.20) and (5.21) for calculations of the reflection functions corresponding tothe shift of cloud upward by 4 km (left panel) and by the increase in its opticalthickness by 10% (right panel). In both cases the initial reflection function, R(λ),is calculated for a set of cloud parameters {hb, ht, τ, ts} ≡ {1, 2, 20, 1} (dottedline in Fig. 5.6). Comparing the exact values of the reflection functions givenby solid lines in Fig. 5.6 to the linear approximation given by symbols, one cansee that the linear approximation works well even for relatively large variationsof the cloud position (+4 km) and the cloud optical thickness (+10%). Thus,the considered examples show that WFs for cloud parameters can be used toestimate variations of the reflection function caused by variations of CTH, CBH,and COT with reasonable accuracy.

5.4.5 Scaling approximation and weighting function for ozonevertical columns

The introduced WFs for cloud parameters offer a simple way to compare thevariations of the reflection function caused by the variation of cloud param-eters and, on the other hand, by the variation of ozone vertical column. Wedemonstrate now that the integrated WFs, Wc(λ), introduced above allow usto obtain the variation of the reflection function caused by the variation of theozone vertical column. Let us assume that the perturbed vertical profile of ozoneconcentration, n′(z), can be expressed as follows:

n′(z) = Cn(z) , (5.22)

where C is a constant. This approximation is widely used in the framework of thedifferential optical absorption spectroscopy (DOAS) and referred to as a scalingapproximation. Integrating both sides of Eq. (5.22) over the entire atmosphere,and taking into account that the corresponding integral is the ozone verticalcolumn, we obtain

V ′ = CV −→ C =V ′

V, (5.23)

where V ′ is the perturbed ozone vertical column corresponding to the numberdensity profile n′(z). Using the obtained expression for the scaling factor C, therelative variation of the ozone vertical profile can be rewritten as follows:

δn(z)n(z)

=n′(z) − n(z)

n(z)= C − 1 =

V ′

V− 1 =

ΔV

V. (5.24)

Thus, according to the scaling approximation the relative variation of the ozonenumber density profile is independent of the altitude and is equal to the rel-ative variation of the ozone vertical column. Substituting δn(z)/n(z) given byEq. (5.24) into Eq. (5.11) and introducing the weighting function for the ozonevertical column as follows:

WV (λ) = Wc(λ)/V , (5.25)

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150 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.7. Relative variation of the reflection function corresponding to the set of cloudparameters {1, 2, 20, 1} caused by the variations of ozone vertical column and cloudparameters; 1, decrease of ozone vertical column by 1%; 2, increase of the cloud opticalthickness by 0.25; 3, shift of the cloud top to 1 km upward; 4, shift of the cloud bottomheight by 1 km upward.

where Wc(λ) is the integrated WF given by Eq. (5.16), we obtain

R′(λ) = R(λ) + WV (λ)ΔV . (5.26)

Summing up all the results obtained, the relative variation of the reflectionfunction caused by the variation of the cloud parameters and the ozone verticalcolumn, can be written as follows:

ΔxR

R=

Wx(λ)R

Δx , (5.27)

where x corresponds either to one of cloud parameters, i.e., top height, ht, bottomheight, hb, optical thickness, τ , or to the ozone vertical column, V .

Fig. 5.7 shows the relative variations of the reflection function expressed inpercentage calculated according to Eq. (5.27) caused by the increasing of cloudtop height by 1 km (Δht), cloud bottom height by 1 km (Δhb), optical thicknessof cloud by 0.25 (Δτ/τ = 1.25%), and by decreasing of ozone vertical columnby 1% (ΔV ≈ −3DU). It can clearly be seen that the variation of the cloudparameters cause significant variation of the reflection function. The impact ofthe cloud optical thickness variation is especially large.

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5 Impact of single and multi-layered cloudiness 151

5.5 Inverse problem

The main goal of the inverse problem solution is to obtain information aboutunknown atmospheric parameters using observations of reflected or transmittedradiation. In the case under consideration the parameters of interest are theozone vertical column and cloud parameters such as cloud top height, bottomheight and cloud optical thickness needed for the correct ozone retrievals. In thissection we briefly describe the retrieval algorithms used in the framework of ournumerical experiments to derive these parameters using the measurements ofthe backscattered solar radiation at the TOA in O3 and O2-A absorption spec-tral bands. The algorithms are based on the numerical solution of the radiativetransfer equation using the discrete ordinates approach.

Formulating the inverse problems, it is suitable to introduce two sets of pa-rameters, i.e., {rf , cf} and {rr, cr}, characterizing the true cloud and the cloudmodel involved in the retrieval process, respectively. The first set of parameters,rf(r), comprises all cloud parameters which will be retrieved employing the cor-responding cloud model. The parameter set, cf , comprises parameters describingtrue cloud or true cloud system which are not the subject of the retrieval process.The vertical profiles of cloud particle radii and LWC/IWC shapes (see Fig. 5.1)as well as the geometrical thicknesses and the distances between clouds belongto the set of parameters cf in the case of a single cloud and a multi-layered cloudsystem. The parameter set cr contains parameters of cloud models which are notthe subject of the retrieval process.

5.5.1 Retrieval of cloud parameters

The number of cloud parameters to be retrieved depends on the cloud modelused to describe the cloudiness solving the corresponding inverse problem. Inthe framework of our numerical experiments we will use several cloud modelssuch as the vertically homogeneous cloud, the vertically inhomogeneous cloud,and the effective Lambertian reflector. The VHC and VIC models are charac-terized by the following parameters: CTH, CBH and COT which belong to theparameter set rr. It is assumed for VHC model that the cloud parameters suchas effective radius of cloud particles and LWC/IWC shape profiles are indepen-dent of the altitude within the cloud. The effective radii of water droplets andice crystals were set to 6 μm and 100 μm, respectively. The vertical profiles ofthese parameters for the VIC model are shown in Fig. 5.8. The effective radii ofwater droplets, ice crystals, and LWC/IWC shape profiles belong to the param-eter set cr in the case of VHC and VIC models. The introduced VIC model canroughly be considered as two coupled homogeneous clouds. Very small valuesof the shape function in the middle of this model allow us to simulate almostcloud-free conditions between upper and lower parts of VIC model. Therefore,this model can be of advantage for retrieval in the case of multi-layered cloudsystems. The LER model is characterized by the spherical albedo and the posi-tion of the Lambertian reflector above the ground. Following Koelemeijer et al.(2001), we will assume that the spherical albedo is fixed for this model and equal

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152 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.8. Vertical profiles of the effective radii (left panel), and ice/water content shapefunction (right panel) for the vertically inhomogeneous cloud model.

to 0.8. The spherical albedo is the single parameter belonging to the parameterset cr for the LER model. Thus, for the LER model we need to estimate onlyone parameter, i.e., the position above the ground. In the following subsectionswe will briefly describe the retrieval algorithms used in the framework of ourinvestigation to derive the cloud parameters.

5.5.1.1 Determination of cloud parameters related to homogeneousand inhomogeneous cloud models

The synthetic reflection function, Rλ(rf , cf ), in the O2-A absorption band spec-tral range can be expressed by the employing in the retrieval process VHC orVIC models as follows:

Rλ(rf , cf ) = Rλ(rr, cr) + Wτ (λ)Δτ + Wt(λ)Δht + Wb(λ)Δhb

+ εl(λ) + εm(λ) , (5.28)

where {rf , cf} and {rr, cr} describe sets of cloud parameters used in the forwardsimulations and in the retrieval process, respectively, εl(λ) is a linearization error,and εm(λ) is a cloud model error. The linearization error contains contributionsof higher-order terms of the Taylor series expansion of the reflection functionwith respect to Δτ , Δht and Δhb. The cloud model error describes the differencebetween reflection functions calculated using cf or cr sets of cloud parameters,respectively, assuming that there is no difference between parameter sets rf

and rr.

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5 Impact of single and multi-layered cloudiness 153

Although the reflection function, Rλ(rr, cr), and WFs in the right-hand sideof this equation are different for VHC and VIC models, we will not explicitly no-tate this for the sake of simplification. Thus, according to Eq. (5.28) we assumethat the difference between synthetic, Rλ(rf , cf ), and the simulated, Rλ(rr, cr),reflection functions can be minimized by a suitable choice of the cloud param-eters, rr, characterizing VHC or VIC models. Taking into account that in thespectral ranges where gaseous absorption is weak, the contribution of CTH andCBH variations to the variation of reflection function is small, we obtain theestimation of COT, neglecting in Eq. (5.28) terms containing Δht and Δhb. Itfollows for the wavelength, λn, where gaseous absorption is weak that

Rλn(rf , cf ) = Rλn

(rr, cr) + Wτ (λn)(τ ′ − τ) + εl(λn) + εm(λn) . (5.29)

Neglecting further the linearization and cloud model errors and solving thisequation with respect to the parameter τ ′, we obtain the estimation of COT as

τ = τ +Rλn

(rf , cf ) − Rλn(rr, cr)

Wτ (λn). (5.30)

The obtained COT, τ , should be considered as a first estimation of the opticalthickness because the impact of the linearization error can be significant. Thiserror can be decreased substantially using the iteration process. Therefore, theimpact of the cloud model error, εm(λ), is the main source of the cloud opticalthickness retrieval error in the framework of our numerical experiments. Thesuggested cloud optical thickness retrieval algorithm has two main advantagesas compared to the existing ones. First, it does not require the time-consumingpreparation of look-up tables which are usually used in the operational cloud op-tical thickness retrieval algorithms (Nakajima and Nakajima, 1995) and, second,it has no restriction relative to the minimal value of τ as comparing to a semi-analytical cloud-retrieval algorithm suggested by Kokhanovsky et al. (2003),which can be employed in the case of optically thick clouds only.

Having estimated the cloud optical thickness, CTH and CBH can be obtainedneglecting in Eq. (5.28) the term containing the variation Δτ , linearization errorand cloud model error. It follows that:

Rλ(rf , cf ) = Rλ(rr, cr) + Wt(λ)Δht + Wb(λ)Δhb , (5.31)

where the simulated reflection function, Rλ(rr, cr) and the WFs are calculatedfor the cloud optical thickness τ given by Eq. (5.30). The estimation of the desiredparameter Δht and Δhb is obtained as a solution of the following minimizationproblem: ∥∥∥ΔRλ − Wt(λ)Δht − Wb(λ)Δhb

∥∥∥2⇒ min , (5.32)

hmint ≤ h′

t ≤ hmaxt , (5.33)

hminb ≤ h′

b ≤ hmaxb , (5.34)

h′b < h′

t , (5.35)

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154 V. V. Rozanov and A. A. Kokhanovsky

where ΔRλ = Rλ(rf , cf )−Rλ(rr, cr) is the difference between synthetic and sim-ulated reflection functions, and additional linear constraints are used to obtain asolution in the physically reasonable range even if the correlation between CTHand CBH weighting functions is considerable. The problem of minimization ofa quadratic function subject to a set of linear constraints on the variables givenby Eqs (5.32)–(5.35) is solved employing the algorithm described by Conn et al.(1999).

We note that in the framework of our numerical experiments the solutionof Eq. (5.29) and minimization of the quadratic form given by Eq. (5.32) arefound iteratively to mitigate the impact of linearization errors. As in the caseof the optical thickness retrieval, the absolute errors of the cloud geometricalparameters determination are mainly due to the impact of a cloud model error.For all retrievals of cloud geometrical parameters we have actually used values ofreflection functions normalized to the value of these functions outside the O2-Aabsorption band.

The described cloud geometrical parameters retrieval algorithm has been sug-gested by Rozanov and Kokhanovsky (2004) to derive the cloud top altitude andthe cloud geometrical thickness using the physical parameterization of the top-of-atmosphere reflection function (Kokhanovsky and Rozanov, 2004). The applica-tions of the algorithm to the retrieval of cloud top height using GOME and SCIA-MACHY data were reported by Rozanov et al. (2004) and Kokhanovsky et al.(2004), respectively. Here, in contrast to the work of Rozanov and Kokhanovsky(2004) based on the asymptotic radiative transfer theory (Kokhanovsky andRozanov, 2004) we use the exact calculation of the reflection function and cor-responding WFs to avoid the impact of parameterization errors. If the cloudtop height is known or estimated from independent measurements the algorithmdescribed above can be used to derive the cloud bottom height. This possibilitywas theoretically investigated by Kokhanovsky and Rozanov (2005) and the al-gorithm was applied to derive the cloud geometrical thickness from GOME data(Rozanov and Kokhanovsky, 2006).

5.5.1.2 Determination of the altitude of the Lambertian reflector

As was pointed out above, employing the LER model with the fixed sphericalalbedo we need to estimate just one parameter only for the case of completelycloudy satellite ground scenes. Therefore, Eq. (5.28) for the synthetic reflectionfunction can be rewritten in this case as follows:

Rλ(ht, cf ) = Rλ(hL, cr) + WL(λ)ΔhL + εl(λ) + εm(λ) , (5.36)

where Rλ(hL, cr) is the simulated reflection function corresponding to the LERposition above the ground, hL, and WL(λ) is the weighting function for LERaltitude. The analytical expression for this WF is given in Appendix A. The es-timation of a single parameter, ΔhL, can be obtained minimizing the quadraticform which characterizes the difference between synthetic and simulated reflec-

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5 Impact of single and multi-layered cloudiness 155

tion functions within the O2-A absorption band:∥∥∥Rλ(ht, cf ) − Rλ(hL, cr) − WL(λ)ΔhL

∥∥∥2⇒ min . (5.37)

The value of ΔhL providing the minimum of the quadratic form given byEq. (5.37) can be obtained analytically as follows:

hL = hL +(Rλ(ht, cf ) − Rλ(hL, cr),WL(λ))

(WL(λ),WL(λ)), (5.38)

where the notation ( , ) is used to define the scalar product of functions.To mitigate the influence of the linearization error, the value of hL is found

iteratively as in the case of cloud geometrical parameters retrieval describedabove. We note that the reflection function and the corresponding WF were cal-culated using solution of the exact radiative transfer equation. This allows us toavoid the error caused by the simplification of radiative transfer processes abovethe position of LER. Such a simplification of the solution of the correspondinginverse problem, i.e., neglecting the scattering processes above LER, was usedby Koelemeijer et al. (2001).

5.5.2 Total ozone column retrieval algorithm

5.5.2.1 Description of the algorithm

The synthetic reflection function in the ozone absorption band, R′(λ, rf , cf ),corresponding to the ozone vertical column V ′ and cloud parameters {rf , cf}can be expressed as follows:

R′(λ, rf , cf ) = R(λ, rr, cr) + WV (λ)ΔV + εl(λ) + εm(λ) + εp(λ) , (5.39)

where R(λ, rr, cr) is the simulated reflection function corresponding to the knownozone vertical column, V , and cloud parameters {rr, cr}, ΔV is the variation ofthe ozone vertical column to be retrieved, εl(λ) is a linearization error, εm(λ) is acloud model error caused by the difference between the cloud parameters cf andcr, and εp(λ) is an error caused by the difference between the cloud parametersrf and rr, which are the subject of the cloud parameters retrieval process.

The estimation of the desired parameter, ΔV , can be obtained as a solutionof the following minimization problem:∥∥∥ΔRλ − WV (λ)ΔV

∥∥∥2⇒ min , (5.40)

where ΔRλ = R′(λ, rf , cf ) − R(λ, rr, cr) is the difference between synthetic andsimulated reflection functions. The value of ΔV providing the minimum of thequadratic form given by Eq. (5.40) can be obtained analytically as follows:

ΔV =

(ΔRλ,WV (λ)

)(WV (λ),WV (λ)

) . (5.41)

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156 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.9. Linear approximation of the reflection function caused by 100% variation ofthe ozone vertical column. Upper left panel: 1,2, exact reflection functions without andincluding ozone absorption for the cloud parameters {5, 6, 20, 1}; 3, linear approxima-tion of the reflection function obtained employing Eq. (5.44); 4, linear approximation ofthe reflection function obtained employing Eq. (5.45). Upper right panel: the same asin the left panel but for cloud parameters {1, 2, 20, 1}. Lower panels: the relative errorsof the reflection function calculations; 1, according to Eq. (5.44) and 2, according to(5.45).

The contribution of the linearization error, εl(λ), in the estimation of ΔVobtained according to Eq. (5.41) can be large if the vertical column V ′ differssignificantly from V . To mitigate the impact of the linearization error one canuse for the retrieval of the absorber concentration the logarithm of the reflec-tion function instead of the reflection function itself. To demonstrate this letus consider the variation of the reflection function and the variation of its log-arithm caused by 100% decreasing of ozone vertical column, i.e., for V ′ = 0(ΔV = −V ). Assuming further that there is no variation of cloud parameters(εm(λ) = 0, εp(λ) = 0), and taking into account that Wc(λ) = V WV (λ) (seeEq. (5.25)), Eq. (5.39) can be rewritten as follows:

R0(λ) = R(λ) − Wc(λ) + εl(λ) , (5.42)

and for the logarithm of reflection function we obtain

lnR0(λ) = lnR(λ) − Lc(λ) + εl(λ) , (5.43)

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5 Impact of single and multi-layered cloudiness 157

where Lc(λ) = Wc(λ)/R(λ) is WF corresponding to the logarithm of reflectionfunction. The linear approximation of the reflection function, R(λ), obtainedemploying Eqs (5.42) and (5.43), i.e., neglecting the linearization errors εl(λ)and εl(λ), respectively, results in

R(λ) = R0(λ) + Wc(λ) , (5.44)R(λ) = R0(λ) eLc(λ) . (5.45)

To estimate the impact of the linearization error we need to compare thereflection functions calculated according to Eqs (5.44) and (5.45) to the ex-act value of R(λ). The corresponding reflection functions in the consideredspectral range and the results of their comparison are shown in Fig. 5.9.The exact reflection functions, R(λ), corresponding to the cloud parameters{hb, ht, τ, ts} = {5, 6, 20, 1} and {1, 2, 20, 1} are shown in the left and right up-per panels of Fig. 5.9, respectively, by solid lines. The approximations of re-flection functions obtained employing Eqs (5.44) and (5.45) are shown in thesame figure by symbols and dotted line, respectively. The corresponding relativeapproximation errors are shown in the lower panels of Fig. 5.9. It can be seenthat employing the linear approximation to the logarithm of reflection functionallows us to reduce significantly the impact of the linearization error. This factmotivates the usage of lnR(λ) instead of R(λ) in the retrieval of ozone verticalprofile (Hoogen et al., 1999) and ozone vertical column (Klenk et al., 1982) usingthe measurements of the backscattered solar radiation.

The variation of cloud parameters can lead to a large variation of the loga-rithm reflection function as well. Fig. 5.10 (left panel) presents the variation oflnR(λ) caused by increasing the cloud optical thickness by 5% (Δτ = 1) andby decreasing the ozone vertical column by 1%. The change of the cloud opticalthickness leads to the change, on the one hand, of the reflection function, and, onother hand, to the change of the photon pathlength distribution. The first effectexplains why the function Δ lnR(λ) (dash-dotted line in Fig. 5.10) has a compo-nent, which is smoothly dependent on the wavelength. The change of the photonpathlength leads to the change of the gaseous absorption. The spectral signatureof this effect is very similar to one caused by the variation of the ozone verticalcolumn. To remove the monotonous component from the total measured signaland to reduce the impact of variations of such parameters as aerosol, clouds,surface albedo etc. on the gaseous absorber vertical column retrieval, the DOASapproach is usually used (Brewer et al., 1973; Platt and Perner, 1980; Richteret al., 1999; Solomon et al., 1987). Employing DOAS technique in combinationwith the weighting functions (WFDOAS) has been suggested by Buchwitz etal. (2000) to derive the vertical column of such gaseous absorbers as CO, CO2,CH4, and N2O using the backscattered solar radiation in the near-infrared spec-tral range. The WFDOAS approach has been further successfully applied to theretrieval of the ozone vertical column from GOME data (Coldewey-Egbers et al.,2005; Weber et al., 2005). According to this approach we rewrite Eq. (5.39) forthe logarithm of reflection function and subtract a low-order polynomial from allterms. Introducing the differential optical depth (DOD), D, and corresponding

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158 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.10. Variations of the logarithm reflection function (left panel) and differentialoptical depth (right panel) caused by decreasing the ozone vertical column by −1%and by increasing the cloud optical thickness by Δτ = 1.

weighting function, L, as follows:

D(λ, rf , cf ) = − lnR(λ, rf , cf ) − Pf (λ) , (5.46)

D(λ, rr, cr) = − lnR(λ, rr, cr) − Pr(λ) , (5.47)

LV (λ) = LV (λ) − Pw(λ) , (5.48)

where Pf (λ), Pr(λ), and Pw(λ) are polynomials of the same order, we obtain

D′(λ, rf , cf ) = D(λ, rr, cr) − LV (λ)ΔV + ε−l (λ) + ε−

m(λ) + ε−p (λ) . (5.49)

Fig. 5.10 (right panel) presents variations of DOD caused by increasing thecloud optical thickness by 5% (Δτ = 1) and by decreasing the ozone verticalcolumn by 1%. It can clearly be seen that the variation of DOD caused by thevariation of the cloud optical thickness (dash-dotted line) is in this case muchsmaller than the variation caused by the 1% decrease of the vertical ozone col-umn. Therefore, subtracting polynomials, we can significantly reduce the impactof variations (that are smoothly dependent on the wavelength) of the reflectionfunction logarithm caused by the variation of cloud parameters.

The expression (5.49) is the basic equation which will be used in the followingsubsection to estimate the impact of the cloudiness on the accuracy of ozonevertical column retrieval. In the framework of our numerical simulations we willassume that there is actually no variation of the ozone vertical column, i.e., thesynthetic and simulated reflection function are calculated for the same ozone

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5 Impact of single and multi-layered cloudiness 159

number density profile. Therefore, Eq. (5.49) results in

D(λ, rf , cf ) = D(λ, rr, cr) − LV (λ)ΔV + ε−m(λ) + ε−

p (λ) . (5.50)

Thus, the difference between DODs caused by the variation of the cloud pa-rameters is interpreted as the variation of the ozone vertical column. In a wayanalogous to the derivation of Eq. (5.41) we obtain

ΔV = − (ΔDλ,LV (λ))(LV (λ),LV (λ))

, (5.51)

where ΔDλ = Dλ(V, rf , cf ) − Dλ(V, rr, cr). Introducing, for the sake of simpli-fication, the ozone VC retrieval operator as follows:

RV = − 1(LV (λ),LV (λ))

(LV (λ), ) , (5.52)

we rewrite Eq. (5.51) in the following operator form:

ΔV = RV [ΔDλ] . (5.53)

The estimation of ΔV obtained according to Eq. (5.53) minimizes the given dif-ference between DODs. The estimated variation of the ozone vertical column willbe considered as an absolute error of the vertical column determination causedby usage of the cloud parameters {rr, cr} instead of the true values {rf , cf}. Inthe following discussion we will use relative errors to characterize the impact ofcloudiness on the ozone vertical column retrieval as defined by

εV =V − V

V=

ΔV

V=

1V

RV [ΔDλ] . (5.54)

This equation provides the nonlinear relationship between the ozone VC retrievalerror and the variation of cloud parameters from their true values given by theparameter set {rf , cf}.

5.5.2.2 Linear estimation of ozone vertical column retrieval errors

In the previous subsections we have discussed retrieval algorithms used to derivecloud parameters and the ozone vertical column. Having retrieved the cloud pa-rameters, the difference between DODs used in Eq. (5.53) to obtain the variationof the ozone VC is represented as follows:

ΔDλ = Dλ(V, rf , cf ) − Dλ(V, rr, cr) , (5.55)

where rr is the set of the retrieved cloud model parameters. Substituting ΔDλ

given by Eq. (5.55) into Eq. (5.54), we obtain

εV =1V

RV

[Dλ(V, rf , cf ) − Dλ(V, rr, cr)

]. (5.56)

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160 V. V. Rozanov and A. A. Kokhanovsky

Clearly, differences in both parameter sets, {rf − rr} and {cf − cr} contributeto the ozone VC retrieval error, εV . However, from the theoretical point of viewit would be of importance to obtain another expression for εV which is additiverelative to the contribution of {rf − rr} and {cf − cr}. To do this we expandDλ(V, rr, cr) in the Taylor series around the true values of cloud parametersgiven by rf . Restricting to linear terms only, we obtain

Dλ(V, rr, cr) = Dλ(V, rf , cr) +∑

Lr(λ)(rr − rf ) , (5.57)

where the summation is carried out over all relevant cloud parameters and Lr(λ)are their weighting functions. This sum contains in the case of the LER modelwith fixed spherical albedo only one parameter, i.e., cloud top height, whereasit includes CTH, CBH, and COT for the VHC and VIC models. SubstitutingDλ(V, rr, cr) given by Eq. (5.57) into Eq. (5.55), we have

ΔDλ = Dλ(V, rf , cf ) − Dλ(V, rf , cr) −∑

Lr(λ)(rr − rf ) . (5.58)

The difference between DODs given by two first terms in this equation is dueto employing a cloud model and the last term describes the contribution of thecloud parameters variation. Substituting further Eq. (5.58) into Eq. (5.54), weobtain

εV =1V

RV

[Dλ(V, rf , cf ) − Dλ(V, rf , cr) −

∑Lr(λ)(rr − rf )

], (5.59)

where RV is the ozone VC retrieval operator given by Eq. (5.52). Thus, the ozoneVC retrieval error consists in the linear approximation of two components. Thefirst component, i.e.,

εm =1V

RV

[Dλ(V, rf , cf ) − Dλ(V, rf , cr)

], (5.60)

describes the error caused by employing in the retrieval process a cloud model(for example, VHC) which does not coincide with the ‘true’ cloud used in thecalculation of synthetic spectra. The second component:

εp = − 1V

RV

[∑Lr(λ)(rr − rf )

], (5.61)

describes the contribution of the cloud parameter errors. Following Rodgers(2000) we will refer to these errors as the modeling error and the forward modelparameter error, respectively.

To illustrate the contribution of the modeling and the forward model param-eter error into the ozone VC retrieval error we consider the selected results ofcalculations εm and εp according to Eqs (5.60) and (5.61), respectively, for asingle ice cloud and for a two-layered water cloud system. Fig. 5.11 shows theozone VC retrieval errors (εm + εp) obtained employing the ice VHC model forthe determination of ozone VC in the case of a vertically inhomogeneous icecloud described by the following set of parameters: {hb, ht, τ, ts} = {5, 7, 5, 2}.

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5 Impact of single and multi-layered cloudiness 161

Fig. 5.11. The ozone VC retrieval error as a function of CTH and CBH variationfrom their true values (hb = 5 km, ht = 7 km). The ice VHC model is employed in theretrieval of ozone VC in the case of the single ice cloud ({hb, ht, τ, ts} = {5, 7, 5, 2}).SZA = 10◦. Each counter line represents the constant value of the ozone VC retrievalerror expressed as a percentage.

To simplify representation we have taken into account the dependence of theozone VC retrieval error on the variation of CTH and CBH from their true val-ues only neglecting the variation of optical thickness (τ = 5). The solid linesin the contour plot of Fig. 5.11 present the constant values of the ozone VCretrieval error expressed as a percentage. The modeling error is caused in thiscase exclusively by the vertical inhomogeneity of the ‘true’ ice cloud (see the ver-tical profiles of the effective radius and shape function given in Fig. 5.1). Thiserror is shown in Fig. 5.11 by the diamond corresponding to the point ΔCTH= ΔCBH = 0 and it is equal to ∼0.09%. Clearly, employing the VHC model forthe retrieval of ozone VC in the case of a two-layered water cloud system canlead to significant increase in the modeling error. This illustrates the contourplot given in Fig. 5.12 for the two-layered water cloud system described by thefollowing set of parameters:

pf =

⎧⎨⎩{hb, ht, τ, ts} ,{6, 7, 5, 1} =⇒ upper cloud ,{1, 2, 10, 1} =⇒ lower cloud .

(5.62)

It follows that in this case the modeling error is almost twenty times larger thanin the previous case and it reaches ∼1.7%. Taking into account the definition ofthe ozone VC retrieval error given by Eq. (5.54), we conclude that employingVHC model in the retrieval of ozone VC in the case of the two-layered cloudsystem leads to the overestimation of ozone VC even if the CTH, CBH, andCOT of the VHC model are in agreement with those of the ‘true’ cloud system.

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162 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.12. The same as in Fig. 5.11 but the water VHC model is employed in theretrieval of ozone VC in the case of the two-layered cloud system.

The obtained result can be easily explained taking into account that the ab-sorption of radiation in the two-layered cloud system is stronger as compared tothe homogeneous single cloud having the same values of CTH, CBH, and COT.The enhancement of absorption is caused by the cloud-free atmosphere betweenupper and lower clouds. Therefore, the lower absorption of VHC model is com-pensated by the increasing of the ozone concentration. The modeling error canbe decreased in the case of a two-layered cloud system employing VIC model inthe retrieval process. The corresponding contour plot is given in Fig. 5.13. It fol-lows that in this case the modeling error is ∼0.4% instead of ∼1.7% as obtainedfor VHC model. This result supports our assumption made in subection 5.5.1that the VIC model can be the of greater advantage for retrievals in the case ofmulti-layered cloud systems.

The ozone VC retrieval errors obtained employing the LER model in the caseof the two-layered water cloud system described by the set of parameters givenby Eq. (5.62) are shown in Fig. 5.14. In this case the VC retrieval error dependson the position of the LER only. It follows from Fig. 5.14 that the modeling error(∼6%) is more than ten times larger than the modeling error corresponding tothe VIC model.

As pointed out above, the ozone VC retrieval error is equal to the modelingerror if in the ozone VC retrieval process the error-free ‘true’ cloud parametersare used. However, since the VC retrieval error is the sum of the modelingerror and the forward model parameters error, it can be decreased choosingappropriate cloud parameters to be used in the VC retrieval process. There areseveral combinations of CTH and CBH values which make it possible to obtainerror-free estimation of the ozone VC employing VHC or VIC models. This

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5 Impact of single and multi-layered cloudiness 163

Fig. 5.13. The same as in Fig. 5.11 but the water VIC model is employed in theretrieval of ozone VC in the case of the two-layered cloud system.

illustrates the red line in the contour plots given in Figs. 5.11–5.13 correspondingto the zero ozone VC retrieval error. In the case of the LER model there is onlyone value of the LER position which leads to error-free ozone VC estimation. Itfollows from Fig. 5.14 that in the considered case of the two-layered water cloudsystem this position of LER should be ∼3.7 km lower than the system cloud topheight (asterisk in Fig. 5.14). Unfortunately such optimal choice of cloud modelparameters requires the knowledge of a ‘true’ cloud system. Nevertheless, theimpact of the modeling error can be reduced using the cloud parameters in theozone VC retrieval process which are obtained employing the same cloud modelin the cloud parameters retrieval process. To demonstrate this we use ΔCTH andΔCBH for the VHC and VIC models and ΔCTH for the LER model obtainedbelow for the corresponding scenarios after retrieval of cloud parameters (seethe CTH retrieval results presented in Fig. 5.19 for the case 4 scenario 2-2).The corresponding ozone VC retrieval errors are shown in Figs. 5.11–5.13 bytriangles. It can be seen that for all considered cases the obtained ozone VCretrieval errors are significantly smaller than the modeling error. In the case ofthe LER model the dependence of ozone VC retrieval error on CTH positionis shown in Fig. 5.14. It follows that using the LER position obtained in thecloud retrieval process (ΔCTH ∼−5 km) the ozone VC retrieval error is ∼−2%(triangle in Fig. 5.14) whereas employing the CTH obtained for the VHC model(ΔCTH ∼−1 km) the ozone VC retrieval error is ∼4.5% (square in Fig. 5.14).This example clearly demonstrates that employing the same cloud model in the

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164 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.14. The contribution of the modeling and forward model parameter errors intoozone VC retrieval error employing LER model in the case of a two-layered water cloudsystem described by the set of parameters given by Eq. (5.62) at SZA = 10◦.

cloud parameters and ozone VC retrieval processes is very important in themitigation of the impact of the modelling error.

5.6 Results of numerical experiments

In this section we describe results of numerical experiments obtained employ-ing VHC, VIC and LER models for the determination of cloud parameters andfor the retrieval of the ozone vertical column. The cloud parameters were ob-tained employing the retrieval algorithms described in subsections 5.5.1.1 and5.5.1.2. The ozone vertical column retrieval errors were calculated according toEq. (5.56). Although the numerical experiments were performed for the solarzenith angles 10◦, 30◦, and 60◦, the preliminary analysis has shown that resultsfor SZA 30◦ are between corresponding results for SZA 10◦ and 60◦ for all consid-ered scenarios. Therefore, to simplify presentation in the following subsections,only results for SZA 10◦ and 60◦ are given.

5.6.1 Single cloud layer

In this subsection we discuss the application of the retrieval algorithms describedabove to deriving the cloud parameters and ozone vertical column in the case

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5 Impact of single and multi-layered cloudiness 165

of single-layer water or ice clouds having different cloud top heights and opticaland geometrical thicknesses.

5.6.1.1 Scenario 1-1: water cloud (τ = 20)

The first scenario corresponds to the water cloud described by the following setof parameters:

pf = {hb, ht, τ, ts} = {i, 1 + i, 20, 1}, i = 1, 2, . . . , 5 . (5.63)

This means that we consider five cases where the water cloud having optical andgeometrical thickness equal to 20 and to 1 km, respectively, is moved upward insteps of 1 km. The retrieval results of the CTH and CBH as well as the errors ofthe ozone vertical column retrieval are shown in the left and in the right panelsof Fig. 5.15, respectively. The positions of the true cloud are shown in the leftpanel of this figure by rectangles. It follows from the right panel of Fig. 5.15that employing the VHC model for retrieval of cloud parameters, the ozonevertical column retrieval errors are smaller than ∼0.05% and they do not showany significant dependence on the cloud position or the solar zenith angle. Alsowe can see that both cloud top (dashed line in the left panel of Fig. 5.15) andbottom height (not shown) are retrieved with good accuracy for the five cloud

Fig. 5.15. Retrieval results for the single-water cloud with the optical thickness τ =20. Left panel: rectangles show the position of ‘true’ clouds; the retrieved CTHs areobtained using the VHC model (dashed lines) and the LER model (dotted lines). Rightpanel: ozone vertical column retrieval errors obtained employing VHC and LER models.

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166 V. V. Rozanov and A. A. Kokhanovsky

positions and solar zenith angles considered. The maximal absolute retrievalerror of the CTH and CBH (difference between retrieved and true value of thecorresponding parameters) is smaller than ∼85 m.

The errors of the ozone vertical column retrieval obtained employing theLER model are considerably larger (up to −2%, see dotted lines in the rightpanel of Fig. 5.15) and show significant dependence on the solar zenith angleand cloud position. This can be explained by the fact that the retrieved effectivecloud top heights employing the LER model underestimate the true CTH (seedotted lines in the left panel of Fig. 5.15). Thus, for the solar zenith angle 60◦

the maximal CTH retrieval error is ∼−750 m and it reaches ∼−1 km for thesolar zenith angle 10◦. The absorption of radiation by ozone in the atmosphereabove the LER depends on its spherical albedo and altitude. The absorption isstronger for the lower position of LER (see dotted line in Fig. 5.5). The lowerposition of the retrieved CTH as compared to true cloud position (see left panelof Fig. 5.15) results in the enhanced absorption. Therefore, to compensate theenhanced absorption the retrieved values of ozone vertical column are smallerthan true ones. This explains the negative sign of the ozone VC retrieval errorsobtained employing the LER model.

5.6.1.2 Scenario 1-2: water cloud with different geometrical andoptical thicknesses

Fig. 5.16 shows the retrieval results corresponding to the water cloud describedby the set of following parameters:

pf = {hb, ht, τ, ts} = {1, 1 + i, 30 × i, 1}, i = 1, 2, . . . , 5 . (5.64)

In this case the position of the cloud bottom is fixed (1 km) and the cloud topheight increases from 2 km to 6 km in steps of 1 km. The COT is increasedaccordingly from 30 to 150 in steps of 30. The positions of the ‘true’ cloudare shown in the left panel of Fig. 5.16 by rectangles. This scenario simulatesthe vertically extended water cloud with constant cloud scattering coefficient∼30 km−1. It can be seen that for COT equals to 30 (case 1) the retrieval resultsobtained employing VHC model are very similar to those shown in Fig. 5.15 forτ = 20. However, the increasing of the cloud optical thickness leads to an increasin the retrieval errors for both CTH and CBH (see dashed lines in the left panelof Fig. 5.16). The maximal retrieval errors correspond to the case 5 (τ = 150)and reach ∼−270 m and ∼−640 m for CTH and CBH, respectively. The retrievalerrors of the ozone VC show a weak dependence on the optical thickness of cloudfor this scenario (see dashed lines in the right panel of Fig. 5.16). The maximalvalue of the VC retrieval error is smaller than ∼0.2%.

The errors in ozone vertical column retrieval obtained employing the LERmodel are considerably larger (up to 2%, see dotted lines in the right panelof Fig. 5.16) and show significant dependence on the solar zenith angle andespecially on the cloud geometrical thickness. The retrieved effective cloud topheights employing the LER model are lower than the true CTH (see dotted lines

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5 Impact of single and multi-layered cloudiness 167

Fig. 5.16. Retrieval results for the vertically extended single-water cloud with constantCBH. Left panel: rectangles show the position of ‘true’ clouds; the retrieved CTHs areobtained using the VHC model (dashed lines) and the LER model (dotted lines). Rightpanel: ozone vertical column retrieval errors obtained employing VHC and LER models.

in the left panel of Fig. 5.16). Thus, for case 1 (τ = 30) the maximal CTHretrieval error is ∼−820 m and it reaches ∼−2.6 km for the case 5 (τ = 150).Although the retrieved effective CTH employing the LER model is systematicallylower than the true CTH as in the previous scenario of the cloud having theconstant optical thickness 20, the sign of the ozone VC retrieval error dependson the cloud optical thickness in this case. Thus, for example, for solar zenithangle 10◦ the LER model leads to the underestimation of ozone vertical columnby ∼−1% for τ = 30 and to the overestimation by ∼2% for τ = 150. This canbe explained by the fact that the ozone absorption in the cloudy atmospheredepends not only on the top height and optical thickness of a cloud but alsoon its geometrical thickness. Comparing the relative ozone absorption for theextended cloud and for LER model presented in Fig. 5.5 by dashed-double-dotted and dotted lines, respectively, we can see that the although the effectiveCTH retrieved employing the LER model is smaller than true CTH, the ozoneabsorption over the LER position is generally smaller as compared to that forthe true cloud. To compensate this the retrieved ozone VC appears to be largerthan true. Therefore, the vertical ozone column is overestimated. This result isin line with the conclusion of Ahmad et al. (2004), although in the cited paperthe ozone vertical column retrieval algorithm was not employed.

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168 V. V. Rozanov and A. A. Kokhanovsky

5.6.1.3 Scenario 1-3: ice cloud (τ = 5)

The importance of the cloud thermodynamic state determination ahead of theozone VC retrieval is demonstrated in Fig. 5.17, where results for an ice cloudwith the optical thickness τ = 5 retrieved using the VHC and LER models areshown. The cloud scenarios are described in this case by the following set ofparameters:

pf = {hb, ht, τ, ts} = {4 + i, 6 + i, 5, 2} , i = 1, 2, . . . , 5 , (5.65)

i.e., an ice cloud of constant geometrical thickness of 2 km is moved upwardin steps of 1 km. The retrieval results for CTH and corresponding ozone VCretrieval errors obtained employing the water VHC model are shown by dashedlines in the left and in the right panels of Fig. 5.17, respectively. It follows thatthe usage of a wrong thermodynamic state in the cloud retrieval model resultsin significant increase in CTH retrieval errors. In this case the retrieved CTHsshow the significant dependence on the solar zenith angle. Thus, for the solarzenith angle 60◦ the retrieved CTHs are larger than true values whereas forthe solar zenith angle 10◦ they are smaller. The maximal CTH retrieval errorsfor these solar zenith angles reach ∼1.7 km and ∼−1.4 km, respectively. The

Fig. 5.17. Retrieval results for the single-layer ice cloud with optical thickness 5. Leftpanel: rectangles show the position of ‘true’ clouds; the retrieved CTHs are obtainedusing the water VHC model (dashed lines), the ice VHC model (dashed-dotted lines),and the LER model (dotted lines). Right panel: ozone vertical column retrieval errorsobtained employing water VHC and LER models.

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5 Impact of single and multi-layered cloudiness 169

significant increase in CTH retrieval errors in this case can be explained by thedifference in the cloud phase function corresponding to the water droplets andice crystals. This results in the significant errors in the retrieved cloud opticalthickness. Thus, for the solar zenith angle 10◦ the retrieved COT is ∼7 and for60◦ it is ∼10.5, i.e., more than twice as large as the true value of COT (τ = 5).The maximal error in ozone VC, however, is less than 1% (see dashed lines inthe right panel of Fig. 5.17).

The information on the correct thermodynamic state of a cloud in the cloudmodel allows us to improve the retrieval results. In the left panel of Fig. 5.17(dash-dotted lines) the retrieved CTHs obtained employing the ice VHC modelare shown. The maximal CTH retrieval error is in this case ∼−550 m, althoughthe cloud optical thickness is retrieved almost without error (Δτ < −0.003). Themaximal CTH retrieval error obtained in the case of the ice cloud is much largerthan that obtained for a water cloud having optical thickness 20 (∼−85 m). Thiscan be explained comparing the liquid water and ice water shape profiles givenin the right panel of Fig. 5.1. It can be seen that the shape profile correspondingto the ice water content shows significantly stronger dependence on the verticalcoordinate inside a cloud. Therefore, the increase in CTH retrieval errors in thecase of ice cloud demonstrates the impact of the vertical inhomogeneity. Nev-ertheless, the ozone VC is retrieved in this case with high accuracy (maximalretrieval error less than ∼0.03%, not shown in Fig. 5.17). The high accuracy ofthe ozone VC retrieval is explained by the fact that the modeling error and CTHerror compensate each other. Indeed, coming back to the counter plot given inFig. 5.11 and taking into account that for case 1 of scenario 1-3 ΔCTH ≈ −550m, ΔCBH ≈ 0.03 km, and Δτ ≈ 0, we can see from this figure that ozone VCretrieval error (shown as a triangle in Fig. 5.11) is almost zero. The consideredexample emphasizes the importance of the information about cloud thermody-namic state for the retrieval of cloud geometrical and optical parameters as wellas for the retrieval of the ozone vertical column.

The retrieved effective CTHs and ozone VC retrieval errors obtained in thecase of an ice cloud employing the LER model are shown in the left and inthe right panels of Fig. 5.17 by dotted lines, respectively. It follows from theleft panel of Fig. 5.17 that the maximal CTH retrieval errors reach ∼−2 kmand ∼−3 km for the solar zenith angles 60◦ and 10◦, respectively. The ozoneVC retrieval errors (dotted lines in right panel of Fig. 5.17) are similar to thoseobtained in the case of a water cloud having optical thickness 20 (see dotted linesin right panel of Fig. 5.15). However, we can see that in the case of an ice cloudthe dependence of the ozone VC retrieval errors on the cloud top height positionis more pronounced than the water cloud especially for the solar zenith angle10◦. The maximal value of VC retrieval error is ∼−2% for case 1 correspondingto ht = 7 km.

5.6.2 Two-layered cloud systems

In this subsection we discuss the retrieval results for three scenarios of two-layered cloud systems. The position of the lower water cloud is fixed for all cases.

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170 V. V. Rozanov and A. A. Kokhanovsky

Its top and bottom height is set to 1 km and 2 km, respectively. The upper cloudhas different optical thicknesses, thermodynamic states and positions relative tothe lower cloud.

5.6.2.1 Scenario 2-1: water cloud system with total optical thicknessτ = 30

Fig. 5.18 shows the retrieval results for the lower cloud having optical thickness20 and an upper cloud characterizing by the following set of parameters:

pf = {hb, ht, τ, ts} = {2 + i, 3 + i, 10, 1} , i = 1, 2, . . . , 5 , (5.66)

i.e., the distance between two water clouds is increased in steps of 1 km. Thedashed lines in the left and in the right panels of Fig. 5.18 correspond to theretrieved CTHs and ozone VC retrieval errors, respectively, obtained employ-ing the water VHC model. It follows from the left panel of Fig. 5.18 that CTHretrieval errors increase with the distance between clouds. This can be easilyunderstood because the VHC model used in the retrieval process does not co-incide with the ‘true’ cloud system. However, the maximal CTH retrieval errorreaches in this case ∼−380 m only and it corresponds to the maximal distance

Fig. 5.18. Retrieval results for two-layered water cloud system with total opticalthickness τ = 30. Left panel: rectangles show the position of “true” clouds; the retrievedCTHs are obtained using VIC model (solid lines), VHC model (dashed lines), andLER model (dotted lines). Right panel: ozone vertical column retrieval errors obtainedemploying VIC, VHC, and LER models.

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5 Impact of single and multi-layered cloudiness 171

between cloud layers (case 5). Although CTH is retrieved quite accurately, thereis a significant bias in the retrieved CBH (not shown in Fig. 5.18). In particular,the retrieved CBH for all cases are at ground level, which is a general featureof retrievals for layered clouds employing the homogeneous cloud model. Thepreliminary results presented by Rozanov et al. (2004) show that this featurecan be used for the detection of multi-layered cloud systems from a satellite.The ozone VC retrieval errors show the dependence on the distance betweencloud layers as well. It follows from the right panel of Fig. 5.18 that employingthe VHC model we overestimate in this case the ozone VC (errors are positive).This is due to the fact that the absorption of radiation below the cloud topheight in the vertically homogeneous cloud is smaller than in the two-layeredcloud system having approximately the same top height. The enhancement ofabsorption in the two-layered cloud system as compared in the homogeneoussingle-layer cloud is caused by the cloud-free atmosphere between the upperand lower clouds. Therefore, greater distance between two cloud layers leads togreater absorption in the two-layered cloud system as compared to in a homo-geneous single cloud. Thus, employing the VHC model, we underestimate theabsorption of radiation inside the cloud and, therefore, overestimate the ozonevertical column. The maximal ozone VC retrieval error is ∼1.3% at 10◦ solarzenith angle for case 5 corresponding to 5 km distance between clouds.

To demonstrate the influence of the cloud model used in the retrieval pro-cess on the retrieved CTH and on the ozone VC retrieval errors we show inFig. 5.18 (solid lines) the retrieval results obtained employing not a verticallyhomogeneous cloud model but rather the inhomogeneous cloud model (VIC).We conclude that this model improves the accuracy of the ozone VC retrieval(see solid lines in the right panel of Fig. 5.18), although the accuracy of CTH re-trieval becomes worse. The CTH retrieval errors increase for the cases of greaterdistances between clouds. The error reaches ∼2.3 km for case 5 at the solarzenith angle 10◦. Nevertheless, the maximal ozone VC retrieval error is smallerthan ∼0.2%. This suggests that there is a possibility of retrieving the ozone VCwith a high accuracy for two-layered cloud systems, even if CTH and CBH arenot retrieved in a correct way.

This is further illustrated in Fig. 5.18, where it is shown that the usage ofthe LER model in the retrieval process (see dotted lines) leads to considerableerrors in CTH, but the maximal relative error of the ozone VC determination iscomparable to that obtained employing the VHC model (see dashed lines in theright panel of Fig. 5.18). This numerical result is in line with our consideration ofthe linear estimation of ozone VC retrieval errors presented in subsection 5.5.2.2.It demonstrates that the large CTH retrieval error (forward model parametererror) significantly compensates the modeling error.

5.6.2.2 Scenario 2-2: water cloud system with total optical thicknessτ = 15

This scenario is used to demonstrate the impact of the total optical thickness ofthe two-layered cloud system on the retrieval results of CTH and the ozone VC.

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172 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.19. Retrieval results for two-layered water cloud system with total opticalthickness τ = 15. Left panel: rectangles show the position of ‘true’ clouds; the retrievedCTHs are obtained using VIC model (solid lines), VHC model (dashed lines), andLER model (dotted lines). Right panel: ozone vertical column retrieval errors obtainedemploying VIC, VHC, and LER models.

The retrieval results for the lower water cloud having optical thickness 10 andan upper cloud characterized by the following set of parameters:

pf = {hb, ht, τ, ts} = {2 + i, 3 + i, 5, 1} , i = 1, 2, . . . , 5 , (5.67)

are shown in Fig 5.19. The employing of VIC model (solid lines in the left andright panels of Fig 5.19) leads to very similar results when compared to scenario2-1 with the total optical thickness of the cloud system τ = 30. VHC modelunderestimates the top height of the cloud system as in the previous scenario.However, the maximal CTH retrieval error is larger in this case and it reaches∼−1.1 km for the solar zenith angle 10◦. Although the top height of the cloudsystem is retrieved less accurately as compared to scenario 2-1 the maximal ozoneVC retrieval error is smaller. It reaches ∼1% at the solar zenith angle 10◦ andcorresponds to the maximal distance between clouds (5 km). The employing ofLER model results in the further significant underestimation of the cloud systemtop height (see dotted lines in left panel of Fig 5.19). It follows that the maximalCTH retrieval error reaches ∼−5.7 km and it corresponds to SZA 10◦ and to themaximal distance between cloud layers. The ozone VC retrieval errors increase ascompared to the scenario with an optically thicker cloud system. The LER modelunderestimates in this case the ozone VC for all considered distances between

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5 Impact of single and multi-layered cloudiness 173

cloud layers. The maximal error is obtained for minimal distance between cloudlayers (up to ∼−3%).

5.6.2.3 Scenario 2-3: ice cloud above a water cloud

The influence of the thermodynamic phase on the retrieval results in the case ofthe two-layered cloud system is shown in Figs. 5.20 and 5.21. In the consideredscenario the lower water cloud has the optical thickness τ = 20 and an upper icecloud is characterized by the following set of parameters:

pf = {hb, ht, τ, ts} = {2 + i, 3 + i, 2, 2} , i = 1, 2, . . . , 5 . (5.68)

The retrieval results obtained employing water VHC and water VIC models foran ice cloud actually on the top of the cloud system are shown in Fig. 5.20. Itfollows from the right panel of Fig. 5.20 that VHC and VIC models produce forthis scenario very similar ozone VC retrieval errors. It can be seen also that ozoneVC retrieval errors show strong dependence on the solar zenith angle, as for thescenario of the single-layer ice cloud discussed in subsection 5.6.1.3. However, themaximal VC retrieval error for this scenario is almost twice as greater that forscenario 1-3 of a single-layer ice cloud and it reaches ∼2% at SZA 10◦ and at the

Fig. 5.20. Retrieval results for two-layered ice-water cloud system with total opticalthickness τ = 22. Left panel: rectangles show the position of ‘true’ clouds; the retrievedCTHs are obtained using the water VIC model (solid lines), the water VHC model(dashed lines), and the LER model (dotted lines). Right panel: ozone vertical columnretrieval errors obtained employing water VIC, water VHC, and LER models.

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174 V. V. Rozanov and A. A. Kokhanovsky

maximal distance between clouds corresponding to 5 km. The large dependenceof the retrieved ozone VC on the solar zenith angle can be explained by verydifferent retrieval results obtained for the cloud optical thickness. The retrievedcloud optical thicknesses for SZA 10◦ and 60◦ are ∼19.3 and ∼29.2, respectively,instead of 22 corresponding to the true optical thickness of the cloud system.Considering now the CTH retrieval results given in the left panel of Fig. 5.20for case 1, we can see that the retrieved CTHs for both SZAs are approximatelythe same. Coming back to Fig. 5.5 illustrating the dependence of the ozoneabsorption in the atmosphere on the optical thickness of a cloud, we can seethat for CTH smaller than ∼5.5 km a cloud with larger optical thickness haslarger absorption. Therefore, due to the difference in the retrieved COT theozone absorption for SZA 60◦ is stronger than for SZA 10◦ and the retrievedozone VC is smaller.

The retrieval results obtained using the LER model are shown in Fig. 5.20 bydotted lines. Although the retrieved CTH strongly underestimates the position ofthe cloud top height of the two-layered system, the maximal ozone VC retrievalerror is comparable to one obtained by using water VHC and water VIC modelsand it reaches ∼−1.8% at SZA 10◦.

Fig. 5.21 shows the retrieval results obtained employing in the retrieval pro-cess ice VHC and ice VIC models. It follows that maximal ozone VC retrieval

Fig. 5.21. Retrieval results for two-layered ice-water cloud system with total opticalthickness τ = 22. Left panel: rectangles show the position of ‘true’ clouds; the retrievedCTHs are obtained using the ice VIC model (solid lines) and the ice VHC model (dashedlines). Right panel: ozone vertical column retrieval errors obtained employing ice VICand ice VHC models.

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5 Impact of single and multi-layered cloudiness 175

errors are smaller in this case compared to those obtained using water VHC andwater VIC models (see solid and dashed lines in the right panel of Fig. 5.20) Themaximal ozone VC retrieval error is smaller than ∼1.2% and ∼0.6% for thesemodels, respectively. This confirms that in the case of the two-layered cloud sys-tem the information about the thermodynamic state of the upper cloud is alsoimportant, as in the case of a single cloud.

5.6.3 Three-layered cloud systems

In this subsection we consider two scenarios for the three-layered cloud systemsputting the thin ice cloud (τ = 2) with CTH equal to 9.5 km and geometricalthickness 0.5 km above the two-layered water cloud system considered in subsec-tion 5.6.2. Thus, scenarios 3-1 and 3-2 are characterized by the additional thinice cloud above the two-layered cloud system described by scenarios 2-1 and 2-2, respectively. Although this case is less realistic, compared to the two-layeredcloud systems considered in the previous subsection, it allows us to investigatethe applicability of the different cloud models in the case of strong verticallyinhomogeneity.

Taking into account that the impact of the thermodynamic state of VIC andVHC models used in the retrieval process was demonstrated in the previous sub-section, we show in Figs 5.22 and 5.23 only retrieval results obtained employingice VIC, ice VHC, and LER models. The retrieved CTHs obtained employingthe ice VHC model are shown in the left panels of Figs 5.22 and 5.23 by dashedlines. It follows that CTH retrieval errors are strongly dependent in this case onthe distance between the middle and upper clouds. If the upper cloud is placednear the middle cloud (case 5) CTH retrieval errors for these scenarios are ap-proximately the same or even somewhat smaller than found for scenarios 2-1and 2-2. However, for the maximal distance between middle and upper cloudscorresponding to 5 km (case 1) the maximal CTH retrieval errors reach ∼−5 kmand ∼−3.5 km for scenarios 3-1 and 3-2, respectively. The retrieved CBH forall cases are at ground level, which is in line with the results obtained for thetwo-layered cloud system. The impact of the ice cloud is much smaller if in theretrieval process the ice VIC model is used. The maximal CTH retrieval erroris ∼2.5 km in this case and it is comparable to one obtained using the waterVIC model for the two-layered cloud system given by scenarios 2-1 and 2-2. TheCTH retrieval results obtained for these scenarios using LER model are shownin the left panels of Figs 5.22 and 5.23 by dotted lines. The retrieved CTHs showfor these scenarios significantly greater dependence on the SZA as compared toscenarios 2-1 and 2-2. It follows that an additional thin ice cloud placed at thetop of a two-layered water cloud system has almost no impact on the retrievedCTH for the SZA 10◦. Therefore, the maximal CTH retrieval error increases inthis case up to ∼−8 km.

The ozone VC retrieval errors obtained for these scenarios are shown inFigs 5.22 and 5.23. It follows from these figures that the ozone VC retrievalerrors obtained using VHC models (see dashed lines) are similar for both sce-narios. However, they are a little bit smaller for scenario 3-2 corresponding to the

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176 V. V. Rozanov and A. A. Kokhanovsky

Fig. 5.22. Retrieval results for three-layered ice-water cloud system with total opticalthickness τ = 32. Left panel: rectangles show the position of ‘true’ clouds; the retrievedCTHs are obtained using the VIC model (solid lines), the VHC model (dashed lines),and the LER model (dotted lines). Right panel: ozone vertical column retrieval errorsobtained employing VIC, VHC, and LER models.

thinner middle and bottom cloud layers. The same is true for VC retrieval errorsobtained employing the VIC model. Generally errors increase with decreasingSZA. Maximal ozone VC retrieval errors reach ∼1.9% and ∼0.7% for VHC andVIC models, respectively, for the scenario 3-1 at SZA 10◦ and maximal distancebetween middle and upper clouds. The impact of the upper thin ice cloud onthe ozone VC retrieval errors are minimal as it is in case of CTH retrieval ifthe ice cloud is placed near the middle clouds. In contrast to VIC and VHCmodels LER model shows significantly different results for scenarios 3-1 and 3-2.Thus, LER model overestimates the ozone VC for scenario 3-1 (maximal error∼1.7% for the minimal distance between middle and upper clouds) whereas itunderestimates VC for scenario 3-2 with maximal error ∼−0.8% correspondingto the maximal distance between the middle and upper clouds.

5.7 Conclusion

The main results of our investigations are summarized in Table 5.2. Althoughthe numerical experiments for each cloud scenario were performed for five dif-ferent positions of a cloud and three solar zenith angles (10◦, 30◦, 60◦), we havepresented in Table 5.2 for each scenario the worst case only. For most scenarios

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5 Impact of single and multi-layered cloudiness 177

Fig. 5.23. Retrieval results for three-layered ice-water cloud system with total opticalthickness τ = 17. Left panel: rectangles show the position of ‘true’ clouds; the retrievedCTHs are obtained using the VIC model (solid lines), the VHC model (dashed lines),and the LER model (dotted lines). Right panel: ozone vertical column retrieval errorsobtained employing VIC, VHC, and LER models.

the worst case corresponds to the higher solar zenith angle (10◦) independent ofthe cloud model used in the retrieval process.

Scenarios 1-1, 1-2, and 1-3 in Table 5.2 present CTH and ozone VC retrievalerrors for the case of a single-layer cloud. It follows that for these scenariosemploying of VHC model leads to very small ozone VC retrieval errors (≤0.2%)if in the retrieval process the correct thermodynamic cloud state is used. Thisdemonstrates that VHC model can be successfully used in the retrieval of cloudgeometrical parameters and ozone VC in the case of a single-layer cloud under theassumption that its thermodynamic state is known. The employing of the LERmodel leads to larger retrieval errors. Depending on the cloud scenario employingthe LER model results in the overestimation or underestimation of ozone VC.The ozone VC retrieval errors for the LER model are in the range −2.4 to +2.3%.The retrieved CTHs employing VHC and LER models are systematically lowerthan true values. However, CTH retrieval errors obtained in the case of the LERmodel are significantly larger than the corresponding error for the VHC model.

Scenarios 2-1, 2-2, and 2-3 in Table 5.2 present CTH and ozone VC retrievalerrors obtained employing VHC, VIC, and LER models in the case of two-layeredwater (2-1, 2-2) and ice-water (2-3) cloud systems. It follows that employing ofVHC model in the retrieval for the case of two-layered cloud system leads to anincrease in CTH and ozone VC retrieval errors. The maximal ozone VC retrieval

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178 V. V. Rozanov and A. A. Kokhanovsky

error reaches ∼1.3% instead of ∼0.2% as compared to the case of a single-cloud.The cloud top height retrieval error is especially large in the case of the thinice cloud placed above the water cloud (scenario 2-3). In this case the VHCmodel underestimates the CTH of the cloud system (ΔCTH ∼−4.5 km). Theemploying of LER model results in further underestimation of the cloud systemCTH. The maximal CTH retrieval error reaches for LER model ∼−5.7 km. Themaximal ozone retrieval error is in the range −1.3 to −3.1% depending on thecloud scenario. Employing the VIC model makes it possible to reduce the ozoneVC retrieval error. It follows that the maximal ozone VC retrieval error is smallerthan ∼0.6% if one employs VIC model in the retrieval process.

The results obtained for the ozone VC retrieval error in the case of a three-layered cloud system (scenarios 3-1 and 3-2) do not differ too much from thoseobtained for the case of a two-layered cloud system. The minimal impact on theozone retrieval accuracy can be obtained employing in the retrieval process VICmodel.

Table 5.2. Maximal cloud top height and ozone vertical column retrieval errorsobtained employing in the retrieval process VHC, VIC and LER models.

Scenario CTH retrieval errors, km Ozone VC retrieval errors, %

number LER VHC VIC LER VHC VIC

1-1 −1.1 −0.1 — −2.1 <0.1 −1-2 −2.6 −0.3 — 2.3 0.2 −1-3 −3.1 −0.6 — −2.4 <0.1 −2-1 −4.8 −0.4 2.3 −1.3 1.3 0.22-2 −5.7 −1.1 2.2 −3.1 1.0 0.22-3 −5.4 −4.5 −0.6 −1.8 1.2 0.6

3-1 −8.1 −5.1 2.4 1.7 1.9 −0.73-2 −7.8 −3.5 2.6 −0.7 1.5 0.4

Summing up all the results obtained, the following conclusions can be for-mulated.

– The maximal ozone VC retrieval errors obtained for all considered scenariosemploying in the retrieval process LER, VHC, and VIC models are 3.1%,1.9%, and 0.7%, respectively.

– The maximal CTH retrieval errors obtained for all considered scenarios em-ploying in the retrieval process LER, VHC, and VIC models are −8.1 km,−5.1 km, and 2.6 km, respectively.

– Employing the VHC and the VIC model in the retrieval process requiresinformation about the thermodynamic state of a cloud system. This infor-mation significantly improves CTH and ozone VC retrieval results. We notethat such information can be obtained using the measurements of reflectedsolar radiation in the spectral range 1500–1700 nm, where the spectral refrac-

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5 Impact of single and multi-layered cloudiness 179

tive index of ice differs considerably from that of water. The correspondingalgorithms and their application to SCIAMACHY data are presented, forexample, by Acarreta et al. (2004) and Kokhanovsky et al. (2006).

– The general feature of employing the VHC model in the retrieval of cloudparameters in the case of multi-layered cloud systems is that the retrievedCBH, in all considered cases, corresponds to ground level. We suggest thatthis feature is used to distinguish the multi-layered and single-layered cloudscenes.

– Employing the VIC model for the retrieval of the ozone VC in the caseof multi-layered cloud system is very promising. However, for some cloudscenarios CTH retrieval error can be relatively large if the VIC model is usedin the cloud parameters retrieval process. This demonstrates that parametersof VIC model which are not involved in the retrieval process (vertical profilesof effective radii and shape function) should be further optimized.

All results presented in this chapter were obtained using the radiative transfersofttware package SCIATRAN 2.1 (Rozanov et al., 2007), which contains notonly the forward radiative transfer solver but also all the expressions neededfor the weighting functions calculations as well as the retrieval blocks for thesolution of the cloud parameters and gaseous absorber vertical column inverseproblems. The SCIATRAN 2.1 is freely available for non-commercial use at thewebsite www.iup.uni-bremen.de/sciatran.

Appendix A

The weighting functions provide a linear relationship between the variation of at-mospheric and surface parameters and the variation of the measured functional.In the case under consideration we assume that the measured functional is thereflected intensity at the top of atmosphere in the direction Ω−

v = {−μv, φv},where −μv and φv define the cosine of the polar observation angle and its az-imuthal angle, respectively. The cosine of the polar angle is measured from thepositive τ -axis (i.e., negative values of μ correspond to the light propagated up-wards). The analytical expressions for WFs used in this chapter were derivedemploying the forward-adjoint approach as presented by Rozanov (2006) andRozanov et al. (2007). Although WFs can be expressed in a different form, wepresent them here in the simplest form containing the total forward and totaladjoint intensities. Other representations of WFs can be easily obtained usingthe substitution approach suggested by Rozanov and Rozanov (2007).

A.1 Gaseous absorber number density WF

WF for the relative variation of the number density of an gaseous absorber isgiven by:

W (λ, z) = −n(z)∫4π

I∗λ(z,Ω)Iλ(z,Ω) dΩ . (A.1)

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180 V. V. Rozanov and A. A. Kokhanovsky

Here, n(z) is the number density of a gaseous absorber, and Iλ(z,Ω) and I∗λ(z,Ω)

are the total forward and total adjoint intensities, respectively. The total forwardintensity, Iλ(z,Ω), is obtained as a solution of the forward radiative transferequation (RTE) (see, for example, Chandrasekhar (1950); Liou (1980); Thomasand Stamnes (1999) for derivation):

μdIλ(τ,Ω)

dτ+ Iλ(τ,Ω) = Jλ(τ,Ω) , (A.2)

and the appropriate boundary conditions are given by:

Iλ(0, Ω) = πδ(Ω − Ω0), μ > 0 , (A.3)

Iλ(τ0, Ω) =Aλ

π

∫Ω+

ρλ(Ω,Ω′)Iλ(τ0, Ω′)μ′ dΩ′, μ < 0 , (A.4)

where Jλ(τ,Ω) is the multiple scattering source function:

Jλ(τ,Ω) =ωλ(τ)

∫4π

pλ(τ,Ω,Ω′)Iλ(τ,Ω′) dΩ′ , (A.5)

τ ∈ [0, τ0] is the optical depth, τ0 is the optical thickness of the medium, μ ∈[−1, 1] is the cosine of the polar angle as measured from the positive τ -axis,φ ∈ [0, 2π] is the azimuthal angle, the variable Ω := {μ, φ} describes the set ofvariables μ ∈ [−1, 1] and φ ∈ [0, 2π], the variable Ω+ := {μ, φ} describes theset of variables μ ∈ [0, 1] and φ ∈ [0, 2π], ωλ(τ) ∈ [0, 1] is the single scatteringalbedo, pλ(τ,Ω,Ω′) is the phase function, δ(Ω −Ω0) = δ(μ−μ0)δ(φ−φ0) is theDirac δ-function, μ0 and φ0 define the cosine of the solar zenith angle and thesolar azimuthal angle at the top of the medium, respectively, Aλ is the sphericalalbedo of the underlying surface, and ρλ(Ω,Ω′) is a function determining theangular reflection properties of an underlying surface.

As demonstrated by many authors (see Rozanov and Rozanov (2007) andreferences therein) the total adjoint intensity can be obtained using a solutionof the following RTE:

μdI ′

λ(τ,Ω)dτ

+ I ′λ(τ,Ω) = J ′

λ(τ,Ω) , (A.6)

and the boundary conditions are given by:

I ′λ(0, Ω) =

δ(Ω − Ωv), μ > 0 , (A.7)

I ′λ(τ0, Ω) =

π

∫Ω+

ρλ(Ω,Ω′)I ′λ(τ0, Ω

′)μ′ dΩ′, μ < 0 , (A.8)

where Ωv = {μv, φv} and δ(Ω − Ωv) = δ(μ − μv)δ(φ − φv). Following Hasekampand Landgraf (2002), we will call this RTE as a pseudo-forward RTE.

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5 Impact of single and multi-layered cloudiness 181

Comparing Eqs (A.6)–(A.8) to Eqs (A.2)–(A.4), it can be seen that the pseudo-forward RTE differs from the corresponding forward RTE by the upper boundarycondition only. If the solution of the pseudo-forward RTE given by Eqs (A.6)–(A.8) is found, the adjoint intensity can be obtained employing the followingsubstitution:

I∗λ(τ, μ, φ) = I ′

λ(τ,−μ, φ) , (A.9)

i.e., the positive directions of propagation should be changed to the negative andvice versa.

A.2 Cloud optical thickness WF

Having obtained the solutions of the forward and pseudo-forward RTEs, theanalytical expressions for all the parameters needed can be derived. Thus, intro-ducing the auxiliary function wi(λ,Ω, z) as follows:

wi(λ,Ω, z) =

⎡⎣ωi(z)4π

∫4π

pi(z,Ω,Ω′)Iλ(z,Ω′) dΩ′ − Iλ(z,Ω)

⎤⎦ I∗λ(z,Ω) ,(A.10)

WF for the absolute variation of the cloud optical thickness, δτi, is obtained:

Wτi(λ) =

1L

ht∫hb

∫4π

wi(λ,Ω, z) dΩ dz . (A.11)

Here, subscripts i = 1 and i = 2 correspond to the water and ice cloud, ωi(z) andpi(z,Ω,Ω′) are the single scattering albedo and the phase function, respectively,L = ht − hb is the geometrical thickness of a cloud.

A.3 Cloud geometrical parameters WFs

Using further the introduced auxiliary function wi(λ,Ω, z) and the expressionfor the cloud optical thickness WF given by Eq. (A.11), WFs for the geometricalparameters of the VHC model are determined via the following expressions:

Wht(λ) = Kei

⎡⎣∫4π

wi(λ,Ω, ht) dΩ − Wτi(λ)

⎤⎦ , (A.12)

Whb(λ) = −Kei

⎡⎣∫4π

wi(λ,Ω, hb) dΩ − Wτi(λ)

⎤⎦ , (A.13)

where Kei is the extinction coefficient of the water (i = 1) or ice crystals (i = 2)which is assumed to be constant within a cloud for VHC model and wi(λ,Ω, hb)and wi(λ,Ω, ht) are values of the auxiliary function at the bottom and top of a

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182 V. V. Rozanov and A. A. Kokhanovsky

cloud, respectively. We note that expressions (A.12) and (A.13) are derived underthe assumption that variations of CTH and CBH do not cause the variation ofthe cloud optical thickness, i.e., the cloud optical thickness remains constant.

Introducing an additional auxiliary function, vi(λ,Ω, z), as follows:

vi(λ,Ω, z) =⎡⎣K ′si

(z)4π

∫4π

pi(z,Ω,Ω′)Iλ(z,Ω′) dΩ′ − K ′ei

(z)Iλ(z,Ω)

⎤⎦ I∗λ(z,Ω) , (A.14)

where K ′si

and K ′ei

(z) are derivatives of the scattering and extinction coeffi-cients, respectively, with respect to the altitude within the cloud, WFs for thegeometrical parameters of VIC model are given by:

Wht(λ) = Kei(ht)∫4π

wi(λ,Ω, ht) dΩ

− 1L

ht∫hb

∫4π

[Kei

(z)wi(λ,Ω, z) + (z − hb)vi(λ,Ω, z)]dΩ dz , (A.15)

Whb(λ) = −Kei(hb)

∫4π

wi(λ,Ω, hb) dΩ

+1L

ht∫hb

∫4π

[Kei(z)wi(λ,Ω, z) − (ht − z)vi(λ,Ω, z)

]dΩ dz . (A.16)

Clearly for a vertically homogeneous cloud we have vi(λ,Ω, z) = 0, Kei(z) =

Kei(ht) = Kei(ht) = Kei , where Kei is a constant value of the extinction coeffi-cient within cloud. Thus, the expressions obtained coincide with Eqs (A.12) and(A.13) in the case of vertically homogeneous cloud as it should be.

A.4 LER altitude (surface elevation) WF

The variation of the surface elevation leads to the variation of the extinction co-efficient. Therefore, to obtain WF for the surface elevation we use the expressionfor extinction coefficient WF in the following form (Rozanov et al., 2007):

We(λ, z) =∫4π

[J(z,Ω) − Iλ(z,Ω)] I∗λ(z,Ω) dΩ . (A.17)

Using Eq. (A.17) the variation of the reflected intensity at the top of atmospherein the direction Ω−

v , δIλ(0,−μv, φv), can be written as follows:

δIλ(0,−μv, φv) =

H∫zs

We(λ, z)δeλ(z) dz , (A.18)

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5 Impact of single and multi-layered cloudiness 183

where zs is the surface altitude above see a level, H is the top of atmospherealtitude, δeλ(z) is the variation of the atmospheric extinction coefficient and theintegration is carried out over the entire atmosphere. To obtain the relationshipbetween the variation of the surface elevation, zs, and variation of the extinctioncoefficient we describe the extinction coefficient in the following form:

eλ(z, zs) = Θ(z − zs) eλ(z) , (A.19)

where Θ(z − zs) is the Heavyside step-function given by

Θ(z − zs) ={

1, z > zs

0, z < zs. (A.20)

Varying Eq. (A.19) with respect to zs, we obtain

δeλ(z, zs) =deλ(z, zs)

dzsδzs =

dΘ(z − zs)dzs

eλ(z) δzs . (A.21)

The derivative of the Heaviside step-function with respect to the argument zs

can be obtained analytically (Korn and Korn, 1968):

dΘ(z − zs)dzs

= −δ(zs − z) , (A.22)

where δ(zs − z) is the Dirac δ-function. Substituting Eq. (A.21) into Eq. (A.18)and taking into account Eq. (A.22), we have

δIλ(0,−μv, φv) = − We(λ, zs) eλ(zs) δzs . (A.23)

Thus, the WF for the surface elevation is obtained as follows:

Wzs(λ) = − We(λ, zs) eλ(zs) . (A.24)

The expression for the partial derivative of the reflected intensity with respectto the surface elevation has been presented by Ustinov (2005) as well. However,although the expression for the extinction coefficient WF presented by Ustinov(2005) is the same as given by Eq. (A.17) (neglecting the thermal emission), theexpression for the surface elevation WF given by (Ustinov, 2005) (see Eq. (5.63))has the opposite sign.

To check the correctness of Eq. (A.24) we consider the simplest case of ra-diation propagation neglecting completely all scattering effects, assuming theLambertian surface and omitting the dependence of all relevant variables on thewavelength. In this case the expression for the reflected intensity at the top ofthe atmosphere in the direction −μv can be obtained analytically as follows:

I(0,−μv) = Aμ0 e−τ0/μ0 e−τ0/μv . (A.25)

Taking into account that the optical thickness of the entire atmosphere is givenby

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184 V. V. Rozanov and A. A. Kokhanovsky

τ0 =

H∫zs

e(z) dz , (A.26)

we can write the variation of the reflected intensity caused by the variation ofthe surface elevation zs as follows:

δI(0,−μv) = −Aμ0 + μv

μv

dτ0

dzse−τ0/μ0 e−τ0/μv δzs . (A.27)

Differencing Eq. (A.26) with respect to zs and substituting the result intoEq. (A.27), we obtain

δI(0,−μv) = Aμ0 + μv

μve(zs) e−τ0/μ0 e−τ0/μv δzs . (A.28)

Thus, the straightforward analytical calculations show that in the case of a non-scattering atmosphere the WF for the surface elevation is given by followingexpression:

Wns = Aμ0 + μv

μve(zs) e−τ0/μ0 e−τ0/μv . (A.29)

As a next step we rewrite Eq. (A.24) for the surface elevation WF for thecase of non-scattering atmosphere. Neglecting the scattering effects, i.e., settingin Eq. (A.17) Jλ(τ,Ω) = 0, we obtain the expression for the extinction coefficientWF in the following form:

We(z) = −∫4π

I(z,Ω)I∗(z,Ω) dΩ . (A.30)

We note that the dependence on the wavelength is omitted. The expression forthe surface elevation WF given by Eq. (A.24) can be rewritten now as follows:

Wzs =∫4π

I(zs, Ω)I∗(zs, Ω) dΩ e(zs) . (A.31)

To calculate the integral over solid angle in this equation we use the analyticalexpressions for solutions of the forward and pseudo-forward RTEs at the bottomof the atmosphere. Neglecting the scattering effects, the solution of the forwardRTE results in

I(zs, μ, φ) ={

πδ(μ − μ0)δ(φ − φ0) e−τ0/μ, μ > 0Aμ0 e−τ0/μ0 , μ < 0

, (A.32)

whereas the solution of the pseudo-forward RTE is given by:

I ′(zs, μ, φ) ={

μ−1δ(μ − μv)δ(φ − φv) e−τ0/μ, μ > 0Aπ−1 e−τ0/μv , μ < 0

. (A.33)

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5 Impact of single and multi-layered cloudiness 185

The integral over the solid angle in Eq. (A.31) can be presented as a sum ofintegrals over the upper and lower hemispheres. The integration over the upperhemisphere results in

∫2π

0∫−1

I(zs, μ, φ)I ′(zs,−μ, φ) dμ dφ = Aμ0

μve−τ0/μ0 e−τ0/μv , (A.34)

and the integration over the lower hemisphere is presented as

∫2π

1∫0

I(zs, μ, φ)I ′(zs,−μ, φ)dμdφ = A e−τ0/μ0 e−τ0/μv . (A.35)

Summing up the contribution from the integration over both hemispheres givenby Eqs (A.34) and (A.35) and substituting the result into Eq. (A.31), we obtain

Wzs= A

μ0 + μv

μve−τ0/μ0 e−τ0/μv e(zs) . (A.36)

This expression is fully equivalent to Eq. (A.29) derived using the analyticaldifferentiation of the reflected intensity with respect to the surface elevation.This confirms the correctness of the derived expression for the surface elevationWF given by Eq. (A.24).

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6 Remote sensing of clouds using linearly andcircularly polarized laser beams: techniques tocompute signal polarization

L. I. Chaikovskaya

6.1 Introduction

Polarization parameters of the radiation field scattered by clouds are needed fordeveloping remote sensing methods for microphysical clouds properties. Usingpolarization, investigators may, firstly, upgrade information capacity of signalsand, secondly, develop polarization-based methods to discriminate a useful sig-nal. The very first calculations to find information content of visible radiationpolarization scattered by a cloud were performed at the beginning of the 1970s(see Hansen, 1971; Hovenier, 1971; Kattawar and Plass, 1972). Hansen with theaid of the adding-doubling method and Kattawar and Plass on the base of theMonte Carlo method evaluated multiple-scattered parameters of polarizationas a supplementary to the radiance of radiation reflected and transmitted byclouds. Their interest was focused on the problem of passive remote soundingof cloud parameters. For the problem of sunlight reflection, the aforementionedcalculations have shown that the polarization degree is more sensitive to cloudmicrostructure than radiance. This is connected with the fact that the peculiari-ties of the angular distribution of polarization are far less subjected to smoothingupon multiple scattering than those of radiance. This underlies the techniquesof studying microphysical properties of clouds and ice clouds composition andof distinguishing ice clouds from water-droplet clouds from the measured degreeof polarization (for example, Hansen and Hovenier, 1974; Breon and Goloub,1998; Chepfer et al., 1998; Kokhanovsky and Weichert, 2002; Mishchenko et al.,2002, 2006; Kokhanovsky, 2003; Liou et al., 2000; Sun et al., 2006; Goloub et al.,2000).

The situation is different for the visible light transmission. Single scatteringby large-sized particles gives a low polarization degree in the forward region. Forthick clouds, multiple scattering (see asymptotic solutions for the first and secondStokes parameters in the weak absorption domain obtained, for example, byRozenberg (1958), Domke and Ivanov (1975), Kuzmina and Maslennikov (1979))leads to a very low polarization. Peculiarities in its angular distribution areobliterated. Generally, the considerable polarization degrees of both transmittedand reflected radiation were observed from the noctilucent clouds in the visible

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192 L. I. Chaikovskaya

(Sassen, 2000). The reason is that ice particles of these clouds are small ascompared to the wavelength of light.

Consideration of the information content of polarization in the problem ofactive sounding with continuous wave (CW) or pulsed lidar systems is of adifferent character. Let us outline investigations of polarization properties oflidar backscatter signals from clouds. For the case of a monostatic lidar withlinear polarization of visible radiation, these properties are known fairly well.The first works where they were considered should be mentioned. The resultsof cloud sounding by Pal and Carswell (1973) showed that multiple scatteringcontributes much to the depolarization ratio, and vertical cloud stratification isclearly revealed by the ratio. It was discussed (Liou and Scotland, 1971) how thedepolarization is affected by the field-of-view (FOV) of a lidar and by the parti-cle number density. For ice clouds, it was found that values of the depolarizationratio are much larger than those for water clouds (Pal and Carswell, 1973; Liouand Lahore, 1974; Sassen, 1974). This feature was used afterwards as a basis ofthe method of distinction between droplets and crystals by measuring the signaldepolarization. However, it should be mentioned that the results of this methodturned out to be qualitative rather than quantitative owing to a noticeable de-pendence of the depolarization ratio on other factors, such as particle sizes andshapes and multiple scattering effects.

Studies of polarization properties of lidar signals from clouds, both backscat-tered and transmitted, were continued in many experiments (for example, Balinet al., 1974; Platt, 1978; Houston and Carswell, 1978; Ryan and Carswell, 1978;Ryan et al., 1979; Samokhvalov and Shamanaev, 1982; Vergun et al., 1988,Sassen, 2000, Winker and Pelon, 2003) and theoretical works (for example, Elo-ranta, 1972; Zuev et al., 1976, 1983; Samokhvalov, 1979, 1980; Bruscaglioni etal., 1995; Mannoni et al., 1996; Kaul et al., 1997a, 1997b; Zege et al., 1998;Mishchenko and Sassen, 1998; Hu et al., 2001; Ishimoto and Masuda, 2002; Op-pel, 2005; Prigarin et al., 2005; Czerwinski, 2006). The theoretical treatment ofbackscattering and propagation of polarized CW and pulsed laser beams throughclouds, where the multiple scattering contributes much to a lidar signal, requiressolving the complex problem of polarized radiation transfer of a narrow beam.Mostly the Monte Carlo method was used as a base for solution of this problem.The first Monte Carlo modelling of a multiple-scattered polarized lidar signal wascarried out by Zuev et al. (1976, 1983). Recently, the Monte Carlo codes havebeen created by a number of research groups over the world (see Bruscaglioni etal., 1995; Mannoni et al., 1996; Kaul et al., 1997b; Ishimoto and Masuda, 2002;Oppel, 2005; Prigarin et al., 2005, Czerwinski et al., 2006). Many theoreticalgroups working in the area of lidar sounding with multiple scattering are coordi-nated by the International Workshop MUSCLE (MUltiple SCattering in LidarExperiments).

Semi-analytical approximate techniques to calculate a multiple-scattered po-larized signal produced by a monostatic lidar have been reported in the litera-ture by Liou and Scotland (1971), Eloranta (1972), Samokhvalov (1979, 1980),Vasilkov et al. (1990), Zege et al. (1998), Zege and Chaikovskaya (1999), Gorod-nichev et al. (2000, 2006), Chaikovskaya (2002), Chaikovskaya and Zege (2004).

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6 Remote sensing of clouds using circularly 193

Let us note techniques and solutions used for clouds. Liou and Scotland, Elo-ranta and Samokhvalov have found analytical solutions for a polarized lidar sig-nal with regard to the first and second orders of scattering applicable for a verysmall field-of-view of a lidar system and small optical depths of clouds. Whenusing the approach by Liou and Scotland, Liou also approximately describedscattering orders higher than the second (see Liou, 1971, 1972). An approxi-mate semi-analytical approach to describe polarization of a lidar return froma cloud with allowance for multiple small-angle forward scattering and singlelarge-angle scattering has been built by the Minsk group (Zege et al., 1998; Zegeand Chaikovskaya, 1999). It is applicable for optical thicknesses less than 5.Theoretical ideas used in this approach will be described below.

Other opportunities for applying the polarization in lidar sounding of cloudsalso were found. Recently, much attention has been paid to the employmentof the polarization properties of returns in sounding with multiple-field-of-view(MFOV) lidar systems (for example, Roy et al., 1999; Kolev et al., 2006). Inthese lidar systems, receivers are equipped with spatial filters in the focal planeto block single-scattered radiation, giving a gate for multiply scattered light.Through the openings, an angular distribution of multiply scattered radiationcan be recorded. The cross-linearly polarized return component produced by anMFOV lidar has been used to retrieve parameters of the particle size distributionof clouds (Roy et al., 1999). Polarization, as assumed, can be also included intosounding methods on the base of a bistatic lidar (Oppel, 2005; Chaikovskayaand Zege, 2003; Czerwinski et al., 2006) and wide-angle lidar (WAIL (Davis etal., 1997; Polonsky and Davis, 2006) is currently working wide-angle lidar).

In a number of lidar systems with a CCD camera, the images that presentangular and/or spatial distributions of variously polarized backscattered radi-ance are recorded (Carswell and Pal, 1980; Dogariu and Asakura, 1993; Cameronet al., 1998; Roy N. et al., 2004; Roy N. and G. Roy, 2006). Measured polar-ized signals with a CCD receiver, with polarizer and analyzer, can be describedby Monte Carlo simulations (for example, Cameron et al., 1998; Oppel, 2005;Oppel et al., 2006) and by approximate solutions (Dogariu and Asakura, 1993;Chaikovskaya and Zege, 2005). At present, these images with the representativetwo-, four- and eight-fold azimuthal structures, which are closely connected tochosen polarizer and analyzer states, are considered as the carriers of informa-tion about a scattering medium. The information contained in angular scatteredintensity patterns of variously polarized light beams is expected to be additionalas compared to the information given by the conventionally measured linear de-polarization ratio under monostatic sounding. The use of the imaging techniquewith polarization is expected to improve methods of cloud diagnosis.

Usually, features of the linear polarization type are considered when investi-gating polarization parameters of backscatter signals from clouds. The situationseems quite natural, as mainly laser sources of linearly polarized (LP) light areinvolved in active sounding. Data on the degree of circular polarization (CP)of a return in the case of cloud sounding by circularly polarized (CP) lightare scarce. For instance, the abovementioned images were obtained includingthe circular polarizer and analyzer (Cameron et al., 1998). In the conventional

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194 L. I. Chaikovskaya

monostatic lidar sounding of clouds circular depolarization measurements weremade by Woodard et al. (1998) (as mentioned by Sassen (2000)). Kaul et al.(2001) measured the CP degree when studying backscattering matrices of crys-tal constituents in the atmosphere. They considered the circular depolarizationratio to be a more convenient and correct parameter for characterization of crys-tal clouds with preferred orientation of ice plates and columns as compared tothe linear depolarization ratio. Gorodnichev et al. (1998, 1999) presented a the-ory of CP light reflection and propagation as applied to media with large-scaleinhomogeneities. Look and Chen (1994) measured parameters of CP light scat-tered over 0 to 90 degrees by a water-suspension of latex particles. The effectsthat are likely to occur in time-resolved backscattering of CP light were theo-retically studied by Kim and Moscoso (2002). Hu et al. (2003) proposed to usethe circular polarization for water/ice discriminating from a spaceborne lidar.Mishchenko and Hovenier (1995) have calculated and compared circular and lin-ear depolarization ratios for single backscattering of light by randomly orientednonspherical particles (see also Mishchenko et al (2006)).

Water clouds are macroscopically isotropic multiply scattering media that arecharacterized by the diagonal backscattering matrix. Ice clouds may consist bothof randomly oriented crystals and of horizontally oriented large-sized plates andcolumnar crystals (Sassen, 2000). For lidar sounding of clouds in the zenith ornadir, the backscattering matrix is also diagonal. In the case of inclined soundingpath, the backscattering matrix of horizontally oriented crystals has nonzerooff-diagonal elements (Kaul et al., 1997a). Besides, preferable orientation of theparticles symmetry axis along some direction in the horizontal plane is observedfrom time to time under lidar sounding of crystal clouds (see, for example, themeasurements by Kaul et al. (1997a, 2004) of backscattering matrices of crystalsin the atmosphere). In this case, some or all off-diagonal matrix elements arenot zeros even while sounding along the vertical path. This means that a formof the backscattering matrix serves to identify the orientation of ice particles.Some approaches have been proposed (Kaul et al., 1997a, 2004; Romashov andRakhimov, 1993) to determine a direction and degree of particles orientationfrom measured elements of the backscattering matrix.

Currently, methods of active polarized remote sensing of clouds continue toprogress. The development of theoretical techniques to describe the polarizationof multiple-scattered lidar signals from clouds and studies of signals informationcontent remains a live issue in modern cloud research. Of interest are techniquesapplicable to various lidar systems such as CW and pulsed lasers, MFOV, imagedetecting and bi-static lidars, and lidars with circularly polarized light. Successhas been recently achieved in the development of approximate semi-analyticaltechniques for calculation of polarized lidar signals including multiple scattering(Vasilkov et al., 1990; Zege et al., 1998; Zege and Chaikovskaya, 1999; Gorod-nichev et al., 2000, 2006; Chaikovskaya and Zege, 2004). The known advantagesof semi-analytical solutions are the secured tractability and rapid computationas well as their usability in algorithms for solving inverse problems.

The present chapter reviews the simplified fast techniques for computingmultiple-scattered LP and CP signals in monostatic lidar sounding of clouds

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6 Remote sensing of clouds using circularly 195

obtained at the Institute of Physics, Minsk, Belarus, during the last decade.These techniques were checked by comparison with Monte Carlo algorithms,which are used for the tasks of polarized lidar sounding, and the adding-doublingcode (Zege et al., 1999), and good agreement was obtained. They are basedon the use of approximate vector equations to derive semi-analytical solutions.Techniques to calculate of backscatter polarized signals were built with takinginto account multiple light scattering in the near-forward directions and singlescattering in the near-backward directions. In the literature, conceptually similarapproaches have been reported by Vasilkov et al. (1990) to the remote sensing ofthe ocean and by Gorodnichev et al. (2006) as applied to scattering media withlarge-scale inhomogeneites.

A key point in the development of the semi-analytical computation techniquesfor the tasks of polarized lidar sounding with multiple scattering is the use of thetheory called the approximate analytical vector theory (AAVT) which providessimpler approximate transfer equations (Zege and Chaikovskaya, 1996, 2000)and allows solving tasks to be fundamentally simplified (Zege and Chaikovskaya,1999; Chaikovskaya and Zege, 2004). This theory is stated in section 6.2, andlater (sections 6.3 and 6.4) its efficiency in computing data on laser light depolar-ization in the sounding of clouds is demonstrated. Computations and compara-tive consideration of the formation of both LP and CP visible light depolarizationfor a water cloud are performed. Results of development of the semi-analyticalcomputation technique for the transmission problem with incident wide LP andCP beams are presented in section 6.3. This technique may be employed to in-terpret data of water clouds sounding with a spaceborne lidar in the geometryof transmission. Section 6.4 deals with backscattering of LP and CP pulses froma water cloud. Features of water cloud sounding with CP beams versus thosewith LP beams are highlighted.

6.2 Basic theory

The axisymmetric geometry is considered. It is assumed that a plane-paralleloptically isotropic scattering medium is illuminated by arbitrary polarized lightalong the internal normal to the upper boundary. In the Cartesian co-ordinatesystem XY Z with the Z-axis set along the internal normal, its origin O being atthe upper boundary, partial-angular distribution of the Stokes vector of scatteredradiation field is defined through the following variables. The position of anobservation point is characterized by the radius-vector R = (z, r), where r =(r, ϕ) is the projection of R onto the plane z = const., with r = |r| beingthe distance from the Z-axis, and ϕ being the azimuth angle measured fromthe radial plane XOZ. The unit vector n(μ, ψ) gives the radiation direction,μ = cosϑ, ϑ and ψ are the polar and azimuth angles in the local co-ordinatesystem X ′Y ′Z ′ with the origin O′ at the point R, the axes X ′, Y ′ and Z ′ beingparallel to the system XYZ axes. It holds that ψ = 0 in the meridian planeX ′O′Z ′. The difference φ = ψ − ϕ can be used instead of ϕ or ψ.

An external light source that emits along the Z -axis is characterized byan axially symmetric spatial-angular power diagram Φscr(R0,n0) and by the

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196 L. I. Chaikovskaya

Stokes vector P (with P1 = 1). The incident Stokes vector is S(R0,n0) =PΦscr(R0,n0). For a given n0(μ0, ψ0), P is defined relative to the orts (l, r)with the ort l projection onto the plane z = 0 parallel to the OX direction. TheStokes vector of multiply scattered radiation is defined by

S(R,n) =∫

dR0

∫dn0G(R,n;R0,n0)L(−ψ0)PΦsrc(R0,n0), (6.1)

where G(R,n;R0,n0) is the 4× 4 Green’s matrix satisfying the vector radiativetransfer equations (VRTEs):

B{G(R,n)

}=

σs

∫ ∫dn′ Z(n,n′) G(R,n′) + Eδ(R − R0) δ(n − n0). (6.2)

Here,B = n∇R + σe, (6.3)

∇R is the gradient, σe and σs, the extinction and scattering coefficients, E, the4 × 4 unit matrix, δ(R − R0) and δ(n − n0), the delta functions,

Z(n,n′) = L(π − χ2) F (x) L(−χ1), (6.4)

the phase matrix (Chandrasekhar, 1960). In Eq. (6.4),

F (x) =

⎛⎜⎜⎝a1(x) b1(x) 0 0b1(x) a+(x) + a−(x) 0 00 0 a+(x) − a−(x) b2(x)0 0 −b2(x) a4(x)

⎞⎟⎟⎠ (6.5)

is the single scattering matrix (SSM) of an optically isotropic scattering medium(Van de Hulst, 1961),

x = cos θ = μμ′ +√

1 − μ2√

1 − μ′2 cos (ψ − ψ′) , (6.6)

θ is the scattering angle, and L(χ), the rotation matrix of the form

L(χ) =

⎛⎜⎜⎝1 0 0 000 l(χ)

00

0 0 0 1

⎞⎟⎟⎠ , l(χ) =(

cos 2χ −sin 2χsin 2χ cos 2χ

). (6.7)

The rotation angles χ1 and χ2 (Chandrasekhar, 1960) are constituted by themeridian planes [n′ × z′] and [z′ × n], respectively, being the reference planesfor the phase matrix with the local single scattering plane [n′ × n] which isthe reference plane for the SSM. Note, the argument z for the extinction andscattering coefficients and for the phase matrix and SSM of a medium will beomitted whenevery unnecessary.

The axial symmetry of the single scattering imposes the following conditionson the SSM:

bj ∼ θ2, j = 1, 2, and a− ∼ θ4, if θ � 1, (6.8)

bj ∼ (π − θ)2, j = 1, 2 and a+ ∼ (π − θ)4, if π − θ � 1, (6.9)

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6 Remote sensing of clouds using circularly 197

where the tilde above defines a reduced (divided by a1(x)) element. Expressions(6.8), (6.9) are easily obtained from the expansion of the matrix F (x) in thegeneralized spherical functions (GSF) (Domke, 1975; Hovenier and Mee, 1983).They show absence of a LP plane rotation in the forward and backward, respec-tively, scattering directions (at θ = 0 and θ = π, the matrix F (x) is strictlydiagonal (Van de Hulst, 1961)) and smallness of a LP plane rotation near thesedirections.

Because the set of diagonal elements of the SSM is self-transformation ofthe incident Stokes parameters and the set of off-diagonal elements describesconversion of one polarization type to another they differ in an angular patterntype (Eqs (6.8), (6.9)) and quantitatively. The sets of diagonal Fii(x) and off-diagonal bj(x) elements can be considered as generating the leading and small,respectively, quantities in a scattering problem (Zege and Chaikovskaya, 1985,1996; Chaikovskaya, 1991). When the phase function a1(x) is strongly forward-elongated, the functions a+(x) and a4(x) are also strongly forward-elongated.For such media, one can believe that the functions a1(x), a+(x) and a4(x) gen-erate the leading quantities while the function a−(x) along with the off-diagonalelements bj(x), (j = 1,2) (Eq. (6.8)) provide the small quantities of a scatteringproblem (Zege and Chaikovskaya, 1985, 2000; Chaikovskaya, 1991). To illustratethe aforesaid we show in Fig. 6.1 the angular patterns of the sets of elementsa1(x), a+(x), a4(x) (a) and bj(x), a−(x) (b) over the forward scattering regionfor the water cloud model C1 (Deirmendjan, 1962).

6.2.1 Matrix describing the light field produced by a normallyincident beam

The scattered Stokes-vector definition (6.1) can be redefined. The rotation ma-trix L [−(ψ − ψ0)] can be taken out of the Green’s matrix. This is evident inthe limiting case of a monodirectional illumination (μ0 = 1). The single scat-tering source of the transfer equation (6.2) will be defined by the phase ma-trix Z(μ, μ0 = 1, ψ, ψ0) = F (μ) L [−(ψ − ψ0)] (see Eq. (6.4), where χ2 = 0 andχ1 = ψ−ψ′, if μ′ = 1) and, therefore, G(R,n)= G(z, r, μ, φ) L[−(ψ−ψ0)]. For thenormally incident light beam with nonzero but small angular dispersion, the fac-tor L [−(ψ − ψ0)] stands in the Green’s matrix as well. To prove this, one can usethe phase matrix expanded in the Fourier series in the azimuth ψ−ψ0 in terms ofthe GSF P l

nm(μ) and P lnm(μ0) (Domke, 1975; Hovenier and van der Mee, 1983):

Z(μ, μ0, ψ−ψ0) =∞∑

m=−∞(−1)m exp(−im(ψ − ψ0))

∞∑l=max{|m|,2}

P l,m(μ) Bl Pl,m(μ0),

(6.10)where P l,m(μ) is a matrix whose (11)- and (44)-elements are the GSF P l

0m(μ),(22)- and (33)-elements are [P l

2m(μ) + P l−2m(μ)]/2, (23)- and (32)-elements are

i[P l2m(μ) − P l

−2m(μ)]/2 and the complex conjugate, respectively, and all otherelements are zeros. The matrix Bl is the commonly known 4 × 4 matrix con-sisting of expansion coefficients of elements of the SSM in the GSF. Considering

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198 L. I. Chaikovskaya

0 20 40 600.1

1

10

100

1000

mat

rix e

lem

ents

(a)(11)(+)(44)

0 20 40 60angle, degree

-1

0

1

2

3

4

mat

rix e

lem

ents (21)

(34)(-)

(b)

Fig. 6.1. Angular patterns over forward scattering region of Cloud C1 (1.064 μm) SSMelements forming leading quantities (a) and small quantities (b) of vector problem.

ϑ0 � 1 and using the GSF property: P lnm(ϑ0 � 1) ∼ ϑ

|n−m|0 , one can transform

the Fourier series (6.10) into the Taylor series in powers of small ϑ0:

Z(μ, μ0, ψ − ψ0) =

{Z(μ, μ0)+

3∑p=1

ϑp0Zp [μ, μ0; cos p(ψ − ψ0), sin p(ψ − ψ0)]

× L [−(ψ − ψ0)] . (6.11)

In the second term, Zp are matrices of the following form: elements numberedby ik = 11, 12, 21, 22, 33, 34, 43, 44 are multiplied by cos p(ψ − ψ0), elementswith ik = 13, 14, 23, 24, 31, 32, 41, 42 by sin p(ψ − ψ0). Therefore, the matricesZp vanish after integrating for (ψ − ψ0).

From the transfer equation (6.2) with the kernel (6.11), it follows that forsmall ϑ0

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6 Remote sensing of clouds using circularly 199

G(R,n;R0,n0)={Gφ(R,n;R0,n0) + GR(R,n;R0,n0;ψ − ψ0)

}L[−(ψ−ψ0)] ,

(6.12)where the matrix GR(R,n;R0,n0;ψ − ψ0) is similar to the second matrix in(6.11) in its dependence on ψ − ψ0, that is, it vanishes after integrating for(ψ − ψ0) (at given φ0). The first term can be defined as

Gφ(R,n;R0,n0) =

2π∫0

dψ0 G(R,n;R0,n0) L(ψ − ψ0). (6.13)

After substitution of the factorized structure of Green’s matrix (6.12) into Eq.(6.1), the rotation matrix L(−ψ) is factored outside the integral sign while therotation matrix L(ψ0) is multiplied by the matrix L(−ψ0) of Eq. (6.1) and givesthe 4 × 4 unit matrix. As was said, the second term vanishes after integratingwith respect to ψ0. In the final analysis the Stokes vector S(R,n) proves to bedetermined by the matrix Gφ(R,n) (6.13) and represented as

S(R,n) = J(R,n) L(−ψ) P, (6.14)

J(R,n) =∫

dR0

∫dn0 Gφ(R,n,R0,n0)Φsrc(R0,n0). (6.15)

The matrix Gφ(R,n) obeys the transfer equation

B{Gφ(R,n)

}=

σs

∫∫dn′ Z(n,n′) Gφ(R,n′) L(φ−φ′)+E δ(R−R0) δ(n−n0),

(6.16)which results from Eq. (6.1) being multiplied by L(ψ − ψ0) and integrated forψ − ψ0, just as in Eq. (6.13).

One can solve the transfer problem with an initial polarized beam normallyincident deriving the matrix Gφ(R,n) (Gφ(R,n) stands for Gφ(z, r, μ, φ)) andnot the Green’s matrix G(R,n). Unlike the Green’s matrix, Gφ(z, r, μ, φ) doesnot contain strongly oscillating factor-functions Lik(ψ − ψ0) in the second andthird columns. This feature is caused by the matrix Gφ(z, r, μ, φ) transformingthe Stokes vector L(−ψ)P into S, i.e., it is, essentially, matrix-defined in theparallel reference planes which are the meridian planes ψ = const. at the pointsR0 and R.

6.2.2 Matrices of propagation and near-backward scattering

Two geometries are of prime significance for laser sounding: the narrow arbi-trary polarized beam propagation through a medium and the near-backwardscattering. Usually, the receiver having the axi-symmetrical spatial-angular pat-tern of sensitivity Φrec(R,n) measures parameters of the near-forward scattered

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200 L. I. Chaikovskaya

radiation or it measures parameters of the near-backward scattered radiation.Also the analyzer is used in both measurements. The analyzer is specified by theStokes vector A whose reference plane is parallel to the reference plane for theincident Stokes vector. The received signal will have the power

W f = AT {∫∫

L(ψ)S(R,n)Φrec(R,n) dR dn}

= AT {∫∫

[L(ψ)J(R,n)L(−ψ)]Φrec(R,n) dR dn }P (6.17)

or

W b = AT {∫∫

L(−ψ)S(R,n)Φrec(R,n) dR dn}

= AT {∫∫

[L(−ψ)J(R,n)L(−ψ)]Φrec(R,n) dR dn }P, (6.18)

respectively, where S(R,n) is the scattered Stokes vector (6.1), L(ψ) and L(−ψ)in the first equalities of Eq. (6.17) and Eq. (6.18), respectively, are the matri-ces that rotate the reference plane for S(R,n) to the one for A. The matricesL(ψ)J(R,n)L(−ψ) and L(−ψ)J(R,n)L(−ψ) of Eq. (6.17) and Eq. (6.18) canbe referred to as propagation matrix and near-backward scattering matrix, re-spectively.

The aforementioned two matrices can be given in particular three-term formswhich clearly show the difference between the matrix elements in magnitudeand in type of the azimuth dependence and are therefore extremely useful fortheoretical studies and interpretation of experimental observations. To provethis, one should preliminarily transform the central 2×2 submatrix of the matrixJ(R,n),

j(R,n) =(

J22(R,n) J23(R,n)J32(R,n) J33(R,n)

), (6.19)

that acts on {Q,U}, using features of the central 2×2 submatrix of phase matrix(Zege and Chaikovskaya, 2000)

d(n,n′) =(

Z22(n,n′) Z23(n,n′)Z32(n,n′) Z33(n,n′)

). (6.20)

From Eqs (6.4)–(6.7),

d(n,n′) = l(π − χ2)(

a+(x) + a−(x) 00 a+(x) − a−(x)

)l(−χ1)

= d+(n,n′) + d−(n,n′), (6.21)

where

d+(n,n′) = l(π − χ2) a+(x) e+ l(−χ1) =(

Z+(n,n′) Z ′+(n,n′)

−Z ′+(n,n′) Z+(n,n′)

), (6.22)

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6 Remote sensing of clouds using circularly 201

d−(n,n′) = l(π − χ2) a−(x′) e− l(−χ1) =(

Z−(n,n′) Z ′−(n,n′)

Z ′−(n,n′) −Z−(n,n′)

), (6.23)

withe± = diag{1,±1}. (6.24)

The ‘plus’ and ‘minus’ phase matrix components d+(n,n′) (6.22) and d−(n,n′)(6.23) are different in that they are expressed through the single scattering func-tions a+(x) and a−(x), respectively, which are of different order for forward andbackward regions (Eqs (6.8) and (6.9)), and obey different rules of permutationwith the 2 × 2 rotation matrix l(χ) (6.7):

l(χ) d+ = d+ l(χ) and l(χ) d− = d− l(−χ). (6.25)

Permutations (6.25) follow from the equalities l(χ) e+ = e+ l(χ) and l(χ) e− =e− l(−χ). One more useful feature of the 2×2 ‘plus’ and ‘minus’ matrices is theirsimple relation to elements of the entire 2 × 2 submatrix d(n,n′) which looks asfollows

d+ =12

(d + ω d ωT

), (6.26)

d− =12

(d − ω d ωT

), (6.27)

where

ω =(

0 11 0

). (6.28)

As can be easily verified, the matrices (6.26) and (6.27) are equivalent to thematrices (6.22) and (6.23), respectively.

The universal properties (6.21)–(6.27) are easily generalized to the 2 × 2central submatrix gφ(R,n) of the matrix Gφ(R,n):

gφ(R,n) =(

G22(R,n) G23(R,n)G32(R,n) G33(R,n)

= g+φ (R,n) + g−

φ (R,n), (6.29)

where

g+φ =

12(gφ+ωgφωT ) =

e+

2

[G22 + G33 G23 − G32−G23 + G32 G22 + G33

≡[

G+ G′+

−G′+ G+

], (6.30)

g−φ =

12(gφ−ωgφωT ) =

e−2

[G22 − G33 G23 + G32−G23 − G32 G22 − G33

≡[

G− G′−

G′− −G−

]. (6.31)

The above ‘plus’ and ‘minus’ components are of different order for forward andbackward regions and enable permutations relatively 2 × 2 rotation matrices asit is given by

l(χ) g+φ = g+

φ l(χ) and l(χ) g−φ = g−

φ l(−χ). (6.32)

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202 L. I. Chaikovskaya

The integration of Eqs (6.29)–(6.31) with the source diagram as a weight, asgiven by Eq. (6.15), yields the submatrix (6.19) of the matrix J(R,n) of theform

j = j+ + j− =(

J+ J ′+

−J ′+ J+

)+(

J− J ′−

J ′− −J−

), (6.33)

where j+ and j− stand for

jν(R,n) =∫

dR0

∫dn0 gν

φ(R,n;R0,n0)Φsrc(R0,n0), ν = (+), (−).

(6.34)From Eqs (6.32) and (6.34), it follows that

l(χ) j+ = j+ l(χ) and l(χ)j− = j− l(−χ), (6.35)

what implies

l(ψ)j l(−ψ) = l(ψ) (j+ + j−)l(−ψ) = j+ + j− l(−2ψ) (6.36)

andl(−ψ)j l(−ψ) = l(−ψ) (j− + j+)l(−ψ) = j− + j+ l(−2ψ). (6.37)

Consequently, the propagation matrix takes the form (see Eq. (6.36))

Jf = L (ψ) J L (−ψ) =

⎛⎜⎜⎝J11 0 0 J140 J+ J ′

+ 00 −J ′

+ J+ 0J41 0 0 J44

⎞⎟⎟⎠

+ L (ψ)

⎛⎜⎜⎝0 J12 J13 0J21 0 0 J24J31 0 0 J340 J42 J43 0

⎞⎟⎟⎠ L (−ψ) +

⎛⎜⎜⎝0 0 0 00 J− J ′

− 00 J ′

− −J− 00 0 0 0

⎞⎟⎟⎠ L (−2ψ) . (6.38)

As follows from Eq. (6.37), the near-backward scattering matrix can be repre-sented as

Jb = L (−ψ) J L (−ψ) =

⎛⎜⎜⎝J11 0 0 J140 J− J ′

− 00 J ′

− −J− 0J41 0 0 J44

⎞⎟⎟⎠

+ L (−ψ)

⎛⎜⎜⎝0 J12 J13 0J21 0 0 J24J31 0 0 J340 J42 J43 0

⎞⎟⎟⎠ L (−ψ) +

⎛⎜⎜⎝0 0 0 00 J+ J ′

+ 00 −J ′

+ J+ 00 0 0 0

⎞⎟⎟⎠ L (−2ψ) . (6.39)

Here, Jf , Jb, J , Jik, J± and J ′± are functions of z, r, μ and φ.

From Eqs (6.38) and (6.39) we see that the matrices Jf and Jb have three-term structures and their terms differ both quantitatively and in the type ofazimuthal dependence. The very axial symmetry of the problem gives rise to

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6 Remote sensing of clouds using circularly 203

these forms. The first submatrices of Jf and Jb are leading ones at ϑ � 1 andπ − ϑ � 1, respectively. Smallness of the second and third submatrices of Jf

and Jb in the near-forward and near-backward, respectively, scattering regions iscaused by the smallness of the single scattering functions bj(ϑ � 1), a−(ϑ � 1)(6.8) and bj(π − ϑ � 1), a+(π − ϑ � 1) (6.9). These appear in the singlescattering sources in the correspondent transfer equations for elements of thesecond and third submatrices of Jf and Jb. The rotation matrices are factorsjust in the second and third terms, which do not contribute to strictly forwardand backward scattering directions. This is closely connected with the fact thata medium produces no rotation of polarization plane in these directions. Ryanet al. (1979) in their experiments on lidar sounding of water clouds paid specialattention to the direction of the polarization of the backscattered light. Themeasurements have shown no changes in it.

In the conventional axisymmetric scheme of lidar sounding, a receiver asa rule makes azimuth-averaging of the spatial and/or angular distribution ofradiation parameters. Let us integrate Eqs (6.38) and (6.39) over spatial azimuthϕ, for a given ψ. The same will be upon integrating over φ, for a given ψ.As a result, the elements (13), (31), (24), (42), (14), (41), (23) and (32) inthe matrix J(z, r, ϑ, φ), which are odd over φ, vanish (Zege and Chaikovskaya,2000):

〈Jf 〉ϕ = L (ψ) 〈J〉φ L (−ψ) =

⎛⎜⎜⎝J11 0 0 00 J+ 0 00 0 J+ 00 0 0 J44

⎞⎟⎟⎠

+L (ψ)

⎛⎜⎜⎝0 J12 0 0J21 0 0 00 0 0 J340 0 J43 0

⎞⎟⎟⎠ L (−ψ) +

⎛⎜⎜⎝0 0 0 00 J− 0 00 0 −J− 00 0 0 0

⎞⎟⎟⎠ L (−2ψ) , (6.40)

〈Jb〉ϕ = L (−ψ) 〈J〉φ L (−ψ) =

⎛⎜⎜⎝J11 0 0 00 J− 0 00 0 −J− 00 0 0 J44

⎞⎟⎟⎠

+L (−ψ)

⎛⎜⎜⎝0 J12 0 0J21 0 0 00 0 0 J340 0 J43 0

⎞⎟⎟⎠ L (−ψ) +

⎛⎜⎜⎝0 0 0 00 J+ 0 00 0 J+ 00 0 0 0

⎞⎟⎟⎠ L (−2ψ) , (6.41)

where the azimuth-averaged submatrix elements Jik and J± are functions of z, rand μ.

Noting that integration on r does not change the general form of these ma-trices, one can conclude that above structures are valid when

(1) a signal is received on the circular area of some radius a perpendicular tothe Z-axis,

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204 L. I. Chaikovskaya

(2) a signal is detected at the axis Z (a → 0),(3) one deals with the scattering problem with an infinitely wide incident beam

(a → ∞ ) .

If Eqs (6.38) and (6.39) are averaged over the angular azimuth ψ, given spatialazimuth ϕ, then, the rotation matrices are represented as L(±ψ) = L(±(ϕ+φ))and L(−2ψ) = L(−2(ϕ + φ)) and the integration of Eqs (6.38) and (6.39) ismade over φ, at given ϕ. The resultant matrix structures will be similar:

〈Jf 〉ψ = L (ϕ) 〈L (φ) J L (−φ)〉φ L (−ϕ)

=

⎛⎜⎜⎝j11 0 0 00 jp,+ 0 00 0 jp,+ 00 0 0 j44

⎞⎟⎟⎠+ L(ϕ)

⎛⎜⎜⎝0 j12 0 0j21 0 0 00 0 0 j340 0 j43 0

⎞⎟⎟⎠ L(−ϕ)

+

⎛⎜⎜⎝0 0 0 00 jp,− 0 00 0 −jp,− 00 0 0 0

⎞⎟⎟⎠ L(−2ϕ) (6.42)

and

〈Jb〉ψ = L (−ϕ) 〈L (−φ) J L (−φ)〉φ L (−ϕ)

=

⎛⎜⎜⎝j11 0 0 00 jb,− 0 00 0 −jb,− 00 0 0 j44

⎞⎟⎟⎠+ L(−ϕ)

⎛⎜⎜⎝0 j12 0 0j21 0 0 00 0 0 j340 0 j43 0

⎞⎟⎟⎠ L(−ϕ)

+

⎛⎜⎜⎝0 0 0 00 jb,+ 0 00 0 −jb,+ 00 0 0 0

⎞⎟⎟⎠ L(−2ϕ), (6.43)

where the submatrix elements are functions of z, r and μ. The structures (6.42)and (6.43) are characteristic of the matrices that give spatial distribution over theplane Z = const. of the Stokes vector of the downward and upward, respectively,propagating radiation fluxes within a certain solid angle.

The one-term form of the near-backward scattering matrix (6.41) is the fol-lowing

〈Jb〉 =

⎛⎜⎜⎝J11 J12 cos 2ψ −J12 sin 2ψ 0J21 cos 2ψ (J− + J+ cos 4ψ) −J+ sin 4ψ −J34 sin 2ψJ21 sin 2ψ J+ sin 4ψ − (J− − J+ cos 4ψ) J34 cos 2ψ0 J43 sin 2ψ J43 cos 2ψ J44

⎞⎟⎟⎠ .

(6.44)The structure of the matrix (6.43) is similar, except that the azimuth ψis replaced by ϕ. In their work Cameron et al. (1998) considered the near-backscattering matrix for the polystyrene sphere suspension and obtained both

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6 Remote sensing of clouds using circularly 205

experimental and Monte Carlo simulated images of elements of this matrix in thereceiving plane with varying incident polarization and analyzer position. It canbe easily checked that the azimuth configuration of images shown in the figurefrom the work by Cameron et al. (1998) is just the universal azimuth structureshown by Eq. (6.44).

Employing the aforementioned matrix forms, one can also interpret otherobservations and Monte Carlo calculations of azimuth configurations of polarizedimages (for example, Carswell and Pal, 1980; Dogariu and Asakura, 1993; RoyN. et al., 2004; Roy N. and G. Roy, 2006; Oppel et al, 2006).

After averaging twice over spatial and angular azimuths (over φ and ψ), thematrices (6.38) and (6.39) are reduced to their first diagonal submatrices

〈Jf 〉ϕ,ψ = diag{

J11, J+, J+, J44}

(6.45)

and〈Jb〉ϕ,ψ = diag

{J11, J−, −J−, J44

}. (6.46)

These gather the leading elements for the near-forward and near-backward scat-tering, respectively.

6.2.3 Simplified transfer equations

The transfer equation for the matrix Gφ(R,n) can be simplified in the case of amedium with a forward elongated phase function, such as a cloud. The methodof simplification was developed earlier for the equation for the Green’s matrix(Eq. (6.2)) (Zege and Chaikovskaya, 1996, 2000). It is analogous in the case ofEq. (6.16). It is basically approximate splitting of systems of four simultaneousequations that define the matrix elements on scalar equations and systems of twoequations. The way of splitting of Eq.(6.16) is the following. The phase matrixis represented as (Zege and Chaikovskaya, 1996):

Z(n,n′) = Z0(n,n′) + Z1(n,n′), (6.47)

where

Z0(n,n′) =

⎛⎜⎜⎝Z11(n,n′) 0 0 00 Z22(n,n′) Z23(n,n′) 00 Z32(n,n′) Z33(n,n′) 00 0 0 Z44(n,n′)

⎞⎟⎟⎠

= L(π − χ2)

⎛⎜⎜⎝a1(x) 0 0 00 a2(x) 0 00 0 a3(x) 00 0 0 a4(x)

⎞⎟⎟⎠ L(−χ1) (6.48)

and

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206 L. I. Chaikovskaya

Z1(n,n′) =

⎛⎜⎜⎝0 Z12(n,n′) Z13(n,n′) 0Z21(n,n′) 0 0 Z24(n,n′)Z31(n,n′) 0 0 Z34(n,n′)0 Z42(n,n′) Z43(n,n′) 0

⎞⎟⎟⎠

= L(π − χ2)

⎛⎜⎜⎝0 b1(x) 0 0b1(x) 0 0 00 0 0 b2(x)0 0 −b2(x) 0

⎞⎟⎟⎠ L(−χ1). (6.49)

As was said, the off-diagonal elements bj generate small quantities of thetransfer problem. Therefore, Z0(n,n′) is the leading term in the phase matrix andis Z1(n,n′) is the small term. Solution of Eq. (6.16) can be similarly expressedas

Gφ(R,n) = Gφ,0(n,n′) + Gφ,1(n,n′), (6.50)

where

Gφ,0(R,n) =

⎛⎜⎜⎝G11(R,n) 0 0 G14(R,n)0 G22(R,n) G23(R,n) 00 G32(R,n) G33(R,n) 0G41(R,n) 0 0 G44(R,n)

⎞⎟⎟⎠φ

, (6.51)

and

Gφ,1(R,n) =

⎛⎜⎜⎝0 G12(R,n) G13(R,n) 0G21(R,n) 0 0 G24(R,n)G31(R,n) 0 0 G34(R,n)0 G42(R,n) G43(R,n) 0

⎞⎟⎟⎠φ

. (6.52)

Here, the matrix Gφ,1(R,n) (6.52) obeys the transfer equations with the singlescattering sources expressed through small elements bj of the SSM. One can seethat the off-diagonal elements (i,2), (i,3) and (2,i), (3,i) (i = 1,4) of the matrixGφ(R,n) are small quantities of the problem concerned.

The representations (6.47)–(6.52) of the phase and Gφ(R,n) matrices allowreformulation of the transfer equation (6.16) in the following fashion:

B{Gφ,0(R,n)

}=

σs

∫∫Z0(n,n′) Gφ,0(R,n′) L(φ − φ′) dn′

+σs

∫∫Z1(n,n′) Gφ,1(R,n′) L(φ − φ′) dn′ + δ(R − R0) δ(n − n0), (6.53)

B{Gφ,1(R,n)

}=

σs

∫∫Z0(n,n′) Gφ,1(R,n′) L(φ − φ′) dn′

+σs

∫∫Z1(n,n′) Gφ,0(R,n′) L(φ − φ′) dn′. (6.54)

Solution of Eqs (6.53) and (6.54) can be in principle accomplished using theiterative procedure in the perturbation method with the aforementioned small

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6 Remote sensing of clouds using circularly 207

quantities Z1(n,n′) and Gφ,1(R,n) (Zege and Chaikovskaya, 1996). To the firstorder of the perturbation method, the matrix Gφ(R,n) is defined as follows.The small second source of Eq. (6.53), which contains the product of small off-diagonal submatrices of the phase and Gφ(R,n) matrices, vanishes. Then thevector equations are split and reduced to the following simplified independentequations. Equations for the (11) and (44) elements are scalar (in what followswe shall omit the subscript ‘φ’ at elements of the matrix Gφ(R,n)),

B {Gii(R,n)}=σs

∫∫ai(n − n′)Gii(R,n′) dn′ + δ(R − R0) δ(n − n0), i=1, 4.

(6.55)The (1,1)-element becomes the Green’s function of the conventional scalar ra-diative transfer equation (RTE). The equation for the central 2 × 2 submatrixg(R,n) (6.29), is of the form

B {g(R,n)} =σs

∫∫dn′ d(n,n′) g(R,n′) l(φ − φ′) + e+ δ(R − R0) δ(n − n0)

(6.56)It will be referred to as the linearly polarized radiation transfer equation (LPRTE). At last,

G14(R,n) ≈ 0 and G41(R,n) ≈ 0 (6.57)

because the phase matrix elements Zik(n,n′) (ik = 14, 41) are the zeros, and thesources in the equations for these elements are the products of small off-diagonalelements of the phase matrix and the basic matrix.

The above equations govern the leading elements of the matrix Gφ(R,n).Vector equations (6.54) for the small submatrix Gφ,1(R,n) become sets of equa-tions with the sources defined through the solutions of the above equations(6.55) and (6.56). The sources that contain the negligible elements G14(R,n)and G41(R,n) (Eq. (6.57)) of the matrix Gφ,0(R,n′) are excluded from Eq.(6.54). As a result, the systems of four simultaneous equations for columns ofthe 4 × 4 matrix Gφ(R,n) are turned into sets of scalar equations and systemsof two equations.

The LP RTE (6.56) can be transformed as follows (Zege and Chaikovskaya,2000). As shown by Eqs (6.21) and (6.29), the phase matrix d(n,n′) can berepresented as sum of ‘plus’ and ‘minus’ submatrices and this property can begeneralized to the solution g(R,n). Equations (6.21) and (6.29) substituted, thetransfer equation (6.56) becomes the equation for the sum g+(R,n)+ g−(R,n).Also, with the help of the matrix ω (6.28), Eq. (6.56) can be reformulated asthe equation for the matrix ωg(R,n)ω = g+(R,n) − g−(R,n). Equation (6.56)being summed up with and then subtracted from the new equation, it takes theform

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208 L. I. Chaikovskaya

B{g+(R,n)

}=

σs

∫∫ [l(−ε(n,n′))a+(n − n′)g+(R,n′)

+ l(ε(n,−n′))a−(n − n′)e−g−(R,n′)]

dn′

+e+δ(R − R0) δ(n − n0), (6.58)

B{g−(R,n)

}=

σs

∫∫ [l(ε(−n,−n′))a+(n − n′)g−(R,n′)

+ l(−ε(−n,n′))a−(n − n′)e−g+(R,n′)]

dn′. (6.59)

Here,−ε(n,n′) = π − [(φ′ − φ) + χ′

1 + χ′2], (6.60)

ε(n,−n′) = [(π − χ′2) + χ′

1 + (π − (φ′ − φ))] − π, (6.61)

ε(−n,−n′) = [(π − χ′1) + (π − χ′

2) + (φ′ − φ)] − π, (6.62)

−ε(−n,n′) = π − [(π − χ′1) + χ′

2 + π − (φ′ − φ)] (6.63)

are the combinations of angles which result from permutations of the subma-trices g+(R,n) and g−(R,n) with the rotation matrix l(φ − φ′) according therules (6.32). Four combinations of angles that are denoted by ε(n,n′), ε(n,−n′),ε(−n,−n′) and ε(−n,n′) are the spherical excesses. All combinations of anglesin the square brackets can be observed on the sphere of unit radius as the sumsof angles of the spherical triangles, the first spherical triangle being formed bythe orts n, n′and z, the second one by orts n, −n′and z, the third one by orts−n, −n′and z and the fourth one by orts -n, n′and z (Zege and Chaikovskaya,2000: Appendix B).

One can easily verify that the equations (6.58) and (6.59) are equivalent tothe transfer equation (6.56). There is an advantage of the new form of LP RTEs.Among the ‘plus’ and ‘minus’ matrices, the first almost totally describes thepropagation of LPR in the forward region, the second prevails in the vicinity ofthe backward direction. This feature is analogous to that of the single scatteringfunctions a+(x) and a−(x). As opposed to the original Eq. (6.56), the new formof LP RTE (6.58) and (6.59) contains the ‘plus’ and ‘minus’ matrices in not onebut in different sources of the equations. This feature makes the new equationsa convenient base for approximate description of the near-forward and near-backward scattering of LP radiation.

As was noted above, in the case of the forward-elongated phase function thefunction a−(x) of single scattering, which is the zero in the forward direction,may be also considered as a small quantity. So, it is possible to treat the problemof LP radiation scattering within the perturbation method, where small param-eters of the problem are the ‘minus’ submatrices of the phase matrix and basicmatrix, i.e., generated by a−(x). Equations (6.58) and (6.59) are simplified tothe first approximation of the perturbation method. The second term in the inte-grand of Eq. (6.58) which contain the product of a−(x) and g−(R,n) is ignored.This results in an independent equation for the ‘plus’ matrix:

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6 Remote sensing of clouds using circularly 209

B{g+(R,n)

}=

σs

∫∫l(−ε(n,n′))a+(n − n′)g+(R,n′) dn′

+e+δ(R − R0)δ(n − n0). (6.64)

Equation (6.59) becomes the equation for g−(R,n) with a source predeterminedby the solution of Eq. (6.64).

A receiver may make azimuth averaging of the spatial distribution of radia-tion parameters. Matrices g+(R,n′) (6.30) and g−(R,n) (6.31), averaged overϕ, i.e., over φ , given ψ, are reduced to the scalars:⟨

g+(R,n)⟩

φ= G+(z, r, μ) e+, (6.65)⟨

g−(R,n)⟩

φ= G−(z, r, μ) e−, (6.66)

where e± = diag{1,±1}. From Eqs (6.58) and (6.59), the equations for G+(z, r, μ)and G−(z, r, μ) follow:

B {G+(z, r, μ)} =σs

2

1∫−1

[〈cos(2ε(n,n′))a+(n − n′)〉φ′−φ G+(z, r, μ′)

+ 〈cos(2ε(n,−n′))a−(n − n′)〉φ′−φ G−(z, r, μ′)] dμ′ + δ(z−z0)δ(r−r0)δ(μ−μ0),(6.67)

B {G−(z, r, μ)} =σs

2

1∫−1

[〈cos(2ε(−n,−n′))a+(n − n′)〉φ′−φ G−(z, r, μ′)

+ 〈cos(2ε(−n,n′))a−(n − n′)〉φ′−φ G+(z, r, μ′)] dμ′. (6.68)

The equation (6.64) is reduced to

B {G+(z, r, μ)} =σs

2

1∫−1

〈cos(2ε(n,n′))a+(n − n′)〉φ′−φ G+(z, r, μ′) dμ′

+δ(z − z0)δ(r − r0)δ(μ − μ0) (6.69)

The transfer equations shown above give convenient approximations of the vectortheory. They are applicable for calculations of the basic matrix in the wholeregion of scattering. The main result is that the leading quantities G11, G44 andg+ or G+(z, r, μ), which are of great practical importance, become defined bymuch simpler equations. The accuracy of these equations is high if a mediumhas a strongly forward elongated phase function (Zege and Chaikovskaya, 1996,2000).

Table 6.1 displays the accuracy provided by three approximations: (6.55),(6.67) and (6.69) in the problems of radiation transmission and reflection bya cloud C1 layer illuminated by a wide beam. The adding code (Zege et al.,1999; Tynes et al., 2001) was used for exact computations of solutions of theexact equation (6.16) and of the approximate equations above listed. It is seen

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210 L. I. Chaikovskaya

Table 6.1. Elements G11, G44, and G+ vs ϑ obtained from rigorous Eq. (6.16) (columns2, 4 and 6) and percentage error in them for using approximate Eqs (6.55) (columns3 and 5), set of equations (6.67) and (6.68) (column 7), and Eq. (6.69) (column 8) invector problems of transmission (ϑ < 90◦) and reflection (ϑ > 90◦) of infinitely widelight beam by cloud C1 (1.064 μm) of optical thickness 5

ϑ(◦) G11 δ11(%) G44 δ44(%) G+ δ+(%) δ+(%)(bj = 0) (bj = 0) (bj = 0) (bj = 0,

a− = 0)

0 116.6 0.00 116.6 0.00 116.6 0 0.0011 0.466 0.04 0.424 0.04 0.426 0.01 0.1533 0.245 0.01 0.200 0.03 0.195 0.03 0.2659 0.150 0.07 0.100 0.04 0.084 0.13 0.3578 0.102 0.10 0.058 0.11 0.041 0.26 0.2796 0.060 0.09 0.018 0.46 0.009 0.72 1.22

114 0.081 0.12 0.023 0.39 0.008 0.80 0.49136 0.088 0.06 0.022 0.07 0.009 0.26 0.02157 0.081 0.11 0.021 0.70 0.008 0.04 0.03169 0.083 0.21 0.022 0.86 0.010 0.00 0.00180 0.109 0.19 0.025 0.82 0 0.60 0.48

that relative errors of the scalar equations (6.55) for G11, G44 and of Eqs (6.67)and (6.68) formulated in the approximation bj = 0 for G+ do not exceed 1%(columns 3,5,7). The independent equation (6.69) for G+, i.e., in the assumptionthat bj = 0 and a− = 0, gives the relative error smaller than 1.3% (column 8).

Still simpler approximations of LP RTE may be formulated for two particularcases: multiple scattering within small angles and near-backward scattering (Zegeand Chaikovskaya, 2000). The spherical excesses defined by Eqs (6.60)–(6.63)provide particular dependences of the kernels of Eqs (6.58), (6.59) and (6.64)on the angles ϑ and ϑ′. Namely, in the case of ϑ � 1 and ϑ′ � 1, i.e., closedirections of orts n,n′ and z, the excess ε(n, n′) is a very small quantity as canbe seen from its determination in the spherical trigonometry:

sinε(n,n′)

2=

sin ϑ2 sin ϑ′

2

cos θ′2

sin(φ′ − φ). (6.70)

The matrix l(−ε(n, n′)) is close to the unit one. For not close directions of ortsn,n′ and z, the elements lik(−ε(n,n′)) are strongly oscillating functions of theazimuth φ−φ′. Thus, the small-angle peak of the first kernel of Eq. (6.58) is lo-cated in the region of close orts (n,n′,z). Dependences of the excesses ε(n,−n′),ε(−n,−n′) and ε(−n,n′) on orts (n,−n′,z), (−n,−n′,z) and (−n,n′,z), respec-tively, are analogous. In each of four integrals of Eqs (6.58) and (6.59), thesmallness of the excess is observed in just those regions of directions n − n′ andn′ of scattering in which the correspondent single scattering function-multipliera+(n−n′) or a−(n−n′) and matrix-multiplier g+(R,n′) or g−(R,n′) contributemost of all. This nice inclusion of rotation matrices into the transfer equation

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6 Remote sensing of clouds using circularly 211

is intimately connected with the sense of ‘plus’ and ‘minus’ matrices that theyare main matrices for the forward and backward, respectively, scattering regionswhere they transform the second and third Stokes parameters defined in paral-lel reference planes. Equations (6.58) and (6.59) are adequate to the physics ofscattering, that is, preserving the plane of vibration when scattering occurs inthe forward and backward directions.

Since the smallness of the excesses displays itself in the regions of maximalcontributions of the corresponding integrands, one may approximate the equa-tion sources by those with zero excesses. When assuming that ε(n,n′) ≈ 0 in Eq.(6.64), the solution g+(R,n′) turns into the scalar Gf

+(R,n)e+ and Eq. (6.64)is reduced to a small-angle equation

B{Gf

+(R,n)}

=σs

∫∫a+(n − n′)Gf

+(R,n′) dn′ + δ(R − R0)δ(n − n0),

(6.71)where Gf

+(R, n) stands for the forward-elongated function, limited to the for-ward region, which governs LP radiance defined as the difference between itsparallel and cross-polarized components.

If we assume that ε(−n,−n′) ≈ 0 and ε(−n,n′) ≈ 0 in Eq. (59), theng−(R,n) = Gb

−(R,n)e− and Eq. (6.59) becomes of a scalar type, its secondsource being determined by the solution of the above small-angle equation (6.71):

B{Gb

−(R,n)}

=σs

∫∫ [a+(n − n′) Gb

−(R,n′) + a−(n − n′) Gf+(R,n′)

]dn′.

(6.72)From Eqs (6.71) and (6.72), the solution of Eq. (6.72) in general form is

Gb−(R,−n;R0,n0)

σs(z)4π

∫dR′

∫dn′

∫dn′′ Gf

+(R′,n′′;R,n) a−(−n′′ − n′)

×Gf+(R′,n′;R0,n0). (6.73)

The function Gb−(R,n) determines the near-backscattering of LP radiance. This

LP radiance is mostly formed at the cost of single scattering at the small angleπ − θ and small-angle propagation and multiple scattering processes before andafter the backward single scattering event when the ‘plus’ function is highlyforward elongated.

Approximate equations (6.71) and (6.72) in contrast to the initial ones (6.64)and (6.59) take no account of the effects of polarization plane rotation that aredescribed by the rotation matrices.

6.3 Polarized light transmission by a cloud

To estimate the effectiveness of polarization parameters of lidar signals in pick-ing up information on optical and microphysical characteristics of clouds, oneshould be able to calculate these parameters in various sounding geometries withmultiple scattering involved. The calculation of angular distributions of the po-larized radiance and polarization degree of laser light transmitted by a cloud is a

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212 L. I. Chaikovskaya

task of great interest. As previously discussed by Polonsky et al. (2001) in theirconsideration of the information content of lidar signals from water clouds, theforward scattering, specifically, the diffraction part, contains most informationon the microstructure of clouds. There exist techniques to retrieve the effectiveradius of particles based on information from multiple forward scattering. Forthese retrieval techniques, use of the polarization characteristics of laser radia-tion is proposed along with the radiance (Roy et al., 1999). That is why precisedescription of the polarization of multiply scattered laser radiation in the near-forward directions becomes extremely important.

6.3.1 Generalization of the multicomponent technique

The theory for the axisymmetric problem of light transmission by a plane-parallelcloud layer of an arbitrary polarized beam falling along the normal (along theZ-axis) can be developed in the following way. The signal power measured bythe receiver with the axis being opposite to the Z-axis through the analyzercharacterized by the Stokes vectorA is defined as

W f = AT

{∫∫Jf (R,n) Φrec(R,n) dR dn

}P, (6.74)

where Jf (R,n) is the propagation matrix (6.38). Here, the reference plane forA is set parallel to that for the polarizer Stokes vector P. The receiver diagramΦrec(R,n) is axisymmetric, which means the receiver makes integration overspatial and angular azimuths. Therefore, the matrix Jf (R,n) (6.38) can bereplaced by the diagonal azimuthally averaged matrix (6.45).

Let an incident beam be linearly polarized: PL = {1, 1, 0, 0}. To find thelinear polarization degree of the transmitted radiation one uses the analyzerwith the Stokes vector AL,⊥ = 0.5{1,−1, 0, 0} and then AL,|| = 0.5{1, 1, 0, 0},and measures the cross and parallel LP power values

W fL,⊥ = AT

L,⊥

{∫∫Jf (R,n) Φrec(R,n) dR dn

}PL (6.75)

and

W fL,|| = AT

L,||

{∫∫Jf (R,n) Φrec(R,n) dR dn

}PL, (6.76)

respectively. Referring to Eq. (6.45), it is easy to get that

W fL,⊥ = 0.5

∫∫[J11(R,n) − J+(R,n)] Φrec(R,n) dR dn (6.77)

andW f

L,|| = 0.5∫∫

[J11(R,n) + J+(R,n)] Φrec(R,n) dR dn, (6.78)

where J+(R,n) and J11(R,n) stand for the functions of transformation of LPradiance and total radiance, respectively. Defining the LP degree, via Eq. (6.15),one has

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6 Remote sensing of clouds using circularly 213

pfL =

W fL,|| − W f

L,⊥W f

L,|| + W fL,⊥

=∫∫ ∫∫

G+(R,n,R0,n0)Φrec(R,n)Φsrc(R0,n0) dR dn dR0dn0∫∫ ∫∫G11(R,n,R0,n0)Φrec(R,n)Φsrc(R0,n0) dR dn dR0dn0

. (6.79)

Arguments of pfL are the registration depth and the polar angle ϑ in the case

of the ring azimuth-averaging receiver diagram. As is clear from Eq. (6.79), theLP degree received forward through the axisymmetric receiver diagram can beinterpreted with the pairs of elements (1,1) and (+) of the basic matrix. Theproblem of the determination of these elements can be solved on the basis of thescalar transfer equation (6.55) at i= 1 for G11(R,n) and the simplified equation(6.69) or (6.71) for G+(R,n) .

In many tasks of lidar sounding, approximate analytical or semi-analyticaltechniques to solve these equations appear most attractive. An approach to ob-tain semi-analytical solutions to the RTE (6.55), at i = 1, as applied to cloudsand generally media with highly forward-elongated phase functions, is describedby Polonsky et al. (2001) and Zege et al. (1993, 1995), with the multicomponentapproach (MCA) forming their basis. It provides the acceptable accuracy andhigh computation speed. Within the MCA, the phase function a1(n−n′) of theRTE is represented as a linear combination of two or more phase functions whoseangular dispersions are very different. Namely, the first phase function has anextremely small angular dispersion while the angular dispersion of the last phasefunction is the largest. In principle, all features of the phase function a1(n − n′)can be taken into account within this approach. A solution is sought as a sum ofterms where the first is the direct transmission and others are scattering com-ponents of small-angle and diffuse type. For the scattering components, a newset of equations is deduced from the initial equation. Within a semi-analyticalsolution, the peak-components are described on the basis of the small-angle orsmall-angle diffusion approximation (SAA or SADA). The relatively diffuse com-ponent can be found by solving the transfer equation formulated in the deltaapproximation with the aid of known approximate techniques (Sobolev, 1956;Ishimaru, 1978; Van de Hulst, 1980; Lenoble, 1985; Zege et al., 1991, 1993, 1995).The phase function of this equation is not very forward-elongated.

The MCA can be generalized to the problem of LP radiation transfer (Eq.(6.69) along with Eq. (6.55), at i = 1). Consider the two-component approach.Representing the phase function as

a1(n − n′) = αaf1 (n − n′) + (1 − α) ad

1(n − n′), (6.80)

one can take the polarized phase function in the form

a+(n − n′) = αaf+(n − n′) + (1 − α) ad

+(n − n′), (6.81)

where

af+(n − n′) = a+(n − n′) af

1 (n − n′) and ad+(n − n′) = a+(n − n′) ad

1(n − n′),(6.82)

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214 L. I. Chaikovskaya

with a+(n − n′) = a+(n − n′)/a1(n − n′). In (6.80) and (6.81), the first termsare the sharp peaks and the second terms are the remaining parts of the phasefunctions a1(n − n′) and a+(n − n′), respectively. The functions af

1 (n − n′) andad1(n − n′) are normalized to unity. After substitution of the phase functions

(6.80) and (6.81) into the RTE (6.55) (i= 1) and LP RTE (6.69), their solutionsare sought in the form

G11 = G011 + GSA

11 + Gd11, (6.83)

G+ = G0+ + GSA

+ + Gd+, (6.84)

where the first terms define the direct transmission and others are scatteringcomponents. Equations (6.55) (i= 1) (azimuthally averaged) and (6.69) aretransformed into two pairs of equations, where the first equations are

B{GSA

11 (z, r, μ)}

=σf

s

2

1∫−1

af1 (μ, μ′)GSA

11 (z, r, μ′) dμ′ + BSAS,11(z, r, μ), (6.85)

and

B{GSA

+ (z, r, μ)}

=σf

s

2

1∫−1

df+(μ, μ′) GSA

+ (z, r, μ′) dμ′+BSAS,+(z, r, μ), (6.86)

where df+(μ, μ′) =

⟨cos(ε(n,n′))af

+(n − n′)⟩

φ′−φ. The functions BSA

S,ν(z, r, μ),

ν = 11, (+), stand for the single scattering sources of the equations. Equations(6.85) and (6.86) govern small-angle scattering, the scattering coefficient beingequal to σf

s = ασs. Differences between the Eqs (6.55) (i= 1), (6.69) and (6.85),(6.86), respectively, are the equations for the second components of solutions.They are written in the delta approximation as follows:

B{Gd

11(z, r, μ)}

=σd

s

2

1∫−1

ad1(μ, μ′)Gd

11(z, r, μ′) dμ′ + Bd

S,11(z, r, μ), (6.87)

B{Gd

+(z, r, μ)}

=σd

s

2

1∫−1

dd+(μ, μ′) Gd

+(z, r, μ′) dμ′+BdS,+(z, r, μ), (6.88)

where dd+(μ, μ′) =

⟨cos(ε(n,n′))ad

+(n − n′)⟩

φ′−φ. The functions Bd

S,ν(z, r, μ),ν = 11,+, stand for the correspondent single-scattering sources. For theseequations, the scattering and extinction coefficients are σd

s = (1 − α)σs andσd

e = (σe − ασs) , respectively. As a result, a problem of strongly anisotropicscattering has been separated on the problem of a pure small-angle scatteringand of not strongly anisotropic scattering.

Suppose that a laser source emits a circularly polarized beam. The receiverdetects the cross and parallel CP power values of a signal in the forward re-gion. Determination of the CP degree of transmitted light in the axisymmetricgeometry is similar to that of the LP degree:

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6 Remote sensing of clouds using circularly 215

pfC =

W fC,|| − W f

C,⊥W f

C,|| + W fC,⊥

=∫∫ ∫∫

G44(R,n,R0,n0)Φrec(R,n)Φsrc(R0,n0) dR dn dR0dn0∫∫ ∫∫G11(R,n,R0,n0)Φrec(R,n)Φsrc(R0,n0) dR dn dR0dn0

. (6.89)

The difference is that the ‘plus’ function is changed to the (44) element. The pairof scalar equations (6.55) (i= 1 and 4) should be solved to find the denominatorand numerator, respectively, in Eq. (6.89). The solution can be performed on thebase of MCA similarly those of the pair of equations (6.55) (i= 1) and (6.69) ofLP light transfer.

To summarize, the theory stated in section 6.2 lays out quite a simple way tofind the LP and CP degrees of light propagated along the Z-axis and multiplyscattered in a medium, and it allows employment and generalization of MCAand other techniques created earlier for the scalar RTE.

6.3.2 Transmission of an infinitely wide beam through water cloud:computation and discussion

Energy and polarization parameters of transmitted radiation were computedby the above-described approach considering a homogeneous cloud layer illumi-nated at the upper boundary by LP and CP infinitely wide beams along thenormal. The evaluated parameters were the radiance G11(τ, ϑ) (τ is the opti-cal thickness of a cloud layer) and polarized radiances G+(τ, ϑ) and G44(τ, ϑ),as well as LP degree pf

L(τ, ϑ) = G+(τ, ϑ)/G11(τ, ϑ) and CP degree pfC(τ, ϑ) =

G44(τ, ϑ)/G11(τ, ϑ) (in Eqs (6.74), (6.79) and (6.89), Φsrc(n0) = δ(n0 − z) andΦrec(n′) = δ(ϑ′−ϑ)). Attenuation and scattering coefficients and phase functionsaν(n−n′) (ν = 1,+, 4), of the cloud C1 model at wavelength λ = 1.064μm wereset. Results of these computations and the developed computation algorithm canbe used in interpreting data of water clouds sounding with a spaceborne lidar inthe geometry of transmission.

The following computation algorithm was applied. The peak of phase functiona1(x) was cut off within the scattering angles from 0◦ to 45◦, phase functionsbeing presented as described above (Eqs (6.80) and (6.81)). The average cosineof the phase function a1(x) after the cutting was 0.446. The transfer equations(6.85), (6.87), similar ones with the phase function a4(n − n′), and equations(6.86), (6.88) after they all have been redefined for the case of an infinitely widebeam and also averaged by azimuth) were solved. The solutions of the equationswith the peak-components of the phase functions were found in the small-angleapproximation (SAA). The SAA on the base of the spherical garmonics methoddeveloped for the case of infinitely wide beam was used. Within it, solutionsGSA

ii (τ, μ), ii = 11 and 44, are series of the Legendre polynomials and functionGSA

+ (τ, μ) is a series of the generalized spherical functions Pn2,2(μ), n = 2,. . .

(Domke, 1975; Hovenier and van der Mee, 1983).To determine diffuse radiance components Gd

11(τ, μ), Gd44(τ, μ), and Gd

+(τ, μ)the approximate approach to solve the RTE described by Sobolev (1956) was

Page 233: Light Scattering Reviews 3: Light Scattering and Reflection

216 L. I. Chaikovskaya

employed (see also Kurchakov, 1960; Kagan and Yudin, 1956). It is valid fornot large values of the phase function average cosine (≤0.5). Preliminarily, thecomparison of results obtained by this approach with the numerical data byvan de Hulst (1980) for reflection and transmission of radiation by homogeneouslayers characterized by the Henyey–Greenstein phase function was made andpretty good agreement was obtained. When applying the approach to the cloudC1 model, the single-scattered part of solution Gd

11(τ, μ) was described exactly bythe known analytical expression, while its multiple-scattered part was obtainedby approximate solution of the RTE under the assumption of phase functionad1(μ, μ′) =

∑n=0,1

α1,nPn(μ)Pn(μ′). This phase function presents the first two

terms of expansion of the true phase function in the Legendre polynomials.The solution was accomplished by two analytical procedures. The first is useof the Edington approximation (Lenoble, 1985) and obtaining a solution forGd

11(τ, μ) in the diffusion approximation (DA), the second is the next iterationof the solution of the RTE with the previous DA solution substituted into thecollision integral. Analogous approaches to find polarized radiances Gd

44(τ, μ)and Gd

+(τ, μ) were developed, in which case the polarized phase functions weresimilarly given in the forms of ad

4(μ, μ′) =∑

n=0,1α4,nPn(μ)Pn(μ′) and ad

+(μ, μ′) =∑n=2,3

α+,nPn2,2(μ)Pn

2,2(μ′), respectively.

Computations of radiance G11(τ, ϑ) and polarization degrees pfL(τ, ϑ) and

pfC(τ, ϑ) are shown in Figs. 6.2 and 6.3. All obtained approximations to G11(τ, ϑ),

pfL(τ, ϑ), and pf

C(τ, ϑ) were checked. For checking, the developed approximatetechnique was compared with simulations according to the exact doubling code(Zege et al., 1999; Tynes et al., 2001). Figures 6.2 and 6.3 also show results ofthis comparison. One can see that the coincidence of the approximate techniquewith the doubling method is very good. Relative error in radiance G11(τ, ϑ) isless than 10%. Absolute errors for the LP degree do not exceed 0.018, 0.09, and0.08 and for the CP degree less than 0.016, 0.016 and 0.045 for values of cloudoptical thickness 1, 5, and 20, respectively.

Figures 6.2(b) and 6.3 demonstrate the interesting feature of a water cloudthat it depolarizes propagated linearly and circularly polarized visible radiationin approximately the same way. This appears to be caused by the fact thatthe phase functions for LP and CP light are alike in the forward hemisphere(Fig. 6.1(a)). Also, the fact is worth attention that the LP and CP degreesdecay very slowly with increasing τ . When τ is less than 10 or so, values of theLP and CP degrees are greater than 0.7 at ϑ < 5◦. This is explained by thedominance of small-angle scattering at these optical thicknesses together withthe feature that no depolarization of LP and CP light occurs at single-scatteringin the forward region (Fig. 6.1(a)).

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6 Remote sensing of clouds using circularly 217

0 20 40 60 80angle, degree

0.01

0.1

1

10

100

1000

radi

ance

1

23

(a)

0 20 40 60 80angle, degree

0

0.2

0.4

0.6

0.8

1

linea

r pol

ariz

atio

n de

gree

(b)1

2

3

Fig. 6.2. Radiance (a) and LP degree (b) of transmitted LP light by cloud layer withoptical thickness 1(1), 5(2), and 20(3) computed by developed approximate technique(signs) and by doubling code (solid curves).

Page 235: Light Scattering Reviews 3: Light Scattering and Reflection

218 L. I. Chaikovskaya

0 20 40 60 80angle, degree

0

0.2

0.4

0.6

0.8

1

circ

ular

pol

ariz

atio

n de

gree 1

2

3

Fig. 6.3. The same as in Fig. 6.2 (b), except for CP degree of transmitted CP light.

Resuming, when polarized visible light propagates through a water cloud, theso-called ‘polarization memory’ (conservation of a polarization degree) exists forboth linear and circular polarizations.

6.4 Polarization of the pulsed lidar return from a cloud

6.4.1 Semi-analytical technique

As previously studied (Polonsky et al., 2001; Zege et al., 1995), small-anglemultiple scattering and large-angle single scattering play a decisive role in theformation of the multiple-scattered part of the lidar backscatter signal powerfrom a cloud down to optical thickness of 4–5. A semi-analytical solution forthe power of a lidar return from a stratified cloud, which includes small-anglemultiple scattering and only single scattering over large angles, was developedthrough use of the backscattering technique and MCA (Zege et al., 1993, 1995).Comparison with other methods (Bissonnette et al., 1995) has shown that theaccuracy of this solution is quite sufficient. On the base of the same physicalmodel of multiple scattering (Zege et al., 1998; Zege and Chaikovskaya, 1999;Vasilkov et al., 1990), the analogous solution to the vector problem of soundingwith a polarized pulsed lidar was obtained

The backbone of the analogous solution to the polarized lidar sounding isthe simplified vector theory (section 6.2). The process of forward propagationand effects of near-backward scattering of radiation are defined by the 4 × 4matrices (6.38) and (6.39), respectively. Often, azimuthally averaged matrices(6.40), (6.41) or (6.42), (6.43) are enough to be used instead of matrices (6.38)

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6 Remote sensing of clouds using circularly 219

and (6.39). For the monostatic sounding, the solution is presented by the diagonalbackscattering matrix:

B = diag {B11, B−, −B−, B44} (6.90)

(cf. Eq. (6.46)). This matrix is expressed through power B11 of a signal andpowers B− and B44 of signal LP and CP components, respectively. The returnLP degree and CP degree due to the incident beam is defined through thesepowers as follows

pL = B−/B11, (6.91)

pC = B44/B11. (6.92)

Corresponding depolarization ratios are

δL =BL,⊥BL,‖

=B11 − B−B11 + B−

(6.93)

andδC =

BC,⊥BC,‖

=B11 − B44

B11 + B44, (6.94)

respectively.Within the model of small-angle multiple scattering and large-angle single

scattering, matrix B (6.90) of the problem of monostatic sounding with a pulsedlidar is defined by

B(t =2zυ

) =υ

2σs

∫dr0

∫dn0Φsrc(r0,n0)

∫dr∫

dnΦrecsrc(r,n)

×∫

dr′∫

dn′′∫

dn′ Gf (z, r′,n′′; 0, r,n)F (−n′′,n′)

4πGf (z, r′,n′; 0, r0,n0),

(6.95)

where t = 2z/υ is the time, z, the sounding depth, υ, the light velocity in amedium, function Φrec

src(r,n) stands for Φrec(r,−n), the axisymmetric spatial-angular pattern of sensitivity of a receiver, and F (−n′′,n′) is the single near-backscattering matrix. Propagation matrix Gf and single near-backscatteringmatrix F of Eq. (6.95) can be approximated by the diagonal matrices (theirdiagonal forms are like those shown in Eqs (6.45) and (6.46)), whose elements arethe leading values in the problems of near-forward and near-backward scattering.Thus, return power B11 and its polarized components B− and B44 can be foundindependently through the similar integrals of the products of three functions(Vasilkov et al., 1990; Zege and Chaikovskaya, 1999):

Bii

(t =

2zυ

)=

υ

2σs

∫dr0

∫dn0 Φsrc(r0,n0)

∫dr∫

dnΦrecsrc(r,n)

×∫

dr′∫

dn′′∫

dn′ Gfii(z, r

′,n′′; 0, r,n)Fii(−n′′,n′)

4πGf

ii(z, r′,n′; 0, r0,n0),

i = 1, 2, 4, (6.96)

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220 L. I. Chaikovskaya

where Fii(−n′′,n′) is single scattering matrix element a1[(−n′′) − n′], at i = 1,element a−[(−n′′)−n′], at i = 2, and a4[(−n′′)−n′], at i = 4 (see, for example,these elements for Cloud C1 in Fig. 6.4), B22 stands for B− and Gf

22 stands forGf

+. The temporal spread of a ”forward” signal is neglected in backscatteringmatrix definition (6.96). The integrand of Eq. (6.96) at i = 2 presents solutionGb

−(0, r,−n; 0, r0,n0) (6.73) of the LP radiation transfer equation.The solution (6.96) for Bii(t) can be simplified by use of the scalar backscat-

tering technique earlier applied to power B11(t) (for example, Katsev et al.,1997). It reduces the multidimensional integrals (6.96) to the much simpler in-tegrals (Vasilkov et al., 1990; Zege and Chaikovskaya, 1999):

Bii(t =2zυ

) =υσs

2

∫dn⊥ Jeff

ii (z, r = 0,n⊥)Fii(n⊥)

4π, i = 1, 2, 4, (6.97)

where

Jeffii (z, r = 0,n⊥) =

∫dr0

∫dn⊥0Φ

effsrc (r0,n⊥0)Geff

ii (z, r = 0,n⊥; 0; r0,n⊥0)

(6.98)is the angular distribution of the (ii)-component of radiance at the beam axis atdepth z in an effective medium with double extinction and scattering coefficientsilluminated by an effective source whose pattern is defined through the patternsof the real source and receiver as Φeff

src (r0,n⊥0) =∫

dr′ ∫ Φsrc(r′,n′⊥)Φrec(r0 +r′,n⊥0 + n′⊥) dn′⊥.

The approximate vector theory (section 6.2) offers three small-angle equa-tions to determine diagonal matrix Geff

ii (z, r = 0,n⊥; 0; r0,n⊥0), among whichthe first and the third (for i = 1 and 4) are defined by Eqs (6.55), i.e., they arescalar equations for radiance and CP radiance, and the second one (for i = 2) isthe similar equation of LP radiation transfer (6.71). These three scalar equationscan be solved in the small-angle approximation (SAA).

6.4.2 Backscattering of linearly and circularly polarized pulses froma water cloud: computation and discussion

Now, let us turn to the calculations (Figs. 6.5 and 6.6) by the above MCA-SAAsemi-analytical technique for the water cloud C1 model. Computed depolariza-tion ratios (Eqs(6.93) and (6.94)) of LP and CP lidar returns from cloud C1at λ = 0.7 μm versus sounding depth τ = σeυt/2 for lidar-cloud distance 2km and receiver field of view (FOV) (2γrec) equal to 3/ and 6/ are presentedin Fig. 6.5. The plots for the linear depolarization ratio of returns give evi-dence of the efficiency of the simplified approach. They show that data ob-tained from this approach are in agreement with the results from the MonteCarlo simulation (Zuev et al., 1976). Figure 6.5 also shows that a CP backscat-ter pulse is characterized by a more rapid growth of the depolarization ratiowith the sounding depth than a LP one. This feature is explained by distinc-tion in the single near-backscattering of LP and CP visible light from water

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6 Remote sensing of clouds using circularly 221

140 160 180angle, degree

-0.7-0.6-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.50.60.7

mat

rix e

lem

ents

(11)(-)(44)

Fig. 6.4. Angular patterns of Cloud C1 (1.064 μm) SSM elements in backward scat-tering region.

0 1 2 3 4 5Optical depth

0

0.2

0.4

0.6

0.8

1

Dep

olar

izat

ion

Fig. 6.5. Linear and circular depolarization ratios, δL(τ) (solid and 0.1-inch dashedlines) and δC(τ) (0.2-inch dashed and dash-doted lines), for receiver FOVs 3′ and6′, respectively, computed via Eqs. (6.93), (6.94), and (6.97). Linear depolarization iscompared with Monte Carlo data (Zuev et al., 1976) (symbols). Lidar-to-cloud distance2 km, source FOV 40′′ and extinction coefficient 0.025 m−1.

Page 239: Light Scattering Reviews 3: Light Scattering and Reflection

222 L. I. Chaikovskaya

0 1 2 3 4 5 6Optical depth

0

0.2

0.4

0.6

0.8

1

LP d

egre

e

(a)

0 1 2 3 4 5 6Optical depth

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

CP

degr

ee

(b)

Fig. 6.6. Polarization degree of linearly (a) and circularly (b) polarized lidar returnsas a function of sounding optical depth computed via Eqs (6.91), (6.92), and (6.97).Lidar-to-cloud distance H is 2 km and receiver FOV 6′(×), 18′(◦), 26′() and 36′(�).Dashed lines, H is 1000 km.

drops: gradient (|a44(π)| − |a44(π − θ)|) / |a44(π)| is somewhat larger than gra-dient [a−(π) − a−(π − θ)] /a−(π) in the vicinity of point θ = π (Fig. 6.4).

The lidar effective footprint being large, dependences of the polarization de-grees of LP and CP backscatter signals on sounding depth are qualitativelydistinctive. Figures 6.6 (a) and (b) show this. Here, the calculated values of po-larization degrees pL (6.91) and pC (6.92), respectively, as functions of opticalsounding depth at receiver FOVs equal to 6′, 18′, 26′ and 36′ for cloud C1 atλ = 1.604 μm are presented. When calculating pC , the CP degree of the inci-dent beam was taken equal to −1. Then, the CP degree of a backscatter signalhas positive sign. It can be seen from data comparison of Fig. 6.6 (a) and (b),that the magnitude of CP degree is less than the linear one. The greater is thevalue of FOV, the less is the CP degree. When the FOV is 6′, the dependencepC(τ) is of constant sign in the depth range considered. With growing FOV,value pC(τ) decreases more quickly with increasing sounding depth and has zerovalue at some depth (at certain time moment). The mentioned peculiarities areclosely related to features of the single near-backscattering, namely, features ofthe angular pattern of the single near-backscattering function for CP radiationF44(π−θ), which are seen in Fig. 6.4. Due to the glory effect, function F44(π−θ)abruptly changes from value F44 = −F11 at θ = π to zero and then changes itssign. The dashed lines in Fig. 6.6 present the polarization degrees of returnsfor spaceborne lidars. In this case, depth-dependence of the polarization degreeis only scarcely affected by the FOV. Dependence of the zero-point of the CPdegree on the FOV is little if any.

Thus, for LP and CP backscatter pulses from a water cloud, it is characteristicthat the circular depolarization grows faster in magnitude with sounding depththan does the linear depolarization. In the case of a large lidar effective footprint,the dependence of the CP degree on sounding depth is characterized by a zeropoint. This feature, which is due to the glory, can be used for cloud remotesensing.

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6.5 Conclusion

From the review presented, for a number of tasks of practical use in the complexproblem of polarized lidar sounding of clouds, fast semi-analytical techniques canbe developed. The complexity of the problem stems from the necessity to solvevector transfer equations involving strongly forward extended phase functions.In many cases, the inclusion of multiple scattering up to very large scatteringorders is needed. The simplified vector theory is a convenient basis for the semi-analytical solutions that account for multiple scattering of arbitrary polarizedradiation due to either localized or wide directional sources. In cloud applica-tions, the approximate VRTEs are characterized by very good accuracy. Note,the simplified vector equations were offered in the 1980s (Zege and Chaikovskaya,1985; Chaikovskaya, 1991) They were discussed at length in two papers (Zegeand Chaikovskaya, 1996, 2000) where they also received conceptualization asthe equations derived to the first approximation of the perturbation method bysmall quantities of the vector problem.

On the basis of the simplified vector transfer equations, semi-analytical tech-niques have been developed for computing the propagation and backscatteringof polarized laser beams with reasonable accuracy (sections 6.3 and 6.4; see alsoZege et al., 1998; Zege and Chaikovskaya, 1999). It should be mentioned thatclose general approaches in the area of polarized lidar sounding of multiply scat-tering media, i.e., approaches based on the use of simplified transfer equationsand approximate scalar techniques, have been developed by Vasilkov et al. (1990)and by Gorodnichev et al. (2006). The semi-analytical solutions presented in sec-tions 6.3 and 6.4 provide tools with which we can analyze the polarization prop-erties of signals from clouds produced by both LP and CP lidars for varying cloudoptical and microphysical characteristics. They may be employed for the creationof inversion techniques. At present, approximate semi-analytical techniques forother geometries of polarized laser sounding of clouds based on the approximatevector theory are under development (Chaikovskaya and Zege, 2005).

Acknowledgment

The author is thankful to Eleonora P. Zege for important discussions related tothis work. Part of this work is supported by INTAS (05-1000008-8024).

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7 LIDORT and VLIDORT: Linearizedpseudo-spherical scalar and vector discreteordinate radiative transfer models for use inremote sensing retrieval problems

Robert Spurr

7.1 Introduction

The modern treatment of the radiative transfer equation (RTE) in plane-parallelmedia dates back to the pioneering work by Ambartsumian and Chandrasekharin the 1940s (Chandrasekhar, 1960; Ambartsumian, 1961). Using a formulationin terms of the Stokes vector for polarized light, Chandrasekhar was able to solvecompletely the polarization problem for an atmosphere with Rayleigh scattering,and benchmark calculations from the 1950s are still appropriate today (Coulsonet al., 1960). The scalar (intensity-only) and vector (with polarization) radiativetransfer equations in one vertical dimension may be solved in a number of ways.These include the doubling–adding method, the discrete ordinates approach, thesuccessive orders of scattering method, Gauss–Seidel iteration, and (not least)the Monte Carlo approach. For a review of solution methods, see for example(Lenoble, 1985). Most solution methods for scalar and vector RTEs divide intotwo camps: the doubling/adding approach and the discrete ordinate method. Fordescriptions of the former, see for example (Hansen and Travis, 1974; de Haan etal., 1987; Hovenier et al., 2004). The well-known scalar DISORT discrete ordinatemodel was developed in the 1980s and released for general use in plane-parallelmulti-layer multiple scattering media (Stamnes et al., 1988a); this was extendedto the vector model VDISORT in the 1990s (Schulz and Stamnes, 2000).

The single-layer plane-parallel ‘slab problem’ has been used to provide bench-marks for scalar and vector radiative transfer results. Siewert and co-workersdeveloped complete solutions for the slab problem with scattering by sphericalparticles, using the spherical harmonics method (Garcia and Siewert, 1986) andthe FN method (Garcia and Siewert, 1989). These benchmarks have been verifiedindependently using a doubling–adding model (Wauben and Hovenier, 1992). Intwo papers appearing in 2000, Siewert revisited the slab problem from a discreteordinate viewpoint, and derived new solutions for the scalar (Siewert, 2000a)and vector (Siewert, 2000b) radiative transfer equations. These solutions usedGreen’s functions for the generation of particular solutions for the beam scat-tering source (Barichello et al., 2000). For the vector problem, Siewert’s analysisshowed that complex eigensolutions for the homogeneous RT discrete ordinate

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equations must be considered. See also Mishchenko et al. (2006) for a detailed re-view of slab problem results, and Rozanov and Kokhanovsky (2006) for selectedapplications.

In this work, we review the scalar LIDORT (LInearized Discrete OrdinateRadiative Transfer) and vector VLIDORT radiative transfer codes developed bythe present author and co-workers over the last few years (Spurr et al., 2001;Spurr, 2002, 2006). These models are based on Siewert’s discrete ordinate up-dates of the slab problem.

In the last decade, there has been increasing recognition of the need for scat-tering RT models to generate fields of analytic radiance derivatives (Jacobians)with respect to atmospheric and surface variables, in addition to the radiancesthemselves. Such ‘linearized’ models are extremely useful in classic inverse prob-lem retrievals involving iterative least-squares minimization (with and withoutregularization) (Rodgers, 2000). At each iteration step, the simulated radiationfield is expanded in a Taylor series about the given state of the atmosphere–surface system. Only the linear term in this expansion is retained, and this re-quires partial derivatives of the simulated radiance with respect to atmosphericand surface parameters that make up the state vector of retrieval elements andthe vector of assumed model parameters that are not retrieved but are sourcesof error in the retrieval.

Although weighting functions can be determined for these applications byfinite difference estimation using repeated calls to the RT model, this processis time-consuming and computationally inefficient. With a linearized RT code,one call is sufficient to return both the simulated radiance field and all relevantJacobians, the latter determined analytically. Aside from the operational gener-ation of weighting functions for different types of remote sensing applications,the linearization facility is tremendously useful for sensitivity studies and errorbudget analyses.

Analytic Jacobians have been a feature of infrared transmittance forwardmodels for many years. Such models are based on Beer’s law of extinction inthe absence of scattering, and the differentiation of exponential attenuations isstraightforward and fast. With the advent of remote sensing atmospheric chem-istry instruments such as GOME (launched April 1995) (ESA, 1995), SCIA-MACHY (launched March 2002) (Bovensmann et al., 1999), GOME-2 (October2006) (Callies et al., 2000) and OMI (July 2004) (Levelt et al., 2006) measuringat moderately high spectral resolution in the visible and ultraviolet, it is neces-sary to use multiple scattering radiative transfer models. Indeed, the retrieval ofozone profiles from GOME measurements (Liu et al., 2005; Voors et al., 2001;Hasekamp and Landgraf, 2001; Munro et al., 1999; Hoogen et al., 1999) hasprovided an important impetus for linearization of radiative transfer multiplescatter models in multi-layer atmospheres.

A number of linearized atmospheric RT scatter models have appeared inrecent years. Initial developments were made for scalar models. One approachto linearization is based on adjoint radiative transfer theory; see for exampleUstinov (2001) and references therein. Adjoint methods were used to developweighting functions for a Gauss–Seidel RT model for ozone profile retrieval

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7 LIDORT and VLIDORT 231

(Landgraf et al., 2001). In other models, Jacobians are derived by perturba-tion analysis (Rozanov et al., 1998; Spurr et al., 2001). The original LIDORTcode (Spurr et al., 2001) generated weighting functions and radiances for the top-of-atmosphere reflectance scenario for a plane-parallel multi-layer atmosphere.In subsequent work, the LIDORT linearization was based on analytic differen-tiation of the scalar discrete ordinate RT theory, and updated to incorporatethe use of Green’s function solution methods (Spurr, 2002). Later models weregeneralized to include a pseudo-spherical treatment of solar beam attenuation,output at arbitrary optical thickness and viewing geometry, and the deploymentof exact single scatter corrections for wide-angle off-nadir viewing in a curved at-mosphere (Spurr, 2003). There is also a detailed treatment of surface parameterJacobians based on BRDF analysis (Spurr, 2004).

The use of scalar radiative transfer (neglecting polarization) can lead to con-siderable errors for modeling backscatter spectra in the UV (Mishchenko et al.,1994; Lacis et al., 1998; Sromovsky, 2005). Studies with atmospheric chemistryinstruments such as GOME, SCIAMACHY and OMI have shown that the treat-ment of polarization is critical for the successful retrieval of ozone profiles fromUV backscatter (Schutgens and Stammes, 2003; Hasekamp et al., 2002). The roleof polarization has been investigated for retrieval scenarios involving importantbackscatter regions such as the oxygen A-band (Stam et al., 1999; Jiang et al.,2003; Natraj et al., 2007). It has also been demonstrated that the use of passivesensing instruments with polarization capabilities can greatly enhance retrievalsof aerosol information in the atmosphere (Mishchenko and Travis, 1997; Deuzeet al., 2000; Hasekamp and Landgraf, 2005b). A number of linearized vector RTmodels have now been developed; these include the LIRA code (Hasekamp andLandgraf, 2002), and VLIDORT (Spurr, 2006).

The LIDORT weighting function methods have also been applied to an end-to-end linearization of the adding method in the Radiant model (Spurr andChristi, 2006), and to a second order of scattering polarized model based on theinvariant imbedding method (Natraj and Spurr, 2007). The LIDORT lineariza-tion techniques have been applied to the CAO-DISORT coupled atmospheric–ocean discrete ordinate code, and it is now possible to generate weighting func-tions with respect to marine constituents such as chlorophyll concentration anddissolved organic matter (Spurr et al., 2007). This has opened the way for a newapproach to simultaneous retrieval of atmospheric and ocean quantities fromSeaWiFS and related instruments.

A description of the LIDORT and VLIDORT models is given in section 7.2.Detailed descriptions of discrete ordinate theory may be found in the literature,for example (Thomas and Stamnes, 1999), and here we will focus on the lineariza-tion of this theory. We start with the underlying radiative transfer equations andgive the linearization definitions. The following four sections deal with the ho-mogeneous vector RTE and its solution, the development of particular integralsfor solar beam scattering, the Green’s function approach for beam scattering andthermal emission sources in the scalar LIDORT model, and lastly, the bound-ary value problem and the post-processing options to deliver fields at arbitraryviewing geometry. Additional sections deal with the BRDF for lower boundary

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232 Robert Spurr

surfaces, the pseudo-spherical approximation, and the use of exact single scattercalculations.

Section 7.3 deals with performance and benchmarking. First, we look at per-formance considerations, including the delta-M approximation and other per-formance enhancements including the ‘solution saving’ and ‘BVP telescoping’options; the latter are labor-saving devices designed to speed up performancethrough the elimination of unnecessary computation. In this section, we alsoreview the Fourier convergence aspects pertaining to the exact treatments ofsingle scattering and direct beam contributions. The second section deals withthe benchmarking of VLIDORT against established results from the literature.

As with most 1-D RT codes, the LIDORT and VLIDORT models are basedon stratification of optically uniform layers. The codes require total layer Inher-ent Optical Property (IOP) inputs; the codes do not distinguish individual tracegas absorbers or particulate scatterers. In section 7.4, we describe the prepa-ration of IOPs, again focusing on the derivation of linearized optical propertyand surface parameter inputs that are necessary requirements for the codes togenerate Jacobians with respect to atmospheric and/or surface properties.

7.2 Description of VLIDORT and LIDORT

7.2.1 Theoretical framework

7.2.1.1 The vector RTE

The atmosphere is divided into separate optically uniform layers in order toresolve variation of inherent optical properties with altitude. The number oflayers L1 is large enough so that the dependence of the single-scattering albedoand scattering phase matrix on optical depth is properly resolved. In this chapterwe use the partial-layer optical thickness x (as measured from the layer upperboundary) as the vertical coordinate. This is related to the cumulative opticaldepth τ as follows. If a point P in layer n has optical thickness x and optical depthτ , then these coordinates are related through the expression τ = x +

∑n−1k=1 Δk,

where Δk (k = 1, . . . , L1) are the whole layer optical thickness values.In this work, we restrict ourselves to scattering for a medium that is ‘macro-

scopically isotropic and symmetric’, with scattering for ensembles of randomlyoriented particles having at least one plane of symmetry (Mishchenko, 2002;Mishchenko et al., 2006).

We start with the basic 1-D vector RTE for plane-parallel scattering in asingle layer:

μ∂

∂xI(x, μ, φ) = I(x, μ, φ) − J(x, μ, φ). (7.1)

The four-vector I is the diffuse field of Stokes components (I, Q, U , V ), withI the total intensity, Q and U describing linearly polarized radiation, and V

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7 LIDORT and VLIDORT 233

characterizing circularly polarized radiation (Chandrasekhar, 1960). The degreeof polarization P of the radiation is:

P = I−1√

Q2 + U2 + V 2 . (7.2)

The vector source term J(x, μ, φ) has the form:

J(x, μ, φ) =ω(x)4π

1∫−1

2π∫0

Π(x, μ, μ′, φ − φ′) I(x, μ′, φ′) dφ′ dμ′ + Q(x, μ, φ). (7.3)

In equations (7.1) and (7.3), φ is the azimuth angle and μ the cosine of thezenith angle; our convention is that μ < 0 for downwelling radiation, and μ >0 for upwelling directions. Also, ω is the single scattering albedo and Π thephase matrix for scattering; in our formulation, these do not depend on theoptical thickness x, and we henceforth drop this dependence. For scattering ofthe attenuated solar beam, the inhomogeneous source term Q(x,μ,φ) is written:

Q(x, μ, φ) =ω(x)4π

Π(x, μ,−μ0, φ − φ0)I0Ta exp[−λx]. (7.4)

Here, μ0 is the cosine of the solar zenith angle; φ0 is the solar azimuth angleand I0 the Stokes vector of the incoming solar beam before attenuation. Inthis chapter, we consider an atmosphere illuminated by natural (unpolarized)sunlight, so that the solar irradiance at TOA is given by Stokes vector I0 ={I0,0,0,0}.

In Eq. (7.4), we are using the pseudo-spherical (P-S) parameterization ofsolar beam attenuation, in which Ta is the transmittance to the top of the layer,and λ is a geometrical factor (the ‘average secant’). In the P-S formulation, allscattering takes place in a plane-parallel medium, but the solar beam attenuationis treated for a curved atmosphere. For plane-parallel beam attenuation, wehave λ = 1/μ0. Details on the pseudo-spherical formulation can be found insubsection 7.2.7.

Matrix Π relates scattering and incident Stokes vectors defined with respectto the meridian plane. The equivalent matrix for Stokes vectors with respect tothe scattering plane is the scattering matrix F. For the type of scattering mediaassumed here, F depends only on the scattering angle Θ between scattered andincident beams (Mishchenko et al., 2006). Matrix Π is related to F(Θ) throughapplication of two rotation matrices L(π − σ2) and L(−σ1) (for definitions ofthese matrices and the angles of rotation σ1 and σ2, see Chandrasekhar (1960)):

Π(μ, φ, μ′, φ′) = L(π − σ2)F(Θ)L(−σ1); (7.5)

cos Θ = μμ′ +√

1 − μ2√

1 − μ′2 cos(φ − φ′). (7.6)

In our case, F(Θ) has the well-known form:

F(Θ) =

⎛⎜⎜⎝a1(Θ) b1(Θ) 0 0b1(Θ) a2(Θ) 0 00 0 a3(Θ) b2(Θ)0 0 −b2(Θ) a4(Θ)

⎞⎟⎟⎠ . (7.7)

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234 Robert Spurr

The upper left entry in this matrix is the phase function and satisfies the nor-malization condition:

12

π∫0

a1(Θ) sin Θ dΘ = 1. (7.8)

7.2.1.2 Azimuthal separation

For the special form of F in Eq. (7.7), the dependence on scattering angle allowsus to develop expansions of the six independent scattering functions in terms ofa set of generalized spherical functions P l

mn(cos Θ) (Mishchenko et al., 2006):

a1(Θ) =LM∑l=0

βlPl00(cos Θ); (7.9)

a2(Θ) + a3(Θ) =LM∑l=0

(αl + ζl)P l2,2(cos Θ); (7.10)

a2(Θ) − a3(Θ) =LM∑l=0

(αl − ζl)P l2,−2(cos Θ); (7.11)

a4(Θ) =LM∑l=0

δlPl00(cos Θ); (7.12)

b1(Θ) =LM∑l=0

γlPl02(cos Θ); (7.13)

b2(Θ) = −LM∑l=0

εlPl02(cos Θ). (7.14)

The sets of six ‘Greek constants’ {αl, βl, γl, δl, εl, ζl} must be specified for eachmoment l in these spherical-function expansions. The number of terms LM de-pends on the level of numerical accuracy. Values {βl} are the phase functionLegendre expansion coefficients as used in the scalar RTE. These ‘Greek con-stants’ specify the polarized-light single-scattering law, and there are a numberof efficient analytical techniques for their computation, not only for sphericalparticles (see, for example, de Rooij and van der Stap (1984)) but also for ran-domly oriented homogeneous and inhomogeneous non-spherical particles andaggregated scatterers (Hovenier et al., 2004; Mackowski and Mishchenko, 1996;Mishchenko and Travis, 1998).

With this representation Eqs (7.9) to (7.14), one can then develop a Fourierdecomposition of Π to separate the azimuthal dependence (cosine and sine seriesin the relative azimuth φ − φ0). The same separation is applied to the Stokesvector itself. In the 1980s, Siewert and co-workers reformulated the azimuthalseparation of the scattering matrix in a convenient analytic manner (Siewert,

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7 LIDORT and VLIDORT 235

1982; Vestrucci and Siewert, 1984), and most vector radiative transfer modelsnow follow this work. We have:

I(x, μ, φ) =12

LM∑l=m

(2 − δm,0)Φm(φ − φ0)Im(x, μ); (7.15)

Φm(φ) = diag{cosmφ, cosmφ, sinmφ, sinmφ}. (7.16)

The phase matrix decomposition is:

Π(μ, φ, μ′, φ′) =12

LM∑l=m

(2 − δm,0)[Cm(μ, μ′) cosm(φ − φ′)

+ Sm(μ, μ′) sinm(φ − φ′)]; (7.17)

Cm(μ, μ′) = Am(μ, μ′) + DAm(μ, μ′)D; (7.18)

Sm(μ, μ′) = Am(μ, μ′)D − DAm(μ, μ′); (7.19)

Am(μ, μ′) =LM∑l=m

Pml (μ)BlPm

l (μ′); (7.20)

D = diag{1, 1,−1,−1}. (7.21)

This yields the following RTE for the mth Fourier component:

μdIm(x, μ)

dx+ Im(x, μ) =

ω

2

LM∑l=m

Pml (μ)Bl

1∫−1

Pml (μ′)Im(x, μ′) dμ′ + Qm(x, μ).

(7.22)

Qm(x, μ) =ω

2

LM∑l=m

Pml (μ)BlPm

l (−μ0)I0Ta e−λx. (7.23)

The phase matrix expansion is expressed through the two matrices:

Bl =

⎛⎜⎜⎝βl γl 0 0γl αl 0 00 0 ςl −εl

0 0 εl δl

⎞⎟⎟⎠ ; (7.24)

Pml (μ) =

⎛⎜⎜⎝Pm

l (μ) 0 0 00 Rm

l (μ) −Tml (μ) 0

0 −Tml (μ) Rm

l (μ) 00 0 0 Pm

l (μ)

⎞⎟⎟⎠ . (7.25)

The ‘Greek matrices’ Bl for 0 ≤ l ≤ LM contain the sets of expansion coefficientsthat define the scattering law. The Pm

l (μ) matrices contain entries of normalizedLegendre functions Pm

l (μ) and functions Rml (μ) and Tm

l (μ) which are relatedto P l

mn(μ) (for details, see, for example, Siewert (2000b)). In the scalar case, weneed only the (1,1) entries in Eqs (7.24) and (7.25).

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236 Robert Spurr

The LIDORT scalar model includes atmospheric thermal emission sources.This formalism is based on Kirchhoff’s law for volume emittance (Thomas andStamnes, 1999); emission is isotropic, and it is only necessary to specify the blackbody Planck function η(x) as a function of optical depth. The appropriate RTEfor the Fourier m = 0 azimuth-independent component is:

μdI(x, μ)

dx+ I(x, μ) =

ω

2

LM∑l=0

Pl(μ)βl

1∫−1

Pl(μ′)I(x, μ′) dμ′ + (1 − ω)η(x) . (7.26)

For a linear regime, in which the Planck functions Hn are specified at the levelboundaries n = 0, 1, 2, . . . , L1 (L1 being the total number of layers) we have:

ηn(x) = Hn−1 + xMn = Hn−1 +x

Δn(Hn − Hn−1). (7.27)

This is the same parameterization used in DISORT (Stamnes et al., 1988a). Inaddition to the linear regime, LIDORT also has quadratic parameterization.

For solutions to Eqs (7.22) and (7.26) in a given layer n, it is only necessary tospecify the layer total optical thickness values Δn, the layer total single scatteralbedo ωn, and the layer 4 × 4 matrices Bnl of expansion coefficients (l beingthe moment number) for the total scattering law; these are the inherent opticalproperties (IOPs). In the scalar case, we have just the phase function expansioncoefficients βnl.

7.2.1.3 Boundary conditions

To complete the calculation of the radiation field in a stratified multilayermedium, we have the following boundary conditions (the Fourier index m hasbeen omitted here):

(I) No diffuse downwelling radiation at top-of-atmosphere (TOA):

I+n (0, μ, φ) = 0. (n = 1) (7.28)

(II) Continuity of upwelling and downwelling radiation fields at intermediateboundaries. If L1 is the number of layers in the medium, then:

I±n−1(Δn−1) = I±

n (0). (n = 2, . . . , L1) (7.29)

(III) A surface reflection condition relating the upwelling and downwelling ra-diation fields at the bottom of the atmosphere:

I−n (Δn, μ, φ) = R(μ, φ;μ′, φ′)I+

n (Δn, μ′, φ′). (n = L1) (7.30)

Here, reflection matrix R relates incident and reflected directions.

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7 LIDORT and VLIDORT 237

The convention adopted here is to use a ‘+’ suffix for downwelling solutions, anda ‘−’ suffix for upwelling radiation. Conditions (I) and (II) are obeyed by allFourier components in the azimuthal series. For condition (III), it is necessaryto construct a Fourier decomposition of the BRDF operator R to separate theazimuth dependence; we discuss this issue in subsection 7.2.8.1. The Lambertiancase (isotropic reflectance) only applies for Fourier component m = 0 and Eq.(30) then becomes (Siewert, 2000b):

I−n (Δn, μ)=2δm,0R0E1

[μ0I0Tn−1 exp (−λnΔn) +

1∫0

I+n (Δn, μ′)μ′ dμ′

]. (7.31)

Here, R0 is the Lambertian albedo, E1 = diag{1, 0, 0, 0}, and Tn−1 exp (−λnΔn)is the whole-atmosphere slant path attenuation for the solar beam in the P-Sapproximation.

7.2.1.4 Jacobian definitions

In this chapter, I shall define atmospheric profile Jacobians (weighting functions)to be normalized analytic derivatives of the Stokes vector with respect to anyatmospheric property ξ defined in layer n:

Kξ(x, μ, φ) = ξ∂I(x, μ, φ)

∂ξ. (7.32)

The Fourier series azimuth dependence is also valid:

Kξ(x, μ, φ) =12

LM∑l=m

(2 − δm,0)Φm(φ − φ0)Kmξ (x, μ). (7.33)

Here and in the sequel, we use the linearization notation:

Lp(yn) = ξp∂yn

∂ξp(7.34)

to indicate the normalized derivative of yn in layer n with respect to variable ξp

in layer p.IOPs are {Δn, ωn, Bnl} for each layer n. For Jacobians, we require an addi-

tional set of linearized optical property inputs {Vn,Un, Znl} defined with respectto variable ξn in layer n for which we require weighting functions. These are:

Vn ≡ Ln(Δn); Un ≡ Ln(ωn); Znl ≡ Ln(Bnl). (7.35)

In subsection 7.4.1 we give an example of input sets {Δn, ωn,Bnl} and theirlinearizations {Vn,Un,Znl} for a typical atmospheric scenario with molecularand aerosol scattering. One can also define weighting functions with respect tothe IOPs themselves: for example, if ξn = Δn, then Vn = Δn.

For surface weighting functions, we need to know how the BRDF matrixoperator R in Eq. (7.30) is parameterized. In subsection 7.2.5, we confine ourattention to the Lambertian case, and discuss the BRDF implementation laterin subsection 7.2.8.

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238 Robert Spurr

7.2.1.5 Solution strategy

The solution strategy has two stages. First, for each layer, we establish discreteordinate solutions to the homogeneous RTE and linearize them. Then we con-sider particular integrals for the solar source term, contrasting the traditionalsubstitution method with the Green’s function approach; all these solutions arealso linearized. Second, for the whole multi-layer atmosphere, we apply boundaryconditions to solve the boundary value problem (BVP) and obtain the completediscrete ordinate field. Then we use this field in the original layer RTEs andapply source function integration in order to establish solutions away from dis-crete ordinate directions. The second stage is also completely differentiable withrespect to atmospheric and surface variables.

The Radiant scalar model (Christi and Stephens, 2004) uses a hybrid of thediscrete ordinate and adding methods. Radiant and LIDORT both develop layerRTE solutions by discrete ordinates; for this stage, the Radiant linearization isbased on the LIDORT approach. Radiant uses the adding method to developthe complete radiation field instead of the BVP approach used in LIDORT. Inthis regard, linearization of global reflection and transmission matrices requiresa different approach (Spurr and Christi, 2006).

In the following subsections, we suppress the Fourier index m unless notedexplicitly, and wavelength dependence is implicit throughout. We sometimessuppress the layer index n in the interests of clarity. For matrix notation, ordinary4×1 vectors and 4×4 matrices are written in bold typeface, while 4N ×1 vectorsand 4N × 4N matrices are written in bold typeface with a tilde symbol (N isthe number of discrete ordinate directions in the half-space).

7.2.2 Homogeneous RTE solutions and their linearization

7.2.2.1 Homogeneous RTE and eigenproblem reduction

We solve Eq. (7.22) without the solar source term. For each Fourier term m, themultiple scatter integral over the upper and lower polar direction half-spaces isapproximated by a double Gaussian quadrature scheme, with stream directions{±μi} and Gauss–Legendre weights {wi} for i = 1, . . . , N . The resulting vectorRTE for Fourier component m is then:

±μidI±

i (x)dx

± I±i (x)

=ωn

2

LM∑l=m

Pml (±μi)Bl

N∑j=1

wj

{Pm

l (μj)I+j (x) + Pm

l (−μj)I−j (x)

}. (7.36)

Eqs (7.36) contain 8N coupled first-order linear differential equations for I±i (x).

These are solved by eigenvalue methods, using the ansatz :

I±α (x,±μi) = Wα(±μi) exp[−kαx]. (7.37)

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7 LIDORT and VLIDORT 239

We define the (4N × 1) vector (superscript ‘T’ denotes matrix transpose):

W±α =

[WT

α(±μ1),WTα(±μ2), . . . ,WT

α(±μN )]T

. (7.38)

Eqs (7.36) are decoupled using Xα = W+α +W−

α and Yα = W+α −W−

α (sum anddifference vectors), and the order of the system can then be reduced from 8N to4N . This gives an eigenproblem for the collection of separation constants {kα}and associated solution 4N -vectors {Xα}, where α = 1, . . . , 4N . The eigenmatrixΓ is constructed from optical property inputs ω and Bl and products of thematrices Pm

l (μj). The eigenproblem is (Siewert, 2000b):

X⊥α Γ = k2

αX⊥α ; ΓXα = k2

αXα; (7.39)

Γ = S+S−; (7.40)

S± =

[E − ω

2

LM∑l=m

Π(l,m)BlA±ΠT (l,m)Ω

]M−1; (7.41)

Π(l,m) = diag [Pml (μ1),Pm

l (μ2), . . . ,Pml (μN )]T ; (7.42)

M = diag [μ1E, μ2E, . . . , μNE] ; (7.43)

Ω = diag [w1E, w2E, . . . , wNE] ; (7.44)

A± = E ± (−1)l−mD. (7.45)

Here, E is the 4×4 identity matrix, and E the 4N ×4N identity matrix. The (⊥)superscript indicates the conjugate transpose. The link between Xα and solutionvectors W±

α comes through the auxiliary equations:

W±α =

12M−1

[E ± 1

kαS+]Xα . (7.46)

Eigenvalues occur in pairs {±kα}. Left and right eigenvectors share the samespectrum of eigenvalues. As noted by Siewert (2000b), both complex- and real-variable eigensolutions may be present. Solutions may be determined with thecomplex-variable eigensolver DGEEV from the LAPACK suite (Anderson et al.,1995). Eigenvectors from DGEEV have unit modulus.

In the scalar case, the formulation of the eigenproblem is simpler. The eigen-matrix is symmetric and all eigensolutions are real-valued. In this case, theeigensolver module ASYMTX (Stamnes et al., 1988b) is used. ASYMTX is amodification of the LAPACK routine for real roots; it delivers only the righteigenvectors. For the vector case, there are circumstances (pure Rayleigh scat-tering, for example) where complex eigensolutions are absent, and one may thenuse the faster ASYMTX routine. We return to this point in subsection 7.3.1.3.

The complete homogeneous solution in one layer is then:

I+(x) = D+4N∑α=1

{LαW+

α exp[−kαx] + MαW−α exp[−kα(Δ − x)]

}; (7.47)

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240 Robert Spurr

I−(x) = D−4N∑α=1

{LαW−

α exp[−kαx] + MαW+α exp[−kα(Δ − x)]

}. (7.48)

Here, D− = diag{D,D, . . . ,D} and D+ = E. These matrices arise fromapplication of symmetry relations (Siewert, 2000b). In the scalar case, thediscrete ordinate homogeneous solution vectors obey the symmetry propertyX±

α (±μi) = X∓α (∓μi), and these D± matrices are not required. The use of

optical thickness Δ − x in the second exponential ensures that solutions re-main bounded (Stamnes and Conklin, 1984). The quantities {Lα,Mα} are theconstants of integration; in LIDORT and VLIDORT, they are determined byapplication of boundary conditions and solution of the resulting BVP.

In Eqs (7.47) and (7.48), some contributions will be complex, some real.It is understood that we compute the real parts of any contributions to theStokes vectors resulting from complex variable expressions. Thus if {kα,W−

α } isa complex eigensolution with associated (complex) integration constant Lα, werequire:

Re [LαW−α e−kαx] = Re [Lα] Re [W−

α e−kαx] − Im [Lα] Im [W−α e−kαx]. (7.49)

From a bookkeeping standpoint, one must keep count of the number of real andcomplex solutions, and treat them separately in the numerical implementation.For clarity of exposition, we have not made an explicit separation of complexvariables, and it will be clear from the context whether real or complex variablesare under consideration.

7.2.2.2 Linearization of the eigenproblem

It turns out that in the vector model, differentiation of the eigenproblem is themost crucial step in the linearization process, and there are several points ofdeparture from the equivalent step in the scalar case. We will therefore discussthis step in some detail.

For the single layer solution, we require derivatives of {kα,W±α } with re-

spect to some atmospheric variable ξ in layer n. From (7.40) and (7.42), theeigenmatrix Γ is a linear function of the single scatter albedo ω and the matrixof expansion coefficients Bl, and its (real-variable) linearization L(Γ) is easy toestablish from chain-rule differentiation:

L(Γ) = L(S+)S− + S+L(S−); (7.50)

L(S±) =

[LM∑l=m

{L(ω)

2Π(l,m)Bl +

ω

2Π(l,m)L(Bl)

}A±ΠT (l,m)Ω

]M−1.

(7.51)In Eq. (7.51), L(ω) = U and L(Bl) = Zl are the linearized IOPs. Next, wedifferentiate both the left and right eigensystems (7.39) to find:

L(X⊥α )Γ + X⊥

α L(Γ) = 2kαL(kα)X⊥α + k2

αL(X⊥α ); (7.52)

ΓL(Xα) + L(Γ)Xα = 2kαL(kα)Xα + k2αL(Xα). (7.53)

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7 LIDORT and VLIDORT 241

We form a dot product by pre-multiplying (7.53) with the transpose vector X⊥α ,

rearranging to get:

2kαL(kα)〈X⊥α , Xα〉 − 〈X⊥

α , L(Γ)Xα〉 = k2α〈X⊥

α , L(Xα)〉 − 〈X⊥α , ΓL(Xα)〉. (7.54)

From the definitions in Eq. (7.39), we have:

〈X⊥α , ΓL(Xα)〉 = 〈X⊥

α Γ, L(Xα)〉 = k2α〈X⊥

α , L(Xα)〉, (7.55)

and hence the right-hand side of (7.54) is identically zero. We thus have:

L(kα) =〈X⊥

α , L(Γ)Xα〉2kα〈X⊥

α , Xα〉. (7.56)

Next, we substitute Eq. (7.56) in (7.54) to obtain the following 4N × 4N linearalgebra problem for each eigensolution linearization:

HαL(Xα) = Cα; (7.57)

Hα = Γ − k2αE; (7.58)

Cα = 2kαL(kα)Xα − L(Γ)Xα. (7.59)

Implementation of Eq. (7.57) ‘as is’ is not possible due to the degeneracy of theeigenproblem, and we need additional constraints to find the unique solution forL(Xα). The treatment for real and complex solutions is different.

Real solutions. The unit-modulus eigenvector normalization can be expressed as〈Xα, Xα〉 = 1 in dot-product notation. Linearizing, this yields one equation:

L(Xα)Xα + XαL(Xα) = 0. (7.60)

The solution procedure uses 4N −1 equations from (7.57), along with Eq. (7.60)to form a slightly modified linear system of rank 4N . This system is then solvedby standard means using the DGETRF and DGETRS LU-decomposition rou-tines from the LAPACK suite.

Complex solutions. In this case, Eq. (7.57) is a complex-variable system for boththe real and imaginary parts of the linearized eigenvectors. There are 8N equa-tions in all, but now we require two constraint conditions to remove the eigen-problem arbitrariness. The first is Eq. (7.60). The second condition is imposedby the following DGEEV normalization: for that element of an eigenvector withthe largest real value, the corresponding imaginary part is always set to zero.Thus for an eigenvector X, if element Re[XJ ] = max{Re[Xj ]} for j = 1, . . . , 4N ,then Im[XJ ] = 0. In this case, it is also true that L(Im[XJ ]) = 0. This is thesecond condition. The solution procedure is then: (1) in Eq. (7.57) to strike outthe row and column J in matrix Hα for which the quantity Im[XJ ] is zero, andstrike out the corresponding row in the right-hand vector Cα; and (2) in the re-sulting 8N −1 system, replace one of the rows with the normalization constraintEq. (7.60). L(Xα) is then the solution of the resulting linear system.

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242 Robert Spurr

Scalar model . The above vector linearization procedure is not applicable to scalarLIDORT code. This is because the eigensolver ASYMTX has no adjoint solution,so there is no determination of L(kα) as in Eq. (7.56). Instead, LIDORT usesthe complete set in Eq. (7.57) in addition to the constraint Eq. (7.60) to form asystem of rank N + 1 for the unknowns L(kα) and L(Xα).

Having derived the linearizations L(kα) and L(Xα), we complete this sectionby differentiating the auxiliary result in Eq. (7.46) to establish L(W±

α ):

L(W±α ) =

12M−1

[∓L(kα)

k2α

S+ ± 1kα

L(S+)]Xα +

12M−1

[E ± 1

kαS+]

L(Xα).

(7.61)Finally, we have linearizations of the transmittance derivatives in Eqs (7.47) and(7.48):

L(exp[−kαx]) = −x {L(kα) + kαL(x)} exp[−kαx]. (7.62)

Here, x and Δn are proportional for an optically uniform layer, so that

Ln(x) =x

ΔnLn(Δn) =

x

ΔnVn. (7.63)

7.2.3 Solar sources: particular integrals and linearization

In the initial version of VLIDORT, solar beam solution particular integrals ofthe vector RTE are established using the traditional substitution method, ratherthan the Green’s function approach. This is mainly for bookkeeping reasonsassociated with the use of complex and real variables. In the scalar LIDORTcode, both methods are available.

7.2.3.1 Chandrasekhar substitution particular integral

Referring to Eq. (7.23), inhomogeneous source terms in the discrete ordinatedirections are:

Qmn (x,±μi) =

ω

2

L∑l=m

Pml (±μi)BnlPm

l (−μ0)I0Tn−1 exp(−λnx). (7.64)

Here Tn−1 is the solar beam transmittance to the top of layer n, and in thepseudo-spherical approximation, λn is the average secant. Particular solutionsmay be found by substitution:

I±(x,±μi) = Zn(±μi)Tn−1 exp[−λnx], (7.65)

and by analogy with the homogeneous case, we define the 4N × 1 vectors:

Z±n =

[ZT

n (±μ1),ZTn (±μ2), . . . ,ZT

n (±μN )]T

. (7.66)

We decouple the resulting equations by using sum and difference vectors G±n =

Z+n ±Z−

n , and reduce the order from 8N to 4N . We obtain the following 4N ×4Nlinear-algebra problem:

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7 LIDORT and VLIDORT 243

A(2)n G+

n = C(2)n ; (7.67)

A(2)n = λ2

nE − Γn; (7.68)

C(2)n =

[S−

n Q+n + λnQ−

n

]M−1; (7.69)

Q±n = ω

LM∑l=m

Π0(l,m)BlA±ΠT (l,m)M−1; (7.70)

Π0(l,m) = [Pml (−μ0),Pm

l (−μ0), . . . ,Pml (−μ0)]

T. (7.71)

This system has some similarities to the eigensolution linearization in Eqs (7.57–7.60). It is also solved using the LU-decomposition modules DGETRF andDGETRS from LAPACK; the formal solution is G+

n =[A(2)

n

]−1C(2)

n . The par-ticular integral is completed through the auxiliary equations:

Z±n =

12M−1

[E ± 1

λnS+

n

]G+

n . (7.72)

In the vector model, the particular solution consists only of real variables.

Linearizing the particular solution. For the linearization, the most importantpoint is the presence of cross-derivatives: in a fully illuminated atmosphere, theparticular solution is differentiable with respect to atmospheric variables ξp inall layers p ≥ n. The solar beam is transmitted through layers p ≥ n before scat-tering in layer n, so transmittance factor Tn−1 depends on variables ξp in layersp > n. Similarly, the average secant λn (in the pseudo-spherical approximation)depends on variables ξp for p ≥ n. In addition, the solution vectors Z±

n de-pend on λn, so their linearizations contain cross-derivatives. Linearization of thepseudo-spherical approximation is treated below in subsection 7.2.7.1, and thisestablishes the quantities Lp(Tn−1) and Lp(λn) ∀ p ≥ n. For the plane-parallelcase, Lp(λn) ≡ 0, since λn = −1/μ0 (constant).

Next, the eigenmatrix Γn is constructed from optical properties only definedin layer n, so that Lp(Γn) = 0 ∀ p �= n. Differentiation of Eqs (7.67–7.71) yieldsa related linear problem:

A(2)n Lp(G+

n ) ≡ C(3)np = Lp(C(2)

n ) − Lp(A(2)n )G+

n ; (7.73)

Lp(A(2)n ) = −δpnLp(Γn) + 2λnLp(λn)E; (7.74)

Lp(C(2)n ) = δnp

[Ln(S−

n )Q+n + S−

n Ln(Q+n ) +

1λn

Ln(Q−n )]

− Lp(λn)λ2

n

Q−n ; (7.75)

Ln(Q±n ) =

LM∑l=m

[UnΠ0(l,m)Bl + ωnΠ0(l,m)Znl

]A±ΠT (l,m)M−1. (7.76)

In Eq. (7.75), the quantity Ln(S−n ) comes from (7.51). Equation (7.73) has the

same matrix A(2)n as in Eq. (7.67), but with a different source vector on the

right-hand side. The solution is then found by back-substitution, given that the

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244 Robert Spurr

inverse of the matrix A(2)n has already been established for the original solution

G+n . Thus Lp(G+

n ) =[A(2)

n

]−1C(3)

np . Linearization of the particular integral isthen completed through differentiation of the auxiliary equations (7.72):

Lp(Z±n ) =

12M−1

[E ± 1

λnS+

n

]Lp(G+

n ) ∓ 12λ2

n

M−1

×[λnδpnLp(S+

n ) − Lp(λn)S+n )]G+

n . (7.77)

This completes the RTE solution determination and the corresponding lineariza-tions with respect to atmospheric variables. The treatment for the scalar case issimilar; see, for example, van Oss and Spurr (2002).

7.2.3.2 Green’s function methods in LIDORT

In the discrete ordinate reformulation of the slab problem (Siewert, 2000a), theparticular solution is expressed in terms of the infinite-medium Green’s functionsolution for the RTE (Barichello et al., 2000). The generalization to a collectionof optically uniform strata was developed for the scalar LIDORT model, and thengiven a complete linearization treatment (Spurr, 2002). Here we summarize themain equations and discuss the linearization.

The Green’s function particular integral may be expressed as a linear com-bination of homogeneous solution vectors through (keeping the explicit layerindex):

G±n (x) =

N∑α=1

[A−

nαC−nα(x)X∓

nα + A+nαC+

nα(x)X±nα

]; (7.78)

A−nα =

1〈Rn〉

N∑j=1

wj

[Q−

njX+njα + Q+

njX−njα

]; (7.79)

A+nα =

1〈Rn〉

N∑j=1

wj

[Q+

njX+njα + Q−

njX−njα

]; (7.80)

〈Rn〉 =N∑

j=1

μjwj

[X+

njαX+njα − X−

njαX−njα

]; (7.81)

Q±ni =

(2 − δm0)ωn

2

LM∑l=m

Pml (±μi)βnlP

ml (−μ0). (7.82)

Here, terms {A±nα, 〈Rn〉} depend on vectors {X±

nα,Q±n }; these terms are inde-

pendent of optical thickness x in a given layer n, and they do not depend on anyquantities outside this layer. The optical thickness dependency is driven by theattenuation qn(x) of the term and is expressed through multipliers C±

nα(x):

C+nα(x) = e−xknα

∫ x

0e+yknαqn(y) dy; (7.83)

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7 LIDORT and VLIDORT 245

C−nα(x) = e+xknα

∫ Δn

x

e−yknαqn(y) dy. (7.84)

For the solar source, qn(x) = I0Tn−1 exp[−λnx], the attenuated beam flux inthe pseudo-spherical average secant formulation. This exponential form makesthe integrals easy to evaluate:

C+nα(x) = I0Tn−1

e−xknα − e−xλn

λn − knα; (7.85)

C−nα(x) = I0Tn−1

e−xλn − e−Δnλn e−(Δn−x)knα

λn + knα. (7.86)

In order to solve the boundary value problem, particular integrals must be de-fined at the upper and lower boundaries of the layers, and this will require thewhole-layer multipliers (setting I0 = 1 in the following):

C+nα(Δn) = Tn−1

e−Δnknα − e−Δnλn

λn − knα; C−

nα(0) = Tn−11 − e−Δnλn e−Δnknα

λn + knα.

(7.87)Linearization of the quantities A±

nα, 〈Rn〉 may be done by chain rule differentia-tion, based on results already derived above; the end-points of the differentiationare the linearized IOP inputs Un,Znl. There are no cross-derivatives from lay-ers p �= n. Linearization of the multipliers is also straightforward; we give oneexample here:

Lp

[C+

nα(Δn)]=Lp(Tn−1)

e−Δnknα −e−Δnλn

λn − knα− C+

nα(Δn)λ − kα

[Lp(λn)−δnpLn(kα)]−

Tn−1e−Δnknαδnp [ΔnLn(kα)+kαLn(Δn)]−e−Δnλn [ΔnLp(λn)+δnpλnLn(Δn)]

λn − knα.

(7.88)This result depends upon the pseudo-spherical linearizations Lp(Tn−1), Lp(λn),∀p ≥ n}, and the linearized IOP Vn. One of the advantages of the Green’sfunction method is that the particular integral can be written down in closedform. Another advantage is that the solution remains bounded whenever thesecant parameter λn equals one of the separation constants knα. In the LIDORTmodel, a Taylor-series expansion for the multipliers is invoked whenever |εnα| <0.001, where εnα = λn −knα. Referring to Eq. (7.85), the basic discrete ordinatemultiplier, we find:

C+nα(x) = I0Tn−1x e−xknα

[1 − xεnα

2+

x2ε2nα

6+ O(ε3

nα)]. (7.89)

It is harder to establish a solution by the substitution method in this limitingcase, as the linear algebra system becomes degenerate.

One more remark is in order here. The exponential optical thickness pa-rameterization of beam attenuation in the average secant approximation is veryconvenient for solving the RTE. However, for geometrically or optically thick

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246 Robert Spurr

atmospheric layers with illumination at high solar zenith angle, the approxima-tion loses some accuracy. This was investigated in (Spurr, 2002), where it wasshown that better approximations to solar beam attenuation are obtained byusing exponential-polynomial or exponential-sine parameterizations. For bothcases, the multiplier integrals may be evaluated in closed form and the resultingGreen’s function RTE solutions determined. We return to this point in subsec-tion 7.2.7.4.

7.2.4 Thermal sources: particular integrals and linearization

In this section, we determine the Green’s function particular integral of the RTEin the presence of atmospheric thermal emission sources. This solution is newand we have adopted it in favor of the substitution approach used in the originalLIDORT work (Spurr et al., 2001) and in the DISORT formalism (Stamnes etal., 1988a). We also present a linearization of this solution with respect to theatmospheric profile variables.

Green’s function formulae in the previous section are still applicable, butnow the source is isotropic thermal emission. The source function is qn(x) =(1 − ωn)ηn(x), where ηn(x) is the black body emission in layer n. We write:

G±n (x) =

N∑α=1

[A−

nαC−nα(x)X∓

nα + A+nαC+

nα(x)X±nα

]. (7.90)

For an isotropic source, we have:

A±nα =

δm0ωn

21

〈Rn〉

N∑j=1

wj

[X+

nαj ∓ X−nαj

]. (7.91)

The quantity 〈Rn〉 has already been defined. With the linear regime for ηn(x)in Eq. (7.27), the Green’s function multiplier integrals in Eqs (7.83) and (7.84)are straightforward:

C+nα(x) =

(1 − ωn)knα

[(Hn−1 − Mn

knα

)(1 − e−xknα) + xMn

]; (7.92)

C−nα(x) =

(1 − ωn)knα

[(Hn−1 +

Mn

knα

)(1 − e−yknα) +

(x − Δn e−yknα

)Mn

].

(7.93)Here, y = Δn − x, and Mn = Hn − Hn−1 from Eq. (7.27). Linearization of thissolution is straightforward; in particular for the multiplier differentiation, we al-ready know Ln(knα), and Ln(x) = x, Ln(Δn) = Δn, and Ln(ωn) = Un. Note that(in contrast with the solar case), there are no cross-derivatives; in other words:

Lp

[C+

nα(x)]

= 0 for p �= n. (7.94)

We discuss the post-processing aspects of this solution in subsection 7.2.6.

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7 LIDORT and VLIDORT 247

7.2.5 Boundary value problem

In this section, we return to the vector problem with solar sources. From sub-section 7.2.3, the complete Stokes vector discrete ordinate solutions in layer nmay be written:

I±n (x) = D±

4N∑α=1

[LnαW±

nα e−knαx + MnαW∓nα e−knα(Δn−x)

]+ Z±

n Tn−1 e−λnx.

(7.95)Quantities Lnα and Mnα are constants of integration for the homogeneous solu-tions, and they are determined by the imposition of three boundary conditionsas noted in subsection 7.2.1.3. For boundary condition (I), we have I+

n (0) = 0for n = 1, which yields (T0 = 1):

D+4N∑α=1

[LnαW+

nα + MnαW−nαKnα

]= −Z+

n . (7.96)

For boundary condition (II), the continuity at layer boundaries, we have:

D±4N∑α=1

[{LnαW±

nαKnα + MnαW∓nα

}−{LpαW±

pα + MpαW∓pαKpα

}]= −Z±

n Tn−1Λn + Z±p Tp−1. (7.97)

In Eq. (7.97), p = n+1. For surface condition (III), staying for convenience withthe Lambertian reflection condition in Eq. (7.31), we find (for layer n = L1):

D−4N∑α=1

[LnαV−

α Knα + MnαV+α

]= Tn−1Λn

[−U− + 2R0μ0E1I0

]. (7.98)

Here we have defined the following auxiliary quantities:

V±α = W±

nα − 2R0ET1 MΩW∓

nαE1; (n = L1) (7.99)

U− = Z−n − 2R0ET

1 MΩZ+n E1; (n = L1) (7.100)

E1 = diag{E1,E1, . . . ,E1}; (7.101)

Knα = e−knαΔn ; Λn = e−λnΔn . (n = 1, . . . , L1) (7.102)

Application of Eqs (7.96–7.98) yields a large, sparse banded linear system withrank 8N × L1. This system consists only of real variables, and may be writtenin the symbolic form:

Φ ∗ Ξ = Ψ. (7.103)

Here Ψ is constructed from the right-hand side variables in Eqs (7.96–7.98) andΦ is constructed from suitable combinations of V±

α ,W±nα and Knα. The vector

Ξ of integration constants is made up of the unknowns {Lnα, Mnα} and will be

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248 Robert Spurr

partitioned into contributions from real and complex parts. For a visualizationof the BVP in the scalar case, see Spurr et al. (2001).

The solution to Eq. (7.103) proceeds first by the application of a compres-sion algorithm to reduce the order and eliminate redundant zero entries. LU-decomposition is then applied using the banded-matrix LAPACK routine DGB-TRF to find the inverse Φ−1, and the final answer Ξ = Φ−1 ∗Ψ is then obtainedby back-substitution (using DGBTRS). For the slab problem (no intermediateboundaries), boundary condition (II) is absent; the associated linear problem isthen solved using the LAPACK DGETRF/DGETRS combination.

Linearizing Eq. (7.103) with respect to a variable ξp in layer p, we obtain:

Φ ∗ Lp(Ξ) = Ψ′p ≡ Lp(Ψ) − Lp(Φ) ∗ Ξ. (7.104)

We notice that this is the same linear-algebra problem, but now with a differentsource vector Ψ′

p on the right-hand side. Since we already have the inverse Φ−1

from the solution to the original BVP, back-substitution gives the linearizationLp(Ξ) = Φ−1 ∗ Ψ′

p of the boundary value constants. Although this linearizationis straightforward in concept, there are many algebraic details arising with chainrule differentiation required to establish Lp(Ψ) and Lp(Φ) in Eq. (7.104).

7.2.6 Post processing: source function integration

7.2.6.1 Solution with substitution

The source function integration technique is used to determine solutions at off-quadrature polar directions μ and at arbitrary optical thickness values in themulti-layer medium. The technique has been demonstrated to be superior tonumerical interpolation (Thomas and Stamnes, 1999). In the RTE Eq. (7.22) forpolar direction μ, the multiple scatter integral is approximated by a quadraturesum, in which the Stokes vector I(x, μ′) is replaced by the discrete ordinatesolution in Eq. (7.95). It is then possible to integrate over optical thickness on alayer to layer basis to build the radiation field. In the vector code, we retain onlythe real part of the Stokes vector obtained as a result of repeated combinationsof complex-variable entities. Here, we note down the principal results for theupwelling field.

The solution in layer n at direction μ for optical thickness x (as measuredfrom the top of the layer) is given by:

I−n (x, μ) = I−

n (Δ, μ) e−(Δ−x)/μ + H−n (x, μ) +

(Z−

n (μ) + Q−n (μ)

)E−

n (x, μ).(7.105)

The first term is the upward transmission of the lower-boundary Stokes vectorfield through a partial layer of optical thickness Δ − x. The other three con-tributions together constitute the partial layer source term from scattered lightcontributions. The first of these three arises from integration of the homogeneoussolution contributions and has the form:

H−n (x, μ) =

4N∑α=1

[LnαX+

nα(μ)H−+nα (x, μ) + MnαX−

nα(μ)H−−nα (x, μ)

], (7.106)

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7 LIDORT and VLIDORT 249

where we have defined the following auxiliary quantities:

X±nα(μ) =

ωn

2

LM∑l=m

Pml (μ)Bnl

N∑j=1

wj

{Pm

l (μj)X±nα(μi) + Pm

l (−μj)X±nα(−μi)

};

(7.107)

H−−nα (x, μ) =

e−xknα − e−Δnknα e−(Δn−x)/μ

1 + μknα; (7.108)

H−+nα (x, μ) =

e−(Δn−x)knα − e−(Δn−x)/μ

1 − μknα. (7.109)

Here, H−±nα (x, μ) are the homogeneous solution multipliers for the upwelling field;

they arise from the optical thickness integration. The other two layer source termcontributions in Eq. (7.105) come from diffuse and direct solar source scatter-ing respectively. In this case, all variables are real numbers, and the relevantquantities are:

Z−n (μ) =

ω

2

LM∑l=m

Pml (μ)Bnl

N∑j=1

wj

{Pm

l (μj)Z−n (μj) + Pm

l (−μj)Z−n (−μj)

};

(7.110)

Q−n (μ) =

ω(2 − δm0)2

LM∑l=m

Pml (μi)BnlPm

l (−μ0)I0; (7.111)

E−n (x, μ) = Tn−1

e−xλn − e−Δnλn e−(Δn−x)/μ

1 + μλn. (7.112)

Expressions (7.106), (7.107) and (7.110) have counterparts in the scalar code; themultipliers are the same for both codes. Multiplier expressions (7.108), (7.109)and (7.112) have appeared a number of times in the literature. Similar expres-sions can be written for post-processing of downwelling solutions.

Linearization. These source term quantities depend upon the IOPs {Δn, ωn,Bnl}, the pseudo-spherical beam transmittance quantities {Tn, λn}, the homo-geneous solutions {knα, X±

nα}, the particular solutions Z±n , and the BVP integra-

tion constants {Lnα,Mnα}. Derivatives of all these contributions have alreadybeen established with respect to variable ξp in layer p, and the linearizationproceeds by careful chain-rule differentiation. The end-points are the linearizedIOPs {Vn,Un,Znl} from Eq. (7.35). We confine ourselves to two remarks. First,for linearization of the homogeneous post-processing source term in layer n,there is no dependency on any quantities outside of layer n; in other words,Lp[H−

n (x, μ)] ≡ 0 for p �= n. Second, the particular solution post-processingsource terms in layer n depend on optical thickness values in all layers aboveand equal to n through the presence of the average secant and the solar beamtransmittances, so there will be cross-layer derivatives.

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250 Robert Spurr

7.2.6.2 Source function integration with Green’s function solutions

In this case, we have particular integrals expressed in terms of Green’s functionvariables, and the layer-by-layer optical depth integration will introduce a newset of multipliers for these particular integrals. Here we confine our attention toaspects of the Green’s function implementation for solar sources; the analogoustreatment for the post-processed thermal follows a similar path (not given here,but this is part of the LIDORT model). We summarize the main results; moredetails may be found in Spurr (2002).

The Green’s function post–processed solution in layer n at direction μ foroptical thickness x is:

I−n (x, μ) = I−

n (Δ, μ) e−(Δ−x)/μ+H−n (x, μ)+U−

n (x, μ)+Q−n (μ)E−

n (x, μ). (7.113)

The homogeneous contribution H−n (x, μ) is the scalar equivalent of Eq. (7.106),

Q−n (μ) is the scalar version of Eq. (7.111), and E−

n (x, μ) is given by Eq. (7.112).The only new contribution to the partial layer source term has the form:

U−n (x, μ) =

N∑α=1

[A+

nαX+nα(μ)D−+

nα (x, μ) + A−nαX−

nα(μ)D−−nα (x, μ)

]. (7.114)

Here, the A±nα functions are defined in Eqs (7.79) and (7.80), and the layer

source-function integration for the Green’s function solution has the effect ofintroducing the integrated upwelling Green’s function multipliers:

D−±nα (x) =

ex/μ

μ

Δn∫x

C±nα(t) e−t/μ dt. (7.115)

A similar result applies to the downwelling processed field, and this will yieldtwo more post-processed multipliers:

D+±nα (x) =

e−x/μ

μ

x∫0

C±nα(t) et/μ dt. (7.116)

For the average secant pseudo-spherical formulation of the solar beam source,we may use Eqs (7.85) and (7.86) for C±

nα(x) to evaluate these integrals, and theresults are:

D−+nα (x)=

I0Tn−1

λn−knα

[e−xknα −e−Δnknα e−y/μ

1 + μknα− e−xλn − e−Δnλn e−y/μ

1 + μλn

]; (7.117)

D−−nα (x)=

I0Tn−1

λn+knα

[e−Δnλn

e−y/μ−e−yknα

1 − μknα+

e−xλn −e−Δnλn e−y/μ

1 + μλn

]; (7.118)

D+−nα (x)=

I0Tn−1

λn+knα

[e−Δnλn

e−yknα −e−Δnknα e−x/μ

1 + μknα+

e−xλn −e−x/μ

1 − μλn

]; (7.119)

D++nα (x)=

I0Tn−1

λn−knα

[e−xknα −e−x/μ

1 − μknα− e−xλn −e−x/μ

1 − μλn

]. (7.120)

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7 LIDORT and VLIDORT 251

Here we have written y = Δ – x. For whole-layer multipliers we require D+−nα (Δn)

and D++nα (Δn) for the downwelling field, with D−−

nα (0) and D−+nα (0) for the up-

welling field.As with the discrete ordinate multiplier C+

nα(x), there are limiting values fortwo of these post-processing multipliers, obtained when εα = |λn − knα| becomesless than some small number. For the whole-layer post-processing multipliers,we find the following Taylor series expansions (suppressing the layer index forclarity):

D++nα (Δ) = I0Tn−1

(c+0 + c+

1 εα + c+1 ε2

α + O(ε3α)); (7.121)

D−+nα (0) = −I0Tn−1

(c−0 + c−

1 εα + c−1 ε2

α + O(ε3α)). (7.122)

Additional coefficients are defined by (the layer index n is understood):

c+0 =

ρα

μ

[Δ e−Δkα − ρα(e−Δkα − e−Δ/μ)

]; (7.123)

c−0 =

σα

μ

[Δ e−Δkαe−Δ/μ − σα(1 − e−Δkα e−Δ/μ)

]; (7.124)

c+1 = ρα

[c+0 − Δ2

2μe−Δkα

]; c−

1 = σα

[−c−

0 − Δ2

2μe−Δkα e−Δ/μ

]; (7.125)

c+2 = ρα

[c+1 +

Δ3

6μe−Δkα

]; c−

2 = σα

[−c−

1 +Δ3

6μe−Δkα e−Δ/μ

]; (7.126)

ρα =μ

1 − μkα; σα =

μ

1 + μkα. (7.127)

Similar expressions pertain to partial layer multipliers.For the linearization, the only new features are the derivatives of the post-

processing multipliers. These may be obtained by differentiating Eqs (7.117) to(7.120). For details, see Spurr (2002).

7.2.7 Spherical and single-scatter corrections

7.2.7.1 Pseudo-spherical approximation

The pseudo-spherical (P-S) approximation assumes solar beam attenuation fora curved atmosphere. All scattering takes place in a plane-parallel situation.The approximation is a standard feature of many radiative transfer models. Wefollow the formulation in Spurr (2002). Figure 7.1 provides geometrical sketchesappropriate to this section. It has been shown that the P-S approximation isaccurate for solar zenith angles up to 90◦, provided the line-of-sight is not toofar from the nadir (Dahlback and Stamnes, 1991; Caudill et al., 1997; Rozanovet al., 2000).

We take points Vn−1 and Vn on the vertical (Fig. 7.1, upper panel); then therespective solar beam transmittances to these points are:

Tn−1 = exp

[−

n−1∑k=1

sn−1,kΔk

]; Tn = exp

[−

n∑k=1

sn,kΔk

]. (7.128)

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252 Robert Spurr

Here, sn,k is the path distance geometrical ratio equal to the path distance cov-ered by the beam to Vn as it traverses through layer k divided by the correspond-ing vertical height drop (geometrical thickness of layer k). At TOA, T0 = 1. Inthe average secant approximation, the transmittance to any intermediate pointbetween Vn−1 and Vn is parameterized by:

T (x) = Tn−1 exp[−λnx] , (7.129)

where x is the vertical optical thickness measured downwards from Vn−1 and λn

the average secant for this layer. Substituting (7.129) into (7.128) and settingx = Δn we find:

λn =1

Δn

[n∑

k=1

sn,kΔk −n−1∑k=1

sn−1,kΔk

]. (7.130)

In the plane-parallel case, we have λn = μ−10 for all n.

Linearization. We require derivatives with respect to an atmospheric propertyξk in layer k. The basic linearized IOP input is the normalized derivative Vn =Ln[Δn]. Applying the linearization operator to (7.129) and (7.130), we find:

Ln[λn] =Vn

Δn(sn,n − λn) ; Ln[Tn] = 0; (7.131)

Lk[λn] =Vk

Δn(sn,k − sn−1,k) ; Lk[Tn] = −Vksn−1,kTn; (k < n) (7.132)

Lk[λn] = 0; Lk[Tn] = 0; (k > n) (7.133)

For the plane-parallel case, we have:

Lk[λn] = 0 (∀k,∀n); Lk[Tn] = −VkTn

μ0(k < n); Lk[Tn] = 0 (k ≥ n).

(7.134)

7.2.7.2 Exact single-scatter solution in the average-secantapproximation

In VLIDORT and LIDORT, we include an exact single-scatter computationbased on the Nakajima–Tanaka procedure (Nakajima and Tanaka, 1988). Thiscorrection procedure is also present in DISORT Version 2.0 (Stamnes et al.,2000). Without it, the internal single-scatter computation in VLIDORT willuse a truncated subset of the complete scattering law information, the numberof usable Greek coefficient matrices Bl being limited to 2N − 1 for N discreteordinate streams.

A more accurate computation results when the post-processing calculationof the truncated single-scatter contribution (the term Q−

n (μ)E−n (x, μ) in Eq.

(7.105), for example) is suppressed in favor of an accurate single scatter compu-tation, which uses the complete phase function or scattering matrix. This is the

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7 LIDORT and VLIDORT 253

so called TMS procedure (Nakajima and Tanaka, 1988). A related computationhas been implemented for the doubling-adding method (Stammes et al., 1989).

The (upwelling) post-processed solution in stream direction μ is now written(cf. Eq. (7.105)):

I−n (x, μ) = I−

n (Δ, μ) e−(Δ−x)/μ + H−n (x, μ)

+(Z−

n (μ) + Q−n,exact(μ)

)E−

n (x, μ), (7.135)

Q−n,exact(μ) =

ωn

4π(1 − ωnfn)Πn(μ, μ0, φ − φ0)I0. (7.136)

Note the presence in the denominator of the expression (1 − ωnfn) which isrequired when the delta-M approximation is in force; fn is the delta-M truncationfactor for the (1,1) elements of the Greek matrices (see subsection 7.2.1.2). Fromsubsection 7.2.1.2, Πn is obtained from the scattering matrix Fn(Θ) throughapplication of rotation matrices. There is no truncation: Πn can be constructedto any degree of accuracy using all available unscaled Greek matrices Bnl. Thescalar model treatment is similar.

Linearization. Chain-rule differentiation of Eq. (7.136) yields the linearizationof the exact single-scatter correction term. Since the elements of Πn consist oflinear combinations of Bnl, the linearization Ln(Πn) is straightforward to writedown in terms of the inputs Znl = Ln(Bnl).

7.2.7.3 Sphericity along the line-of-sight

For nadir-geometry satellite instruments with wide-angle off-nadir viewing, onemust consider the Earth’s curvature along the line of sight from the ground to thesatellite. This applies to instruments such as OMI on the Aura platform (swath2600 km, scan angle 114◦ at the satellite) (Levelt et al., 2006) and GOME-2(swath 1920 km) (EPS/METOP, 1999). Failure to account for this effect canlead to errors of 5–10% in the satellite radiance for TOA viewing zenith anglesin the range 55–70◦ (Caudill et al., 1997; Rozanov et al., 2000; Spurr, 2003).

In subsection 7.2.7.2, scattering was assumed to take place along the nadir, sothat the scattering geometry Ω ≡ {μ0, μ, φ−φ0} is unchanged (at least for non-refractive media) along the vertical. Figure 7.1 (lower panel) shows the geometryfor the single-scattering outgoing sphericity correction along the line of sight. Ina non-refractive atmosphere, the solar zenith angle, the line-of-sight zenith anglesand the relative azimuth angle between the incident and scattering planes willvary along path AB, but the scattering angle Θ is constant for straight-linegeometry. For layer n traversed by this path, the upwelling Stokes vector at thelayer-top is (to a high degree of accuracy) given by:

I↑(Ωn−1) ∼= I↑(Ωn)T (Ωn) + Λ↑n(Ωn) + M↑

n(Ωn). (7.137)

Here, I↑(Ωn) is the upwelling Stokes vector at the layer bottom, T (Ωn) the layertransmittance along the line of sight, and Λ↑

n(Ωn) and M↑n(Ωn) are the single-

and multiple-scatter layer source terms respectively. The transmittances and

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254 Robert Spurr

Fig. 7.1. Upper panel: Pseudo-spherical viewing geometry for scattering along thezenith AC. Lower panel: Line-of-sight path AB in a curved atmosphere, with viewingand solar angles changing along the path from A to B.

layer source terms are evaluated with scattering geometries Ωn at positions Vn.Equation (7.137) is applied recursively, starting with the upwelling Stokes vectorI↑BOA(ΩL1) evaluated at the surface for geometry ΩL1 , and finishing with the

field I↑TOA(Ω0) at top of atmosphere.

The single-scatter layer source terms Λ↑n(Ωn) may be determined through an

exact single-scatter calculation (cf. Eq. (7.136)) allowing for changing geomet-rical angles along the line of sight. To evaluate the multiple-scatter sources, we

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7 LIDORT and VLIDORT 255

can run VLIDORT in ‘multiple-scatter mode’ successively for each of the geome-tries from ΩL1 to Ω1, retaining only the appropriate multiple scatter layer sourceterms, and, for the first VLIDORT calculation with the lowest-layer geometryΩL1 , the surface upwelling Stokes vector I↑

BOA(ΩL1).For L1 layers in the atmosphere, we require L1 separate calls to VLIDORT,

and this is much more time-consuming that a single call with geometry ΩL1 (thiswould be the default in the absence of a line-of-sight correction). However, sincescattering is strongest near the surface, the first VLIDORT call (with geometryΩL1) is the most important as it provides the largest scattering source termM↑

L1(ΩL1).

An even simpler line-of-sight correction is to assume that all multiple scattersource terms are taken from this first VLIDORT call with geometry ΩL1 ; in thiscase, we require only the accurate single-scatter calculation to complete I↑

TOA.This approximation is known as the ‘outgoing’ sphericity correction; it requiresvery little extra computational effort. The sphericity correction can also be setup with two calls to VLIDORT with the start and finish geometries ΩL1 and Ω1;in this case, multiple-scatter source terms at other geometries are interpolated atall levels between results obtained for the two limiting geometries. In the scalarcase, accuracies for all these corrections were investigated in Spurr (2003).

So far, we are still using the average secant approximation to compute exactfirst-order scattering contributions Λ−

n (μn) = Q−n,exact(μn)E−

n (Δn, μn) for theline-of-sight zenith direction μn. As noted already, this approximation loses ac-curacy for high SZA and wide slant paths, and we now look briefly at a moreprecise formulation of the single-scatter calculation.

7.2.7.4 A more accurate outgoing sphericity correction

In this subsection, the exposition applies to the scalar intensity, but the treat-ment is the same for the VLIDORT implementation. First, we recast the primaryscatter RTE in terms of the vertical height coordinate z:

μ(z)dI(z)dz

= εnI(z) + εnΨnTn(z). (7.138)

Here, 4πΨn = I0σnPn(Θ) for phase function Pn(Θ) and scattering coefficientσn in layer n; εn is the extinction coefficient for the layer (also constant). Theattenuation Tn(z) is a function of z. From (non-refractive) geometry, the viewingzenith angle θ(z) is related to z through:

sin θ(z) =(Re + z0) sin θ0

Re + z. (7.139)

Here, Re is the Earth’s radius and the subscript ‘0’ indicates values at TOA.Thus, since μ(z) = cos θ(z), we can change the variable in (7.138) to get:

sin2 θdI(θ)dθ

= knI(θ) + knΨnTn(θ). (7.140)

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256 Robert Spurr

Here, kn = εn(Re +z0) sin θ0. An integrating factor for this differential equationis kn cot θ(z), and the whole-layer solution is then:

I↑n−1 = I↑

n ekn(cot θn−cot θn−1) + Jn; (7.141)

Jn = −knΨn e−kn cot θn−1

θn∫θn−1

dθTn(θ) ekn cot θ

sin2 θ. (7.142)

The integral in Eq. (7.142) can be done to a very high degree of accuracy bynumerical summation. The TOA upwelling intensity is computed according tothe recursion (L1 is the total number of layers in the atmosphere):

I↑0 = I↑

surfaceCL1 +N∑

n=1

JnCn−1; (7.143)

Cn =n∏

p=1

exp[kp(cot θp − cot θp−1)]; C0 = 1. (7.144)

Linearization. We consider differentiation with respect to the inherent opti-cal properties (IOPs) defined for each layer – the extinction and scatteringcoefficients εn and σn and the phase function expansion coefficients βnl. Thelinearization operator for variable ξp in layer p is defined in the usual way:Lp(yn) = ξp∂yn/∂ξp.

If ξp = εp, then Lp(kn) = δnpkn from the above definition of kn, andLp(Ψn) = 0. Differentiating (7.141) and (7.142) with respect to εp yields:

Lp[I↑n−1] = {Lp[I↑

n] + I↑nknδnp(cot θn − cot θn−1)} ekn(cot θn−cot θn−1)

−kn(1 − kn cot θn−1)Ψn e−kn cot θn−1

θn∫θn−1

dθTn(θ) ekn cot θ

sin2 θ

−knΨne−kn cot θn−1

⎡⎢⎣kn

θn∫θn−1

dθTn(θ) ekn cot θ cot θ

sin2 θ+

θn∫θn−1

dθLp[Tn(θ)] ekn cot θ

sin2 θ

⎤⎥⎦ .

(7.145)The only new quantity here is Lp[Tn(θ)] in the final integral. To evaluate this,we note that the attenuation of the solar beam to a point z with zenith angle inlayer n can be written as:

Tn(θ) = exp

[−

NMAX∑p=1

dnp(θ)εp

]. (7.146)

Here, dnp(θ) are geometrical distances, independent of the optical properties. Itfollows that:

Lp[Tn(θ)] = −dnp(θ)εpTn(θ). (7.147)

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7 LIDORT and VLIDORT 257

All integrals in the last line of Eq. (7.145) can again be done accurately usingnumerical summation. For most geometrical situations, dnp(θ) = 0 for all layersp > n; this corresponds to points on the line of sight that are illuminated fromabove. In this case, the attenuation does not depend on extinction coefficients inlayers below n, and hence Lp[Tn(θ)] = 0 for p > n. However, there are situations(near the top of the atmosphere for a wide off-nadir viewing path and a highsolar zenith angle) in which some points along the line of sight are illuminated bydirect sunlight coming from below the horizontal. In this case, the solar path hasgone through a tangent height in the atmosphere, and dnp(θ) is not necessarilyzero for p > n.

For linearization with respect to other optical properties, the situation issimpler. Defining now a linearization operator Lq = σq∂/∂σq for derivativeswith respect to the scattering coefficient σq in layer q, the only non-vanishingterm arising in the linearization of Eq. (7.142) is Lq[Ψn] = Ψnδnq:

Lq[I↑n−1] = Lq[I↑

n] ekn(cot θn−cot θn−1) − δnqknΨn e−kn cot θn−1

θn∫θn−1

dθT (θ) ekn cot θ

sin2 θ.

(7.148)Similar considerations apply to linearization with respect to phase function mo-ments βnl.

7.2.8 Surface reflectance

7.2.8.1 BRDFs as a sum of kernel functions

A scalar three-kernel bidirectional reflectance distribution function (BRDF)scheme was implemented in LIDORT (Spurr, 2004). The BRDF ρtotal(μ, μ′,φ−φ′) is specified as a linear combination of (up to) three semi-empirical kernelfunctions:

ρtotal(μ, μ′, φ − φ′) =3∑

k=1

Rkρk(μ, μ′, φ − φ′;bk). (7.149)

Here, (θ, φ) indicates the pair of incident polar and azimuth angles, with theprime indicating the reflected angles. The Rk are linear combination coefficientsor ‘kernel amplitudes’, while the kernels ρk(θ, θ′, φ − φ′;bk) are derived fromsemi-empirical models of surface reflection for a variety of surfaces. For eachkernel, the geometrical dependence is known, but the kernel function dependson the values taken by a vector bk of pre-specified parameters.

A well-known example is the single-kernel Cox–Munk BRDF for glitter re-flectance from the ocean (Cox and Munk, 1954a, 1954b); the kernel is a combi-nation of a Gaussian probability distribution function for the square of the wavefacet slope (a quantity depending on wind-speed W ), and a Fresnel reflectionfunction (depending on the air–water relative refractive index mrel). In this case,vector bk has two elements: bk = {W , mrel}. For a Lambertian surface, there

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258 Robert Spurr

is one kernel: ρ1 ≡ 1 for all angles, and coefficient R1 is just the Lambertianalbedo.

In order to develop solutions in terms of a Fourier azimuth series, Fouriercomponents of the total BRDF are calculated through:

ρmk (μ, μ′;bk) =

12π

2π∫0

ρk(μ, μ′, φ;bk) cosmφdφ. (7.150)

This integration over the azimuth angle from 0 to 2π is done by double numericalquadrature over the ranges [0,π] and [−π,0]; the number of BRDF azimuthquadrature abscissa NBRDF is set to 50 to obtain a numerical accuracy of 10−4

for all kernels considered in Spurr (2004).Linearization of this BRDF scheme was reported in Spurr (2004), and a

mechanism developed for the generation of surface property weighting functionswith respect to the kernel amplitudes Rk and to elements of the non-linearkernel parameters bk. It was shown that the entire discrete ordinate solution isdifferentiable with respect to these surface properties, once we know the followingkernel derivatives:

∂ρtotal(θ, α, φ)∂bp,k

=∂ρk(θ, α, φ;bk)

∂bp,k; (7.151)

∂ρtotal(θ, α, φ)∂Rk

= ρk(θ, α, φ;bk). (7.152)

The amplitude derivative Eq. (7.152) is trivial. The parameter derivative Eq.(7.151) depends on the empirical formulation of the kernel in question, but allkernels in the LIDORT BRDF scheme are analytically differentiable with respectto their parameter dependencies.

Remark . In VLIDORT, the BRDF is a 4 × 4 matrix linking incident and re-flected Stokes 4-vectors. The scalar BRDF scheme outlined above has been fullyimplemented in VLIDORT by setting the {1,1} element of a 4 × 4 vector kernelρk equal to the corresponding scalar kernel function ρk; all other elements arezero.

7.2.8.2 Ocean glitter kernel function

For ocean glitter, we use the well-known geometric-optics regime for a singlerough-surface redistribution of incident light, in which the reflection function isgoverned by Fresnel reflectance and takes the form (Jin et al., 2006):

ρCM (μ, μ′, φ − φ′,m, σ2) = r(θr,m).1

μμ′ |γr|4P (γr, σ

2)D(μ, μ′, σ2); (7.153)

Here, σ2 is the slope-squared variance (also known as the MSS or mean slopesquare) of the Gaussian probability distribution function P (γ, σ2) which hasargument γ (the polar direction of the reflected beam); r(θ,m) is the Fresnel

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7 LIDORT and VLIDORT 259

reflection for incident angle θ and relative refractive index m, and D(μ, μ′, σ2)is a correction for shadowing. The two non-linear parameters are σ2 and m. Wehave the usual Cox–Munk empirical relation (Cox and Munk, 1954a):

σ2 = 0.003 + 0.00512W, (7.154)

in terms of the wind speed W in m/s. A typical value for m is 1.33. The MSSGaussian is:

P (γ, σ2) =1

πσ2 exp[− γ2

σ2(1 − γ2)

]; (7.155)

The shadow function of Sancer (1969) is widely used, and is given by:

D(α, β, σ2) =1

1 + Λ(α, σ2) + Λ(β, σ2); (7.156)

Λ(α, σ2) =12

([(1 − α2)

π

]1/2σ

αexp

[− α2

σ2(1 − α2)

]− erfc

σ√

(1 − α2)

]).

(7.157)Both the Gaussian function and the shadow correction are fully differentiablewith respect to the defining parameters σ2 and m. Indeed, we have:

∂P (α, σ2)∂σ2 =

P (α, σ2)σ4

[α2

(1 − α2)− σ2

]. (7.158)

The shadow function can be differentiated in a straightforward manner. Thecomplete kernel derivative with respect to σ2 is then:

∂ρCM (μ, μ′, φ − φ′,m, σ2)∂σ2 = r(θr,m)

1μμ′ |γr|4[

∂P (γr, σ2)

∂σ2 D(μ, μ′, σ2) + P (γr, σ2)

∂D(μ, μ′, σ2)∂σ2

] . (7.159)

VLIDORT has a vector kernel function for sea-surface glitter reflectance, basedon the specification in (Mishchenko and Travis, 1997); this kernel has also beencompletely linearized with respect to the MSS (Natraj and Spurr, 2007).

With this formulation of linearized input for the glitter kernel, LIDORT andVLIDORT are thus able to deliver analytic weighting functions with respect tothe wind speed. This is important for remote sensing instruments with a glitterviewing mode; an example is the Orbiting Carbon Observatory (Crisp et al.,2004). Note that it is possible to use other parameterizations of the MSS (Zhaoand Toba, 2003) in this glitter formalism.

7.2.8.3 Land surface BRDF kernels

LIDORT has an implementation of a set of five semi-empirical MODIS-type ker-nels applicable to vegetation canopy (Wanner et al., 1995); each such kernel must

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260 Robert Spurr

be used in a linear combination with a Lambertian kernel. Thus for example, aRoss-thin BRDF surface type requires a combination of a Ross-thin kernel anda Lambertian kernel:

ρtotal(θ, α, φ) = c1ρRossthin(θ, α, φ) + c2. (7.160)

Linear factors c1 and c2 are not independent, and are specified in terms ofbasic quantities of the vegetation canopy. The kernels divide into two groups:those based on volume scattering empirical models of light reflectance (Ross-thin,Ross-thick), and those based on geometric-optics modeling (Li-sparse, Li-dense,Roujean). See Wanner et al. (1995) and Spurr (2004) for details of the kernelformulae.

LIDORT also has implementations of two other semi-empirical kernels forvegetation cover; these are the Rahman (Rahman et al., 1993) and Hapke(Hapke, 1993) BRDF models. Both kernels have three non-linear parameters,and both contain parameterizations of the backscatter hotspot effect. Here isthe Hapke formula:

ρhapke(μi, μj , φ) =ω

8(μi + μj){(1 +

Bh

h + tanα

)(2 + cos Θ) +

(1 + 2μi)(1 + 2μj)(1 + 2μi

√1 − ω

) (1 + 2μj

√1 − ω

) − 1

}.

(7.161)In this equation, the three non-linear parameters are the single scattering albedoω, the hotspot amplitude h and the empirical factor B; μi and μj are the direc-tional cosines, and Θ is the scattering angle, with α = 1

2Θ.The important point to note here is that all these kernels are fully differ-

entiable with respect to any of the non-linear parameters defining them. Fordetails of the kernel derivatives, see (Spurr, 2004). It is thus possible to generateanalytic weighting functions for a wide range of surfaces in the models. Surfacereflectance Jacobians have also been considered in other linearized RT models(Landgraf et al., 2002; Ustinov, 2005).

7.2.8.4 The direct beam correction for BRDFs

For BRDF surfaces, the reflected radiation field is a sum of diffuse and direct(‘single-bounce’) components for each Fourier term. One can compute the directreflected beam with a precise set of BRDF kernels rather than use their trun-cated forms based on Fourier series expansions. This exact ‘direct beam (DB)correction’ is done before the diffuse field calculation. Exact upwelling reflection(assuming plane-parallel beam attenuation) to optical depth τ may be written:

I↑REX (μ, φ, τ) = I0ρtotal(μ, μ0, φ − φ0) exp

[−τatmos

μ0

]exp

[−(τatmos − τ)μ

].

(7.162)Here, τatmos is the whole atmosphere vertical optical depth. For surface propertyJacobians, we require computation of the derivatives of this DB correction with

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7 LIDORT and VLIDORT 261

respect to the kernel amplitudes and parameters; this follows the discussion insubsection 7.2.8.1. For atmospheric profile weighting functions, the solar beamand line-of-sight transmittances in Eq. (7.162) need to be differentiated withrespect to variables ξp varying in layer p.

7.2.8.5 Surface emission in the LIDORT model

In addition to the surface reflection of diffuse and direct radiation, there is asurface emission source term which is present for Fourier component m = 0:

I−n,emission(Δn, μ) = δm,0κ(μ)B(Tg). (7.163)

Here, Tg is the surface emission temperature, and B(Tg) the Planck function.The emissivity is given by Kirchhoff’s law:

κ(μ) = 1 − 2

1∫0

μ′ρ0(μ, μ′) dμ′. (7.164)

Here, ρ0(μ, μ′) is the azimuth-independent component of the total BRDF kernelFourier expansion. For the Lambertian surface with albedo R, we have κ(μ) =1−R for all directional cosines. Note that for anisotropically reflecting surfaces,the emissivity Eq. (7.164) will have derivatives with respect to kernel amplitudesRk and kernel parameters bk.

7.3 Performance and benchmarking

7.3.1 Performance considerations

7.3.1.1 The delta-M approximation

In the scalar model, sharply peaked phase functions are approximated as a com-bination of a delta-function and a smoother residual phase function. This isthe delta-M approximation (Wiscombe, 1977), which is widely used in discreteordinate and other RT models. The delta-M scaled IOP inputs are:

Δ = Δ(1 − ωf); ω = ω(1 − f)(1 − ωf)

; βl =βl − f(2l + 1)

(1 − f). (7.165)

The delta-M truncation factor is:

f =β2N

(2N + 1). (7.166)

In VLIDORT, Legendre coefficients βl appear as the (1,1) entry in matrixBl. In line with the scalar definition in terms of the phase function, we take thetruncation factor f as defined Eq. (7.166), and adopt the following scaling for

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262 Robert Spurr

the six entries in Bl. Four coefficients (αl, βl, ζl and δl) will scale as βl in Eq.(7.165), while the other two coefficients γl and εl scale as γl = γl/(1 − f). Thisspecification can also be found in (Chami et al., 2001) where a more detailedjustification is presented. Scaling for the optical thickness and single scatteralbedo in Eq. (7.165) is not changed in the vector model. Linearizations of Eqs(7.165) and (7.166) are straightforward, and these are discussed in Spurr (2002)for the scalar model.

7.3.1.2 Multiple solar zenith angle facility

Both models are able to generate results for multiple solar geometries at onecall. In solving the RTE, the first step to determine solutions of the homoge-neous equations does not need to be repeated for each solar beam source. Thehomogeneous solution is solved once only; then for each solar beam geometry g,we generate a set of particular integral solutions Pg for our multi-layer atmo-sphere.

The boundary value problem (BVP) has the form AXg = Bg, where Xg

is the vector of integration constants appropriate to solar beam with geome-try g, Bg is the source term vector consisting of contributions from the set ofparticular solutions Pg, and the banded tri-diagonal matrix A contains onlycontributions from the RTE homogeneous solutions. The inverse matrix A−1

can be determined once only, before the loop over solar geometry starts. Thisis the most time-consuming step in the complete solution for the RT field, andonce completed, it is straightforward and fast to set the integration constantsXg = A−1Bg by back-substitution.

Convergence of the Fourier azimuth series for the radiation field depends onthe solar beam angle. We keep track of the convergence separately for each SZA;once the field at our desired output angles and optical depths has convergedfor one particular SZA, we stop further calculation of Fourier terms for thisSZA, even though solutions at other SZAs still require further computation ofFourier terms. The multiple SZA feature provides a very substantial performanceenhancement for VLIDORT, particularly in view of the increased time taken overthe eigenproblem (complex roots) and the much larger BVP matrix inversioncompared with the scalar code.

7.3.1.3 Eigensolver usage

We have already noted differences between the LAPACK solver DGEEV andthe condensed version ASYMTX as used in LIDORT and DISORT. DGEEVmust be used for any layers with scattering by aerosols or clouds, since therewill be complex roots in this case. ASYMTX only deals with real symmetriceigenmatrices, and does not deliver adjoint solutions.

It turns out that, aside from additional elements down the diagonal, theeigenmatrix Γn in layer n consists of blocks of 4 × 4 matrices of the formPlm(μi)BnlPT

lm(μj), where the Plm and Bnl matrices were defined in Eqs (7.25)and (7.24) in subsection 7.2.1.2 (μi are the discrete ordinates). Since P and PT

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7 LIDORT and VLIDORT 263

are symmetric, then Γn will be symmetric if Bnl is. Thus, Γn will be symmetricif the Greek constants εl in Bnl are zero for all values of l. This is a specialcase that is satisfied by the Rayleigh scattering law; here, we can use the faster‘real-only’ ASYMTX package. For aerosols and clouds we require the complexeigensolver DGEEV from LAPACK. The policy in VLIDORT is to retain botheigensolvers and use them as appropriate – if any of the Greek constants εl inBnl is non-zero for a given scattering layer, then we will choose the complexeigensolver in that layer. For an application with a few particulate layers in anotherwise Rayleigh-scattering atmosphere, both eigensolvers will be required.

7.3.1.4 Solution saving

In DISORT and earlier LIDORT versions, the models contained full computa-tions of all RTE solutions in all layers and for all Fourier components. Solutionsare computed regardless of the scattering properties of the layer. In solutionsaving , numerical computations of homogeneous and particular solutions areskipped in the absence of scattering. If there is no scattering for a given Fouriercomponent m and layer n, then the RTE solution is trivial – it is just attenuationacross the layer with transmittance factor Tn(μ) = exp[−Δn/μ], where μ is anypolar direction and Δn is the layer optical thickness. It follows that, if there areN discrete ordinates μj in the half-space, the jth homogeneous solution vectorhas components {Xj}k = δjk, and the separation constants are μ−1

j . Particularsolution vectors are set to zero, since there is no solar beam scattering. Sourcefunction integration required for post-processing is then a simple transmittancerecursion using transmittances Tn(μ). Linearizations of RTE solutions in anynon-scattering layer are zero, and linearized solutions in adjacent scattering lay-ers will be transmitted with factors Tn(μ). We note that if this transmittancepropagation passes through layer n for which a linearization L[Δn] exists, thenthe linearization will pick up an additional term L[Tn(μ)] = −μ−1Tn(μ) L[Δn].

Rayleigh scattering has a P(Θ) = cos2 Θ phase function dependency on scat-tering angle Θ. There is no scattering for Fourier components m > 2; solutionsaving then applies for m > 2. For an atmosphere with Rayleigh scattering anda limited number of aerosol or cloud layers, there will be a substantial reduc-tion in RTE solution computations when the solution saving option applies, andconsequently a marked improvement in performance. In general, the phase func-tion has a Legendre polynomial expansion Φ(Θ) =

∑βλPλ(cos Θ) in terms of

moment coefficients βλ. For a discrete ordinate solution with N streams, thephase function is truncated: β2N−1 is the last usable coefficient in the multiplescatter solution. In the delta-M approximation, β2N is used to scale the problemand redefine the βλ for 0 ≤ λ ≤ 2N−1. Solution saving occurs when βλ = 0for m ≤ λ ≤ 2N − 1; there is then no scattering for Fourier component m andhigher.

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264 Robert Spurr

7.3.1.5 BVP telescoping

For some Fourier component m, we consider a single active scattering layer nwith non-trivial RTE solutions; all other atmospheric layers have no scattering(the extension to a number of adjacent active layers is easy). Working with thescalar RTE, the integration constants Ln and Mn appear through the discreteordinate solution (a plane-parallel solution has been assumed):

I±(x, μi) =N∑

α=1

[LnαX±

inα e−knαx + MnαX∓inα e−knα(Δn−x)

]+ G±

in e−x/μ.

(7.167)For the non-scattering layers, we have X±

ipα = δiα and G±ip = 0 for all layers

p �= n. In this case the downwelling and upwelling solutions are:

I+pj(x) = Lpj exp[−x/μj ]; (7.168)

I−pj(x) = Mpj exp[−(Δp − x)/μj ]. (7.169)

Integration constants propagate upwards and downwards through all non-scat-tering layers via:

Lp+1,j = Lpj exp[−Δp/μj ]; (7.170)

Mp−1,j = Mpj exp[−Δp/μj ]. (7.171)

If we know constants Ln and Mn for the active layer n, then constants forall other layers will follow by propagation. We now write down the boundaryconditions for layer n. At the top of the active layer, we have:

N∑α=1

[LnαX+

inα + MnαX−inαΘnα

]+ G+

in = Ln−1,iCn−1,i; (7.172)

N∑α=1

[LnαX−

inα + MnαX+inαΘnα

]+ G−

in = Mn−1,i. (7.173)

At the bottom of the active layer, we have

N∑α=1

[LnαX+

inαΘnα + MnαX−inα

]+ G+

inΛn = Ln+1,i; (7.174)

N∑α=1

[LnαX−

inαΘnα + MnαX+inα

]+ G−

inΛn = Mn+1,iCn+1,j . (7.175)

We have used the following abbreviations:

Θnα = exp[−knαΔn], Δn = exp[−ηnΔn], Cnj = exp[−Δn/μj ]. (7.176)

We now consider the top and bottom of atmosphere boundary conditions.At TOA, there is no diffuse radiation, so that Lp = 0 for p = 1 and hence by

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7 LIDORT and VLIDORT 265

Eq. (7.170) for all p < n. At BOA, the Lambertian reflection condition onlyapplies to Fourier m = 0; for all other components there is no reflection, and soin our case Mp = 0 at BOA and hence by Eq. (7.171) for all p > n. With theseconditions Eqs (7.172) and (7.175) become:

N∑α=1

[LnαX+

inα + MnαX−inαΘnα

]= −G+

in; (7.177)

N∑α=1

[LnαX−

inαΘnα + MnαX+inα

]= −G−

inΛn. (7.178)

This is a 2N -system for the desired unknowns Ln and Mn (there is actually noband-matrix compression for a single layer). For the layer immediately above n,we use (7.173) to find Mn−1 and for remaining layers to TOA, we use (7.171).Similarly for the layer immediately below n, we use (7.174) to find Ln+1 andfor remaining layers to BOA, we use (7.170). In this way, we have reduced ortelescoped the BVP, so that we only need to solve for integration constants inone active layer rather than developing the BVP linear algebra system for alllayers.

If the telescoped BVP is written as AY = B, then the corresponding lin-earized problem may be written ALk[Y] = B* = Lk[B] − Lk[A]Y; the k sub-script refers to the layer for which weighting functions are required. The lin-earized telescoped BVP is essentially the same problem with a different sourcevector, and the solution may be found be back-substitution, since the matrixinverse A−1 is already known from the original telescoped BVP solution. Con-struction of the source vector B* depends on the RTE solution linearizations;clearly if k = n there will be more contributions to consider than if k < n.However the linearized boundary conditions for B* are essentially the same asthose noted for the full atmosphere problem – the only thing to remember isthat the upper boundary is the same as TOA but with the first layer active, andthe lower boundary is the same as BOA but with the last layer active.

7.3.1.6 Convergence with exact single scatter and direct beamcontributions

As noted above, the Nakajima–Tanaka TMS correction provides an exact cal-culation of the single scatter contribution using an unlimited number of (non-delta-M scaled) phase function or phase matrix expansion moments. This cor-rection replaces the truncated single-scatter terms that would emerge from thepost-processed solution of the discrete ordinate field. In the DISORT code,TMS is implemented by first taking away the truncated SS term from thealready-computed overall field, and replacing it with the exact term: I =IMS+SS + ISSexact − ISStrunc ; Fourier convergence is applied to I. In LIDORT,the unwanted truncated SS term is simply omitted from the start, with only thediffuse field being computed: I = IMS + ISSexact . An improvement in Fourierconvergence can be obtained by applying TMS first and including ISSexact right

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266 Robert Spurr

from the start in the convergence testing for I. The rationale here is that theoverall field has a larger magnitude with the inclusion of the ISSexact offset, sothat the addition of increasingly smaller Fourier terms in the diffuse field willbe less of an influence on the total radiation. Convergence is faster with thesmoother diffuse field, and the number of separate Fourier terms can be reducedby up to a third in this manner.

A similar consideration applies when the DB correction (subsection 7.2.8.4) isin force for BRDF surfaces. As with the exact single-scatter case, I↑

REX (μ, φ, τ)in Eq. (7.162) should be added to the total field just after calculation of theazimuth-independent Fourier term, and before the higher-order Fourier are com-puted and the total radiance field examined for convergence. This is an importantperformance enhancement for ocean glitter scenarios.

7.3.2 VLIDORT validation and benchmarking

7.3.2.1 Checking against the scalar code

VLIDORT is designed to work equally with Stokes 4-vectors {I,Q,U ,V } and inthe scalar mode (I only). The first validation task for the vector model is to runit in scalar mode and reproduce results generated independently from the scalarLIDORT model. A set of options can be used to test the major functions of themodel (the real RT solutions, the boundary value problem and post process-ing) for the usual range of scenarios (single layer, multilayer, arbitrary opticalthickness and viewing angles, plane-parallel versus pseudo-spherical, etc.). Thisbattery of tests is very useful, but of course it does not validate the Stokes-vectorsolutions and in particular the complex variable treatment.

7.3.2.2 The Rayleigh slab problem

A first validation was carried out against the Rayleigh atmosphere results pub-lished in the Coulson, Dave and Sekera (CDS) tables (Coulson et al., 1960).These tables apply to a single-layer pure Rayleigh slab in plane-parallel geome-try; the single-scattering albedo is 1.0 and there is no depolarization consideredin the scattering matrix. Tables for Stokes parameters I, Q and U are given forthree surface albedos (0.0, 0.25, 0.80), a range of optical thickness values from0.01 to 1.0, for seven azimuths from 0◦ to 180◦ at 30◦ intervals, some 16 viewzenith angles with cosines from 0.1 to 1.0, and for 10 solar angles with cosinesfrom 0.1 to 1.0. With the single-scattering albedo set to 0.999999, VLIDORTwas able to reproduce all these results to within the levels of accuracy specifiedin the introduction section of the CDS tables.

7.3.2.3 Benchmarking for aerosol slab problems

The benchmark results noted in Siewert (2000b) were used; all eight outputtables in this work were reproduced by VLIDORT. The slab problem used a

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7 LIDORT and VLIDORT 267

solar angle 53.130◦ (μ0 = 0.6), with single-scatter albedo ω = 0.973527, surfacealbedo 0.0, total layer optical thickness of 1.0, and a set of Greek constantsas noted in Table 1 of Siewert (2000b). Output was specified at a number ofoptical thickness values from 0 to 1, and at a number of output streams. 24discrete ordinate streams were used in the half space for the computation. Anadditional benchmarking was done against the results of (Garcia and Siewert,1989) for another slab problem, this time with albedo 0.1. With VLIDORT set tocalculate using only 20 discrete ordinate streams in the half space, Tables 3–10 in(Garcia and Siewert, 1989) were reproduced to within one digit of six significantfigures. This result is noteworthy because the radiative transfer computationsin (Garcia and Siewert, 1989) were done using a completely different radiativetransfer methodology (the so-called FN method).

7.3.2.4 Weighting function verification

For Jacobians, validation can be done by using a finite difference estimate of thepartial derivative (ratio of the small change in the Stokes vector induced by asmall change in a parameter in one layer). However, there are pitfalls associatedwith this procedure (quite apart from the arbitrariness and time-consuming na-ture of the exercise). In certain situations, a small perturbation of one or moreof the Greek constants can give rise to a set of eigensolutions which cannot becompared (in a finite-difference sense) with those generated with the originalunperturbed inputs.

7.4 Preparation of inputs

7.4.1 Example: specification of atmospheric IOP inputs

For a Stokes vector computation, VLIDORT requires the IOP input set {Δn, ωn,Bnl} for each layer n, where Δn is the total optical thickness, ωn the total single-scatter albedo, and Bnl the set of Greek matrices specifying the total scatteringlaw. For Bnl in Eq. (24), the six Greek constants {αl, βl, γl, δl, εl, ζl} must bespecified for each moment l in the spherical-function expansion of the phasematrices. The values βl are the traditional phase function expansion coefficients,the ones that appear as inputs to the scalar version; they are normalized to 4π.

As an example, we consider an atmosphere with Rayleigh scattering by airmolecules, some trace gas absorption, and scattering and extinction by aerosols.If in a single layer, the Rayleigh scattering optical depth is σRay and trace gascolumn density C and absorption cross-section αgas , and aerosol scattering andextinction optical thickness values by εAer and σAer, then the total IOPs are:

Δ = Cαgas + σRay + εaer ; ω =σaer + σRay

Δ; Bl =

σRayBl,Ray + σaerBl,aer

σRay + σaer.

(7.179)

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268 Robert Spurr

In terms of the depolarization ratio ρ, the only non-zero Greek matrix coefficientsfor Rayleigh scattering are:

β0 = 1; δ1 =3(1 − 2ρ)

2 + ρ; α2 =

6(1 − ρ)2 + ρ

; β2 =(1 − ρ)2 + ρ

; γ2 = −√

6(1 − ρ)2 + ρ

(7.180)Aerosol quantities must in general be derived from a suitable particle scatteringmodel (Mie calculations, T-matrix methods, etc.). We consider a two-parameterbimodal aerosol optical model with the following combined optical property defi-nitions in terms of the total aerosol number density N and the fractional weight-ing f between the two aerosol modes:

Δaer = Neaer ≡ N [fe1 + (1 − f)e2]; (7.181)

ωaer =σaer

eaer≡ fz1e1 + (1 − f)z2e2

eaer; (7.182)

βl,aer =fz1e1β

(1)l + (1 − f)z2e2β

(2)l

σaer. (7.183)

Here, e1, z1 and β(1)l are the extinction coefficient, single-scatter albedo and Leg-

endre expansion coefficient for aerosol type 1; similar definitions apply to aerosoltype 2.

In a linearized model, we also require linearized IOPs, normalized partialderivatives {Vξ,Uξ,Zl,ξ} of the original IOPs with respect to layer parametersξ. These parameters may be elements of the retrieval state vector, or they may besensitivity parameters (not retrieved, but sources of uncertainty in the retrieval).As an example, we will assume that the retrieval parameters are the trace gascolumn density C, the total aerosol density N and the bimodal ratio f ; all otherquantities in the above definitions are sensitivity parameters. For the gas density,we find:

VC = Cσgas ; UC = −ωCσgas

Δ; Zl,C = 0. (7.184)

For the linearized aerosol IOPs with respect to N and f we find (we have justconsidered one of the Greek-matrix elements for simplicity):

N∂Δ∂N

= N∂Δaer

∂N= Δaer ; f

∂Δ∂f

= f∂Δaer

∂f= fN(e1 − e2); (7.185)

N∂ω

∂N=

Nσaer − ωΔaer

Δ; f

∂ω

∂f=

fN [(z1e1 − z2e2) − ω(e1 − e2)]Δ

; (7.186)

N∂βl

∂N=

Nσaer (βl,aer − βl)Nσaer + σRay

; f∂βl

∂f=

fN(z1e1 − z2e2) (βl,aer − βl)Nσaer + σRay

. (7.187)

The linearized IOP inputs in Eqs (7.185). (7.186) and (7.187) are necessary forthe RT models to generate Jacobians with respect to macroscopic (or bulk)quantities such as N and f . We remark that at least for spherical particles,it is also possible to define linearized IOPs with respect to microscopic aerosol

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7 LIDORT and VLIDORT 269

quantities such as the complex refractive index and particle size for monodisperseaerosols and parameters such as effective radius and particle size distributioneffective variances for polydisperse particulates. A ‘linearized Mie’ program willdeliver these IOPs, and such a program can be used in conjunction with RTmodeling to investigate retrieval of aerosol microscopic properties in a consistentanalytic manner; see, for example, (Hasekamp and Landgraf, 2005b). LinearizedMie algorithms have been developed by the present author and others (Graingeret al., 2004; Hasekamp and Landgraf, 2005a).

7.4.2 Surface and other atmospheric inputs

For the pseudo-spherical approximation, LIDORT and VLIDORT require knowl-edge of the earth’s radius Rearth and a height grid {zn} where n = 0 to n = L1(the total number of layers); heights must be specified at layer boundaries withz0 being the top of the atmosphere. This information is sufficient if the at-mosphere is non-refracting. If the atmosphere is refracting, it is necessary tospecify pressure and temperature fields {pn} and {tn}, also defined at layerboundaries. The refractive geometry calculation inside VLIDORT is based onthe Born–Wolf approximation for refractive index n(z) as a function of height:n(z) = 1 + α0p(z)/t(z). The user must specify factor α0. In the refractive case,the models have an internal fine-layering structure to deal with repeated appli-cation of Snell’s law. In this regard, the user must specify the number of finelayers to be used for each coarse layer.

For BRDF input, it is necessary for the user to specify up to three amplitudecoefficients {Rk} associated with the choice of kernel functions, and the non-linear parameter vectors {bk}. For example, if the BRDF is a single Cox–Munkfunction, it is only necessary to specify the wind speed (in meters/second) andthe relative refractive index between water and air. Fourier component speci-fication is done numerically by a double Gauss–Legendre quadrature over theintervals [−π, 0] and [0, π], and for this, it is necessary to specify the number ofBRDF azimuth quadrature abscissa NBRDF . The choice NBRDF = 50 is suffi-cient to obtain numerical accuracy of 10−4 in this Fourier component calculation.For surface property weighting functions, we need only specify whether we re-quire weighting functions with respect to {Rk} and/or to the components ofvectors {bk}.

For thermal emission input, the current specification in LIDORT requiresthe Planck function to be input at layer boundaries, the surface emission Planckfunction is separate. A convenient routine for generating the integrated Planckfunction (in W m−2) was developed as an internal routine in the DISORT code(Stamnes et al., 2000); this can be used outside the LIDORT environment to gen-erate the required Planck functions. For thermal emission alone, Planck functionsare specified in physical units. For solar sources only, output is normalized tothe input solar flux vector (which can be set to arbitrary units). For calculationswith both sources, the solar flux must be specified in physical units.

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270 Robert Spurr

7.5 Concluding remarks

In this chapter, we have reviewed the multi-layer multiple-scattering discreteordinate radiative transfer models LIDORT and VLIDORT. We have focused inparticular on the linearization capacity of the models: the ability to generate an-alytic weighting functions of the radiation field with respect to any atmosphericor surface parameter. Both models have a fully linearized pseudo-spherical ca-pability to deal with solar beam attenuation in a curved atmosphere. We havealso discussed the implementation of exact single-scatter calculations (includinglinearizations). VLIDORT has been validated against a number of benchmarksin the literature. We have discussed a number of performance aspects, includingthe multiple SZA facility, and the use of time-saving devices such as solutionsaving.

VLIDORT Version 2.1 has all the capabilities of its scalar counterpart LI-DORT Version 3.2. Both codes have been streamlined and reorganized so thatinputs and outputs are consistent. Both these models are available from theauthor at RT Solutions, along with documentation.

Acknowledgments

Funding for the LIDORT and VLIDORT work has come from a number ofsources. The main one has been a series of six Ozone SAF Visiting ScientistGrants from the Finnish Meteorological Institute, spread over the 6-year period1999–2005. Funding has also come from two grants from the European SpaceAgency to work on GOME total ozone retrieval algorithms (2003, 2004), and agrant from NASA to work on LIDORT applications (2003). From 2005 onwards,the VLIDORT research has been funded in part through the Orbiting CarbonObservatory Project at the Jet Propulsion Laboratory, California Institute ofTechnology, under contracts with the National Aeronautics and Space Adminis-tration. The LIDORT family of models continues to receive support from NASAGSFC and SSAI, and other institutions in the USA and Europe.

User feedback is always helpful for radiative transfer developments. In thisregard, the author would like to thank Knut Stamnes (Stevens Institute forTechnology), Jukaa Kujanpaa (Finnish Meteorological Institute), Vijay Natraj(CalTech), Colin Seftor (SSAI), Mick Christi (Colorado State University), NickKrotkov (NASA) and Roeland van Oss (KNMI) for some very helpful user feed-back. The author would like to thank European colleagues Piet Stammes, Jo-han de Haan, Diego Loyola, Werner Thomas, Stefano Corradini, and Michelvan Roozendael, and American colleagues Wei Li, Xiong Liu, Thomas Kurosu,Kelly Chance, Randall Martin, Sasha Vassilkov, Hartmut Boesch, and EugeneUstinov. The author is grateful to Rowan Tepper for help with the manuscriptpreparation.

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Liu, X., K. Chance, C.E. Sioris, R.J.D. Spurr, T.P. Kurosu, R.V. Martin, and M.J.Newchurch, 2005: Ozone profile and tropospheric ozone retrievals from the globalozone monitoring experiment: Algorithm description and validation. J. Geophys.Res., 110, D20307, doi:10.10.29/2005JD006240.

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Natraj, V., and R. J. D. Spurr, 2007: A linearized approximately spherical two orders ofscattering model to account for polarization in vertically inhomogeneous scattering-absorbing media, J. Quant. Spectrosc. Radiat. Transfer, in press.

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Part III

Bi-directional Reflectance of Light from Naturaland Artificial Surfaces

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8 Bi-directional reflectance measurements ofclosely packed natural and preparedparticulate surfaces

Hao Zhang and Kenneth J. Voss

8.1 Introduction

One of the most general ways to quantify the surface reflectance of a mediumis by use of the bi-directional reflectance distribution function (BRDF) (Hapke,1993). The BRDF gives the reflectance of a surface as a function of illuminationgeometry and viewing geometry and is required in many scientific and engineer-ing disciplines. For example, in satellite remote sensing, measurements takenfrom spaceborne sensors are affected significantly by sun–target–sensor geome-try. Since the reflectances of most land surfaces such as soil, snow and vegetationare anisotropic, and atmospheric scattering is also anisotropic, the same surfaceviewed at different times of the day, or from different directions, may appear tohave a different reflectance. To compare measurements carried out under differ-ent illumination and viewing conditions, the angular properties of the groundsurface reflectance must be taken into account (Royer et al., 1985) in additionto the atmospheric effects (Gordon, 1997).

The BRDF of a natural particulate layer is determined by many competingfactors such as the optical properties of the individual particles, the packingcondition of these particles including the surface roughness and the surroundingmedium physical properties. These properties are determined or affected by thegeological and biological processes that formed and shaped the particulate sur-face. Thus studies of the BRDF characterizations of particulate surfaces providea powerful tool in remote sensing applications.

This chapter consists of seven sections. In section 8.2, basic BRDF-relatedquantities are defined; then we give a brief review of the BRDF instrumentationwith an emphasis on our in situ BRDF-meter and its calibration. In section 8.4we summarize several major BRDF modeling efforts from the 1980s onward. Sec-tion 8.5 describes the controlled laboratory BRDF measurements of reflectancefrom a layer with monodispersed spherical particles and comparison with BRDFmodels. In section 8.6 we present the in situ BRDF data of benthic sediment sur-faces. Finally in section 8.7 we demonstrate the wetting liquid complex refractiveindex effects on particulate BRDF.

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280 Hao Zhang and Kenneth J. Voss

8.2 Definitions of bi-directional reflectance and relatedquantities

Figure 8.1 displays the typical configuration used to describe the scattering ofradiation by a flat surface with normal direction n . Greek characters θ and φrepresent the zenith and azimuth angles, respectively. The subscripts ‘i’ and ‘r’stand for incident and reflected quantities, respectively. Collimated irradiance,Ei(θi,φi), is incident onto the surface element. The reflected radiance, in a spe-cific direction is Lr(θr,φr). The bi-directional reflectance (BDR) is defined asthe ratio of Lr(θr,φr) to Ei(θi,φi) (measured relative to a surface perpendicularto the beam) on a unit area of the surface (Hapke, 1993):

BDR(θi, φi; θr, φr) =Lr(θr, φr)Ei(θi, φi)

. (8.1)

For a surface which is rotationally symmetric, the BDR may be expressed asBDR(θi, θr, g) where g is the phase angle defined by:

g = cos−1(cos θi cos θr + sin θi sin θr cos(φr − φi)). (8.2)

It follows that at θi = θr = 0 (see Fig. 8.1) g=0. Also we have:

g = 180◦ − Θ

where Θ is the scattering angle usually used in light scattering applications.This BDR(θi, θr, g) expression convention will be used throughout this chapter.The bi-directional reflectance distribution function is the ratio of the radiance

Fig. 8.1. Radiometric and geometric quantities used for BRDF definitions.

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8 Bi-directional reflectance measurements of particulate surfaces 281

reflected by a surface into a given direction to the collimated power incident ona unit area of the surface (Hapke, 1993):

BRDF (θi, θr, g) =Lr(θr, φr)

Ei(θi, φi) cos(θi)=

BDR(θi, θr, g)μ0

, (8.3)

whereμ0 = cos(θi). (8.4)

is the cosine of incident zenith angle.One of the most useful analytic and mathematically simple expressions for

BDR can be derived for the perfect Lambertian reflector (Hapke, 1993)

BDRL(θi, θr, g) =μ0

π. (8.5)

A Lambertian reflector has a constant radiance when viewed from any direction.When interpreting experimental data it is often more instructive to use the

reflectance factor (RF) rather than the bi-directional reflectance. The RF isdefined as the ratio of the BDR of a sample to that of a perfect Lambertiansurface:

RF (θi, θr, g) =BDR(θi, θr, g)

μ0/π= πBRDF (θi, θr, g) =

πLr

μ0E. (8.6)

One of the advantages of using RF rather than the bi-directional reflectance isthat it is easily compared to a perfect Lambertian surface.

It should be noted that in the literature the reflectance factor defined byEq. (8.6), may appear under different names, e.g., the bi-directional reflectionfunction (BRF) (Mishchenko et al., 1999), bidirectional reflectivity (Stamnes etal., 2000), and the reflection function (van de Hulst, 1980; Kokhanovsky, 2006).

Both the BRDF and RF obey the Helmholtz reciprocity principle (Hapke,1993):

BRDF (θi, θr, g) = BRDF (θr, θi, g), (8.7)

RF (θi, θr, g) = RF (θr, θi, g), (8.8)

if the surface is laterally uniform and thermal radiation and polarization effectsare neglected. In this chapter we describe BRDF measurements in the visiblewavelength range and only unpolarized scattering is considered, and thus thermalemission and polarization effects are neglected. The reciprocity property of theBRDF is widely used in testing scattering models since they must satisfy thisrelationship to be correct.

Another useful quantity is the hemispherical reflectance, or the plane albedo,given by

A(μ0) = π−1∫ 2π

0

∫ π/2

0RF (θr, φr) cos θr sin θrdθrdφr, (8.9)

for a specific incident direction.

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282 Hao Zhang and Kenneth J. Voss

8.3 BRDF models

The BRDF models in the literature may be classified into three broad categoriesbased on their theoretical basis: empirical models, geometrical optics (GO) mod-els and radiative transfer (RT) models. Empirical models are either based purelyon observations, or on simplified physical principles of GO and RT theory. Oneof the great advantages of the empirical BRDF models is that they are ana-lytic and most have few free parameters. For this reason the empirical modelsare widely used. For example, the Phong model (Phong, 1975) has two termsdescribing specular and diffuse reflectivity and, although none of the parame-ters in this model has a physical meaning, it has been found to be successfulin describing the BRDF of a rough metallic surface. The Walthall et al. BRDFmodel (Walthall et al., 1985) uses a three-term polynomial to parameterize theBRDF and has been found adequate for describing many vegetation canopiesand bare soil surfaces. This model has found numerous applications in remotesensing. As suggested by its name, the GO BRDF models are based on the prin-ciples of GO and are presented either in an analytical form (e.g., Li and Strahler,1986) or as numerical results derived by ray-tracing techniques (Peltoniemi andLumme, 1992; Shkuratov and Grynko, 2005). The RT BRDF models are eitherapproximations (Hapke, 1993; Kokhanovsky, 2006) or strict numerical solutionsof the radiative transfer equation (RTE) (Stamnes et al., 1988; Mishchenko et al.,1999). Recently we carried out laboratory measurements of the BRDF for welldescribed surfaces and compared these measurements with RT BRDF models(Zhang and Voss, 2005), we will concentrate on five popular BRDF RT models.As mentioned in section 8.2, our discussions are limited to unpolarized scatter-ing, and thus all BRDF models presented are unpolarized ones. We only brieflyoutline the RF format of these models since a detailed summary of model formatscan be found in the original literature.

8.3.1 Hapke’s isotropic multiple-scattering approximation (HIMSA)

The HIMSA model (Hapke, 1993) solves the single-scattering exactly and ap-proximates the multiple-scattering term by Chandrasekhar’s H functions (Chan-drasekhar, 1960). Two semi-empirical factors, the ‘hot-spot’ and the surfaceroughness correction, are incorporated into the model. The closed form of theHIMSA model is one of the reasons that it has gained so much popularity inremote sensing community. In terms of RF and neglecting the surface roughnessfactor, the HIMSA model is expressed as

RFHIMSA(μ0, μ, g) ="0

41

μ0 + μ{[1 + B(g)]P (g) + H(μ)H(μ0) − 1}, (8.10)

whereμ = cos θr, (8.11)

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8 Bi-directional reflectance measurements of particulate surfaces 283

H(x) is Hapke’s approximation of Chandrasekhar’s H function (Hapke, 2002)

H(x) ≈ 1

1 − "0x(r0 +

1 − 2r0x

2ln

1 + x

x

) . (8.12)

"0 is the single-scattering albedo (SSA), r0 is the diffusive reflectance (Hapke,1993). B(g) describes the opposition effect at small phase angle and is given by

B(g) ≈ B0

1 + (1/h) tan(g/2), (8.13)

where h is related to both the transparency of the individual particles and theporosity of the packed layer.

8.3.2 Hapke’s anisotropic multiple-scattering approximation(HAMSA)

In this improved version of the Hapke’s model (Hapke, 2002), the single-scattering term remains the same as in the HIMSA whereas the multiple-scattering part is replaced by a more anisotropic term by solving the Ambart-sumian’s nonlinear integral equation for reflectance (Hapke, 2002). Without thehotspot and surface roughness terms, the HAMSA is given by

RFHAMSA(μ0, μ, g) ="0

41

μ0 + μ[p(g) + M(μ0, μ)], (8.14)

where

M(μ0, μ) = C(μ0)[H(μ)−1]+C(μ)[H(μ0)−1]+Δ[H(μ)−1][H(μ0)−1], (8.15)

C(x) = 1 +∞∑

n=1

AnbnPn(x), (8.16)

Δ = 1 +∞∑

n=1

A2nbn (8.17)

An = 0, for n even

An =(−1)

n+12

n

1 × 3 × 5 × ... × n

2 × 4 × 6 × ... × (n + 1), for n odd (8.18)

and the ‘bn’s are the Legendre expansion coefficients of the phase function:

p(g) = 1 +∞∑

n=1

bnPn(cos g). (8.19)

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284 Hao Zhang and Kenneth J. Voss

8.3.3 Lumme–Bowell’s (LB) model

The LB model (Lumme and Bowell, 1981; Bowell et al., 1989) assumes thesingle-scattering term comes from three distinct parts: scattering from individualparticles, shadowing and surface roughness and may be expressed as

RFLB(θi, θr, g) = RFSingle + RFMultiple

="0

41

μ + μ0[2p(g)ΦS(g)ΦR + H(μ,"∗

0)H(μ0, "∗0) − 1] , (8.20)

where "∗0 is the scaled single-scattering albedo according to the ‘similarity rela-

tions’

"∗0 =

1 − 〈cos g〉1 − 〈cos g〉"0

"0, (8.21)

where 〈cos g〉 is the asymmetry parameter of the phase function p(g) and H maybe given by Eq. (8.12). ΦR is the surface roughness correction factor given by

ΦR =1 + (1 − q)ρξ

1 + ρξ, (8.22)

where q is the fraction of the surface covered with ‘holes’ and ρ is the mean slopeof a hole on a rough surface

ρ =l

l0= tan(θ), (8.23)

with l the depth and l0 the radius of a hole, respectively.

ξ =

√μ2 + μ2

0 − 2μμ0 cos g

μμ0. (8.24)

ΦS is the shadowing correction factor given by

ΦS(y) = yey

∫ 1

0t2y−1 e−yt dt ≈ y + 3/4

y + 3/2. (8.25)

For the case of packed spheres, y is given by

y =f

2.38μ + μ0

(μ2 + μ20 − 2μμ0 cos g)1/2 , (8.26)

where f is the volume density (or filling factor) of the layer. ρ is 1.2 for f = 0.64and 1.1 for f = 0.60.

8.3.4 Mishchenko et al.’s BRF algorithm (MBRF)

In the MBRF algorithm (Mishchenko et al., 1999), the single-scattering phasefunction is expanded as a Fourier series

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8 Bi-directional reflectance measurements of particulate surfaces 285

p(μ, μ0, φ) = p0(μ, μ0) + 2mmax∑m=1

pm(μ, μ0) cosmφ, (8.27)

where the Fourier components of the mth order are given by

pm(μ, μ0) = (−1)mnmax∑n=m

bnPnm0(μ)Pn

m0(μ0), (8.28)

and bn are coefficients in the expansion of the phase function

p(Θ) = 1 +nmax∑n=1

bnPn(cosΘ), (8.29)

Pn(cosΘ) is the Legendre polynomial which is the special case of the generalizedspherical functions Pn

mn(μ)Pn(x) = Pn

00(x). (8.30)

The intensity of the reflected radiation from a flat surface is defined as:

L(μ, ϕ) = μ0R(μ, μ0, φ)Ei

π, (8.31)

where Ei is the collimated incident flux per unit area perpendicular to the inci-dent beam and R(μ, μ0, φ) is the RF. R(μ, μ0, φ) is expanded as Fourier seriesin azimuth

R(μ, μ0, ϕ) = R0(μ, μ0) + 2mmax∑m=1

Rm(μ, μ0) cosmϕ. (8.32)

Once the pm(μ, μ′) are known, the coefficients Rm(μ, μ0) can be found usingAmbartsumian’s nonlinear integral equation and the RF is

RFMBRF = πL

Ei

1μ0

= R. (8.33)

This algorithm neglects close-packing effects. However, it does not involve anyother approximations, other than polarization, and is valid for arbitrary semi-infinite plane parallel media.

8.3.5 The DISORT model

DISORT (Stamnes et al., 1988) solves the RTE by the discrete ordinate methodand starts with the expansion of the radiance in terms of a Fourier series and thesingle-scattering phase function in a series of Legendre polynomials. By takingadvantage of addition theorem for spherical harmonics, the RTE is split into2M independent integro-differential equations and is further transformed intoa system of ordinary differential equations (ODE). With the integration termreplaced by a Gaussian quadrature summation, this system of ODE is solvedwith appropriate boundary conditions. This enables the determination of BRDFfor a plane-parallel layer of an arbitrary thickness. DISORT also neglects close-packing effects.

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286 Hao Zhang and Kenneth J. Voss

8.3.6 Some remarks on the models

Under the single-scattering approximation both the Hapke and the LB modelsreduce to a simple expression

RFSingle ="0

41

μ + μ0p(g). (8.34)

For isotropic scatters with p(g) = 1, Eq. (8.34) gives the Lommel–Seeliger law(Hapke, 1993).

The HIMSA model was criticized by Mishchenko for systematically re-trieving negative asymmetry parameters when applied to planetary surface re-flectance data, even though ice, snow and soil are forward-scattering particles(Mishchenko, 1994). Hapke (1996) demonstrated that if the range of the phaseangle in reflectance measurements is small, both the HIMSA and strict RTEcan retrieve the wrong single-scattering parameters; however, he argues that,if the phase angle can be extended to 120◦ and above, the HIMSA is able toretrieve the correct parameters. Mishchenko and Macke (1997) pointed out thatthe HIMSA seriously underestimates the multiple-scattering, especially for highalbedo surfaces, and thus the model violates energy conservation.

Both HIMSA and LB models assume that, for closely packed particulate me-dia, the phase function does not include the diffraction peak. This is equivalent totreating the forward-scattering as indistinguishable from directly transmitted ra-diation, or, because the gaps are small, assumes diffraction is both small and dif-fuse. Thus, in principle, when introducing the Mie phase function into these twomodels the diffraction peak should be removed. However, this is not necessarysince the reflectance region does not include the direct forward-scattering andthe isotropic multiple-scattering terms in these two models do not contain anyquantities derived from the phase function. In the HAMSA model it is not clearlystated whether the diffraction peak should be removed from Eqs (8.14)–(8.19).

Mishchenko (1994) performed the Percus–Yevick structural factor calcula-tions and found that diffraction may be neglected when the filling factor f , whichis the ratio of the volume taken by the particles and the sum of the individualparticle volume, roughly exceeds 0.2. He found that the effect of packing is espe-cially significant at Θ ≤ 0.4λ/R0, where λ and R0 are wavelength and particleradius, respectively. Mishchenko and Macke (1997) also emphasized, however,that there is no critical value of f before which the diffraction contribution is50% (f = 0) and after which is 0 (f = 1). In the current work the closely packedspheres have filling factors above 0.5; however, 0.4λ/Ro is only about 0.1◦. Thusthe full Mie phase functions were supplied to the five models.

8.4 BRDF instrumentation

8.4.1 General considerations

When measuring the radiance reflected from a flat surface, one has the optionof either illuminating a larger area than the sensor field of view (FOV) at alltimes (over-illumination), or illuminating a smaller area than the sensor FOV

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8 Bi-directional reflectance measurements of particulate surfaces 287

(over-view). Obviously, for field goniometers using natural illumination such asthe sun (e.g., Warren et al., 1998; Sandmeier, 2000) the illuminated area is largeand the over-illumination condition is always satisfied. However, for laboratorygoniometers, using either a laser or a lamp as light source, the option of over-view is often favored for the following reasons. First, over-viewing an illuminatedsample area improves the collection efficiency and makes the sensor alignmentless critical. Second, if the illumination spot is non-uniformly illuminated, es-pecially when using a laser as the light source, viewing only a portion of thebeam may cause large errors. Third, for a given illumination zenith angle, as theviewing angle varies, the illuminated area is constant. The major disadvantage ofthe over-view scheme is that the detector signal from a Lambertian-like surfacedecreases as cosine of the viewing angle, due to a smaller fraction of the viewedarea being illuminated.

The optimal size of the illuminated area is determined by the dimension ofthe individual components of the particulate surface. For example, if a surface iscomposed of nearly spherical particles, modeling results show that a light spot10 times larger than the diameter of the individual particles can be regarded asan infinite plane wave (Tsai and Pogorzelski, 1975).

The complete hemispherical coverage of the incident and viewing directions isanother important factor to be considered. It is well known that many particulatesurfaces display an enhanced backscattering peak, or hotspot (Hapke, 1993),and a strong forward-scattering peak, and these phenomena are more prominentat oblique incident angles than at normal incidence. Hence it is important tosample the BRDF at both large viewing and incident zenith angles. Furthermore,surface roughness conditions that reveal packing information of the particles inthe surface layer (Hapke, 1993) usually occur at large viewing angles.

There are many situations that angular resolution needs to be consideredespecially when fine structures are expected in the BRDF data. For example,the hotspot is proposed to have two distinct physical origins: shadow hidingand coherent backscattering. Distinguishing these effects requires polarizationmeasurements within 2◦ of the backscattering peak (e.g., Shkuratov et al., 1994;Mishchenko et al., 2006). Another example is the BRDF of layers with trans-parent spherical grains for which rainbow structures can be detected if the ra-diometer has a typical angular resolution better than 3◦.

A high signal-to-noise ratio (SNR) of the detection system is critical in ob-taining high-quality BRDF data. For BRDF apparatus using a CCD as a de-tector, the CCD array is usually cooled below ambient temperature by usinga thermoelectric (TE) pad (Voss et al., 2000). For goniometers using discretesensors, phase-sensitive techniques can be used to improve the SNR (Voss andZhang, 2006).

8.4.2 An in situ BRDF-meter

Figure 8.2 shows the schematics of a BRDF-meter (Voss et al., 2000) whichis capable of both underwater in situ and laboratory measurements. The lightsource is composed of three colors of light-emitting diodes (LEDs), red (658 nm),

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Fig. 8.2. Configuration of the BRDF instrument. Only one out of eight illuminationfibers and one out of 107 viewing fibers are shown.

green (570 nm), and blue (475 nm), located at angles from 0◦ to 65◦ in zenith.The illuminated area ranges from a 1.5 cm diameter circle at 0◦ incidence toa 3.8 cm by 1.5 cm ellipse at 65◦. To collect the reflected light, 107 viewingfibers with fixed viewing angles located from 5◦ to 65◦ in zenith and from 5◦ to345◦ in azimuth bring light to a cooled Apogee AP260 CCD array camera. Acomputer sequentially turns on and off the eight illumination fibers. Dark noisesuppression is achieved by both setting the thermoelectric cooler embedded inthe camera at 0◦C to suppress thermal noise, and by subtracting dark framescollected both before and after a sample is measured. A three-color sequencemeasurement consists of 30 pictures, including three dark images with no LEDturned on at the beginning, eight data images for red (658 nm), eight green(570 nm) and eight blue (475 nm) at the each illumination angle, and three darkimages in the end.

To calibrate the measurements, eight full measurement sets of a LabsphereSpectralon plaque with nominal 99% reflectance are made. When underwatermeasurements are needed, such calibration measurements are done with boththe plaque and BRDF-meter submerged in water. Between each measurementset the plaque is rotated 90◦ to eliminate any orientation biases of the plaquesurface.

Overall, the calibration is done by taking the ratio of the measured reflectancein a given direction to that which a 99% Lambertian reflector would have, thusthe data presented in this form are RF. Both the dry plaque and the plaquesubmerged in water have been measured (Voss and Zhang, 2006). The measured

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8 Bi-directional reflectance measurements of particulate surfaces 289

RF of the spectralon plaques were fitted to three-term polynomials (with fittingerrors in parenthesis):

RF bare = 1.04(±0.00) − 1.52(±0.23) × 10−5θ2r − 3.14(±0.45) × 10−9θ4

r , (8.35)

RF sub = 1.13(±0.00) − 3.85(±0.28) × 10−5θ2r − 5.34(±0.55) × 10−9θ4

r , (8.36)

where θr is in degrees. A more complete description of the calibration process isdescribed in Zhang (2004).

8.4.3 A simple goniometer

The in situ BRDF-meter described in subsection 8.4.2 has a limited numberof viewing positions and an angular resolution limited to about 5◦. In order toperform measurements with higher angular resolutions, a simple gonio scatteringmeter, as shown in Fig. 8.3, was built. An unpolarized He-Ne laser of 632.8-nm wavelength serves as the light source. After a beam expander, the light ispolarized by a linear polarizer. Next the linearly polarized beam is split into twoorthogonal polarized beams by a polarizing beam-splitting cube, with one goingto the monitoring detection electronics and the other to the sample. By rotatingthe polarizing cube both p- and s-polarized incident beams can be obtained. Theviewing tube consists of a narrow band interference filter, a focusing lens and a

Fig. 8.3. Schematic of the goniometer setup.

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290 Hao Zhang and Kenneth J. Voss

photodiode with electronics. The viewing optics configuration gives an angularresolution of 2.9◦, which is determined by the ratio of the aperture diameter(1.57 cm) to the radial distance of the aperture to the sample plane (31 cm).

The viewed area is a circle with diameter of 36 mm. For the goniometer atnormal illumination, the circular illuminated spot on a flat sample is 13 mm indiameter and at 60◦ incidence the spot is elongated to 26×13 mm. As discussedearlier when the beam radius is about 10 times larger than the sphere radius, theincident beam can be regarded as infinite plane wave. Since the largest particlesused in this work have diameter of 1 mm, this condition is satisfied for both thegonio- and the BRDF-meter. The minimum phase angle that can be measuredis about 8◦ due to mechanical interference between the viewing tube and theincident optics.

Both the monitoring and viewing sensors are Hamamatsu S8745 Si photo-diodes with preamplifiers. The output voltages from these two photodiodes areamplified and sent to a National Instrument DAQ-700 data acquisition card con-nected to a laptop computer’s PCMCIA socket. The ratio of the viewing channeland the monitor channel was going to be used to eliminate effects due to powerfluctuation of the laser. However, it was found that if sufficient warm-up time forthe He-Ne laser is allowed, the output voltage from the viewing channel is stableand using the ratio adds unnecessary noise. Before and after each measurement,dark signals (typically around −0.005 V in the viewing channel) are recordedand the average is subtracted from the measurement. The measured RF on aLabsphere nominal 99% reflectance plaque agrees with the Multiangle ImagingSpectroRadiometer (MISR)’s data (Bruegge et al., 2001) within 2%.

This goniometer configuration allowed the incident light to be either p- ors-polarized while the unpolarized scattered radiance was collected. To comparewith the RTE algorithms introduced in section 8.3, the average of these twoincident polarizations was taken to obtain the unpolarized case. For reflectancedata, the sample measurement was divided by the measurement of a Labspherecalibration plaque with nominal 99% reflectance to obtain the RF and was cor-rected for the non-Lambertian properties of the Spectralon plaque. An improvedversion of this goniometer is described in Voss and Zhang (2006).

8.4.4 An example of the calibration measurements

Figure 8.4 shows the RF measured by the BRDF-meter at 0◦ and 65◦ incidence,with the average RF, (a) and (b), and the normalized standard deviation (STD),(c) and (d). In the contour plots, the zenith angle is linearly proportional to thedistance from the center (0◦, 0◦). φ is arranged such that 0◦, measured from thesource of the illumination, is in the top center of the graph. Light reflected backtoward the source would be represented toward the top of this graph. Light thathas been reflected in the forward direction is towards the bottom of the graph.The left and right side of the images should be symmetric for a perfect surface,but with real data the images are not exactly symmetric. The plus signs in theplot show the measurement viewing locations, upon which the contour graphhas been based. They are distributed throughout the left and right side of the

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8 Bi-directional reflectance measurements of particulate surfaces 291

Fig. 8.4. Contour plots of the RF of a Labsphere calibration plaque at (a) 0◦ and(b) 65◦-incidence. The markers in the 0◦-incidence plot are viewing fiber locations; (c)and (d) are the standard deviation between the measurement rotations of a Labspherecalibration plaque at 0◦ and 65◦-incidence.

coordinate system, giving a measure of surface symmetry. The measurements arealso concentrated in the forward (specular) and backward (‘hotspot’) directions,as these are often areas of rapid change in natural samples. The data indicatethat at normal incidence the surface of the plaque is approximately Lambertianwith 10% fall-off from nadir to 65◦ viewing angle. As the incident direction movesoff from nadir, the plaque becomes more anisotropic, which turns into specularat 65◦ incidence. The standard deviations between measurements are smallerthan 1% on such a surface.

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8.5 Controlled BRDF measurements on prepared packedsurfaces and comparisons with models

In order to understand the connection between the single particle optics and theoptics of a packed layer, a series of controlled laboratory measurements for mediawith spherical particles were carried out. Since the single-scattering quantitiesof these Mie scatters can be easily introduced into the radiative transfer models,the comparisons between the model predictions and measurements may providea good validation of the models from the prediction point of view.

8.5.1 Samples and single-scattering quantities

Two different types of spherical particles are employed in this work: polystyrenespheres with nominal diameter of 200 μm from Duke Scientific Corporation(Catalog Number 4320) and silicon glass spheres with nominal diameter of 600μm from Whitehouse Scientific (Catalog Number MS0589). These spheres werechosen for the following reasons: (1) both are commercially available and NISTtraceable; (2) particle sizes are commensurate with that of the major sedimentparticles that we have measured in the field (see section 8.6); (3) the Mie phasefunctions of both kinds of spheres have a steep feature near the rainbow region.This last characteristic may be used to trace the single-scattering features inreflectance spectroscopy of a packed surface. Mie calculations were done withthe spher.f Mie code available online (www.giss.nasa.gov/∼crmim) (de Rooijand van der Stap, 1984; Mishchenko et al., 2002). The Duke 4320 spheres havea Gaussian particle size distribution (PSD) given by:

n(D) =1√2πσ

e−(D−D0)2

2σ2 , (8.37)

where D0 = 197.0 μm and σ = 6.1 μm, provided by Duke Scientific. Fig-ure 8.5(a) shows the p- and s-polarized phase functions, assuming a refractiveindex n of 1.59 at 633 nm and PSD given by Eq. (8.37). The size distribution ofthe Whitehouse Scientific WH0589 sample is fit by lognormal distribution

n(D) =1

w0De−(

ln DD0

)22(ln σ)2 , (8.38)

with w0 = 0.022, D0 = 587.6 μm and σ = 1.0. The Mie phase functions areplotted for silicon glass in Fig. 8.5(b) assuming n = 1.52 and a PSD given byEq. (8.38). It is well known that the rainbow features, the fine-scale oscillations,are strongly polarized in the direction perpendicular to the scattering plane (e.g.,Adam, 2002), or the s-polarization direction, as can be seen from Fig. 8.5.

8.5.2 Some parameters of packed surfaces and measurement results

In order to supply the RTE algorithms such as the DISORT, the optical thicknessτ of a layer is needed. For the large spheres used in this work the optical thicknessis roughly

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8 Bi-directional reflectance measurements of particulate surfaces 293

Fig. 8.5. Mie phase functions for PSD Eq. (8.37) n = 1.59 (a) and PSD Eq.(8.38)n = 1.52 (b): s-polarization and p-polarization stand for electric field of the incidentirradiance perpendicular and parallel to the scattering plane, respectively. The unpo-larized Mie phase function is their average. The phase angle 180◦ corresponds to theforward-scattering. Additional results of calculations are given in Table 8.1.

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Table 8.1. Summary of the parameters used in Mie and RTE calculations. f is thefilling factor; s is the geometrical depth of the layer; τ is the optical thickness; 〈cos θ〉is the asymmetry parameter; �0 is the single-scattering albedo; the number of streamsused in DISORT is 100

Samples f s(mm) τ 〈cos θ〉 �0

Duke 4320 0.60 10 88.5 0.80 0.999

WH0589 0.54 15 41.25 0.82 0.990

τ =3fs

2reff, (8.39)

where s is the layer’s geometric thickness, reff is the effective radius defined asthe ratio of the third to the second moment of PSD, and f is the filling fac-tor. The filling factors for Duke 4320 and WH0589 spheres have been evaluatedto be 0.63 and 0.54, respectively, by estimating the amount of spheres in sam-ple holders and the volume they occupy. The resultant optical thicknesses forDuke 4320 and WH0589 sphere layers are 88.5 and 41.25, respectively. We havedemonstrated with DISORT calculations that these τ values are within 0.5% (forDuke 4320) and 0.2% (for WH0589) of the asymptotic values for infinitely thicklayers (τas > 1000). In principle, the precise values of τas could be determinedby measurements of layers with different thicknesses (Zhang et al., 2003). How-ever, in reality many difficulties would limit the validity of such measurements.These difficulties include, but not limited to, the accurate determination of alayer’s thickness for packed tiny spheres, the validity of Eq. (8.39) for closelypacked layers, and the limited amount of spheres. For these reasons, we usedthe method described above to demonstrate that our layers may be deemed asinfinitely thick for reflectance measurement. The SSAs for the spheres are takento be 0.999 for Duke 4320 and 0.99 for WH0589, respectively, to account for anynon-zero imaginary refractive index (ni) effects (Zhang and Voss, 2005).

When this work was being carried out, the goniometer allowed the incidentlight to be either p- or s-polarized while the unpolarized scattered radiance iscollected. To compare with the RTE models summarized in section 8.3, theaverage of these two incident polarizations is calculated to get the unpolarizedincident light case.

Figure 8.6 shows the unnormalized RF of a 10 mm thick layer of the 200 μmspheres with p- and s-polarized incidences at 60◦ zenith. The unnormalized RFis the ratio of the radiance scattered from the sample and that scattered fromthe calibration plaque (Zhang and Voss, 2005). When compared with the cor-responding Mie phase functions shown in Fig. 8.5(a), one can see that the re-flectance curves for the two orthogonal polarizations resemble their respectiveMie phase functions. Besides the strong rainbow peak that appears in the s-polarization, a peak around 100◦ phase angle exists (or 40◦ viewing angle inthis configuration) which is the second-order rainbow (g = 102◦ for n = 1.59,see, for example, Adam, 2002). The peaks above 70◦ viewing angle, however,

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8 Bi-directional reflectance measurements of particulate surfaces 295

Fig. 8.6. Unnormalized RF from a 10 mm thick 200-μm sphere layer for p- and s-polarization incidences at 60◦ incidence. The viewing zenith angle in this configurationis phase angle minus 60◦ (see Fig. 8.3 for configurations).

must be attributed to surface roughness effects since they are not consistent inrepeated measurements with different surface realizations. The p-polarization,on the other hand, exhibits only a shoulder around the rainbow region andremains featureless throughout the rest of the region, closely resembling the p-polarization single-scattering phase function (Fig. 8. 5(a)). It is also obvious thatthe steep feature on the larger phase angle side of the rainbow, present in theMie phase functions for both polarizations, are preserved in the RF. However,some of the Mie features present in s-polarization such as the peaks around 40◦

phase angle disappear in the RF. These peaks are washed out by the strongmultiple-scattering.

Figure 8.7 is a comparison of the models with the goniometer reflectancedata for the 200 μm spheres at three illumination angles. In order to comparethe models, we first neglect the backscattering (or shadowing) factors and thesurface roughness factors in the Hapke and LB models. Also for clarity we useDISORT to represent the strict numerical RTE model as we found it virtuallyidentical to MBRF. It can be seen that the DISORT results are very close tothe experimental curve at phase angles from 15◦ to between 55◦ and 110◦ de-pending on illumination angle. The upper value beyond which DISORT eitherunderestimates or overestimates the measured RF is near phase g = 55◦ forθi = 0◦; g = 70◦, for θi = 35◦; g = 110◦, for θi = 60◦. All three approximatemodels have larger errors than DISORT. Compared to the HIMSA, the HAMSA

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296 Hao Zhang and Kenneth J. Voss

Fig. 8.7. Comparisons of goniometric measurement, DISORT, LB model, HIMSA andHAMSA for a 10 mm thick (τ = 88.5) 200-μm sphere layer. Incident zenith angles are(a) 0◦, (b) 35◦ and (c) 60◦ (see Fig. 8. 3 for configurations). Note the phase angle isdisplayed logarithmically to better display the rainbow region.

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is a better approximation in the backscattering region, but is up to 10% higherthan measurements in the region where DISORT works well. The LB model isalways low in all phase angle ranges, which is attributed to the similarity relationtransformed albedo (Eq. (8.21) used in the H function. If, the unscaled SSA isused in the LB model, one returns to Eq. (8.10) which is the same as the Hapkemodel without the surface roughness and hotspot factor.

Figure 8.8 shows the same comparisons for the 600 μm spheres. For thissample, all models other than the LB model predict much higher values thanmeasurements. Although the LB model seems to be relatively good for the 600μm spheres, in the 200-μm sphere case the LB model was shown to have amultiple-scattering contribution that was too low. The big difference between theexperimental data and DISORT results could possibly be attributed to errorsfrom (1) partial non-sphericity of the 600-μm spheres; (2) incorrect estimateof "0; (3) insufficient number of spheres in a relatively small sample holder;or (4) close packing effects. The first error source was identified by visuallyobserving the spheres with a 15× eyepiece and noticing quite a few non-sphericalgrains including spheroids, broken grains, and even some that appeared colored.This could also be seen from the gonio data where the rainbow feature is smallat normal incidence and basically does not exist at 60◦-incidence, indicatingnon-spherical grains or larger absorption than estimated. For "0, although inprinciple one can vary the input of ni and "0 to find the best values to fit thedata, it is not helpful from the predictive point of view. For the third possibility,it is pointed out in section 8.4 that our light spot sizes can be regarded as infiniteplane-parallel beams according to numerical simulations (Tsai and Pogorzelski,1975), no criterion of how RTE would work in the case studied is available to ourbest knowledge. The estimated filling factor 0.54 for this sample is well belowthe lower limit of the typical ‘random close packing’ value 0.6 (Torquato, 2002),thus the sampled scattering volume might not be statistically large enough andlocal packing structures could effect the scattering patterns. In fact it was foundthat the 600-μm spheres have larger sample-to-sample variations than the 200-μm spheres in repeated measurements. More experimental results are needed tosettle these questions.

8.5.3 Some discussions on controlled BRDF measurements

Many single-scattering features are retained in packed layers, even though thepacked surfaces have filling factors higher than 0.5, a considerable optical thick-ness, and single-scattering in general contributes only a small fraction to thetotal scattered power. Figure 8.9 shows the RF of the measurement data forthe 10 mm thick layer and the single-scattering approximation predicted by Eq.(8.34) for the 200-μm spheres at three incident angles. It can be seen that interms of intensity, the single-scattering RF is several tens of percent of the go-nio measurement within the rainbow and drops down to a few percent outsideof it. This low fraction remains quite flat until phase angle 100◦ (for 35◦ and60◦ incidences) where it starts to climb to about 10% and further to nearly70% around the grazing angle (for 60◦ incidence). However the RF minimum is

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Fig. 8.8. Comparisons of goniometric measurement, DISORT, LB model, HIMSA andHAMSA for a 15 mm thick (τ = 41.25) 600-μm sphere layer. Incident zenith angles are(a) 0◦ and (b) 60◦ (see Fig. 8.3 for configurations). Note the phase angle is displayedlogarithmically to better display the rainbow region.

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Fig. 8.9. Contributions of single-scattering (Eq. (8.34)) to total reflectance at (a)normal, (b) 35◦ and (c) 60◦ incidence for a 10 mm thick layer of the 200-μm spheres.Also shown is the gonio measurement. Note both axes are displayed logarithmically tobetter display the angular structures.

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on the order of 0.8. This may semi-quantitatively explain why, for the 200-μmspheres, the peaks in the Mie phase function around 40◦ phase angle have beentotally washed out while those around 100◦ are evident in the RF (Fig. 8.6). Thisalso shows that since both the rainbow and the grazing regions consist of largersingle-scattering contributions they are also more sensitive to surface roughnesscaused by packing structures.

The reflectance data for both samples (Fig. 8.7 and Fig. 8.8) show that duringthe progression from normal to oblique incidence, a peak in the forward directiongrows but is never as strong as the backscattering peak except at 60◦ incidence.This demonstrates that intrinsically forward-scattering particles, when in aggre-gate, can look backscattering in reflectance measurements (Mishchenko, 1994;Mishchenko and Macke, 1997), thus inverting reflectance data to retrieve single-scattering quantities should be done cautiously.

Even for the measurements for which the strict RTE has partial success(for the 200-μm spheres shown in Fig. 8.7), the backscattering peaks are about10% higher than predicted in the smallest phase angle region (∼8◦). Due tomechanical interference, the gonio device used in this measurement could onlydetect scattered radiance at phase angles larger than 7◦, thus opposition effectswhich are normally observed in the phase angle range of 2◦–7◦ were not detectedhere. This backscattering range is perhaps among the most poorly understoodin radiative transfer theory (Shkuratov et al., 2002, 2005), as the numericalRTE solutions do not agree with each other well in this region. Neither theHapke hotspot function (Eq. (8.13)) nor the Lumme–Bowell shadowing factor(Eq. (8.25)) can predict this enhancement. Since the LB shadowing factor ismonotonically decreasing from a value of 0.5 at 0◦ phase angle, it cannot increasethe RF values as predicted by DISORT. While Hapke’s hotspot function canincrease the RF value’s single-scattering portion at 0◦ phase angle by a factorup to 2, its B0 parameter in Eq. (8.13) appears to be hard to predict.

8.5.3.1 Diffraction

As introduced in section 8.3, both the Hapke and LB models treat diffraction asundistinguishable from the incident flux, thus the diffraction peak should be re-moved when using Mie phase functions. In order to evaluate the accuracy of thisassumption, we performed the so-called δ-N approximation computations (Wis-combe, 1977; Thomas and Stamnes, 1999). This operation separates the phasefunction p into the sum of a delta-function to replace the forward-scattering peakand a truncated phase function b∗ thus

p(cosΘ) ≈ 2ηδ(1 − cosΘ) + (1 − η)2N−1∑l=0

(2l + 1)b∗l Pl(cosΘ), (8.40)

whereb∗l =

bl − η

1 − η, (l = 0, ..., 2N − 1) (8.41)

η = b2N . (8.42)

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8 Bi-directional reflectance measurements of particulate surfaces 301

Since in this work the DISORT calculations were done with stream numbers of100 (see Table 8.1), here we take 2N = 100. In other words, we supply DISORTwith the input phase function

ptrunc(cosΘ) ≈99∑

l=0

(2l + 1)(bl − b100)Pl(cosΘ), (8.43)

but keep the optical thickness unchanged. This would have the effect of removingthe diffraction peak for a single particle.

Figure 8.10 is the comparison of DISORT, δ-N truncated Mie phase functionsupplied DISORT (DISORT Delta-N) and HAMSA for the 200-μm spheres. Thisexample demonstrates that (1) the HAMSA agrees well with the diffraction-removed numerical RTE solution over a rather large phase angle range – theimprovement is very good in the backward direction and the overall agreementis the best at 35◦-incidence; (2) treating diffraction as un-scattered may not bea good approximation for this specific example, as HAMSA overestimated theRF through much of the phase angle range for the 200-μm spheres. This specificexample shows that the diffraction peak may have a significant effect on themultiple-scattering RF.

We also applied the LB roughness correction to the single-scattering termin DISORT and found that although the correction factor further reduces theDISORT values in the backscattering direction, it improves the agreement inthe grazing angles (Zhang and Voss, 2005). Since single-scattering contributestens of percent in these two regions, applying the roughness correction to single-scattering changes the intensities significantly. A predictive enhanced backscat-tering theory is needed to accurately describe the hotspot in the measurements.

8.6 In situ BRDF measurements on benthic sedimentfloors

8.6.1 Typical features of benthic sediment BRDF

In situ BRDF measurements were carried out for submerged carbonate sedimentsat six sites in the vicinity of Lee Stocking Island, Bahamas, during the 1999Coastal Benthic Optical Properties (CoBoP) field experiment. These samplesare typical of carbonate sediments in shallow tropical coastal zones. BRDF datawere collected at each sediment site by averaging three to five measurements inwhich the instrument was rotated or moved to an adjacent spot for each separatemeasurement. Since a flat sample surface is required for the BRDF-meter, whentaking data, the instrument either sat on naturally flat floors (Norman’s Yellowand Norman’s White), locally flat long-wavelength wave structures (HorseshoeReef D and Ooid Shoal), or an artificially flattened surface (Horseshoe Reef C).The measured RF varied, ranging from nearly Lambertian to very anisotropic.To be representative, data chosen from two sites (Rainbow South and Norman’sYellow) showing two extremes are displayed here.

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302 Hao Zhang and Kenneth J. Voss

Fig. 8.10. Comparisons of HAMSA, DISORT and DISORT Delta-N for a 10 mm thick(τ = 88.5) 200-μm layer. Incident zenith angles are (a) 0◦. (b) 35◦ and (c) 60◦. Noteboth axes are displayed logarithmically to better display the angular structures.

Figure 8.11 shows the RF for the Rainbow South Sand sample at three illumi-nation angles (0◦, 35◦ and 65◦), for red (658 nm) and blue (475 nm) light, whileFig. 8.12 shows the associated plots of the normalized standard deviation. Fig-ures 8.13 and 8.14 are the corresponding graphs for Norman’s Yellow sediments.From Figure 8.11 it can be seen that at normal illumination the Rainbow SouthRF is nearly Lambertian, with less than 10% deviation from a perfectly diffuse

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8 Bi-directional reflectance measurements of particulate surfaces 303

Fig. 8.11. Contour plots of the RF of Rainbow South. In these plots the upper halfis light backscattered to the receiver, while the lower half is closer to the speculardirection. The left and right sides would be perfectly symmetric for a uniform surface;the deviation from this symmetry illustrates the non-uniformity of the surface.

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304 Hao Zhang and Kenneth J. Voss

Fig. 8.12. Normalized standard deviation of the RF of Rainbow South

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8 Bi-directional reflectance measurements of particulate surfaces 305

Fig. 8.13. Contour plots of the RF of Norman’s Yellow.

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306 Hao Zhang and Kenneth J. Voss

Fig. 8.14. Normalized standard deviation of the RF of Norman’s Yellow.

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8 Bi-directional reflectance measurements of particulate surfaces 307

Table 8.2. Sampling sites and sediment characteristics; ‘est.’ indicates visual estimate

Site Sediment description Average Commentsgrain size(dia. in mm)

Norman’s Yellow grapestone sand with 1.10 2 m water depth,thick yellow film yellow algal film on

sedimentNorman’s White grapestone sand 0.88 2 m water depth,

turbated area withlittle algae

Rainbow South migrating ooid sand 0.43 2 m water depth,high current, sandwaves

Ooid Shoal A migrating ooid sand 0.60 1–2 m water depth,high current, sandwaves

Ooid Shoal B migrating ooid sand 0.60 1–2 m water depth,high current, sandwaves

Horseshoe Reef A skeletal sand, with lots N/A 10 m water depth,broken shells with large sand

wavesHorseshoe Reef B skeletal sand, with brown N/A 10 m water depth,

film with large sandwaves

Horseshoe Reef C skeletal sand, molluscs 0.5 (est.) 10 m water depth,dominant (est.) grain size varies in

sand wavesHorseshoe Reef D skeletal sand, molluscs 1–2 (est.) 10 m water depth,

dominant (est.) grain size varies insand waves

reflector. When the illumination angle becomes oblique at 35◦ and 65◦, two non-Lambertian features appear in BRDF. The first is an enhanced reflectance inthe backward direction (toward the top), the ‘hotspot’. This is observed in manyparticulate surfaces (Hapke, 1993). This enhancement is often caused by the lackof shadows in the backward direction. The other feature is the enhancement inthe forward, or specular, reflectance. This feature is not present in most of thesamples that have been measured, but is obvious in this sample. It also appearsin some of the artificial particulate surfaces, such as spectralon (McGuckin et al.,1996; Bruegge et al., 2001). For the Rainbow South sample, the hotspot causedan enhancement of the reflectance by a factor of 2 between the backscatteringdirection and nadir, while the specular component is a 50% enhancement be-tween nadir and the specular maximum. Figure 8.12 shows that the standarddeviation between averaged samples is small, mostly less than 10%. This is acombination of measurement precision (for a very uniform, homogeneous sur-

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308 Hao Zhang and Kenneth J. Voss

face, such as the spectralon plaque, the precision error is less than 1%, as shownin Fig. 8.4), sample-to-sample variation, and sample measurement conditions(sample height, residual slope). Overall the data presented here are a very goodrepresentation of the Rainbow South RF, since even for θi = 65◦ (Fig. 8.12) thestandard deviation is less than 20%.

The Normans Yellow (Fig. 8.13) sample is among the sediments with largestgrain sizes. In the normally illuminated case, the first contrast with RainbowSouth sand is the much lower overall reflectance. This sample was visually darkerthan the Rainbow South sand, and, as can be seen, the RF values are much lower.The next difference between the two samples is that the RF is somewhat lessLambertian, even for data taken at normal incidence. As the sample grain sizeincreased there was a tendency for the RF to decrease with increasing θr. Even so,the decrease to the edge of the measurements is only approximately 0.06 in RF,or 25%. The RF for θi = 65◦ shows a common feature as the grain size increases.The hotspot has increased by a factor of 3 between θr = 0◦ and θr = 65◦ alongthe direction of φ=0. In this sample the reflectance in the specular directionhas not increased. With the larger grain size, and less homogeneous surface,the standard deviation between the individual measurement samples is larger.Although at normal incidence the standard deviation reaches 30% and at 35◦-and 65◦-incidence the standard deviation reaches 60% at the edge, the deviationsin the absolute reflectance values are still less than 0.07. So while there is greatervariability between the samples, the absolute magnitude of the variation is stillnot large. For the most part the standard deviation is approximately 10–20% fornormal illumination, and 20–30% for the illumination at 65◦.

As can be seen from the samples shown in Figs. 8.11 and 8.13, the absoluteRF values vary spectrally, but the shape of the BRDF does not. This feature ispresent for all samples and leads us to simplify the simple model fitting proce-dures described in subsection 8.6.2.

8.6.2 A simple model for sediment BRDF

In order to be incorporated into the radiative transfer models, the analytic ex-pression of sediment BRDF is needed and should be well-behaved up to 90◦ il-lumination and viewing zenith angles. We start with an empirical BRDF model(Walthall et al., 1985) which will be referred to as the Walthall Model (WM) inthis section:

RF (θi, θr, φ) = (C0 + C1θi + C2θ2i ) + (B0 + B1θi)θr cosφ

+(A0 + A1θi + A2θ2i )θ2

v, (8.44)

where all the angles are in degrees, and Ci, Bi, and Ai are constants empiricallyfit to the data. This equation can be fit to the data in a straightforward way(Zhang et al., 2003; Zhang, 2004). Since the hotspot was found throughout thedata set, an extra term describing the enhanced backscattering is added to theWM. This hotspot function is exponentially dependent on the phase angle g(Eq. (8.2)). The difference between the WM and the measured RF (θi, θr, φ) is

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8 Bi-directional reflectance measurements of particulate surfaces 309

calculated for each viewing and incident angle, then this residual is fit by theequation:

RF ′(θi, θr, φ) = W0 + W1 exp(−W2g) (8.45)

Since a hotspot is only evident in the data with incident angles ≥ 25◦, this fittingis done for θi ≥ 25◦. Once W0, W1 and W2 are found for each θi angle, they arefit with a linear equation which varies with θi. The final hotspot correction is:

RF ′(θi, θr, φ) = W00 + W01θi + (W10 + W11θi) exp(−(W20 + W21θi)g) (8.46)

Finally for one of the samples, the residuals showed a peak in the speculardirection which could be fit with another function of the same general shape asthe hotspot correction, but in the forward direction

RF”(θi, θr, φ) = (W30 + W31θi) exp(−(W40 + W41θi)g′) (8.47)

where g′ is the angle between the view direction and the specular direction givenby

g′ = cos−1(cos θi cos θr − sin θi sin θi cos(φr − φi)). (8.48)

For other samples, while a few seemed to have a small specular component,adding this function did not reduce the residuals. Finally the total RF is givenby

RF total = RF + RF ′ + RF ′′. (8.49)

RF data from the nine sites as listed in Table 8.2 are fitted with the above model.We found that the RF of a specific sample when normalized to some factor, suchas RF (θi = 0◦, θr = 45◦, φ) was spectrally invariant, within the experimentalerror (Zhang et al., 2003). Thus we normalized each data set by this factor,averaged the normalized color data together, and then fit the equation to thisdata set. We chose RF (θi = 0◦, θr = 45◦, φ) as the normalization parameterbecause this measurement can be obtained in great spectral detail by otherinstrumentation (e.g. Mazel, 1997). Table 8.3 shows the sample RF (0◦, 45◦, φ)used to normalize the data to find the above mentioned model parameters.

Table 8.3. RF(0◦, 45◦, φ) used in sample normalization

Sample Blue Red

Norman’s Yellow 0.121 0.218Norman’s White 0.299 0.390Rainbow South 0.420 0.488Ooid Shoal A 0.439 0.558Ooid Shoal B 0.441 0.560Horseshoe A 0.241 0.344Horseshoe B 0.283 0.392Horseshoe C 0.291 0.386Horseshoe D 0.209 0.418

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310 Hao Zhang and Kenneth J. Voss

Table 8.4. Model parameters for samples from nine sites

Parameter Norman’s Yellow Norman’s White Rainbow South Ooid Shoal A

A0 −5.44 × 10−5 −6.32 × 10−5 −8.99 × 10−7 −1.23 × 10−4

A1 5.58 × 10−7 8.73 × 10−7 9.84 × 10−8 6.00 × 10−7

A2 1.41 × 10−8 8.48 × 10−9 7.89 × 10−9 1.97 × 10−8

B0 −5.56 × 10−5 −1.89 × 10−5 −1.92 × 10−5 3.50 × 10−4

B1 9.51 × 10−5 9.21 × 10−5 4.29 × 10−5 1.69 × 10−4

C0 1.10 1.16 1.04 1.19C1 1.42 × 10−4 1.89 × 10−3 2.76 × 10−3 1.73 × 10−3

C2 −1.31 × 10−4 −1.06 × 10−4 −6.94 × 10−5 −2.22 × 10−4

W00 0.00 −3.41 × 10−1 −5.51 × 10−2 −2.32 × 10−1

W01 0.00 5.94 × 10−3 1.87 × 10−3 6.63 × 10−3

W10 −8.78 × 10−1 −8.53 × 10−1 0.00 −1.49W11 4.19 × 10−2 4.36 × 10−2 1.32 × 10−2 6.37 × 10−2

W20 3.00 × 10−2 8.84 × 10−3 7.76 × 10−2 4.92 × 10−2

W21 4.00 × 10−4 6.96 × 10−4 0.00 2.56 × 10−4

W30 −7.69 × 10−1 0.00 0.00 0.00W31 1.96 × 10−2 0.00 0.00 0.00W41 8.37 × 10−4 0.00 0.00 0.00

Parameter Ooid Horseshoe Horseshoe Horseshoe HorseshoeShoal B Reef A Reef B Reef C Reef D

A0 −5.53 × 10−6 −1.00 × 10−5 −3.26 × 10−5 −2.23 × 10−5 −3.95 × 10−5

A1 2.35 × 10−7 −4.84 × 10−7 6.24 × 10−7 4.92 × 10−7 5.64 × 10−7

A2 7.92 × 10−9 5.63 × 10−9 4.91 × 10−9 5.53 × 10−9 7.80 × 10−9

B0 5.58 × 10−5 −3.86 × 10−4 2.78 × 10−4 −4.32 × 10−5 −1.61 × 10−4

B1 3.95 × 10−5 6.58 × 10−5 5.53 × 10−5 6.94 × 10−5 6.22 × 10−5

C0 1.03 1.06 1.10 1.07 9.91 × 10−1

C1 3.61 × 10−4 3.22 × 10−3 1.03 × 10−3 2.81 × 10−3 −5.69 × 10−3

C2 −1.76 × 10−5 −1.10 × 10−4 −4.39 × 10−5 −9.55 × 10−5 −1.08 × 10−5

W00 −1.63 × 10−1 −4.28 × 10−2 −5.77 × 10−1 0.00 −1.04 × 10−1

W01 2.78 × 10−3 1.45 × 10−3 7.25 × 10−3 0.00 1.86 × 10−3

W10 0.00 0.00 0.00 −7.66 × 10−1 0.00W11 1.22 × 10−2 1.82 × 10−2 2.18 × 10−2 3.39 × 10−2 2.24 × 10−2

W20 0.00 6.25 × 10−2 0.00 4.92 × 10−2 1.37 × 10−2

W21 6.5 × 10−4 0.00 6.66 × 10−4 0.00 6.93 × 10−4

W30 0.00 0.00 0.00 0.00 0.00W31 0.00 0.00 0.00 0.00 0.00W41 0.00 0.00 0.00 0.00 0.00

The resulting model parameters for the samples are shown in Table 8.4.To reconstruct the measurements from the model parameters it is necessaryto insert the parameters from Table 8.4 in Eqs (8.45)–(8.48), then multiply bythe reflectance factors listed in Table 8.3. In general, for other wavelengths, ifRF (0◦, 45◦, φ) is known, then the BRDF can be determined for that wavelength.

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8 Bi-directional reflectance measurements of particulate surfaces 311

Fig. 8.15. Model reconstructed RF of the Rainbow South sample (red light).

To evaluate the performance of this simple model fit, we may look at boththe fitted data and the residuals, or the difference between the model and data,at the specific data points measured. As an example, the model reconstructedRF is shown in Fig. 8.15 and the percentage difference from measurements forred and blue colors of Rainbow South are shown in Fig. 8.16. It can be seen

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312 Hao Zhang and Kenneth J. Voss

Fig. 8.16. Average percentage difference at eight illumination angles for the RainbowSouth sample.

that for this sample the WM fitted RF represents the real RF quite well. Theresiduals are less than the standard deviation between the measurements. Thusthe empirical model does a very good job fitting the measured BRDF. For theRainbow South sample it can be seen that the residuals grow in the speculardirection. However we did not add a specular component to the model becauseit was found that this tended to increase the residuals elsewhere, thus reducingthe overall quality of the fit.

Figure 8.17 shows the plot of the model reconstructed Norman’s Yellow RFat 0◦ and 80◦ illuminations by red light. It is clear that the WM is well behavedat viewing angles up to 90◦ and at higher incident angles, thus it can be used inradiative transfer modeling of benthic surfaces (Mobley et al., 2003).

8.7 Effects of translucent grains and pore liquid complexrefractive index on particulate BRDF

In this section we demonstrate, with laboratory BRDF measurements, the effectsof translucent (not opaque) particle concentrations in a particulate layer andthe effects of wetting liquid absorption coefficient on the BRDF. Twomey et al.(1986) (TBM) used the enhancement of forward-scattering due to a reductionin refractive index contrast to explain the wetting induced darkening effect.Lekner and Dorf (1988) (LD) gave an alternate explanation based on Angstrom’sGO model (Angstrom, 1925). For the LD theory the diffuse reflectance froma rough surface is reflected at the air–liquid interface, and less light escapesbecause of total reflection. Both the TBM and the LD models qualitativelyexplain the darkening of sand and soil. As our samples cover a wide range ofoptical properties, from very low to very high albedos and from totally opaque

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8 Bi-directional reflectance measurements of particulate surfaces 313

Fig. 8.17. Model RF for Norman’s Yellow at normal and 80◦ incidences (red light).

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314 Hao Zhang and Kenneth J. Voss

to totally transparent grains, we attempted to differentiate physical effects ineach of the models and have found additional factors which are important incontrolling the surface brightness (Zhang and Voss, 2006).

8.7.1 Sample descriptions

The following samples have been used in measurements:

• Opaque shallow water sediments: ooid sands from near Lee Stocking Island,Exumas, Bahamas with smooth, round grains, a lustrous surface, and diame-ters between 0.25 and 0.5 mm (Sample A and Sample B); broken shells witha size distribution between 0.125 to 0.25 mm (Sample G); large (1–2 mm)and rough platelets (Rough).

• Beach sands and soil: volcanic black beach sand from the Big Island, Hawaii(Volcanic); sand from the beach at Crandon Park, Miami (Crandon); sandfrom the beach at the University of Miami’s Rosenstiel School of Marine andAtmosphere Science (RSMAS ), and soil particles from University of Miami’sGifford Arboretum (Soil). These samples were chosen because they containvaried concentrations of translucent particles.

• Broken glass: black silica sand used as cigarette urn sand at the Universityof Miami (Black Sand); non-absorbing broken glass obtained by crushingFisherbrand microscope glass slides (Glass) (Catalog number 12-550C).

These samples have plane albedo from nearly 0 (Volcanic) to about 0.7 (Sam-ple G). For Crandon, RSMAS and Soil particles, the grains were sieved to obtaina size distribution of 0.25–0.5 mm in diameter. For Volcanic and the Black Sandthe size selection was between 0.5 and 1 mm as this was the dominant size. ForGlass the grains were passed through a 1 mm mesh sieve, as this was the endmember of the Sample A–Glass mixture described in the next section.

8.7.2 Effects of translucent particle concentrations on wetting

Figure 8.18 shows the dry and water-wetted RF of three samples, SampleA, Sample G and Rough, at normal and 65◦-incidence. Positive phase anglescorrespond to 0◦ ≤ φ ≤ 180◦ and the negative phase angles correspond to180◦ ≤ φ ≤ 360◦. One can see that at normal illumination Sample A is the mostLambertian while Rough is the most anisotropic among the three. At 65◦ illumi-nation, both Sample A and Sample G show a forward-scattering peak in additionto the hotspot, while Rough has only a strong and broad hotspot. At normalincidence, wetting the surface reduces the overall reflectance and makes the sur-face more Lambertian. All three samples have small variations in wetted RFranging from 3% (Sample A) to 7% (Rough) for phase angles from 0◦ to 65◦. At65◦ incidence, wetting with water decreases the backscattering peak but causesa relative increases in the forward-scattering peak. Even for Rough the forwardscattering peak becomes larger than the hotspot. In agreement with the TBMtheory, wetting has increased forward-scattering and decreased backscattering.

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8 Bi-directional reflectance measurements of particulate surfaces 315

Fig. 8.18. Dry and water-wetted three benthic sediment samples at θi = 0◦ and 65◦.Open squares are dry and solid squares are water-wetted.

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316 Hao Zhang and Kenneth J. Voss

Fig. 8.19. Plane albedos of dry, water-wetted and glycerin-wetted Sample A.

Although significant changes occurred in the RF after wetting with water, thesurface brightness does not decrease significantly. According to TBM, wettingwith larger refractive index liquid should further darken the surface. In Fig. 8.19we plot Sample A’s dry, water- and glycerin-wetted directional plane albedos (Eq.(8.9)) at eight illumination angles. Obviously wetting with glycerin (n = 1.47)does not enhance the darkening effect compared with water, and only decreasesthe albedo by approximately 10%. Note the slightly higher albedo values of theglycerin-wetted layer are caused by the specular reflectance produced by residualglycerin on the layer surface, due to its viscid nature. According to TBM, surfaceswith either very high or very low albedos will have little wetting effect. However,Sample A has a dry plane albedo value of 0.6, and hence is expected to have amore appreciable wetting effect.

In Fig. 8.20 we display the dry, water-wetted and glycerin-wetted planealbedos of six samples containing quartz-like translucent particles. The translu-cent particle concentrations for Crandon, RSMAS and Soil are estimated fromoptical stereo-microscope images and are found to be 36%, 50% and 80%, respec-tively. Figure 8.20 clearly shows that (1) wetting with water has greatly reducedthe surface brightness and (2) except for Black Glass, the reduction dependson the wetting liquid refractive index. Wetting with glycerin caused a biggerdarkening effect than water for most of the samples.

To further understand the effect of the translucent particles, we dispersedGlass in Sample A with varied concentrations from 0 (pure Sample A) to 100%(pure Glass) by volume. In addition to dry and wet measurements, we also mea-sured the totally submerged BRDF, as opposed to simply wetting the surface.Totally submerged means that the samples are measured completely underwa-

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8 Bi-directional reflectance measurements of particulate surfaces 317

Fig. 8.20. Plane albedos of dry, water-wetted and glycerin-wetted six samples.

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318 Hao Zhang and Kenneth J. Voss

ter, with no water–air interface between the sample and the measuring device. Asubmerged sample would not have an obvious interface (thus LD would predictno change), but would still have an increased forward-scattering effect (TBM).

We first look at the glass concentration effects on wetting. Figure 8.21 showsthe RF at normal and 65◦ incident angles, for both dry and wet mixtures. Onecan see that the progression from pure Sample A to pure Glass are opposite fordry and wet mixtures. For the dry samples, increasing the Glass concentrationincreases the RF especially in the forward-scattering direction (g > 90◦, at 65◦-incidence). However, for the wet mixtures, increasing the Glass concentrationdecreases the RF. This can be better perceived by plotting the plane albedovariation versus Glass concentration at normal and 65◦ incidences, as shownin Fig. 8.22. Obviously, increasing glass concentrations can indeed lead to anenhanced darkening effect for the water-wetted mixture. For wet samples, a cleardecrease in the plane albedo from pure Sample A to pure Glass occurs at thetwo illumination angles. For the dry case, for concentrations of Glass < 50%, theplane albedo does not change significantly. Above 50% the plane albedo clearlyincreases. Thus the difference between wet and dry albedos clearly increases withincreasing translucent particle concentration.

We have found additional factors such as individual particle surface rough-ness play a role in the wetting effect, as simply knowing translucent particleconcentrations alone may not accurately predict the albedo of a mixture (Zhangand Voss, 2006).

As shown in Fig. 8.22, Sample A has a submerged plane albedo at bothnormal and 65◦-incidence that is 16% higher than the corresponding dry albedo.As the glass concentration increases, the submerged brightness decreases, the‘cross-over’ point is at 80% for normal and 90% for 65◦ incidence, respectively.This higher submerged plane albedo effect was verified with more opaque benthicsamples (Zhang and Voss, 2006) and is similar to the effect seen with a spectralonplate (Voss and Zhang, 2006). Our measurements of a submerged spectralonplaque also show that the RF (θi = 0◦, θv < 55◦) is higher when the plaqueis submerged than when dry. Qualitatively, we explain this effect in terms ofthe immersion effect seen when plastic diffusers are placed in the water (Tylerand Smith, 1970). In this case there is an increase in the amount of radiancebackscattered into the medium, because of the decrease in reflectance at theinterface between the particulate surface and the medium (water). It is likely thisalso causes the change in the shape of the RF, as radiance which has reflectedback into the medium is not available to be backscattered into the particulatesurface and then re-emitted at larger angles.

8.7.3 Effects of the wetting liquid absorption coefficient

To quantify the wetting liquid absorption coefficient effects on BRDF, we wettedseveral typical sediment samples using absorbing gel with nine different dye con-centrations. These two samples are chosen because, as shown in subsection 8.7.2,they represent grains having small and big wetting effects. The dye liquids had

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8 Bi-directional reflectance measurements of particulate surfaces 319

0.7

0.66

0.64

0.64

0.62

0.6

0.58

0.56

0.56

0.86

0.8

0.8

0.8

0.74

0.74

0.68 0.68

0.68

0.62

0.62

0.56 0.56

0.56

0.52

0.5

0.48

0.46

0.44

0.42 0.7

0.64

0.64

0.64

0.58

0.58 0.52

0.52

0.52

0.46

0.4

0.4

Fig. 8.21. RF versus the Glass concentration and phase angle: (a) normal incidence,dry; (b) normal incidence, wet; (c) 65◦ incidence, dry; and (d) 65◦ incidence, wet.

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320 Hao Zhang and Kenneth J. Voss

Fig. 8.22. Dry, water-wetted and submerged in water plane albedos of Sample A–Glassmixtures: (a) normal incidence and (b) 65◦-incidence.

absorption coefficients between 0.254 cm−1 and 40.80 cm−1 at 657 nm (Voss andZhang, 2007).

The RF for Sample B when dry, water-wetted, and wetted by three absorbingsolutions at normal and 65◦ incidences are displayed in Fig. 8.23. At normalincidence, when going from dry to wet, the RF decreases by approximately 15%and becomes more Lambertian. When the absorption coefficient of the wettingliquid increases, the RF decreases and becomes more Lambertian until at very

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8 Bi-directional reflectance measurements of particulate surfaces 321

0.60

0.55

0.50

0.45

0.40

0.35

0.30

RF(

θ i= 0

¡,θv,g

)

-40 0 40Phase angle (deg)

1.0

0.8

0.6

0.4

0.2R

F(θ i=

65¡

,θv,g

)100500-50-100

Phase angle (deg)

1.06

1.04

1.02

1.00

0.98

0.96

0.94

0.92

RF(

θ i= 0

¡,θv,g

)/RF(

θ i= 0

¡,θv=

5¡,g

)

-40 0 40Phase angle (deg)

1.4

1.2

1.0

0.8

0.6

RF(

θ i= 6

5¡,θv,g

)/RF(

θ i= 6

5¡,θv=

5¡,g

)

100500-50-100Phase angle (deg)

Fig. 8.23. The RF of Sample B when dry (open circles), wetted with water (opensquares) and wetted with three solutions with absorption coefficients of 6.5 cm−1 (tri-angles), 10.49 cm−1 (solid squares) and 40.80 cm−1 (solid circles). (a) Normal incidence,(b) 65◦-incidence, (c) normal incidence and normalized, (d) 65◦-incidence and normal-ized. Positive phase angles correspond to 0◦ ≤ φ ≤ 180◦ and the negative phase anglescorrespond to 180◦ ≤ φ ≤ 360◦.

large absorption (40.80 cm−1) the RF at nadir is lower at larger viewing anglesby 5%. To better display this feature we normalized the RF’s to the RF value ata minimum phase angle. Specifically, for normal incidence we picked up a viewingfiber located at 5◦ zenith and −135◦ azimuth which gives a phase angle of 5◦;for 65◦ incidence, we picked up the fiber at 65◦ zenith and −5◦ azimuth which

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322 Hao Zhang and Kenneth J. Voss

RF(

θ i= 0

¡,θv,g

)

RF(

θ i= 6

5¡,θv,g

)

RF(

θ i= 0

¡,θv,g

)/RF(

θ i= 0

¡,θv=

5¡,g

)

RF(

θ i= 6

5¡,θv,g

)/RF(

θ i= 6

5¡,θv=

5¡,g

)

Fig. 8.24. Same as Fig. 8.23, but for Crandon.

gives a phase angle −4.5◦. Figure 8.23(c) and (d) show the normalized RF at 0◦-and 65◦-incidences, respectively. At 0◦ incidence this increased side-scatteringeffect has not been previously observed in our field or laboratory BRDF data.At 65◦-incidence, wetting by water makes the RF forward-scattering in contrastto the backscattering dry RF; however, increasing the absorption coefficient of

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8 Bi-directional reflectance measurements of particulate surfaces 323

Fig. 8.25. Plane albedo at θi = 60◦ of Sample B and Crandon versus the wettingliquid absorption coefficient.

the wetting liquid only seems to lower the overall RF values but preserves theangular pattern (Fig. 8.23(b) and (d)).

Measurement results for Crandon which contains about 36% translucentquartz-like grains are shown in Fig. 8.24. One can see that the additional darken-ing effect caused by absorbing liquids is not as significant as in Sample B. Whengoing from dry to water-wetted, the surface brightness has already decreasedby nearly 50%. Thus wetting the surface with more absorbing solutions doesnot cause as great an additional significant effect. However, the increased sidescattering at normal incidence is still obvious though not as great as Sample B(see Fig. 8.23).

The relationships between the plane albedo at θi = 60◦, A(60◦), and wet-ting liquid absorption coefficient for Sample B and for Crandon are plotted inFig. 8.25. Hapke (1993) showed that A(60◦) is approximately the same as thediffuse reflectance. It is seen that for Sample B A(60◦) decreases nonlinearly asthe wetting liquid absorption coefficient increases; for Crandon, the decrease isalso nonlinear although the greater wetting-induced darkening effect makes thisnonlinearity less significant.

We found that the behavior of A(60◦) versus the absorption coefficient inFig. 8.25 was similar to r0 versus the particulate bulk absorption coefficient inradiative transfer calculations of particulate layers of spherical grains (Voss andZhang, 2007). This similarity indicates that it would be difficult to separatethe effects of particle bulk absorption coefficient and the pore liquid absorptioncoefficient from the plane albedo data alone.

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324 Hao Zhang and Kenneth J. Voss

For benthic sediments, we have found that only the first few layers of asediment determine the RF and plane albedo of a surface (Zhang et al., 2003). Asseen here, the absorption coefficient of the interstitial material must be very high(> 100 m−1) to have a significant effect. Thus it is unlikely that the plane albedo,or RF, of a surface would provide unambiguous information on the absorptioncoefficient of the interstitial liquid in a natural sediment. Conversely, for liquidswith absorption coefficients <10 m−1, one needs not consider the absorptioncoefficient of the interstitial liquid when modeling or predicting the reflectancefrom natural surfaces.

8.8 Concluding remarks

The controlled laboratory measurements of the BRDF on near-monodispersespherical particles show that when intrinsically forward-scattering particles areaggregated they can look very backscattering in BRDF measurements. Thusapplying approximate RTE models to retrieve single-scattering properties fromreflectance measurements should be done with caution. We also found that nu-merical solutions of the RTE (DISORT and MBRF) can predict the BRDF overa large phase angle range, especially at oblique incidence, for 200 μm diameterpolymer spheres. However the backscattering direction cannot be predicted wellby any of the RTE models. Prominent features of single-scattering, such as therainbow, are preserved even when the closely packed sphere layers have fillingfactors larger than 0.5. However, multiple-scattering tends to wash out some ofthe smaller sharp features present in single-scattering.

When benthic sediment samples are illuminated normal to the surface theyappear to be almost Lambertian. The deviation from a Lambertian reflector isbiggest for the sediments with the larger grain sizes. As the illumination anglemoves from the normal the samples become increasingly non-Lambertian. Thedominant feature in the BRDF is the hotspot, or the enhanced backscattering.Here the BRDF increases by varying degrees over the nadir value, with the largesthotspots occurring for the large sizes of particles. The hotspot magnitude alsoincreases with increasing illumination angles. We provide an empirical model forsediment particles that has been shown to represent the sediment data withina standard deviation of the sample variation. This model is well behaved atviewing angles out to 90◦, and thus can be used in radiative transfer modelsto describe benthic reflectance. With this model a realistic bottom reflectancecan be incorporated into the radiative transfer algorithm to improve the lightfield predictions in shallow waters. Since measurements shown here cover a rangeof sediment types and sizes, the model could be applied in other environmentswhere the sediment has similar physical properties.

Through the series of measurements of wetted, submerged, and dry partic-ulate surfaces we have shown that the BRDF/plane albedo depends on manyfactors and the wetting effect cannot simply be attributed to the particle/wettingliquid refractive index contrast. Two additional parameters, the fraction of

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translucent particle concentrations and the individual grain microscopic sur-face roughness, must also be considered in natural samples, and can be veryimportant. Specifically, the more translucent particles in a particulate layer, thebrighter it is when dry, and the darker it is when wet. Thus the translucent par-ticle concentration can be a large proportion of the wetting-induced darkeningeffect. The results may have applications in remote sensing, as when a particulatesurface is seen to have significant variation in reflectance between dry and wet,it may indicate that such a surface contains translucent particles such as quartzgrains. The surface roughness of the particles is also important, as low-albedoparticles with a rough surfaces tend to have a larger wetting effect.

The overall effects of wetting with an absorbing liquid are to decrease theplane albedo and make the surface appear more Lambertian. Since for opaquesediments the first few layers determine the BRDF and plane albedo of a surface,the absorption coefficient of the interstitial material must be very high (> 100m−1) to have a significant effect. Thus it is unlikely that the BRDF of a surfacewould provide unambiguous information on the absorption coefficient of theliquid in a natural sediment. Conversely, for liquids with an absorption coefficient<10 m−1, one need not consider the absorption coefficient of the interstitial liquidwhen modeling or predicting the reflectance from natural surfaces. Modelingefforts on multiple-scattering of light by particles in an absorbing light scatteringmedium are needed to quantitatively explain these phenomena.

Acknowledgement

The authors are indebted to the late Albert Chapin for help in instrumentationand measurement. This work was supported by the Office of Naval ResearchOcean Optics program.

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Angstrom, A., 1925: The albedo of various surfaces of ground, Geogr. Ann., 7, 323–342.Bowell, E., B. Hapke, K. Lumme, J. Peltoniemi and A. W. Harris, 1989: Application

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Hapke, B., 2002: Bidirectional reflectance spectroscopy 5. The coherent backscatteropposition effect and anisotropic scattering, Icarus, 157, 523–534.

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Lekner, J. and M. C. Dorf, 1988: Why some things are darker when wet, Appl. Optics,27, 1278–1280.

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Mazel, C. H., 1997: Diver-operated instrument for in situ measurement of spectralfluorescence and reflectance of benthic marine organisms and substrates, Opt. Eng.,36, 2612–2617.

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Mishchenko, M. I., 1994: Asymmetry parameters of the phase function for denselypacked scattering grains, J. Quant. Spectrosc. Radiat. Transfer, 52, 95–110.

Mishchenko, M. I., and A. Macke, 1997: Asymmetry parameters of the phase functionfor isolated and densely packed spherical particles with multiple internal inclusionin the geometric optics limit, J. Quant. Spectrosc. Radiat. Transfer, 57, 767–794.

Mishchenko, M. I., J. M. Dlugach, E. G. Yanovitskij and N. T. Zakharova, 1999: Bidi-rectional reflectance of flat, optically thick particulate layers: an efficient radiativetransfer solution and applications to snow and soil surfaces, J. Quant. Spectrosc.Radiat. Transfer, 63, 409–432.

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Mishchenko, M. I., L. D. Travis and A. A. Lacis, 2006: Multiple Scattering of Light byParticles: Radiative Transfer and Coherent Backscattering, New York: CambridgeUniversity Press.

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Royer, A., P. Vincent and F. Bonn, 1985: Evaluation and correction of viewing angleeffects on satellite measurements of bidirectional reflectance, Photogramm. Eng.Remote Sens., 51, 1899–1914.

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Shkuratov Yu. G. and Y. S. Grynko, 2005: Light scattering by media composed of semi-transparent particles of different shapes in ray optics approximation: consequencesfor spectroscopy, photometry, and polarimetry of planetary regoliths, Icarus, 173,16–28.

Shkuratov, Yu. G., K. Muinonen, E. Bowell, K. Lumme, J. I. Peltoniemi, M. A.Kreslavsky, D. G. Stankevich, V. P. Tishkovetz, N. V. Opanasenko, L. Y. Melku-mova, 1994: A critical-review of theoretical-models of negatively polarized-light scat-tered by atmosphereless solar-system bodies, Earth, Moon and Planets, 65, 201–246.

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Shkuratov Yu. G., A. Ovcharenko, E. Zubko, O. Miloslavskaya, R. Nelson, W. Smythe,K. Muinonen, J. Piironen, V. Rosenbush and P. Helfenstein, 2002: The oppositioneffect and negative polarization of structural analogs for planetary regoliths, Icarus,159, 396–416.

Shkuratov, Yu. G., D. G. Stankevich, D. V. Petrov, P. C. Pinet, A. M. Cord, Y. H.Daydou and S. D. Chevrel, 2005: Interpreting photometry of regolith-like surfaceswith different topographies: Shadowing and multiple-scattering, Icarus, 173, 3–15.

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Effects of pore liquid absorption coefficient, J. Quant. Spectrosc. Radiat. Transfer,105, 405–413.

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9 Light scattering from particulate surfaces ingeometrical optics approximation

Yevgen Grynko and Yuriy G. Shkuratov

9.1 Introduction

Measurements of light scattered from particulate surfaces provide informationabout the composition and structure of the surfaces. An obvious way to charac-terize the scattering properties is to consider how the brightness and polarizationof scattering depend on the wavelength λ of incident light and the geometry ofobservations. The geometry is often characterized by the phase angle α whichis defined as the source-object-detector angle. Instead of α the scattering angleθ = π − α is used also. The plane defined by the light source, scattering objectand detector is called the scattering plane. The method of optical remote sens-ing of particulate surfaces is based on the measurements of the characteristicsas functions of λ and α. The problem of theoretical interpretation of this kindof data is not solved at present. Numerical modeling based on the geometricoptics (GO) approximation can be efficient for some practical applications. Inchapter we give an introduction to the past and current status of the theoreticalmethods and GO simulation results achieved for media consisting of particleslarge compared to the wavelength of incident light.

9.1.1 Practical tasks in optical remote sensing

Remote observations of natural surfaces are conducted in different ways: fromspacecrafts and with ground-based facilities. Multiangular and multispectraldata are collected and interpreted then to retrieve the physical and chemicalproperties of the surface. However, the problem of interpretation is inverse andill-posed and is not solved at present. In order to interpret the observational dataone should conduct exhaustive measurements and have an adequate theoreticalmodel of light scattering by a given surface. As this condition is not fulfilledthe existing remote sensing data for the Earth and the Solar System bodies re-main ambiguous from the diagnostic point of view. Therefore, it is importantto develop a rigorous model of light scattering from a particulate surface andsimultaneously to examine the applicability of the existing approximations.

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330 Yevgen Grynko and Yuriy G. Shkuratov

Fig. 9.1. Microphotographs of lunar regolith particles: agglutinate (a), orthopyroxenegrain (b), pieces of glass (c, d) [50].

The main mechanisms which determine angular dependencies of photometricand polarimetric parameters of particulate surfaces are the shadow-hiding effect,single scattering, and coherent and incoherent multiple scattering. Each of themplays a different role at different phase angles. Development of a rigorous modelcombining all scattering mechanisms appears to be a too difficult task. However,a lot of theoretical work has been done to study these factors separately. So fara satisfactory theory of multiple scattering exists only for media consisting of in-dependent scatterers with known scattering indicatrices [24]. There are attemptsto develop a theory for densely packed media formed by spheres with sizes com-parable with the wavelength of light (e.g., [77]). Unfortunately such models aretoo complicated and cumbersome to be used in practice. In addition, the sphereis not an appropriate shape to approximate natural particles that have randomshape and external and internal structure.

GO approximation allows the construction of a simple but powerful modelfree of many shortcomings. The only limitation arises from the condition for theparticle size and surface curvature radius: they must be much larger than thewavelength of incoming radiation.

There are many examples where average sizes of particles of natural mediaare large as compared to the wavelength. Some of the Earth soils and sandsare presented by large particles with random irregular shape and with differentvalues of optical constants. Various types of snow also are mixtures of largeregular crystals or irregular ice grains. Aerosol components of the Earth andplanetary atmospheres can contain large particles of irregular as well as regular(e.g., transparent ice crystals) shape.

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9 Light scattering from particulate surfaces 331

Fig. 9.2. Fragment of a microphotograph of Martian surface. Crop size is ≈ 1.5 cm(http : //marsrovers.jpl.nasa.gov).

Similarly, the surfaces of planets and their satellites are covered with regolith.This is a porous layer of particles, which is a result of continuous meteoriticbombardment of the primordial surface.

Lunar regolith is the most studied type of an extraterrestrial soil. Samplesof lunar regolith have been delivered with space missions. Laboratory analysishas shown that the characteristic size of the lunar mineral and glassy grains(Fig. 9.1) is ≈ 100 μm [50] (with a maximum of the mass distribution around60–70 μm [37]). Another example is the Martian sands. Recently Mars roversSpirit and Opportunity equipped with microscopes obtained a series of imagesof the Martian surface at small scales. The characteristic particle sizes appearedto be of the order of tens of micrometers (Fig. 9.2). Indirect estimations from thedata of the ground-based spectroscopic observations of Mercury show averageparticle size on its surface of around 30 μm [80].

The examples presented give a ground for the use of GO as a first approxi-mation in the theoretical analysis of light scattering in particulate media. This isthe only computational method applicable for particles without upper size limit.

Phase dependencies of brightness and polarization obtained at observationsof natural surfaces have details that can be used for retrieving the properties ofparticulate surfaces and constituent particles. First of all it is a backscatteringeffect expressed in the brightness increase with decreasing phase angle at moder-

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332 Yevgen Grynko and Yuriy G. Shkuratov

Fig. 9.3. Integral phase functions of intensity and polarization for the Moon [64].

ate and small α (see Fig. 9.3). Linear polarization degree demonstrates negativebranch at small phase angles. Laboratory measurements of powder samples showthat such observable parameters as the half-width of the brightness oppositionpeak, the inversion angle of polarization αinv and the minimum value of the neg-ative polarization |Pmin| strongly depend on the physical properties of particlesand structure of the surface. A detailed analysis of the observational data nearopposition can be found in a recent review [64].

Polarization at large phase angles also contains information about the surfaceproperties. In this case the diagnostics is based on the Umov effect that is ananti-correlation between albedo A and polarization degree near the polarizationmaximum Pmax. Such an anti-correlation is typical for a wide set of natural andartificial dielectric particular surfaces. Photopolarimetric observations of lunarsurface at large phase angles show almost linear correlation between log Pmax

and log A [10,11,54]. The deviations from the general trend caused by the varia-tions of the surface properties can be employed for remote sensing analyses [54].Another important parameter is the position of the polarization maximum αmax

which also changes for different regions on the lunar surface [10].One of important objectives of GO simulations is a detailed analysis of pho-

tometric and polarimetric phase curves for particulate surfaces. The simulationsin the range of small phase angles are of special interest, as they can clarify therelative influence of shadowing and incoherent multiple scattering on the bright-ness opposition surge. Umov effect diagnostics can also be based on the lightscattering modeling.

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9 Light scattering from particulate surfaces 333

A wide field for modeling applications is interpretation of the reflectance spec-troscopy data. This is one of the most important methods in the remote sensingof the Solar System bodies. It is based on the detection of the reflectance spectraof surfaces with different mineralogical composition. Parameters of the absorp-tion bands specific for different materials as well as overall spectral slope arerelated to composition. GO light scattering modeling appears to be a powerfultool for analysis and interpretation of such parameters. For instance, reflectancespectroscopy is applied for estimates of the optical constants of the materialsof particulate surfaces (e.g., [31]). The obtained wavelength dependencies of thereal and imaginary parts of the complex refractive index can be used then todetermine concentrations of chemical elements. Laboratory measurements pro-vide connection of these characteristics with the optical constants of the analogminerals. Theoretical modeling helps to make a transfer from reflectance to thevalues of the complex refractive index and particle sizes.

9.1.2 Principle and history of the ray tracing method

The basis of the GO approach is very simple. It is assumed that the energyincident on a particle splits into reflected and refracted components accordingto Fresnel’s equations. Formulation of the main points of GO can be foundelsewhere (see, for example, [88]) Obviously computer modeling based on raytracing is an efficient way to study light scattering in the GO approximation.

Ray tracing was used for the first time by Descartes in the 17th centuryto understand the formation of the primary and secondary rainbows. A typi-cal ray tracing computer algorithm begins by defining a particle or a spatialarrangement of particles in the computer memory. The particle sizes, shapes,optical constants, and their spatial orientations are given. Then the system is‘illuminated’ with a large number of rays. These rays are multiply reflected andrefracted on their way. Besides, they can be partly absorbed within the particle,if it is semitransparent. Each ray is traced from the first to the last interactionwith the particulate surface. The interaction is described by means of the Fresnelformulas and Snell’s law, though, in principle, more complicated scattering phe-nomena, like diffraction on particle facets [5] or interference of waves [85,86] canalso be taken into account. Note, that for large particles Fraunhofer diffractionis important only in the small-angle region (θ ≤ λ/r, where θ is the scatteringangle, λ is the wavelength, r is the particle radius).

However, even with such simplifications the light scattering problem remainscomplex enough. There have been several attempts of analytical studies of mul-tiple scattering inside large semitransparent particles with irregular shape, butthey all are approximate. Using shadowing theory, Schiffer and Thielheim [51]developed simplified analytical forms for some components of scattering: singleexternal reflection, multiple scattering from a rough surface and light transmit-ted by a particle. Mukai et al. [44] assumed large absorbing spherical particlewith rough surface. Using 1-D radiative transfer theory, they considered multi-ple scattering by a surface and integrated the result over the sphere. In recentdecades many papers have been devoted to the computer ray tracing simulations

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334 Yevgen Grynko and Yuriy G. Shkuratov

of scattering by single particles. The first results for the scattering indicatrix ofregular crystals were obtained in [7, 26, 29, 30, 35, 41, 76, 81, 82]. Computer simu-lations of scattering by particles with irregular shapes have been carried out in[14,16,17,42,47].

In [34, 39] large particles with multiple randomly positioned inclusions wereconsidered. In these models the inclusions scatter light isotropically or accordingto Mie theory, while rays are traced in the usual way. Yang and Liou [85–87]developed a hybrid method for scattering by hexagonal ice crystals, incorporatingwave interference in the GO approach. Their model uses ray tracing technique tofind the near field on the particle surface or in the volume within the particle. Forthe traced rays their phases are kept and the interference of different trajectoriesis taken into account. The near field is transformed then to the far field onthe basis of the electromagnetic equivalence theorem. The method gives goodagreement with the exact solution (finite-difference time domain method) forthe phase function at size parameters x = 2πr/λ larger than ≈ 20, though itis very time-consuming and was realized so far only for simple regular shapes.In [5] and [25] ray tracing models taking into account diffraction on facets wereproposed.

Geometric optics was also applied to fit experimental measurements of thesingle scattering matrix for some dust samples. In [79] scattering angle depen-dences of non-zero scattering matrix elements of mineral aerosols (red clay,quartz, volcanic ash) were fitted with curves calculated in the GO approxima-tion using Gaussian random shapes. In [45] a GO model incorporating randomGaussian shapes was used to fit the experimental data for Saharan dust particles.

At the present time investigations of light scattering in the geometric opticsapproximation by means of computer modeling are being actively developed notonly for studying of individual particle properties, but also for the simulation ofthe scattering characteristics of aggregate particles [20,43,73] and rough surfaceswith different structures and topographies [15,18,19,46,48,62,63,65,69–74].

9.1.3 Problems of analytical accounting for multiple scattering inparticulate media

If the number of particles in the scattering system is more than one then mul-tiple scattering component must be taken into account. For particulate surfacesformed by high albedo constituent particles multiple scattering plays an impor-tant role and must be calculated as accurately as possible. Despite the fact thatthere are efficient and accurate methods for scattering by single particles withsizes up to several wavelengths there is no exact solution for scattering in denseparticulate media with any size of particles. Usually radiative transfer theory isused as an approximation [24]. In general, an extinction coefficient E and angularscattering coefficient G(α) in the radiative transfer equation (RTE) are intro-duced. E describes extinction of radiation during propagation in the medium.G(α) represents angular dependence of scattering by elementary volume. If thesevalues are determined, the RTE can be solved.

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9 Light scattering from particulate surfaces 335

Strictly speaking, the RTE approach is hardly applicable to dense media. Thesingle scattering indicatrix is identified as the indicatrix of an isolated particle,which is by definition determined at infinitely distant light source and receiver.However at distances comparable with the particle size, the spatial distributionof the scattered field is different from that at infinity. In natural powders particlesare packed, having little space between each other. The second problem is thecase of surfaces with low albedo, when the main contribution in the scatteredflux is given from the few upper layers of particulate medium. Under such acondition it is difficult to define the elementary volume dV which is necessaryfor application of the classical radiative transfer theory. Another shortcoming isthe set of parameters in the RTE models. They describe optical properties of themedium volume but not the material of particles. One should make additionalnon-trivial steps to establish a connection between the properties of the mediumand the material (e.g., [31, 68]).

One of the earliest radiative transfer theories dealing with the parameters ofthe particle substance is the model of multiple reflections and transmissions in apile of semi-transparent slabs. This model is one-dimensional and allows calcula-tion of integral spectrophotometric properties of particulate surfaces. The idea isto approximate multiple scattering in a realistic medium with light propagationin the pile with slab width equal to characteristic particle size. Equations for thereflectance and transmittance of the pile were derived by Stokes [75]. This modelwas used to describe light scattering by powder-like media at the condition thatthe width of the plates is approximately equal to the average size of the powderparticles (e.g., [3]). Later this 1-D model was substantially refined. For example,Fresnel’s coefficients, averaged out over scattering angles from 0◦ to 90◦, wereused to characterize the reflectance and transmittance for interfaces in the pile[23, 38, 53, 61]. This takes into account the total internal reflections that occurin real semitransparent particles. To show the simplicity of the 1-D analyticalapproach we quote here the resulting formula for the integral albedo derived in[61]:

A =1 + ρ2

b − ρ2f

2ρb−

√√√√(1 + ρ2b − ρ2

f

2ρb

)2

− 1 (9.1)

where ρb and ρf are functions of the optical constants of material and size andpacking density of particles. An examination of the 1-D radiative transfer modelby means of comparison with 3-D ray tracing has been made in [65]. This testhas proved its practical usefulness.

It is clear that 1-D models cannot provide angular dependencies of intensityof scattered light. That is why it is hard to apply the 1-D albedo in the analysis ofthree-dimensional problems of light scattering. The 1-D albedo is often identifiedwith the integral (hemispherical) reflection coefficient (albedo) of particulatesurfaces.

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336 Yevgen Grynko and Yuriy G. Shkuratov

9.1.4 Range of applicability of ray tracing

The great advantage of the GO approximation is that it has almost no limita-tions on the particle shape. However, its range of applicability is limited by theminimum size parameter xmin and requirements to the curvature radii of thesurface. The question of how xmin depends on different factors remains open.One can easily estimate xmin for perfect spheres by means of the Mie theory. Fora good agreement of phase functions xmin must be of the order of several hun-dreds [22]. However, GO appears to be more appropriate for smaller nonsphericalparticles than spherical ones. For spherical or other highly symmetric shapes in-terference effects are much more important than in case of an arbitrary-shapedparticle; they are not included in the GO. So far the GO approximation has beenchecked against T-matrix computations of regularly shaped particles: randomlyoriented spheroids and circular cylinders. Macke et al. [33] considered prolateand oblate spheroids of different sizes from x = 10 to 60 with complex refractiveindex m = 1.394 + i0.00684. It was found that for a size parameter of 60, GOand T-matrix phase functions agree fairly well over the entire range of scatter-ing angles. At the same time linear polarization appears to be more sensitiveto the errors introduced by GO approximation. At x = 60 spheroidal polariza-tion curves show significant difference at intermediate phase angles. Wielaard etal. [83] modified T-matrix approach so that efficient computations of scatteringby nonspherical particles with size parameters exceeding 100 became possible.With the improved model they considered two types of shape: randomly ori-ented monodisperse oblate spheroids (x = 85) and circular cylinders (x = 125).Calculations were made for non-absorbing (m = 1.31+ i0) and weakly absorbing(m = 1.31+ i0.003) particles. The authors conclude that from the phase functionpoint of view the considered particles are already in the GO domain. GO andT-matrix differences in the scattering matrix elements F22, F33, F44, F12 andF34 are noticeably larger than those in the phase function. Introduction of weakabsorption at the same value of the size parameter improves agreement. In gen-eral, approximate (GO) and accurate (T-matrix) computations for the cylindersare in better agreement than those for spheroids.

One can see that increasing particle irregularity tends to make ray tracingcalculations more accurate for smaller particles. Most exact methods for lightscattering are restricted to quite simple particle geometries and for truly irregularshapes such a test has not been done yet. The case of medium can be evenless sensitive to the GO shortcomings as multiple scattering tends to smoothout peculiar effects produced by single particles. However, this statement is notproved yet either.

Up to now the attempts at theoretical interpretation of the observationalremote sensing data have been based on rather rough assumptions and approx-imations. On the other hand, in last decade a large number of theoretical andexperimental papers have demonstrated a significant progress in the field of lightscattering by isolated particles and particulate media. In this review we collectrecent results on the photometry, polarimetry and spectroscopy of particulate

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surfaces in the GO approximation in order to give a general view of the problemfrom which one can decide on the future theoretical studies.

In section 9.2 we describe an example of a geometrical optics computer modelincluding methods of the medium generation and ray tracing. In sections 9.3, 9.4and 9.5 we present a series of results of simulations carried out with the modelfor, respectively, the shadow-hiding effect and single and multiple incoherentscattering in particulate media with different parameters.

9.2 Computer modeling

Aspects of the computer realization of a complete ray tracing model are consid-ered in this section. We present an algorithm for the generation of a particulatemedium consisting of particles with random irregular shape. Then we describea procedure of ray tracing in a semi-infinite medium followed by a comparisonof the modeling results and laboratory photopolarimetric measurements.

9.2.1 Particulate medium generation and description of irregularshapes

The generation and quantitative description of arbitrary-shaped particles is oneof the most important problems in computer modeling. There are many classesof random shapes. For instance, this can be a prism with random lengths of thediagonals, or a roughly faceted ellipsoid, or a particle with a very irregular shapeclose to an aggregate of a few glued particles, or a sphere ‘spoiled’ with randomfluctuations of its radius and so on. The latter is the so-called Gaussian particledescribed in [42, 47] and is used at present by different authors (e.g., [45, 46]).The Gaussian particles are assumed to be stochastically deformed spheres, theradius of which follows to the log-normal statistic law; the correlation function ofdistances between the particle conventional center and its surface is also definedto be Gaussian. There is also a class of particles with a regular shape in average,but with the surface complicated by small-scale ripples defined by a randomfunction.

Different random shape classes may show different optical properties. Thatis why choosing the model for particles in each specific problem, one should beguided by a priori data or, at least, by estimates of the shape class of studiedparticles. In the case of quasi-regular particles that are close to spheres, ellipsoids,and cubes, the particles can be described analytically and then approximatedby a different number of triangular facets.

For particles with random irregular shape which is the case for constituents ofnatural powders the generating technique is more complicated. To generate themwe use a method based on the utilization of a random Gaussian field (RGF).The method is described below in detail.

At first, a 3-D random field h(x, y, z) with certain properties, which arecharacterized by the probability distribution W(h) and the correlation function

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338 Yevgen Grynko and Yuriy G. Shkuratov

Fig. 9.4. A sample of 3-D discrete volume filled with particles.

q(Δx, Δy, Δz) is generated in the computer memory. For example, Gaussianstatistics can be applied for this purpose. A Gaussian random field is character-ized with the following probability density function

W = limm→∞

1σm

√2mπmD

exp

⎛⎝− 12Dσ2

m∑k,j=1

Dk,jhkhj

⎞⎠ (9.2)

and the correlation function

di,k = q(li,k) = exp(−l2i,k/l20), (9.3)

where σ is the dispersion of values h(x, y, z), D is the determinant of the covari-ance matrix of the hk and hj at m locations, di,k is the element of covariancematrix, Dk,j is the signed minor and li,k is the distance between the points i andk, and l0 is the correlation radius.

Such a random field can be considered as a 4-D single-valued topographywith the Gaussian statistics of heights, respectively. Then, this relief is dissectedby a 3-D hyperplane at a certain level that is parallel to the topography averageplane. It means that the values of the field in every point of the space (x, y,z) are compared with a constant C. It is regarded that, if h(x, y, z) > C, thereis substance of a particle in the point (x, y, z) and emptiness in the oppositecase. The altitude of the level over the average plane influences the shape ofparticles and density of their distribution. The altitude can be considered as anindependent parameter. Finally, after such a procedure a 3-D medium is formed,consisting of particles with random shape and different sizes (see Fig. 9.4). Thuswe have in the computer memory a realization of a finite volume filled with RGFparticles. One can use it as a medium sample for a further ray tracing procedure

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9 Light scattering from particulate surfaces 339

0 10 20 30 40 50

L , a rbitra ry un its

0

2000

4000

6000

8000

N

Fig. 9.5. Size distribution of particles generated by means of a random Gaussian field.

or consider the particles as independent realizations of one RGF particle anddetermine its average scattering matrix elements. The surface of particles ispresented by a succession of triangular facets for which the laws of reflectionand refraction and the Fresnel formulas are applicable.

A function C(z) can be applied in the comparison procedure instead of aconstant and a sample with varying particle density in the vertical directionρ(z) is obtained in this case. The main parameter of such a gradient mediumwill be the width of the transition layer τ where the density decreases from unityto zero. τ can be determined by the choice of the function C(z). Large values ofτ mean a more complicated structure of the surface; small values correspond toa smooth topography.

This idea previously was suggested for the case of a 3-D single-valued stochas-tic surface in [48, 55, 56] and has been implemented, for example, in [73] tostudy the shadow-hiding effect in media consisting of randomly shaped parti-cles. The above technique has also been applied to generate surfaces with ran-dom multiple-valued topography [15] and individual particles [16,17] particulatemedia [19,65,73].

To characterize random particles generated as described above we investi-gated their statistical properties, e.g., the size distribution and the deviation oftheir irregular shape from a spherical one. The maximal elongation has been cho-sen as a parameter of particle size. Distribution of the size approximately followsPoisson’s law (Fig. 9.5). To describe the deviation of particles from a sphere thecharacteristic variation of local slopes is used. To determine this parameter onecan find the center of a particle, then, put a vector from the center to each facetand calculate the angle between this vector and the normal to the facet. Theangle β (it can also be called the deviation angle) averaged over all facets of theparticle is directly related to the particle shape: it is equal to zero in the case of a

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340 Yevgen Grynko and Yuriy G. Shkuratov

0 20 40 60 80β,

0

1000

2000

3000N

sphero ida l

com pact

f lu ffy

Fig. 9.6. Shape distribution over parameter β of particles generated by means of therandom Gaussian field. Vertical lines show conventional division of particles into threetypes of shape.

Fig. 9.7. Samples of RGF particles of three types of shape: spheroidal (a), irregular‘compact’ (b) and ‘fluffy’ (c).

spherical particle and approaches to 90◦ for a complicated shape. In Fig. 9.6 thedistribution of the deviation angle is presented. One can conventionally divideall generated particles into three types of shape: spheroidal (β < 25◦), irregular‘compact’ (25◦ < β < 70◦) and ‘fluffy’ (β > 70◦). Fig. 9.7 shows samples of eachtype.

An important value depending on particle size, which explicitly enters intothe absorption exponent law, is the characteristic distance that a ray passesbetween two facets inside a particle. In the ray tracing calculations we use aparameter of the characteristic pathlength l, introducing it as the median of sta-tistical distribution of pathlengths for certain orders of scattering in a particle.Obviously, it depends on the particle shape and the scattering order. Fig. 9.8exemplifies plots of the number of pathlengths in equal bins normalized by thetotal number of pathlengths for different numbers of internal reflections (ordersof scattering). The length of a particle, i.e. the maximal possible pathlength, istaken here to be 1 and divided into 30 bins. Each type of particle is illustratedwith five curves. Solid lines correspond everywhere to the sum of all significantorders of scattering except external reflection. Numbers 0–3 on the plots signify,

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9 Light scattering from particulate surfaces 341

0 0.2 0.4 0.6 0.8 1pathlength

0.04

0.08

0.12

0.16

num

ber

of p

athl

engt

hs all orders0123

(a) Perfect spheres

0 0.2 0.4 0.6 0.8 1pathlengths

0.02

0.04

0.06

0.08

num

ber

of p

athl

engt

hs

(b) Rough spheres

0 0.2 0.4 0.6 0.8 1pathlength

0.04

0.08

0.12

num

ber

ofpa

thle

ngth

s

(c) Cubes

0 0.2 0.4 0.6 0.8 1pathlength

0

0.02

0.04

0.06

0.08

0.1

num

ber

of p

athl

engt

hs

(d) RGF particles

Fig. 9.8. Statistical distributions of pathlengths for spheres formed with 30 000 and100 facets, (a) and (b), cubes (c), and RGF particles (d). The numbers correspond to0, 1, 2, and 3 internal reflections in particles.

respectively, the scattering component that passes through particles withoutinternal reflection and the components correspond to 1, 2, and 3 internal re-flections. The statistical distributions appear to be very different for particlesof different shapes. As can be seen, for perfect spheres large pathlengths domi-nate in the distribution. Short pathlengths become feasible when spheres are notperfect. Owing to total internal reflections close to the apices, cubes are charac-terized with a very wide distribution of pathlengths revealing the two maxima,one of them being located at zero. RGF particles have a unimodal distributionwith the average l (in terms of particle size) near 0.18. We note that in this caseall scattering orders have locations close to maxima.

As a particle with an arbitrary shape and size is generated, it is placed ran-domly within a rectangular parallelepiped. The upper side of the parallelepipedis the medium border. The lateral and bottom planes are cyclically closed [70];that is, if a ray leaves the parallelepiped volume, for example, through the bot-tom plane, it comes into the same volume from the opposite side. This is aneffective way to simulate a semi-infinite medium using a finite volume.

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342 Yevgen Grynko and Yuriy G. Shkuratov

ie

Fig. 9.9. Model cube of particles and illumination/observation geometry.

The medium is characterized with the following parameters: volume fractionof particles ρ (packing density), complex refractive index of the material m =n + ik, and average particle size d (or a size distribution law). Simple uniformdistribution results in a maximum density of ρ ≈ 0.1 for irregularly shapedparticles. Fig. 9.9 shows an example of a cubic volume filled with N = 5000faceted RGF particles at ρ = 0.1.

The packing density of the laboratory samples is estimated within ρ ≈ 0.05–0.50 for the very upper layer of the particulate substrate [52,54] depending on theparticle size. To generate particulate media with densities higher than ρ ≈ 0.1,we use a simple isometric inflation of particles to reach higher densities [19].Generation begins with a medium with a density ρ ≈ 0.1 as described above.We inflate the particles progressively until each particle touches at least oneneighboring particle. We then perform a number of random rotations until theparticle becomes free. Then the inflation continues. If after a given number ofrotations the particle is still touching its neighbor a vector pointing to the neigh-bor particle is determined and this particle is moved in the opposite direction. If

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9 Light scattering from particulate surfaces 343

it becomes free it continues inflating. As the packing density gets higher a par-ticle may intersect more than one neighbor. In this case the direction of motionfor this particle is determined from the sum of vectors pointing to the touchingparticles. Every manipulation with every particle is followed by a special routinechecking all possible intersections. The procedure stops when the needed packingdensity is reached. Compact irregular particles can be packed more easily thanthose with a complicated shape. The maximal packing density we achieved wasρ ≈ 0.4.

There is also another way of packing, appropriate if constituent particlesare opaque [73]. One can generate several independent realizations of the par-allelepiped filled with particles at low packing density and then superimposedone volume on another up to the required density. In this process the overlap ofsome particles from different realizations occurs. One regards such overlappingparticles just as new ones with irregular shape eliminating their common parts(Boolean summing). To take into account the change of packing density dueto this overlap, the Monte Carlo method is used to calculate the final packingdensity of the modeled medium.

9.2.2 Ray tracing algorithm

Once a sample of a medium has been generated in the computer memory itis ‘illuminated’ with a large enough number of rays. The rays are launchedfrom random points uniformly distributed in the plane that is perpendicular tothe direction of incidence expressed through the angle of incidence i (Fig. 9.9).Each ray is traced from facet to facet until a stopping condition occurs. In ourcalculations we used 106–107 rays.

Matrix formalism for interaction of radiation with an interface in the geo-metrical optics limit is described elsewhere (e.g., [17,41,42]). As we noted above,particles are characterized with complex refractive index m = n + ik. The re-fractive index of the surrounding medium which is considered as non-absorbingis taken to be unity. We also assume the imaginary part of the refractive indexof particles to be very small and to have negligible influence on reflection andrefraction (examination of ray tracing in absorbing media is given in [8]). Thusthe value k can affect results only through absorption.

The interaction of a ray with the particle surface results in two new rays:are transmitted and reflected ones. The sum of their intensities is equal to theintensity of the initial ray. The Fresnel formulas give the intensity of each of theserays and Snell’s law provides their propagation directions. Splitting rays everytime they meet the particle surface would be ineffective, since their numberincreases in geometric progression with scattering order. Therefore we used amore effective technique. We randomly choose between the two possibilities (tobe refracted or reflected) and use only one of them, treating intensities as thecorresponding choice probability [14,17].

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344 Yevgen Grynko and Yuriy G. Shkuratov

The Stokes parameters of the reflected and refracted rays in the correspon-dent coordinate system can be obtained from

Ir = R · K · Ii, (9.4)

It = T · K · Ii, (9.5)

where Ii is the Stokes vector of the incident ray, K is the matrix of rotationto the plane of incidence, and R and T are Fresnel’s reflection and refractionmatrices, respectively. In explicit form [4]

K =

⎛⎜⎜⎝1 0 0 00 cos 2ω sin 2ω 00 − sin 2ω cos 2ω 00 0 0 1

⎞⎟⎟⎠ , (9.6)

where ω is the rotation angle of the scattering plane for transition from one facetto the other and

R =12

⎛⎜⎜⎝r||r∗

|| + r⊥r∗⊥ r||r∗

|| − r⊥r∗⊥ 0 0

r||r∗|| − r⊥r∗

⊥ r||r∗|| + r⊥r∗

⊥ 0 00 0 2Re(r||r∗

⊥) 2Im(r||r∗⊥)

0 0 −2Im(r||r∗⊥) 2Re(r||r∗

⊥)

⎞⎟⎟⎠ , (9.7)

T =n2 cos θt

2n1 cos θi

⎛⎜⎜⎝t||t∗|| + t⊥t∗⊥ t||t∗|| − t⊥t∗⊥ 0 0t||t∗|| − t⊥t∗⊥ t||t∗|| + t⊥t∗⊥ 0 0

0 0 2Re(t||t∗⊥) 2Im(t||t∗⊥)0 0 −2Im(t||t∗⊥) 2Re(t||t∗⊥)

⎞⎟⎟⎠ , (9.8)

where n1,n2 are the proper indices of refraction (n1 = 1 and n2 = n, or n2 =1 and n1 = n), θi and θt are the angles of incidence and refraction, r‖, r⊥, t‖,t⊥ are the amplitude Fresnel coefficients. The matrix in (9.8) is prefixed with afactor required to obey the energy conservation law.

Note that equation (9.7) is also applicable to total internal reflection withparallel and perpendicular components having different electromagnetic phases[6]:

tanδ||2

= −

√sin2 θi − (n2/n1)2

(n2/n1)2 cos θi, (9.9)

and

tanδ⊥2

= −

√sin2 θi − (n2/n1)2

cos θi. (9.10)

In this case the matrix R is simplified to

R =

⎛⎜⎜⎝1 0 0 00 1 0 00 0 cos

(δ|| − δ⊥

)sin(δ|| − δ⊥

)0 0 − sin

(δ|| − δ⊥

)cos

(δ|| − δ⊥

)⎞⎟⎟⎠ . (9.11)

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9 Light scattering from particulate surfaces 345

To characterize absorption of a semitransparent particle we use the absorp-tion parameter τ = 4πkl/λ, where l is the characteristic length of light prop-agation on the way between two facets inside a particle λ is the wavelength.The pathlength l (and hence τ) is determined for a particle with the ray tracingtechnique. We treat the absorption, as it has been for Fresnel’s formula: exp(−τ)is a choice probability for rays to be absorbed at in-particle propagation. Forexample, if the random number generator gives a value larger than exp(−τ) thenthe ray is absorbed and the tracing of this ray is stopped. Thus in our schemethe absorption decreases the initial number of rays. Theoretically at τ = 0 allincoming rays should leave the studied scattering system, providing its unitaryintegral albedo. In practice, however, one can never achieve this, as any ray trac-ing procedure deals with a finite number of scattering orders and a portion ofrays always remains in the light scattering system. We controlled this numberpermitting only 1–2% of the remnant rays.

The Monte Carlo ray tracing lasts until the ray has been absorbed or hasleft the surface after a sequence of interactions with particles or the number ofinteractions exceeded a certain cutoff value restricting calculations of scatter-ing orders. After that the scattering matrix Fik for a given trajectory can bedetermined.

The emerging rays are collected depending on the value which is interestingfor us: surface integral albedo or all non-zero elements of the scattering matrix

Fig. 9.10. Samples of model and real quartz particles.

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346 Yevgen Grynko and Yuriy G. Shkuratov

0 40 80 120 160 α, °

0

0.1

0.2

P

1

2

3

4

I

Laboratory measurements

Best fit: m = 1.5 + 2⋅10-6i, ρ = 0.4, <d> = 220 μm

Fig. 9.11. Results of computer simulations and laboratory measurements. Upper andlower panels correspond to intensity and polarization degree dependencies.

(including the reflectance coefficient and linear polarization degree) at differentphase angles.

In order to determine angular dependencies the phase angle range is dividedinto a number of angular bins. Then, for each bin, elements Fik correspondingto the ray scattered into this bin are summed. The number of rays normalizedby the solid angle of a given bin is by definition the intensity of scattered lightat the bin. The polar regions (at very small and very large phase angles) of thespherical coordinate system have small solid angles and accumulation of raysgoes slowly there. As a result, it is necessary to use a large number of initialrays, since we need to get high enough precision for all directions of scatteringand study separate scattering components.

Reflectance of a surface at a given phase angle is defined as a ratio of thebin intensity corresponding to arbitrary k (or τ) and k = 0. To some extentthis simulates comparison with the Lambertian surface. The integral albedo is

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9 Light scattering from particulate surfaces 347

defined in the same way for the whole hemisphere. Calculation of reflectance fora set of wavelengths at given photometric geometry gives a spectrum.

We have thoroughly tested the model. In particular, the algorithm was veri-fied on the reciprocity principle expressed in the vector form.

Despite the approximate nature of ray tracing our model can produce satis-factory results in comparison with laboratory measurements of a powder sample.We verified the computer model, comparing model plots with corresponding dataof photometric and polarimetric measurements of a quartz sand with the aver-age size of particles near 220 μm. The measurements were carried out with thelaboratory photopolarimeter of the Astronomical Institute of Kharkov NationalUniversity [66]. A red color filter with λeff = 0.63 μm was used. In Fig. 9.10 weshow two SEM photos of the quartz particles that are compared with the modelparticles. Fig. 9.11 demonstrates the model and experimental plots that revealgood coincidence. The measurements and model calculations were carried out atan observing angle close to 70◦; the incident angle change provides the phase an-gle range from 0◦ to 150◦. For the best model fit we use n = 1.5 and k = 2 ·10−6,and λ = 0.63 μm. The imaginary part was found after small variations. We notethat the model ‘feels’ well the packing density parameter and at ρ < 0.4 it isimpossible to find satisfactory intensity and polarization fits at the same valueof k. Small differences seen in Fig. 9.11 can be related to diffraction effects anddiscrepancy of the faceted and real rough surface of particles. Thus our modelsuggests more or less adequate description of light scattering by surfaces at leastsimilar to quartz sand.

The next sections present the results of the computer modeling for theshadow-hiding effect in the systems of opaque particles as well as single andmultiple incoherent scattering for particles with zero and non-zero absorption.

9.3 The shadow-hiding effect and multiple scattering insystems of opaque particles

Shadow-hiding is a geometric optics effect. It plays the main role in the forma-tion of the photometric properties of powdered surfaces and planetary regoliths.Its contribution is important at all phase angles, even in the case of surfacesconsisting of weakly absorbing particles. Let a particulate surface with opaqueparticles that have a mat surface be illuminated with a parallel beam. Each prop-agating ray is stopped and gets scattered by a particle at a certain depth (Fig.9.12). The strictly reversed direction of scattering becomes preferred becausethis path has been passed already by rays without interruption. Therefore theangular dependence of the scattered intensity has a maximum at backscattering.The main factor that influences the shadowing effect is the packing density ofthe medium.

Analytical calculation of the effect appears to be a difficult task, since itplays a role at different topography scales from the order of the wavelength tothe size of the planetary body. Several semi-analytical models of the shadow-hiding effect exist. The most popular are the Hapke [24] and the Lumme and

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348 Yevgen Grynko and Yuriy G. Shkuratov

ksc kincα

Fig. 9.12. A scheme illustrating the effect of shadowing in a particulate medium.Arrows show wave vectors of incident and scattered radiation.

Bowell [32] models, though recently a rigorous theory of the shadow-hiding effectfor pre-fractal rough surfaces has been suggested [62].

The Hapke model uses five parameters that characterize scattering propertiesand the structure of the surface and are assumed to be free. It is often used forthe interpretation of photometric observations. However, different combinationsof the set of parameters sometimes produce close results [12] which makes theproblem of interpretation more difficult. This means that the model parametersdepend on each other and a change of one of them can be compensated bychanging the other ones. The reason is that the model is approximate and basedon a series of assumptions.

A lot of work has been done in this field by the Kharkov light scatteringgroup. Using both analytical and numerical methods the shadow-hiding effecthas been studied in systems with a discrete random structure. In particular,shadowing was considered on surfaces with a single-valued relief, taking intoaccount correlation of propagation of the incident and emergent rays [55,56,69].Clusters of particles and powder-like surfaces were also studied [43,72,73]. Suchan important property of natural surfaces as fractal structure was also taken intoaccount in studies [58,59,62,69]. Attempts have been made to consider randommultiple-valued topography [15,56] and regolith-like surfaces with different typesof topographies [63].

According to the results of this work one can make some conclusions.Backscattering from powders depends on the packing density of particles. Thelower the density the more pronounced the opposition peak. This is true for bothmono- and polydispersed media. Both cases lead to similar phase curves at equalpacking densities. The photometric properties of clusters of particles depend alsoon the number of particles in the cluster. A smaller number of particles at thesame packing density results in a weakening of the opposition peak [43].

To illustrate some of the above results we show a few plots. Phase-anglefunctions in Figs 9.17–9.19 are calculated for particulate media with differentproperties. Constituent particles are assumed to be opaque and to have diffuselyscattering surfaces.

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9 Light scattering from particulate surfaces 349

9.3.1 Ray tracing modification

For technical reasons we use below so-called reversed ray tracing suggested byStankevich and Shkuratov [70,72,73]. This implies that parallel rays are tracedfrom the observer into a medium and then, after scattering, from the medium inthe light-source direction. The start points of initial rays are distributed statis-tically uniformly in a plane perpendicular to the direction of observation. Eachray is traced from its starting point to the first medium particle that is on itsway. Then from the point of intersection the ray is traced in the light-sourcedirection. If the ray can reach the light source, it contributes to intensity of lightscattered from the medium in the first order. In the opposite case the given pointis shadowed and hence the ray does not contribute to the intensity. Thus, whatwe calculate using such an algorithm is the average intensity of scattered light:

I(i, ε, δ) =1N

N∑j=1

Ij(i′j , ε′j , δ

′j), (9.12)

where Ij(i′j , ε′j , δ

′j) is the intensity of the j-th ray emerged from the intersection

point, i′j , ε′j , δ

′j being, respectively, the local incident, emergent, and azimuth

angels, and i, ε, δ are the angles of the same type with respect to the globalnormal of the medium. The summation is made over illuminated points only.With the Lambertian law of scattering, formula (9.12) yields:

IL(i, ε, δ) =A

N

N∑j=1

cos i′j , (9.13)

where A is the albedo of the particle surface.Accounting for higher scattering orders in this ray tracing algorithm turns

out to be difficult because of the exponential increase of ray number with growthof the order. One can avoid such a branching by tracing only one ray at eachstage after interaction with the n-th particle. Initially this gives a high error butaveraging over many events makes the total result exact enough [72, 73]. Theaverage intensity scattered by a surface in this case is equal to

I =1N

⎡⎣AN∑

j=1

fj(i′j , ε′j , δ

′j) + A2

N∑k=1

fk(i′′k , ε′′k , δ′′

k ) + A3N∑

l=1

fl(i′′′l , ε′′′l , δ′′′

l ) + ...

⎤⎦ .

(9.14)In Eq. (9.14) all sums have clear physical sense – these are contributions of

scattering orders to the full intensity of rays coming from the medium at A = 1.Using the Lambertian law as an indicatrix of the particle surface elements, wecan re-write (9.14) as

IL(i, ε, δ) =1N

⎛⎝AN∑

j=1

cos i′j + A2N∑

k=1

cos i′′k + A3N∑

l=1

cos i′′′l + ...

⎞⎠ . (9.15)

This formula is used in the final ray tracing procedure.

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350 Yevgen Grynko and Yuriy G. Shkuratov

9.3.2 Results of simulations

9.3.2.1 Light reflection from multiple-valued random topography

Here we present the results of the ray tracing for random multiple-valued to-pographies having density gradient [15]. We restricted ourselves to the singlescattering approximation and the following geometry of illumination/observationi = α, e = 0◦. The method of auxiliary random field described above is used forsurface sample generation. Random fields with Gaussian (Eq. (9.2)) and frac-tal statistics are applied in the algorithm. To define surfaces with the fractalstatistics we use the following structure function:

|h(r(x, y) + Δ(x, y)) − h(r(x, y))| ∝ [Δ(x, y)]3−D (9.16)

Fractal surfaces are characterized with the fractal dimension parameter Dthat determines its structure independently of the function C(z) [15]. In Fig.9.13 phase functions of reflectance (brightness relative to the Lambertian surfaceat normal illumination/observation) for different values of D are shown. Thefunction C(z) is the same for all curves, i. e. samples with the same transitionlayer width τ are taken. As one might expect, larger D leads to more fluffy surfacestructure and makes the shadowing effect more prominent, which is expressedin the clearly seen opposition surge.

Fig. 9.14 shows similar data for a surface with Gaussian statistics and dif-ferent τ . It is expressed in the correlation radii units l0 (see formula (9.3)). One

0 10 20 300.2

0.4

0.6

0.8

1.0

1

2

3

α, °

Ref

lect

ance

fact

or

Fig. 9.13. Phase curves of reflectance forrandom topography with fractal statisticsand fractal dimension equal to D = 2.1(1), D = 2.5 (2) and D = 2.9 (3).

0 10 20 300.3

0.4

0.5

0.6

0.7

0.8

1

2

3

Ref

lect

ance

fact

or

α, °

Fig. 9.14. Phase curves of reflectance forrandom topography with Gaussian statis-tics and transition layer width equal toτ = l0 (1), τ = 3l0 (2) and τ = 7l0 (3).

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9 Light scattering from particulate surfaces 351

-90 -60 -30 0 300.0

0.2

0.4

0.6

0.8

1.0B

righ

tnes

s

Longitude, deg

Mariner 10Model calculations

Fig. 9.15. Mariner 10 data for Mercurylongitude brightness distribution [24] andmodel fit at some choice of parameter τ .

Latitude, degB

right

ness

Mariner 10Model calculations

Fig. 9.16. Mariner 10 data for Mercurylatitude brightness distribution [24] andmodel fit at some choice of parameter τ .

can see that growing τ is accompanied by an increase of surface roughness andthe opposition surge again becomes more prominent, as in the fractal case.

Calculations with such an approximate model can nevertheless fit the ob-servational photometric data. Figs 9.15 and 9.16 demonstrate this. Comparisonof the brightness distribution over the disk of planet Mercury with the modelproves its adequacy. These data were obtained during the Mariner 10 missionat phase angle α = 77◦. For the model fit the Gaussian surface with the best-fitvalue of τ was chosen. From this comparison one can conclude that the shadow-hiding effect plays the main role in the formation of the brightness distributionover the disk of a planet with a rough surface.

9.3.2.2 Multiple light scattering in media

Here we study photometric effects reproduced in simulations of multiple scat-tering in media formed by opaque particles. More information can be found ina series of cited papers [63,73,74].

Results of calculations of phase functions for two semi-infinite media withdifferent packing densities, ρ = 0.1 (lines) and ρ = 0.3 (points), are presentedin Fig. 9.17. So-called ‘mirror’ geometry is used which means i = e = α/2(see Fig. 9.9) at the principal scattering plane. The Lambertian indicatrix forthe particle surface elements with the albedo A = 1 is used. As one can see,the contribution of high orders of scattering drops quickly even at unit albedoof the surface element. The reason for this reduction is ray escape from themedium. This escape is partly caused by the backscattering tendency of the

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0 30 60 90 120 150 180Phase angle, degrees

0.01

0.10

1.00

Ref

lect

ance

1

23

4 5 6

Fig. 9.17. Phase-angle functions of scattering orders from the first to the sixth fora semi-infinite medium consisting of RGF particles with unit surface albedo and theLambertian surface element indicatrix at i = e = α/2 (mirror geometry) and differentpacking densities: ρ = 0.1 (lines) and ρ = 0.3 (points).

integral Lambertian indicatrix that is finally formed at scattering of rays frommany surface elements of each particle. If the albedo of the surface elementsis less than 1, the decrease in the contributions with the growth of scatteringorders will be more rapid.

The opposition surge is observed only for the first order of scattering andis related to the correlations between incident and emergent rays in a particu-late medium close to zero phase angle. The form of this surge depends stronglyon the packing density of the medium. It becomes wider and less prominentwith increasing ρ. The dependence for the second order of scattering also has aninteresting feature: a weak maximum at a phase angle near 100◦. The phase de-pendencies for higher orders do not reveal small-phase-angle backscattering andhave a very inert behavior close to zero phase angle. At α > 90◦ the contributionof higher orders of scattering exceeds that of the first order. The influence of ρon high orders of scattering is comparatively small and is almost invisible in thefifth and sixth orders. Their indicatrices tend to follow the cosine law. In thiscase the corresponding rays lose information about medium properties (i.e. thecorrelation between propagations at incidence and emerging) and form diffuseflux.

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9 Light scattering from particulate surfaces 353

0 30 60 90 120 150 180Phase angle, degree

0.01

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lect

ance

i = ε = α/2, ω = 1, ρ = 0.1

1

2

3

4 5 6

Fig. 9.18. Phase-angle functions of the first six scattering orders for semi-infinitemedia consisting of spheres (points) and irregular particles (lines) at mirror geometryat ρ = 0.1. Unit particle surface albedo and the Lambertian surface element indicatrixis assigned [73].

It is interesting to study also the influence of the particle shape on photo-metric properties of such media. For comparison, media consisting of sphericalparticles are used. Figs 9.18 and 9.19 present phase-angle dependencies of thecontributions of the first six scattering orders that form reflectance of mediawith ρ = 0.1 and ρ = 0.3. The mirror geometry of illumination/observation isused again at A = 1. As one can see, in this case the contributions of scatter-ing diminish with the growth of scattering order too. The curves for the firstorder almost coincide and at small phase angles the opposition effect is clearlyseen. The most noticeable difference is observed in the second scattering or-der. At α < 100◦ random irregular particles make a larger contribution to thereflectance. At ρ = 0.1, the difference for higher orders becomes insignificant(Fig. 9.18) and for the sixth-order curves it is absent. However, at ρ = 0.3 thedifferences turn out to be more prominent and are clearly seen for all ordersconsidered.

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354 Yevgen Grynko and Yuriy G. Shkuratov

0 30 60 90 120 150 180Phase angle, degree

0.01

0.10

1.00

Ref

lect

ance

i = ε = α/2, ω = 1, ρ = 0.3

1

2

3

4 5 6

Fig. 9.19. Phase-angle functions of the first six scattering orders for semi-infinitemedia consisting of spheres (points) and irregular particles (lines) at mirror geometryat ρ = 0.3. Unit particle surface albedo and the Lambertian surface element indicatrixis assigned [73].

9.4 Single scattering component. Transparent andsemitransparent particles

In this section we use ray tracing to study phase curves of all non-zero elementsof the scattering matrix for several classes of randomly shaped particles. Onecan find more extended coverage of this subject in [17]. We note, however, thata portion of the curves in [17] have poor statistics (some plots of the elementsF22/F11, F33/F11, F34/F11, F44/F11) and here we present their improved ver-sions.

Examples of particles used in the simulations are shown in Fig. 9.20. Thefirst class is presented with faceted spheres that have a different number N offlat facets which varies in our studies from 100 to 30 000. The second class isbinary faceted spheres (contacting components). The third class is faceted ellip-soids, for which the ratio of their axes varies as well as the value N . The fourthclass is presented with cubes of ‘spoiled’ forms, when the length of cube edgesslightly varies. Finally, the fifth class is randomly shaped particles generated byan auxiliary random Gaussian field.

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9 Light scattering from particulate surfaces 355

Fig. 9.20. Examples of randomly shaped particles used for calculations of scatteringmatrices. Roughly faceted sphere (a) and ellipsoid (b), irregular cube (c) and irregularRGF particle (d).

9.4.1 Faceted spheres

We start our consideration with spheres of different numbers of facets. In Fig.9.21 the element F11 of scattering matrix and the ratios −F12/F11, F22/F11,F33/F11, F34/F11, F44/F11 versus α are presented for such spheres with the realpart of refractive index n = 1.5 and imaginary part k = 0. Sums over all signifi-cant orders of scattering are taken; usually this includes a few hundreds of orders.As one can see, the curves corresponding to N = 10 000 and 30 000 are similarfor all studied parameters, i.e. on average a spherical particle formed with morethan 10 000 flat facets, can be considered as a rather perfect sphere. When N issmaller, changes are observed. Thus, for F11 at N = 1000 the glory surge disap-pears and, moreover, instead of the surge an opposition ‘anti-spike’ is developed.At N = 100, neither the glory nor the first rainbow are observed.Weakening ofoscillations with the decrease of N is clearly seen for the ratio −F12/F11. Thisratio corresponds to the definition of linear polarization degree of scattered ra-diation, when particles are illuminated with unpolarized light. So, as one can seein Fig. 9.21, the small negative polarization branch, which is observed for perfectspheres near the backscatter direction, α = 0◦, disappears for roughly facetedparticles. The surge of positive polarization corresponding to the first rainbowgoes down quickly with decreasing N. The negative polarization branch at largephase angles weakens for roughly faceted spheres too.

Spheres do not depolarize the incident light at all [4,6]. The curves of the ratioF22/F11 presented in Fig. 9.21 are in agreement with this. Thus, at N = 10 000and 30 000 the ratio F22/F11 is almost equal to 1. There are deviations only atvery small phase angles that are probably due to poor statistics at these angles.For non-perfect spheres, N = 1000 and 100, the ratio F22/F11 varies ratherstrongly. These particles depolarize light noticeably. The ratio F33/F11 is related

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0.8

F11

-F12/F 11

N=100N=1000N=10000N=30000

F33/F11F22/F11

F34

/F11

F44/F11

Fig. 9.21. The scattering matrix element F11 and ratios −F12/F11, F22/F11, F33/F11,F34/F11, F44/F11 versus phase angle (sums over all significant orders of scattering) forspherical particles approximated by different number of facets N with the refractiveindex n = 1.5 and absorption coefficient k = 0.

to orientation of the linear polarization plane of scattered light. We see in Fig.9.21 that the angle dependences of this ratio correlate with the correspondingcurves −F12/F11. The ratio F34/F11 is responsible for mutual transformation oflinear and circular polarization. Our calculations show that this transformationis not observed in the range 120◦–180◦ and the backscattering direction. Theratio F44/F11 describes the change of circular polarization. It has different signsfor perfect and roughly faceted spheres near the backscattering direction. TheF33/F11 and F44/F11 curves are similar; noticeable difference is observed onlyat small α.

In [17] one can find a detailed study of the scattering matrix elements cal-culated for the dominant components of scattering: (1) the forward refractionand (2) the single and (3) double internal reflection. The first order of scatter-

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9 Light scattering from particulate surfaces 357

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-0.4

0

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0.8

F11 -F12/F 11

Singlen=1.3n=1.8

F33/F 11F 22/F 11

F 34/F11

F 44/F11

Binaryn=1.3n=1.8

Fig. 9.22. The scattering matrix element F11 and ratios −F12/F11, F22/F11, F33/F11,F34/F11, F44/F11 versus phase angle for single sphere and two similar touching spheresapproximated by N = 30 000 facets with different values of n at k = 0.

ing is not considered as this is the well-known single Fresnel reflection from theexternal surface of a particle.

9.4.2 Binary spheres and faceted ellipsoids

Light scattering by arbitrary-shaped particles comparable in size to the wave-length of incident radiation is often simulated by that of ellipsoids and aggregatesof spheres [40]. We study here this approximation in the geometric optics ap-proach.

We consider binary touching spheres (bispheres) as a model of an irregularparticle. The spheres have perfect shape and they are approximated by N =30 000 facets. Comparison of the scattering matrix elements for single and binaryspheres with k = 0 show a high similarity of the curves (see Fig. 9.22). Thisindicates that the contribution of the inter-particle scattering component to

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358 Yevgen Grynko and Yuriy G. Shkuratov

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0

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0 30 60 90 120 150 180α, °

-0.8

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0.4

0.8

F11

-F 12/F11

F 33/F11F22/F11

F34/F11

F44/F11

N=100N=1000N=10000N=30000

Fig. 9.23. The scattering matrix element F11 and ratios −F12/F11, F22/F11, F33/F11,F34/F11, F44/F11 versus phase angle for ellipsoids approximated by different numberof facets N with d = 4/5 at n = 1.5 and k = 0.

the total flux of scattered light is small. An exception is observed for the ratioF22/F11, which is related to the depolarization ability of scattering objects. Theratio noticeably deviates from unity (as it should be) at small α. This generallyconfirms the result obtained with the DDA approximation for small particles [89]:it is not a sufficiently good approach to model irregular particles with systemsof spheres, as the optical properties of single spheres substantially dominate thetotal scattering.

Another example of irregularly shaped particles is faceted ellipsoids. In thiscase we have two parameters to describe the particle shape: the ratio of ellipsoidaxes d and number of facets N. An ellipsoid is oblate if d < 1 and it is prolate ifd > 1. In Fig. 9.23 the same phase dependencies are given for ellipsoids approx-imated by different number of facets N with axis ratio d = 4/5 at n = 1.5, and k= 0. Sums over all significant orders of scattering are presented. The curves havebeen averaged over orientations. Different features in the intensity and polariza-

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9 Light scattering from particulate surfaces 359

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1000

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0.2

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1

-0.4

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0 30 60 90 120 150α, °

0

0.2

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0 30 60 90 120 150 180α, °

-0.4

0

0.4

0.8

F11 -F12/F 11

F33/F11

F22/F 11

F34/F11 F44/F11

δ = 0.000δ = 0.025δ = 0.050δ = 0.200

Fig. 9.24. The scattering matrix element F11 and ratios −F12/F11, F22/F11, F33/F11,F34/F11, F44/F11 versus phase angle (sums over all significant orders of scattering) forparticles with shapes randomly deviating from cube with n = 1.5 and k = 0.

tion dependencies are blurred as N decreases. At N = 1000 they even disappear.In the maximum the ratio −F12/F11 is three times lower for roughly facetedellipsoid with N = 100, than for a well approximated one. Plots for differentscattering orders are qualitatively similar to those for spherical particles, and wedo not show these here. We note that all studied parameters, the element F11and ratios −F12/F11, F22/F11, F33/F11, F34/F11, F44/F11, for roughly facetedspheres and ellipsoids at d = 4/5 are very similar.

9.4.3 Perfect and ‘spoiled’ cubes

To obtain particles of irregular shape we used perfect cubes deforming themin the following way. Each vertex of a cube is randomly moved in a randomdirection. The shift has zero average and certain amplitude δ that is given in thelength of the cube edge. We varied δ from 0 (perfect cube) to 0.2 (very irregularparticle). As in the case of other particles irregular cubes are approximated with

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360 Yevgen Grynko and Yuriy G. Shkuratov

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0

0.1

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0

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0 30 60 90 120 150α, °

0

0.1

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0 30 60 90 120 150 180α, °

-0.8

-0.4

0

0.4

0.8

F11 -F12/F 11

β = 18°β = 33°β = 49°

F 33/F11

F22

/F11

F34

/F11

F44/F11

Fig. 9.25. The scattering matrix element F11 and ratios −F12/F11, F22/F11, F33/F11,F34/F11, F44/F11 versus phase angle (sums over all significant orders of scattering) forRGF particles with different degrees of irregularity β at n = 1.5 and k = 0.

triangle facets (see Fig. 9.20). Sums over all significant orders of scattering arepresented in Fig. 9.24 that illustrates how F11 and the remaining parameters,being functions of α, depend on random deviations from a cube at n = 1.5 andk = 0. The ideal cube gives forward and backward scattering brightness spikesas well as a strong negative polarization branch at small α. Backscattering isa manifestation of the so-called retro-reflector effect [78]. The effect is quicklydegraded with deviation of particles from cubical shape. These results are ingood quantitative agreement with [41]. Perfect cubes can depolarize incidentlight, having the maximum of the depolarization ability at α ≈ 30◦. When thecubes are deformed their depolarization ability rapidly grows. For the deformedcubes the forward and backward scattering spikes are reduced and become wider.The negative polarization branch vanishes. Independently of the parameter δ thecubes are able to very effectively transform linear into circular polarization andvice versa at intermediate angles of scattering (see F34/F11 curves in Fig. 9.24).

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9 Light scattering from particulate surfaces 361

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0

0.04

0.08

0.12

0.16

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-0.4

0

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0 30 60 90 120 150α, °

0

0.1

0.2

0.3

0 30 60 90 120 150 180α, °

-0.4

0

0.4

0.8

F11

-F12/F11

n=1.3n=1.5n=1.8n=2.0

F33/F 11

F22/F11

F34/F11

F44/F11

Fig. 9.26. The scattering matrix element F11 and ratios −F12/F11, F22/F11, F33/F11,F34/F11, F44/F11 versus phase angle for RGF particles with different n at k = 0.Averaging over particle shapes was made.

We note that the F33/F11 curves are similar to the F44/F11 curves at all δ (exceptδ = 0.2). Close to the backscattering direction the particles with non-zero δ cansignificantly depolarize light that initially is circularly polarized.

9.4.4 RGF particles

Fig. 9.25 shows the results of our calculations of the element F11 and ratios−F12/F11, F22/F11, F33/F11, F34/F11 for RGF particles with different degreesof nonsphericity β at n = 1.5 and k = 0; sums over all main orders of scatteringare used. We note that the RGF particles with high β have no backscattering ef-fect. As β grows the maximum of the polarization curve (−F12/F11) diminishes;all irregularities are smoothed; the negative branch of polarization in the for-ward scatter direction almost disappears. For perfect spheres the ratio F22/F11is almost equal to unity at all phase angles, but for the RGF particles this de-pendence essentially varies with angle, even at small values of β. This means

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362 Yevgen Grynko and Yuriy G. Shkuratov

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α, °

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α, °

-0.8

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0

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0.8

F11 -F 12/F11

k=0.000k=0.002k=0.004k=0.006

F33/F11

F22/F 11

F34/F 11 F44/F11

Fig. 9.27. The scattering matrix element F11 and ratios −F12/F11, F22/F11, F33/F11,F34/F11, F44/F11 versus phase angle for RGF particles with different k at n = 1.5.The ratio of the mean ray pathlength within the particle to the wavelength L/λ = 17.Averaging over particle shapes was made.

that RGF particles are able to depolarize light very considerably. The nonspher-ical shape of particles makes the ratios F33/F11 and F44/F11 unequal, while forspheres they are equal over the whole range of phase angles. It is interesting tonote also that for all the scattering parameters the phase angle curves revealgood qualitative resemblance between roughly faceted spheres (N = 100) andRGF particles with β = 18◦.

Figs 9.26 and 9.27 present variations of the element F11 and ratios −F12/F11,F22/F11, F33/F11, F34/F11, F44/F11 as functions of phase angle for RGF parti-cles with different n and k. These results were averaged over particle shapes. Thegrowth of n increases the total reflection of the particle surface. The forwardlyrefracted surge progressively widens and diminishes with growing n. Decreasingn produces not only more forward scattering, but also deeper negative polar-ization branch at phase angles greater than 130◦. The effect of absorption is

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9 Light scattering from particulate surfaces 363

more evident. The intensity of scattered light decreases with increasing k. Withgrowth of k the degree of positive polarization grows and the negative polariza-tion branch at large α disappears (see curves −F12/F11). This is explained bythe fact that the light externally reflected from the particle becomes a largerfraction of the total scattered light. This is a manifestation of the familiar Umoveffect. The same effect is observed for faceted spheres and cubes [17]. It is in-teresting to note the qualitative similarity of all investigated characteristics ofthe RGF particles, roughly faceted spheres (ellipsoids) and irregular cubes withδ = 0.2.

9.5 Incoherent multiple scattering

In this section we study multiple scattering in media consisting of transparentand semi-transparent particles. Our ray tracing simulations cover some aspectsof the photopolarimetry and reflectance spectroscopy of particulate surfaces.

9.5.1 Photometric and polarimetric phase curves

We begin with the simulation of photometric and polarimetric phase curves ofmedia at different values of τ for the particle material. Figs 9.28–9.31 show phasecurves of the normalized intensity (F11) and polarization degree (−F12/F11) forthree media, each composed of particles with different shapes (spheres, cubes,and RGF particles) at several values of τ . All phase curves of intensity arenormalized at 30◦. As can be seen, the photometric and polarimetric phasecurves depend significantly on τ . The narrow opposition spikes of spherical andcubic particles decrease with growth of τ , being conspicuous even at τ = 3.2.The decrease can be anticipated, as both the spikes are formed with internallyreflected components. These spikes are due to the glory and retroreflector effectsfor spheres and cubes, respectively. Near the phase angle α = 17◦, the primaryrainbow produced by spheres can be seen. The relative amplitudes of the rainbowsurge, both for intensity and polarization, decrease with the increase in τ , asshould happen.

At very large phase angles the positive branch of polarization grows quicklywith increasing τ . This is in agreement with the Umov effect. It is ubiquitouslyobserved for natural and artificial particulate dielectric surfaces, including theplanetary regoliths (e.g., [54]).

At high τ the difference in phase curves of the polarization degree of me-dia composed of particles of different shapes disappears because single externalreflection dominates the light flux leaving the media. Wide surges seen in theintensities at phase angles less than 15◦ when τ = 10 are caused by the shadow-hiding effect. Of special interest is the transformation of the negative polarizationbranch of the media composed of cubes with changing τ . The inversion anglefor such a medium is almost 70◦ for non-absorbing particles and it quickly goesto zero when τ tends to 1. We note that the media composed of RGF parti-cles do not have negative polarization, at least at this observation/illumination

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364 Yevgen Grynko and Yuriy G. Shkuratov

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100

I

0 30 60 90

α, °

-0.4

-0.2

0

0.2

0.4

PSpheresCubesRGF

τ = 0.10

Fig. 9.28. Phase curves for normalizedintensity and linear polarization degreefor media consisting of perfect spheres,cubes, and RGF particles at n = 1.55,ρ = 0.1, and τ = 0.1. Incident rays fallnormally on the surfaces.

1

10

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I0 30 60 90

α, °

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P

SpheresCubesRGF

τ = 1.05

Fig. 9.29. Same as Fig. 9.28 forτ = 1.05.

1

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I

0 30 60 90

α, °

0

0.2

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SpheresCubesRGF

τ = 1.75

Fig. 9.30. Same as Fig. 9.28 for τ = 1.75.

1

I

0 30 60 90

α, °

0

0.2

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P

SpheresCubesRGF

τ = 3.25

Fig. 9.31. Same as Fig. 9.28 for τ = 3.25.

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9 Light scattering from particulate surfaces 365

90 120 150 180

α, °

2

4

6

8

10

P, %

simulations (ρ = 0.1)simulations (ρ = 0.3)experiment

Fig. 9.32. Laboratory polarization phase functions of glass powder [52] and results ofcomputer simulations for two packing densities, ρ = 0.1 and 0.3 at zero absorption.

geometry, regardless constituent particles are absorbing or not. This result isin contradiction with a semi-empirical model of Wolff [84] who considered dou-ble Fresnel reflection as a mechanism for the negative polarization branch atbackscattering.

In 1986 polarimetric measurements of particulate surfaces that were carriedout with a large-phase-angle photopolarimeter at Kharkov Astronomical Ob-servatory showed that surfaces composed of fluffy glass powders with particlesizes of the order of 10 μm produce a narrow secondary maximum, or at least aledge, at angles between approximately α = 160◦ and 170◦ [52] and a wavelengthcentered near λ = 0.48 μm. An example of the polarimetric measurements ofpowdered transparent glass with an average particle size near 12 μm is shownin Fig. 9.32 (solid triangles). The sample, comprising a thick layer of glass pow-der, was sifted onto a plane substrate forming a porous (fairy castles) structure.As can be seen, the measured curve reveals a ledge with a hint of a secondarymaximum at forward scattering near α = 170◦.

Carrying out ray-tracing simulations with models of particulate surfaces wehave found a possible reason for the ledge: this is the contribution of single-particle scatter in combination with the shadow-hiding effect. Below we brieflydescribe our results.

In Fig. 9.32 we show numerical simulation data for the phase dependenciesof linear polarization degree for particulate surfaces with packing densities ofρ = 0.1 and 0.3. They are compared with experimental results shown with errorbars. One can see the laboratory and computer experiments to be in qualitative

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366 Yevgen Grynko and Yuriy G. Shkuratov

agreement when ρ = 0.1. The ledge is clearly revealed at ρ = 0.1, but it isnot seen for the dense medium ρ = 0.3. We note the primary maximum to behigher for the denser substrate; thus, at large α the degree of linear polarizationdepends significantly on the packing density.

Two types of rays appear to be responsible for this ledge [19]: the rays re-flected once from the external boundaries of particles and the rays refractedby particles in the forward direction without internal reflections. Rays that un-dergo a single Fresnel reflection have a positive polarization at all phase angles;whereas, rays that undergo two refractions by interfaces tend to be negativelypolarized [17]. The sum of these scattering components at the given geometryof illumination/observation produces the ledge.

One can also explain the dependence of polarization at large α on the packingdensity. The relative contributions of the singly reflected and refracted compo-nents depend on the shadow-hiding effect produced by neighboring particles.This effect is stronger for rays from the lowermost portions of particles; i.e., thesingle-particle scatter from isolated and shadowed particles are different becausethe shadowing selects the portion of the particle illuminated in addition to therays that can traverse freely to the observer. The contribution of the refractedcomponent becomes smaller at high packing density: particles are close to eachother and after single forward refraction very few rays can leave the particulatesubstrate without shadowing and the majority continue to scatter in the medium(see Fig. 9.32).

If the incidence angle is large enough (this allows large phase angles) po-larization maximum can be observed. Thus one can study parameters of theUmov effect. Using our calculations at i = 85◦ we made a diagram showinganti-correlation of the polarization degree Pmax and surface albedo (Fig. 9.33).Analogous data for photopolarimetric observations of the Moon are given forcomparison. The plot reveals some difference of the effect for different shapes

0.01 0.1 1Albedo

10

100

Pm

ax,%

SpheresCubesRGF

Polarimetric observationsof the Moon (Shkuratov & Opanasenko, 1992)

Fig. 9.33. Maximum of polarization vs. surface albedo.

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9 Light scattering from particulate surfaces 367

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F11 -F 12/F 11

F 33/F11F 22/F11

F 34/F11

F 44/F11

SpheresCubesRGF

Fig. 9.34. Phase curves of F11 and five non-zero scattering matrix elements normalizedby F11 for perfect spheres, cubes, and RGF particles at n = 1.55 and k = 0.

of constituent particles. However, almost all types of particles show linear de-pendence of log Pmax on log A. The lunar data significantly deviate from theresults of calculation, since in reality diffraction may decrease polarization inmaximum.

To end up the review of the photopolarimetric studies we make a comparisonof the matrices obtained for individual particles and those obtained for particu-late media consisting of them. Results of these calculations at τ = 0 are presentedin Figs 9.34 and 9.35, respectively, for particles and proper media (ρ = 0.1). Inthe case of media the incident rays fall normally on the medium boundary, i.e.surface. Phase curves of all studied scattering matrix elements for particles andcorresponding media are fairly different. In all cases randomly shaped particlesshow smoother phase-angle behavior. As one can see in these plots the particlesof regular shapes reveal well-detected features, when they are individual as wellas when they compose a medium. For instance, spheres composing media re-

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368 Yevgen Grynko and Yuriy G. Shkuratov

0.1

1

10

100

-0.2

0

0.2

0.4

0.2

0.4

0.6

-0.6

-0.4

-0.2

0

0 30 60

α, °

-0.04

0

0.04

0.08

0 30 60 90

α, °

-0.6

-0.4

-0.2

0

0.2

F11

-F12/F 11

F33/F11

F22/F 11

F 34/F 11

F 44/F11

SpheresCubesRGF

Fig. 9.35. Phase curves of F11 and five non-zero scattering matrix elements normalizedby F11 for media consisting of perfect spheres, cubes, and RGF particles at n = 1.55,ρ = 0.1 and τ = 0. Incident rays fall normally on the surfaces.

veal the glory and first rainbow. This also concerns the cubes that demonstratethe retroreflector effect and the deep negative polarization branch. It should beemphasized once more that all these features clearly manifest themselves forthe case τ = 0, when interparticle scattering most effectively suppresses them.Comparison of Figs 9.34 and 9.35 shows the different phase curves of the other el-ements of the scattering matrix for spheres, cubes, and RGF particles. The RGFparticles show very boring phase curves: no opposition effect and no negativepolarization are observed for individual RGF particles or for media composed ofthese particles.

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9 Light scattering from particulate surfaces 369

0.5 1 1.5 2 2.5 λ, μm

4x10-4

5x10-4

6x10-4

7x10-4

k(λ)

Fig. 9.36. Spectral dependence of imaginary part of the complex refractive index ofthe lunar mare surface material [61].

9.5.2 Spectrophotometry of particulate surfaces

9.5.2.1 Reflectance spectra at different phase angles

Reflectance of a particulate surface and, consequently, its spectrum is depen-dent on observation/illumination geometry. Thus the continuum slope and theparameters of the absorption bands can be different for the same area on a plan-etary surface, if the spectra are taken under different phase angle α. Examplesare spectrophotometric measurements of asteroids Eros [9] and Itokawa [1] insitu. Although there have been conducted special laboratory experiments withregolith structure models [13, 27, 49], measurements of lunar samples [60], andtelescopic observations of the Moon [28], the solution of the problem is not com-plete. The need for interpretation of the space mission data and for planningfuture projects require more detailed study of the role of photometric geometryin the formation of the reflectance spectra. However, there are many unresolvedquestions which should be considered in order to make this interpretation moreaccurate. For instance, it is interesting to estimate contributions of single parti-cles and multiple scattering at different phase angles. Important problems alsoare to bring lunar photometric data to the same geometry of illumination andobservation and accounting for the polarimetric effect on spectra [36,67].

We here use ray tracing simulations to study the phase angle and polarimetriceffects on lunar spectra [21]. We vary the particle size from 25 to 1500 μm. Thepacking density of the particulate surface in all experiments equals ρ = 0.1.Natural powders usually are denser and our algorithm allows packing up to ρ =0.4. However, lower density significantly simplifies simulations and, in general,the parameter ρ plays a small role in the spectral reflectance (e.g., [61]). We didnot intend to precisely simulate an analogue of the lunar regolith. The primarygoal was to study scattering effects in a well determined particulate sample toreveal the main regularities. With regard to the complex refractive index, fork(λ) we used an average dependence for the lunar mare material (see Fig. 9.36)

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370 Yevgen Grynko and Yuriy G. Shkuratov

0.5 1 1.5 2 2.5 λ, μm

0

0.2

0.4

0.6

Ref

lect

ance

10°

150°

120°

90°30°

60°

0.8

1.2

1.6

2

Nor

mal

ized

refl

ecta

nce 10°

150°

120°

90°30°60°

d = 50 μma

b

Fig. 9.37. Spectral dependencies of re-flectance and normalized reflectance atdifferent phase angles for medium withsize of particles d = 50 μm.

0.5 1 1.5 2 2.5 λ, μm

0

0.2

0.4

Ref

lect

ance

10°

150°

120°

90°30°

60°

1

1.5

2

2.5

3

Nor

mal

ized

refl

ecta

nce

10°

150°

120°

90°

30°

60°d = 250 μma

b

Fig. 9.38. Same as Fig. 9.37 for d = 250μm.

[61]. This is of an illustrative character. The plot clearly shows the absorptionbands near 1 μm and 2 μm. The real part of the complex refractive index isconstant, n = 1.6. The angle of incidence in these simulations is i = 70◦. Hence,phase angle α varies within 0–160◦. Scattered intensity is collected in the narrowsector containing a plane perpendicular to the average surface. Calculation ofreflectance for a set of wavelengths at the given photometric geometry gives aspectrum. We study relative (normalized at λ = 0.7 μm) and absolute multi-angular spectral dependencies of the surface reflectance.

Figures 9.37 and 9.38 show normalized (a) and absolute reflectance (b) spec-tra for media consisting of particles with different average size: 50 and 250 μm.At first, we note that the surface reflectance becomes lower with increasing par-ticle size. The slope of spectra changes with phase angle in all plots. It can beeither increasing or decreasing depending on the size of particles. For sizes 25 to100 μm in the range of phase angles ≈ 0–80◦ the slope increases. This can be at-tributed to the so called ‘phase reddening’ that is observed for natural surfaces,the albedo of which is higher in larger wavelengths. This was described in thelaboratory experiments (e.g., [2,13,49,60]) but has not been satisfactorily studied

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9 Light scattering from particulate surfaces 371

0.8 1 1.2 1.4λ, μm

0.98

1

Nor

mal

ized

refl

ecta

nce

d = 50 μm

Phase angle:

10° 30° 60° 90° 120° 150°

0.8 1 1.2 1.4λ, μm

0.96

0.98

1

Nor

mal

ized

ref

lect

ance

Phase angle:

10° 30° 60° 90° 120° 150°

d = 100 μm

Fig. 9.39. Surface reflectance divided by continuum near 1 μm absorption feature atdifferent phase angles. The size of constituent particles is d = 50 μm (a) and d = 100μm.

with numerical methods. For particles with sizes larger than ≈ 250 μm spectralslope decreases monotonously. The reason for the strong increase of reflectanceat very large α is that at near-grazing incident rays transmitted and reflectedforwardly by the particles of the very upper layer dominate over the case ofnormal incidence (see also Fig. 9.11). Being scattered after a few acts of forwardtransmission and reflection such a component experiences minimal absorption.

Changing illumination/observation geometry also influences the depth of theabsorption bands. To illustrate this we plotted spectra divided by the continuumin the wavelength range near the 1 μm absorption feature (see Fig. 9.39). Forapproximation of the continuum we just used a linear dependence between theband wings. It is seen that beginning from approximately α = 60◦ the bandquickly becomes weaker. In the range 10◦–120◦ its depth decreases by a factorof two.

This shows that the reflectance spectra of atmosphereless celestial bodiesshould be interpreted with great precaution and with accounting for observa-tion/illumination geometry.

In Fig. 9.40 one can see the phase dependence of the color ratio C(2.4/1.2μm) more clearly for media with particle sizes from 25 to 1500 μm. Smaller par-ticles produce reddening and have the maximum at moderate phase angles. Themaximum shifts towards small α as the size grows. For particle sizes 100—500μm the color ratios increase is the greatest. For very large particles of the orderof 1 mm the multiply scattered rays are almost completely absorbed inside par-ticles. Only those transmitted through particles and singly externally reflectedby particles of the upper layer survive. This results in a weak phase dependenceof the color ratios; such surfaces have very low reflectance of the order of 1%.

Ray tracing allows a detailed consideration of different factors influence thescattering. In Figs 9.41 and 9.42 we study the observed phenomena using de-composition of the reflected flux into single-particle (a) and multiple-particle

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372 Yevgen Grynko and Yuriy G. Shkuratov

0 30 60 90 120 150α, °

1.2

1.6

2

C(2

.4/1

.2 μ

m)

d, μm 25 50 100 250 500 1000 1500

Fig. 9.40. Phase dependencies of spectral slope C(2.4/1.2 μm) for media with differentparticle sizes d.

0.5 1 1.5 2 2.5 λ, μm

0.4

0.8

1.2

1.6

2

Nor

mal

ized

refl

ecta

nce 10°

150°

120°

90°30°

60°

0.8

1

1.2

1.4

Nor

mal

ized

refl

ecta

nce

10°

150°120°90°

30°60°d = 50 μm

a Single scattering

b Multiple scattering

Fig. 9.41. Normalized spectral depen-dencies of single and multiple scatter-ing components of reflectance at differentphase angles for medium with size of par-ticles d = 50 μm.

0.5 1 1.5 2 2.5 λ, μm

0.5

1

1.5

2

2.5

Nor

mal

ized

refl

ecta

nce

10°

150°

120°

90°30°60°

0.8

1.2

1.6

Nor

mal

ized

refl

ecta

nce

10°

150°

120°

90°

30°60°d = 100 μm

a Single scattering

b Multiple scattering

Fig. 9.42. Same as Fig. 9.41 for d = 100μm.

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9 Light scattering from particulate surfaces 373

d = 50 μm

0.5 1 1.5 2 2.5 λ, μm

0

0.2

0.4

0.6

0.8

Ref

lect

ance

0.5 1 1.5 2 2.5 λ, μm

0

0.1

0.2

0.3

0.4

Ref

lect

ance

0.5 1 1.5 2 2.5 λ, μm

0

0.1

0.2

0.3

0.4

Ref

lect

ance

0.5 1 1.5 2 2.5 λ, μm

0

0.1

0.2

0.3

0.4R

efle

ctan

ce

α = 10° α = 60°

α = 120°

Single

Multiple

Total

α = 150°

Single

Multiple

Total

Single

Multiple

Total

Single

Multiple

Total

Fig. 9.43. Comparison of the contributions of single and multiple scattering compo-nents to the total scattered flux at different phase angles for a medium with size ofparticles d = 50 μm.

(b) scattering components. As can be seen, for multiple scattering spectra theiroverall slopes are much larger than those for single scattering. We also note thatthe phase dependence of the slopes is monotonous in the case of single-particlescattering and has non-monotonous behavior for the multiple scattering compo-nent. Multiple scattering reveals weak dependence on phase angles in the range10◦–60◦. Fig. 9.43 shows that both components have comparable contributionsto the total scattering at all phase angles except very large ones. Fig. 9.44 showsthat this is true for particle sizes of the order of 100 μm. As the size of con-stituent particles grows multiple scattering becomes less important because ofincreasing absorption of individual particles. Thus, at least, for ‘lunar’ values ofk(λ) both components appear to be important and play a significant role in theformation of the reflectance spectra and its behavior with change of the phaseangle.

An explanation can be suggested for the observed behavior of the color ratios.We may consider the total ray pathlength L in a particulate medium between thepoints of entrance and emergence from the surface. The intensity of a transmitted

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374 Yevgen Grynko and Yuriy G. Shkuratov

d = 50 μm

0.5 1 1.5 2 2.5 λ, μm

0

0.02

0.04

0.06

Ref

lect

ance

0.5 1 1.5 2 2.5 λ, μm

0

0.04

0.08

0.12

Ref

lect

ance

0.5 1 1.5 2 2.5 λ, μm

0

0.04

0.08

0.12

0.16

0.2

0.24

Ref

lect

ance

0.5 1 1.5 2 2.5 λ, μm

0

0.1

0.2

0.3

0.4

Ref

lect

ance

α = 30°

Single

Multiple

Total

Single

Multiple

Total

Single

Multiple

Total

Single

Multiple

Total

d = 100 μm

d = 250 μm d = 500 μm

Fig. 9.44. Comparison of the contributions of single and multiple scattering compo-nents to the total scattered flux for media with different particle sizes d, phase angleα = 30◦.

ray is proportional to exp(−4πk(λ)L/λ). The value of L is a function of thephase angle α. These values are different for different orders of scattering. InFig. 9.45 we show the calculated distribution of 〈L(α)〉. For each ray trajectorywe calculated pathlength L and took the average value over all rays collected ineach phase angle bin. As one can see, the average ray pathlength increases inthe range 0◦–80◦ (this corresponds to the increasing spectral slope), reaches amaximum at 70◦, and then decreases at large α.

There is another application of the light scattering modeling in the remotesensing of planetary surfaces. Recently a new approach to the the analysis ofmultispectral polarimetric data has been proposed [67]. It was shown that colorratios C‖(0.65/0.42 μm) = R‖(0.65 μm)/R‖(0.42 μm) and C⊥(0.65/0.42 μm) =R⊥(0.65 μm)/R⊥(0.42 μm) (polarized, correspondingly, parallel and perpen-dicular to the plane of scattering) suggest independent information about thesurface of a planet. The value C⊥ is formed mostly with quasi-Fresnel scatteringand gives the main contribution to the commonly used color-index distributionon the lunar surface. The color ratio C‖ is primarily formed by internal multiple

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9 Light scattering from particulate surfaces 375

0 30 60 90 120 150α, °

1

2

3

4

5

<L

>,

arbi

trar

yun

its

Fig. 9.45. Phase-angle distribution of the average pathlengths 〈L〉 that rays passthrough the particles in the medium between the moments of entrance and emergence.

scattering and is very sensitive to the absorbing properties of regolith particles.Thus, imaging of such color ratios allows mapping of the planetary surface unitswith different chemical and physical properties. Accordingly, it seems to be in-teresting to study spectral dependencies of scattering components polarized indifferent polarization planes. Our model can be applied to do this and to improvethe theoretical basis of the approach.

In Fig. 9.46 we study spectra separated according to the plane of polarizationof the scattered radiation. As expected, the normalized spectra corresponding tothe perpendicular and parallel polarization approximately reproduce the phaseangle behavior of the total scattering spectra. The perpendicular component isless absorbed with the increase of the particle size. The reflectance correspond-ing to this component at large phase angles is higher than that of the parallelpolarized component. This is due to the contribution of the external Fresnelscattering to perpendicular polarization which is absorption-independent. Thisis consistent with the conclusion reached in [67]. We note that for both compo-nents the slope of spectra increases in a similar way with increase of particle size.As for absorption bands, they do not change noticeably from one component tothe other.

9.6 Conclusion

Practical use of the optical remote sensing data obtained for natural particulatesurfaces requires developing methods of interpretation which allows extraction ofinformation about the physical properties from the measured scattering charac-teristics. The interpretation basis may include laboratory experiments, analyticaltheories and numerical simulations. The last approach combines the advantagesof theoretical and laboratory methods. With computer modeling based on the ge-ometric optics approximation one can solve some problems in radiative transfer

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376 Yevgen Grynko and Yuriy G. Shkuratov

0.5 1 1.5 2 2.5 λ, μm

0

0.1

0.2

0.3

Ref

lect

ance

10°

150°

120°

90°

30°60°

0.8

1.2

1.6

2

2.4

Nor

mal

ized

ref

lect

ance

⊥ 10°

150°

120°

90°

30°60°

d = 100 μma

0.5 1 1.5 2 2.5 λ, μm

0

0.1

0.2

0.3

Ref

lect

ance

||10°

150°

120°

90°30°

60°

0.8

1.2

1.6

2

2.4

Nor

mal

ized

ref

lect

ance

|| 10°

150°

120°

90°30°60°

d = 100 μmb

Fig. 9.46. Spectral dependencies of the polarization components R⊥ (a) and R‖ (b)of reflectance and normalized reflectance at different phase angles for a medium withsize of particles d = 100 μm.

in particulate media if the particles are larger than the wavelength of the light.Using our model we have studied photometric and polarimetric phase functionsand reflectance of surfaces formed with random irregular particles.

From the results of our ray tracing simulations we can conclude the following.

1. Modeling of multiple scattering in systems of opaque particles with Lamber-tian surface showed that the contribution of high orders of scattering dropsquickly even at unit albedo. The reason for this reduction is ray escape fromthe medium. If the albedo of the particle surface were less than 1, the de-crease of the contributions with growth of scattering orders would be morerapid.

2. The shadow-hiding effect produces the opposition surge only in the first orderof scattering. The parameters of this surge depend strongly on the packingdensity of the medium. The packing density plays a secondary role for higherorders. A small influence of the shape of particles, if they are opaque, on thephase functions of surfaces is found.

3. No opposition effect and no negative polarization are observed for individualrandomly irregular particles or for media composed of these particles if they

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9 Light scattering from particulate surfaces 377

are transparent or semi-transparent. This means that these phenomena arecaused only by coherent backscattering.

4. All scattering features specific for regular particles, spheres and cubes, (likethe glory, rainbow, and retroreflector effect) are clearly observed in the caseof media even at conservative scattering. On the other hand, media composedof spheres clearly exhibit the depolarization effect, showing a significant roleof multiple scattering. Thus an adequate light scattering model for natu-ral surfaces like planetary regoliths should not be based on the constituentparticles with regular shapes.

5. With ray-tracing simulations, we found a polarization ledge at very largephase angles which had been found before in the laboratory photopolari-metric measurements of light scattered by substrates consisting of semi-transparent particles with sizes significantly larger than the wavelength. Theledge appears to be related to light passing through particles in the upperlayers of the substrates.

6. For all types of media the Umov effect is observed. It can depend on the shapeof particles forming the medium. This feature can be used as a diagnostictool in the remote sensing of Solar System bodies.

7. Our results also reveal a strong dependence of the spectral slope on thephase angle. This dependence is not monotonous: the slope can be eitherincreasing or decreasing depending on the size of particles. Both single andmultiple scattering appear to be important and play a significant role in theformation of the reflectance spectra.The multiple scattering component isresponsible for the non-monotonous phase-dependence of the spectral slope.The different illumination/observation geometry can also influence the depthof the absorption bands.

Acknowledgment

The authors are grateful to Sergey Bondarenko for measurements of quarts pow-der to verify the computer model.

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44. Mukai S., T. Mukai, K. Weiss, R. Zerull, 1982: Scattering radiation by a largeparticle with a random rough surface, Moon and Planets, 26, 197–208.

45. Nousiainen, T., K. Muinonen, P. Raisanen, 2003: Scattering of light by large Saha-ran dust particles in a modified ray optics approximation, J. Geophys. Res., 108,12-1–12-17.

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46. Okada, Y., A. M. Nakamura, T. Mukai, 2006: Light scattering by particulate me-dia of irregularly shaped particles: laboratory measurements and numerical simu-lations, J. Quant. Spectrosc. Radiat. Transfer, 100, 295–304.

47. Peltoniemi, J., K. Lumme, K. Muinonen, W. Irvine, 1989: Scattering of light bystochastically rough particles, Appl. Opt., 28, 4088–4095.

48. Peltoniemi, J., 1992: Radiative transfer in stochastically inhomogeneous media, J.Quant. Spectrosc. Radiat. Transfer, 50, 655–671.

49. Pieters, C. M., S. Pratt, H. Hoffman, P. Helfenstein, J. Mustard, 1991: Bidirectionalspectroscopy of returned lunar soils: detailed ‘Ground Truth’ for planetary remotesensors, LPSC, 22, 1069.

50. Rode, O., A. Ivanov, M. Nazarov, A. Cimbal’nikova, K. Jurek, V. Hejl, 1979:Atlas of Photomicrographs of the Suface Structures of the Lunar Regolith Particles,Akademia, Prague.

51. Schiffer, R., and K. Thielheim, 1982: A scattering model for the zodiacal lightparticles, Astron. Astrophys., 116, 1–9.

52. Shkuratov, Yu., and L. Melkumova, 1986: On a feature in polarization of light scat-tered by the Moon at large phase angles, Astronomical Circular (GAISh, Moscow),1447, 5–7.

53. Shkuratov, Yu., 1987: A model of spectral albedo of atmosphereless celestial bodies,Kinematika I Fizika Nebesnykh Tel, 3, 39–46 [in Russian].

54. Shkuratov, Yu., and N. Opanasenko, 1992: Polarimetric and photometric proper-ties of the Moon: telescope observation and laboratory simulation 2. The positivepolarization, Icarus, 99, 468–484.

55. Shkuratov, Yu. G., and D. G. Stankevich, 1992: The shadow effect for planetarysurfaces with Gaussian mesotopography, Sol. Syst. Res., 26, 201–211.

56. Shkuratov, Yu. G., 1994: Light backscattering by the solid surfaces of celestialbodies: theoretical models of the opposition effect, Sol. Syst. Res., 28, 418–431.

57. Shkuratov, Yu., K. Muinonen, E. Bowell, K. Lumme, J. Peltoniemi, M. A.Kreslavsky, D. G. Stankevich, V. P. Tishkovetz, N. V. Opanasenko, L. Y. Melku-mova, 1994: A critical review of theoretical models for the negative polarization oflight scattered by atmosphereless solar system bodies, Earth, Moon, and Planets,65, 201–246.

58. Shkuratov, Yu. G., 1995a: Photometric properties of physical fractals, Opt. Spec-trosc., 79, 102–108.

59. Shkuratov, Yu. G., 1995b: Fractoids and photometry of solid surfaces of celestialbodies, Sol. Sys. Res., 29, 421–432.

60. Shkuratov, Y. G., L. Y. Melkumova, N. V. Opansenko, D. G. Stankevich, 1996:Phase dependence of the color indices of solid surfaces of celestial bodies, Sol. Sys.Res., 30, 71–79.

61. Shkuratov, Yu., L. Starukhina, H. Hoffmann, G. Arnold, 1999: A model of spectralalbedo of particulate surfaces: implication to optical properties of the Moon, Icarus,137, 235–246.

62. Shkuratov, Yu., D. Petrov, G. Videen, 2003: Classical photometry of prefractalsurfaces, J. Opt. Soc. Am. A, 20, 2081–2092.

63. Shkuratov, Yu., D. Stankevich, D. Petrov, P. Pinet, Au. Cord, Y. Daydou, 2004a:Interpreting photometry of regolith-like surfaces with different topographies: shad-owing and multiple scattering, Icarus, 173, 3–15.

64. Shkuratov, Yu., G. Videen, M. Kreslavsky, I. Belskaya, V. Kaydash, A. Ovcharenko,V. Omelchenko, N. Opanasenko, E. Zubko, 2004b: Scattering properties of plane-

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tary regoliths near opposition, in Photopolarimetry in Remote Sensing, G. Videenet al. (eds.), Kluwer Academic Publishers, Dordrecht, 191–208.

65. Shkuratov, Yu., and Ye. Grynko, 2005: Light scattering by media composed of semi-transparent particles of different shapes in ray optics approximation: consequencesfor spectroscopy, photometry, and polarimetry of planetary regoliths, Icarus, 173,16–28.

66. Shkuratov, Yu., S. Bondarenko, A. Ovcharenko, C. Pieters, T. Hiroi, H. Volten,O. Munoz, G. Videen, 2006: Comparative studies of the reflectance and degree oflinear polarization of particulate surfaces and independently scattering particles,J. Quant. Spectrosc. Radiat. Transfer, 100, 340–358.

67. Shkuratov, Yu., N. Opanasenko, E. Zubko, Ye. Grynko, V. Korokhin, C. Pieters,G. Videen, U. Mall, A. Opanasenko, 2007: Multispectral polarimetry as a tool toinvestigate texture and chemistry of lunar regolith particles, Icarus, 187, 406–416.

68. Simmons, E. L., 1975: Diffuse reflectance spectroscopy: a comparison of the theo-ries, Appl. Opt., 14, 1380–1386.

69. Stankevich, D. G., and Yu. G. Shkuratov, 1992: Numerical simulation of shadowingon a statistically rough planetaty surface, Sol. Sys. Res., 26, 580–589.

70. Stankevich, D., Yu. Shkuratov, K. Muinonen, 1999: Shadow-hiding effect in ingo-mogeneous and layered particulate media, J. Quant. Spectrosc. Radiat. Transfer.,63, 445–458.

71. Stankevich, D. G., and Yu. G. Shkuratov, 2000: The shadowing effect in regolith-type media: numerical modeling, Sol. Sys. Res., 34, 285–294.

72. Stankevich, D. G., Yu. G. Shkuratov, K. O. Muinonen, O. V. Miloslavskaya,2000: Simulation of shadings in systems of opaque particles, Opt. Spectrosc., 88,682–685.

73. Stankevich, D., Yu. Shkuratov, Ye. Grynko, K. Muinonen, 2003: Computer simula-tions for multiple scattering of light rays in systems of opaque particles, J. Quant.Spectrosc. Radiat. Transfer, 76, 1–16.

74. Stankevich, D., and Yu. Shkuratov, 2004: Monte Carlo ray-tracing simulation oflight scattering in particulate media with optically contrast structure, J. Quant.Spectrosc. Radiat. Transfer, 87, 289–296.

75. Stokes, G., 1904: On the intensity of light reflected from or transmitted througha pile of plates, in Mathematical and Physical Papers of Sir George Stokes, vol. 4,Cambridge University Press, London, 145–156.

76. Takano, Y., and K. Jayaweera, 1985: Scattering phase matrix for hexagonal icecrystals computed from ray optics, Appl. Opt., 24, 3254–3263.

77. Tishkovets, V. P., and K. Jockers, 2006: Multiple scattering of light by denselypacked random media of spherical particles: Dense media vector radiative transferequation, J. Quant. Spectrosc. Rad. Trans., 101, 54–72.

78. Trowbridge, T., 1984: Rough-surface retroreflection by focusing and shadowingbelow a randomly undulating interface, J. Opt. Soc. Am. A, 1, 1019–1027.

79. Volten, H., O. Munoz, E. Rol, J. F. de Haan, J. W. Hovenier, K. Muinonen, T.Nousiainen, 2001: Scattering matrices of mineral aerosol particles at 441.6 nm and632.8 nm, J. Geophys. Res., 106, 17,375–17,402.

80. Warell, J., and D. T. Blewett, 2004: Properties of the Hermean regolith: V. Newoptical reflectance spectra, comparison with lunar anorthosites, and mineralogicalmodeling, Icarus, 168, 257–276.

81. Wendling, P., R. Wendling, K. Weickman, 1979: Scattering of solar radiation byhexagonal ice crystals, Appl. Opt., 18, 2663–2671.

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82. Whalley, E., and G. McLaurin, 1984: Refraction halos in the Solar System: 1. Halosfrom cubic crystals that may occur in atmospheres in the Solar System, J. Opt.Soc. Am. A, 12, 1166–1170.

83. Wielaard, D. J., M. I. Mishchenko, A. Macke, B. E. Carlson, 1997: ImprovedT-matrix computations for large nonabsorbing and weakly absorbing nonspheri-cal particles and comparison with geometrical-optics approximation, Appl. Opt.,36, 4305–4313.

84. Wolff, M., 1975: The polarization of light reflected by rough planetary surface,Appl. Opt., 14, 1395–1404.

85. Yang, P., and K. N. Liou, 1995: Light scattering by hexagonal ice crystals: com-parison of finite-difference time domain and geometric optics models, J. Opt. Soc.Am. A, 12, 162–176.

86. Yang, P., and K. N. Liou, 1996: Geometric-optics-integral-equation method forlight scattering by nonspherical ice crystals, Appl. Opt., 35, 6568–3584.

87. Yang, P., and K. N. Liou, 1997: Light scattering by hexagonal ice crystals: solutionsby a ray-by-ray integration algorithm, J. Opt. Soc. Am. A, 14, 2278–2289.

88. Yang, P., and K. N. Liou, 2007: Light scattering and absoroption by nonsphericalice crystals, in Light Scattering Reviews, vol. 1, A. Kokhanovsky (ed.), Springer-Praxis, Berlin, 31–72.

89. Zubko, E., Yu. Shkuratov, K. Muinonen, 2001: Light scattering by composite par-ticles comparable with wavelength and their approximation by systems of spheres,Opt. Spectrosc., 91, 273–277.

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10 Laboratory measurements of reflected lightintensity and polarization for selectedparticulate surfaces

Yuriy G. Shkuratov, Andrey A. Ovcharenko, Vladimir A. Psarev andSergey Y. Bondarenko

10.1 Introduction

Optical measurements in the laboratory are an important source of informa-tion on light scattering by particulate surfaces. One example of such surfaces isplanetary regolith. Laboratory photometric and polarimetric measurements al-low direct verifications of theoretical models of scattering (e.g., Shkuratov et al.,2007a; Zhang and Voss, 2005); they potentially may suggest new regularities thatcan improve theoretical interpretation. Moreover the measurements can suggestnew effects, which require innovative theoretical approaches. For example, photo-metric and polarimetric opposition phenomena, well-known to astronomers andexperimenters (e.g., Lyot, 1929; Oetking, 1966), were long without a satisfactoryexplanation. The effect of coherent backscattering enhancement explanation ofthese phenomena in astronomy was suggested in Shkuratov (1985, 1988) andMuinonen (1989) almost 60 years after their discovery.

This chapter will focus on photometric and polarimetric measurements car-ried out with three instruments constructed at the Astronomical Institute ofKharkov V. N. Karazin National University. In all the instruments we used un-polarized light for illumination to simulate phase dependencies of intensity andlinear polarization degree of light scattered by the regoliths of atmospherelesscelestial bodies. Such a simulation is the main motivation of our laboratorymeasurements. In particular, we have studied fundamental regularities for pho-tometric and polarimetric opposition effects, searching for correlations of theirparameters with albedo of particulate surfaces and average size of particles; thisallows us to reach conclusions concerning the physical properties of planetaryregoliths (e.g., Shkuratov et al., 2004).

We consider measurements carried out at a wide range of phase angles α(α = π−θ, where θ is the scattering angle). We present the experimental compar-ison of the scattering properties of representative particles in air and correspond-ing particulate surfaces. These measurements have been carried out at a widecommon range of phase angles, 7–150◦, of Kharkov (for scattering measurementsof surfaces) and the University of Amsterdam (for scattering measurements ofparticles in air) instruments (Volten et al., 2001; Shkuratov et al., 2006). This

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range includes the positive polarization maximum. Analysis of the maximumpolarization parameters may give a unique opportunity to study compositionalinhomogeneity of particles (Shkuratov et al., 2007b).

We also consider measurements at small phase angles. To model the narrowphotometric and polarimetric opposition spikes in a laboratory is not a simpletask, since very small angular apertures of the light source and the receiver mustbe used for such measurements. The small-phase-angle photometer/polarimeterthat we built (Shkuratov et al., 2002) allows a minimum phase angle of 0.2◦. In-teresting regularities were revealed with this instrument. In particular we havefound systematic dependence of the parameters of negative polarization on par-ticle size.

Numerous small-phase-angle laboratory measurements were made at the JetPropulsion Laboratory using the long-arm goniometric photopolarimetry witha laser as a light source covering a phase angle range of 0.05–5◦ (Nelson etal., 1999). We made intercalibration of our and JPL instruments using the samesamples. This reveals very good coincidence. We note that the JPL limit of 0.05◦

seems still to be too large to model these spikes of transneptunian objects. Be-cause of that we construct the third instrument allowing measurements startingfrom 0.008◦ (Psarev et al., 2006). We here show first results obtained with thisinstrument for very small phase angles.

This survey consists of two basic sections. One of them is devoted to de-scription of the laboratory instruments and samples used including optical andelectronic microphotographs of some samples. The last section represents mea-surements carried out with the three laboratory photopolarimeters of the As-tronomical Institute of Kharkov National University. At the present time all theinstruments are available for measurements in international cooperation.

10.2 Laboratory instruments and samples

The instruments mentioned cover different overlapping ranges of phase angles.All instruments are intercalibrated; they allow measurements in blue and redspectral bands. Two photopolarimeters use lamps as light sources simulating thesolar illumination. One of them working in the range 2–150◦ we call the wide-phase-angle photometer/polarimeter, the other one (0.2–17◦) is named the small-phase-angle photometer/polarimeter. The third instrument using lasers coversthe range of phase angle 0.008–1.6◦ (laser super-small-phase-angle photometer).

10.2.1 The wide-phase-angle photometer/polarimeter

This instrument allows us to measure phase curves of intensity and linear po-larization degree for powdered samples illuminated by unpolarized light (Bon-darenko et al., 2006). An image and scheme of the instrument are shown inFig. 10.1.

We study samples in two spectral bands with λeff = 0.44 μm and λeff = 0.63μm (the bandwidths are approximately 10%). Our polarimetric measurements

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(a) (b)

Fig. 10.1. A view of the Kharkov instrument at a large tilt of the receiver arm (a)and its general scheme (b).

have an accuracy of about 0.05%. The spectral bands are formed by means offilters with accounting for spectral features of the device. Rotation of the armwith the light source (a lamp) can provide reliable phase-angle measurements inthe range 2–150◦. We can use this entire range when the viewing angle is fixed at70◦ from the surface normal (Fig. 10.1 a,b). The scattering plane is perpendicularto the sample surface. A portion of measurements have been carried out at zeroviewing angles. The linear dimension of the powdery samples is approximately10 × 20 mm. The thickness of the samples is about 4–5 mm, which provides agood approximation of a semi-infinite medium. The samples are formed withoutand with compacting to study the influence of surface compression on scatteringproperties. The albedo of the samples is determined at a phase angle of 2◦ withrespect to a compressed Halon sample (Weidner and Hsia, 1981).

10.2.2 The small-phase-angle photometer/polarimeter

This laboratory instrument (photopolarimeter) measures the phase-angle de-pendences of intensity and linear polarization of light (Shkuratov et al., 2002Ovcharenko et al., 2006). The instrument measures in the phase-angle range0.2–17◦ (see Fig. 10.2) using unpolarized light sources (filament lamps). Theinstrument has almost the same spectral bands in comparison with the wide-phase-angle photopolarimeter. This allows mutual calibration and merging ofdata files. The angular diameters of the source and receiver apertures are 0.05◦.

Instrumental (parasitic) polarization is a very important problem in labora-tory polarimetric measurements. Determining and correcting the instrumental

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Fig. 10.2. The small-phase-angle photopolarimeter: general view (a), light source andphotomultiplier blocks (b) and (c).

polarization associated with the light source is the most difficult problem. Thepolarization of the halogen lamp can achieve several percent are points. We com-pensate much of the polarization by inserting a tilted glass plate with thicknessof about 300 μm, decreasing the instrumental polarization to several tenths ofa percent. The rest of the parasitic polarization is compensated with a Lyotdepolarizer. The instrumental polarization of the receiver portion is determinedby positioning a diffusing light source at the sample location. This light sourceis a lamp covered by two frosted glass plates. Measurements obtained by rotat-ing the diffusing calibration light source allow us to determine the instrumentalcomponent of the receiver. Estimates of the instrumental polarization compo-nent show the value of the receiver parasitic polarization to be about 0.03%. Itis very stable (with variations less than 0.001%) and taken into account in dataprocessing.

The small-phase-angle photopolarimeter accepts samples of 60 mm in diame-ter. Changing phase angles is performed by rotating the light source around theaxis that lies on the sample surface; thus the receiver is immobile (see Fig. 10.2).The optical design of the instrument allows very close positions of the lightsource and receiver inlets. The construction of the small-phase-angle photopo-larimeter allows the light source to move on one side; whereas, in the case of thewide-phase-angle photopolarimeter the light source can approach the receiverfrom both sides.

The small-phase-angle photopolarimeter works so that the optical axis of thereceiver is inclined at 5◦ to the normal of the horizontal plane. This permits usto perform investigations of particulate samples as well as to measure colloids inliquids to study light scattering by rarefied media, since the specular flash fromthe liquid surface does not hit the receiver.

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Measuring of scattering properties of a sample is carried out in two passes,with increasing and then with decreasing of phase angles, and their deviationallows for some estimate of instrumental accuracy. If the deviations are signif-icant, data acquisition is repeated. For the small-phase-angle photopolarimeterthe minimal phase-angle step is 0.027◦, and data acquisition can last for up to70 hours to achieve high accuracy. Low-resolution measurements with a step of0.5◦ precede these measurements to locate the phase-angle ranges needed formore detailed study. Typically, the range of interest is the small phase angles.Sample albedo A is measured and determined at α = 2◦ relative to a com-pressed Halon sample that we use as a reflectance standard, as in the case ofthe wide-phase-angle photopolarimeter. It should be emphasize once more thatthe small-phase-angle-range photopolarimeter is calibrated against a compara-ble JPL NASA instrument (Nelson et al., 1999). Both the Kharkov and JPLinstruments are capable of making measurements at phase angles of less thanone degree where photometric phase curves are typically very steep. We findthat the reflectance phase curve measurements with both instruments agree rea-sonably well.

10.2.3 The laser super-small-phase-angle photometer/polarimeter

To decrease the limiting phase angle further, we constructed an instrument tomeasure extremely small phase angles. The final instrument also was constructedin the Astronomical Institute of Kharkov University (see Fig. 10.3(a)). The newinstrument allows for high-quality measurements with a minimum phase angleof 0.008◦. We here present several results of measurements obtained in the laserphotometry regime (Psarev et al., 2006). The regime allows investigations of theopposition effect for complicated surfaces in vertical and horizontal positions inthe phase-angle range 0.008–1.6◦. The extremely small phase angles are feasibledue to small linear apertures of the light source and receiver (photomultiplierHamamatsu H5783-01) and the large distance from the light source and detectorto the scattering surface (samples) that is 25 m. The linear diameter of theapertures is 2 mm. The diameter of the output light beam is 1.4 mm. In themeasurements we use a diode monomodal non-polarized laser (50.0 mW) withwavelength 0.658 μm as the light source. For light detection we use a pinholecamera, a circular cone with a truncated top (see Fig. 10.3(b)). The opticalscheme of the photometer is shown in Fig. 10.4. The linear diameter of thesamples we use is about 7 cm. We change the phase angle by moving the detectorblock. The phase angle resolution is about 0.008◦. The block consists of thepinhole camera with the photomultiplier inside and a coaxial guiding spyglass.The spyglass is needed for aligning the sample after a phase-angle displacementof the detector block. Each sample is measured at least twice at increasing anddecreasing phase angles. Coincidence of these two dependencies is an indicatorof the reproducibility of the measurements.

An important verification is to estimate parasitic light scattering by dustin air for low-albedo samples at extremely small phase angles. To test we useas a sample an optical filter that entirely absorbs light at the laser wavelength

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(a) (b)

Fig. 10.3. A general view of the laser laboratory photometer (a) and the laser andreceiver blocks of the instruments (b).

Fig. 10.4. Optical scheme of the laser photometer: (1) laser, (2) total reflection prism,(3) pinhole camera, (4) main photomultiplier, and (5) correction spyglass.

(see Fig. 10.5(a)). We tilt the filter so the specular reflection is diverted fromthe detector. Thus we obtain the signal due to the dust in air (Fig. 10.5(b)).Although in the laboratory we use a clean-room environment, the false spike isnoticeable. We take this spike into account in data processing. It is especiallynecessary for dark samples.

To avoid problems with laser speckle pattern we move samples during mea-surements providing good averaging. The sample is mounted on a moveablespring hanger. For powdered samples we use a deflection mirror or large prism

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(a) (b)

Fig. 10.5. Sample block of the laser photometer (a). Measurements of the instrumentdust contribution to the opposition effect (b).

of total reflection (10×10 cm) that allows measurements of horizontally locatedsamples.

All measurements are carried out in a dark room. The problem of eliminatingparasitic illumination is important. The presence of such illumination is verifiedby digital control shots with a Canon EOS 300D camera at the highest sensitiv-ity for different phase angles. The measurement includes estimates of the totallight background. In addition, the background-illumination test includes mea-surements of the light background from the small totally reflecting prism that isused in the optical scheme to steer the laser beam.

10.2.4 Samples

We here present samples used to study the influence of albedo, particle size, andparticle shape on the brightness opposition surge and the negative polarizationbranch. We also study the role of single particle scattering in the formation ofthe opposition phenomena.

The photometric and polarimetric properties of particle size separates ofvery absorbing (boron carbide, B4C) and very bright (alumina, Al2O3) materi-als from MICROGRIT were investigated. We measured samples of each materialcovering the particle size range of approximately 3–30 μm. The cumulative dis-tributions of particle sizes for the samples and other information can be foundat (http://www.microgrit.com). The distributions are rather wide and overlap.The samples were prepared by sieving the powders on plane substrates. Exam-ples of photomicrographs of samples with particle size near 30 μm are shown inFig. 10.6. The samples reveal a very complicated topography.

We used samples of natural mineral particles to experimentally compare thephase functions of intensity and degrees of linear polarization for particles inair and on a substrate. Detailed information about the physical properties of

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(a) (b)

Fig. 10.6. Examples of optical images of samples consisting of particles with a sizeof 30 μm. The samples have very different albedo: 92% for Al2O3 (a) and 6% for B4C(b).

the mineral samples can be found in Munoz et al. (2000), Volten et al. (2001),and Shkuratov et al. (2006). In Fig. 10.7 we show examples of SEM imagesof fine feldspar powder and powdered olivine with particle size <65 μm. Thefeldspar and olivine powders presented in Fig. 10.7 consist of particles withsharp angular shapes forming flat facets (Shkuratov et al., 2006). Large olivineparticles are covered with small roughness and dust particles. We also measuredthree particle-size separates of powdered red glass KC-17; the shape of the glassparticles resembles that of the feldspar particles.

(a) (b)

Fig. 10.7. Electronic photomicrographs of feldspar (a) and olivine (b).

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(a) (b)

Fig. 10.8. Electronic photomicrographs of smoked MgO (a) and carbon soot (b).

(a) (b)

Fig. 10.9. Electron photomicrographs of samples used. Silica spheres with the diameter≈ 1.0 μm (a) and iron spheres with average diameter 2 μm (b).

For measurements with the laser super-small-phase-angle photometer we usedvery fine powders of smoked MgO and carbon soot. Electronic microphotographsof these samples are shown in Fig. 10.8. As can be seen the MgO sample containsnumerous small cubic particles, whereas carbon soot particles are very smallaggregates of arbitrary shapes.

We also studied opposition spikes of small dielectric and iron spheres. Elec-tron photomicrographs of the samples are shown in Fig. 10.9: silica spheres withdiameters ≈ 1.0 μm (a) and iron spheres with average diameter 2 μm (b).

10.3 Results of measurements

In this section we present examples of photopolarimetric measurements with thethree instruments described above, studying (1) the influence of albedo and par-ticle size on photometric and polarimetric opposition effects, (2) the contributionof single-particle scattering to the formation of the opposition effects, and (3)the extremely narrow opposition spikes of dielectric and metal powders.

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10.3.1 Albedo and particle size effects

In Fig. 10.10 we present phase curves of the intensity (normalized at 0.2◦) anddegree of linear polarization of light scattered from particle-size separates of thebright (Al2O3) and dark (B4C) powders. The curves were obtained with thetwo instruments, small-phase-angle (small circles) and wide-phase-angle pho-topolarimeters (large triangles), and the data obtained with these different in-struments show very good coincidence.

(a) (b)

(c) (d)

Fig. 10.10. Phase curves of normalized intensity (upper panels) and degree of negativepolarization (lower panels) for samples of alumina and boron carbide with averageparticle size close to 3 μm (a), 7 μm (b), close to 13 μm (c), and 30 μm (d).

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As can be seen, dark and bright samples demonstrate very different opposi-tion effects. For fine powders consisting of small 3-μm and 7-μm particles, thenegative polarization branches are relatively wide with the the inversion angleαinv close to 14◦ and Pmin near −0.7% (αinv is the angle where the sign ofpolarization reverses). For the corresponding alumina particle-size fractions thenegative polarization |Pmin| is very small, 0.15%.

The behavior of the brightness spike is also very different for the dark andbright surfaces. Indeed, the dark powders display almost linear phase curves withthe surge amplitude near 45% at the phase angle range 0–20◦. Figure 10.6 showsthat particles, independent of their albedo, constitute a clump topography thatis clearly seen in an optical microscope. For dark boron carbide the topographyeffectively generates a shadow-hiding effect that is responsible for the brightnesssurge.

The bright alumina samples reveal very prominent non-linear behavior atphase angles < 4◦. In this case the coherent enhancement of backscatter is likelythe main contributor. However the nearly linear decrease of the normalized inten-sity at phase angles greater than 4◦ indicates the influence of the shadow-hidingeffect despite the high albedo of the alumina samples.

The bright samples reveal very shallow negative polarization branches. Con-versely the dark samples have prominent negative polarization. For boron carbidesamples, with increase of particle sizes from 3 μm to 30 μm, the minimum of thenegative polarization branch shifts toward zero phase angle. This accompaniesa gradual increase of |Pmin| and the branch acquiring a more asymmetric, beak-like shape. Measurements of particle-size separates of powdered red glass KC-17suggest an interesting evolution of phase curve with particle size: the larger thesize, the smaller the inversion angle and the smaller the |Pmin| (see Fig. 10.11).

The amplitude of the brightness surge gradually decreases for samples ofalumina and boron carbide in the range of phase angles 0–20◦ as particle sizeincreases. This is due to weakening of the shadow-hiding effect, since with theparticle size increase the particulate surfaces become more uniform. This effectis well known and is related to weakening of the role of cohesion (van der Waals)forces that form ‘fairy castles’ structures. The gradual decrease of the oppositionsurge amplitude is accompanied by a narrowing of the intensity peak. In theboron-carbide samples, the intensity peak does not have linear response at smallphase angles as particle size increases. In Figs 10.10, it also is interesting thatthe intensities of the bright and dark materials intersect near α = 5◦ almostindependently of particle size.

10.3.2 The contribution of single light scattering

Comparative photometric and polarimetric laboratory measurements of parti-cles in air and on a substrate (particulate surface) at large phase angles areimportant. For instance, they allow one to estimate the contributions of singleand multiple scattering to the formation of the positive polarization maximum(Shkuratov et al., 2007a). They also provide an opportunity to verify the ap-plicability of classical radiative transfer theory to calculate light scattering by

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394 Y. G. Shkuratov, A. A. Ovcharenko, V. A. Psarev and S. Y. Bondarenko

Fig. 10.11. Polarimetric phase curves of particle size separates of powdered red glassKC-17. Measurements were carried out in red and blue light. Albedo A is determinedat α = 2◦. Three average particle sizes are studied: 2, 5, and 30 μm.

particulate surfaces. Experimental comparison of scattering properties of repre-sentative particles in air and corresponding particulate surfaces has been carriedout with the equipment at Kharkov National University (for scattering measure-ments of surfaces), and the equipment located at the University of Amsterdam(for scattering measurements of particles in air) (Munoz et al., 2000; Volten etal., 2001). We have compared the light-scattering measurements for the rangeof phase angles, 7–150◦, that is common for both the instruments. We haveshown the backscatter effect of particulate surfaces to be due partially to thecontribution of single scattering. In addition it was shown that the negative po-larization of the surfaces is a remnant of the negative polarization of the singlescattering by the particles that constitute the surfaces (Shkuratov et al., 2004,2006). Partially this concerns the positive polarization maximum (Shkuratov etal., 2007b).

Figure 10.12 presents measurements of intensity and polarization degree atthe two wavelengths for particles in air and particulate surfaces before and aftercompression for feldspar and olivine powders. In the case of particles in air wedeal with independent single-particle scattering. The scattering demonstrates

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10 Laboratory measurements of reflected light intensity 395

(a) (b)

Fig. 10.12. Photometric and polarimetric phase curves for particulate surfaces (com-pressed and uncompressed) and aerosol particles measured at λ = 0.44 μm and 0.63μm for the feldspar (a) olivine (b) powders. The intensity is normalized at the phaseangle 10◦.

strong forward and a relatively small backward increase in intensity. All thecompressed samples also reveal prominent forward scattering. The uncompressedsamples mainly do not show this feature. This is consistent with the shadow-hiding effect.

As can be seen in Figs 10.12 the samples demonstrate positive polarizationbranches with maxima at large phase angles. For bright uncompressed surfacesthe branches are very shallow as compared to those of the aerosol particles inair. Generally polarization phase dependencies of uncompressed surfaces quali-tatively resemble those of single particles, suggesting that for the uncompressedsurfaces the positive polarization is probably dominated by single-particle scat-tering. Bright particulate surfaces have lower values of the positive polarizationmaxima because multiple scattering suppresses polarization. The polarizationmaximum of single-particle scattering is located at smaller phase angles thanthat of the particulate surfaces. This also can be related to multiple scatteringin the particulate surfaces.

Figure 10.13(a) presents a correlation of the inversion angle αinv of particu-late surfaces and corresponding αinv of independently scattering particles. Thiscorrelation unambiguously indicates that the negative polarization branches ob-served for the particulate surface may be remnants of the corresponding branchesarising at single-particle scattering. Almost all points lie below the line with 45◦

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396 Y. G. Shkuratov, A. A. Ovcharenko, V. A. Psarev and S. Y. Bondarenko

(a) (b)

Fig. 10.13. The inversion angle of particulate surfaces as a function of the correspond-ing inversion angle of independently scattering particles. Measurements were carriedout in blue and red light when the viewing angle is zero (a) and 70◦ (b). The angleαinvis given in degrees. The line with 45◦ inclination is given.

inclination, showing that multiple scattering in particular surfaces can reducethe inversion angle. It should be emphasized that in the case of Fig. 10.13(a)we use data for standard observation/illumination geometry, when the viewingangle is zero. There is also a correlation between values of αinv for tilted viewing(Fig. 10.13(b)). In this case all points lie above the line with the 45◦ inclinationpassing through the 0,0 point. This means that for particulate surfaces there isan additional mechanism of negative polarization at tilted viewing (see below).

Prominent differences are observed for polarimetric curves of compressedand uncompressed samples. The compression is made with a glass plate, and thesample porosity decreases by nearly a factor of two, while the sample microtopog-raphy becomes much smoother. After compression the polarimetric curves havemuch more prominent maxima, shifting to larger phase angles. As for increasingmaximum values, this is partly due the compression smoothing the topographyand suppressing the multiple scattering on the surfaces, which provides a largercontribution from the single-particle scattering. However, for some surfaces themaximum values for the compressed samples can be noticeably higher than thosefor the aerosol particles, showing that the positive polarization branch is notformed with the single-particle scattering only. At large phase angles, when theincident and emergent angles are close to grazing, coherent single scattering canbe observed, if electromagnetic phase shifts between rays scattered by differentsurface particles is small enough. As can be seen in Fig. 10.14, the larger thephase angle, the smaller the shift, which can provide constructive interference.It can be shown that the condition for constructive interference is

h1 + cosα

cos i� λ,

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10 Laboratory measurements of reflected light intensity 397

Fig. 10.14. The interference of singly scattered rays at large phase angles.

where h is the difference of heights, λ is the wavelength, α and i are the phaseangle and angle of incidence, respectively (see Fig. 10.14). When this conditionholds, the particulate surfaces appear relatively smooth and scatter light as anearly plane surface whose reflections are described by the Fresnel formulas thatpredict a high polarization degree for the phase angle range.

All the surfaces also demonstrate a prominent backscattering effect at thistilted observation geometry. The brightness phase curves of the surfaces andaerosol particles almost coincide with each other at small phase angles ofless than 20◦. Uncompressed particulate surfaces show negative polarizationbranches; however, the depths of the branches are less than those of the aerosolparticles. As was noted, this effect is partly a remnant of the negative polariza-tion of the single scattering by the particles that constitute the surfaces. On theother hand, it can be related to a pure geometric factor as the compressed tiltedsurfaces reveal negative polarization branches deeper and wider at small phaseangles than before compression (e.g., Fig. 10.13(b)).

There is a simple mechanism that can be responsible for the negative po-larization at slanting view. Figure 10.15 shows double scattering in the surfaceplane at tilted geometry. Scatterers 1 and 2 lie in the scattering plane, whereasscatterers 3 and 4 lie in a plane perpendicular plane. If we consider small singlescatterers whose polarization tends to be positive over the entire phase-anglerange, the double-scattering events 0→3 and 0→4 result in a negative polar-ization, whereas the double-scattering events 0→1 and 0→2 result in positivepolarization. Obviously for large tilts the negative polarization produced by thedouble-scattering events 0→3 and 0→4 dominates the positive polarization orig-inated with the double scattering 0→1 and 0→2 (when the tilt is close to 90◦

the double scattering 0→1 and 0→2 produce almost zero polarization). This in-

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398 Y. G. Shkuratov, A. A. Ovcharenko, V. A. Psarev and S. Y. Bondarenko

Fig. 10.15. Double-scattering mechanism of the negative polarization for tilted par-ticulate surfaces.

terpretation is consistent with the fact that the negative polarization branchesare more prominent for compressed (smoothed topography) samples.

10.3.3 Opposition spikes

First we show the results of our measurements of three samples with signifi-cantly different albedo and structure: a very bright smoked MgO surface, a veryfluffy powder consisting of very small particles of SiO2 (the particle size is about10 nm), and a very dark smoked carbon soot surface (see Fig. 10.8). We use thicklayers of materials, and all three surfaces are very fluffy, especially the surfaceof the SiO2 sample. Each point of the phase curves in Figs 10.16 and 10.17 isthe result of averaging three measurements. The duration of each measurementis 2 s. The curves for MgO and carbon soot are in good agreement with similardependencies obtained for the same samples using the small-phase-angle pho-topolarimeter working in the range of phase angles 0.2–17.2◦. The bright anddark samples have albedo A = 99% and 2.5%, respectively.

Figure 10.16(a) shows that the MgO sample demonstrates a very prominentopposition spike at phase angles less than 0.4◦. This spike undoubtedly is relatedto the coherent backscattering enhancement effect that is ubiquitously observedin nature (e.g., Shkuratov et al., 2004). This spike is similar to what was observedfor some Kuiper Belt objects and icy satellites (Belskaya et al., 2006). The darksample of carbon soot does not show the brightness opposition surge; its phasecurve is almost linear in the range 0.008–1.5◦ (see Fig. 10.16(a)). This featurecan be explained in terms of the shadow-hiding effect, although a contributionof the coherent backscatter enhancement cannot be entirely excluded. The fluffy

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10 Laboratory measurements of reflected light intensity 399

(a) (b)

Fig. 10.16. Normalized photometric phase curves for magnesia (MgO), ultra-dispersequartz (SiO2), and carbon soot (a) and mixtures of chalk and carbon soot (b) at λ =0.658 μm. The normalizing angle is 1.4◦.

(a) (b)

Fig. 10.17. Photometric phase curves for silica spheres (a) and iron particles (b).Open circles and points designate measurements, respectively, for thick and thin layers.Solid lines correspond to calculations with Mie theory. Crosses represent measurementsfor thick layer (semi-infinite layer) of coarse iron particles. All the dependencies arenormalized at 1.4◦. The insert demonstrates the Mie theory curve at wider range ofphase angles.

SiO2 sample demonstrates the narrowest opposition spike starting almost from0.1◦. The amplitude of this spike is similar to that of MgO.

Figure 10.16(b) shows our results for mixtures of white chalk and carbonsoot. These mixtures were prepared by dry-grinding chalk with the additionof soot to provide samples of different albedo. As can be seen, the larger thesample albedo, the more prominent the opposition effect. This is consistent withthe mechanism of coherent backscatter enhancement.

Figure 10.17 illustrates photometric phase-angle dependencies for particulatesurfaces consisting of dielectric and metal spherical particles with different di-ameters. The sample composed of small dielectric spheres shows a typical phasecurve resembling that for white powders, like chalk and MgO. A monolayer of

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400 Y. G. Shkuratov, A. A. Ovcharenko, V. A. Psarev and S. Y. Bondarenko

the spheres does not reveal the opposition spike, being in agreement with ourLorenz–Mie theory calculations for the silica (m = 1.46 + 0i ) spheres (seeFig. 10.17(a), line). The same is observed for iron spheres (m = 1.5 + 1.2i).We note that the large iron particle sample with a rough surface reveals a veryprominent opposition spike at phase angles less than 0.5◦. This is perhaps dueto the shadow-hiding effect.

Excepting the monolayer, no particulate surface in our measurements exhibitsa phase curve rounding at zero phase angle, as is predicted by rigorous modelsof coherent backscattering. It is apparent that we have not reached the phaseangle at which this rounding occurs.

10.4 Conclusion

We report the results of experiments designed to increase our understanding ofthe influence of particle size on the photometric opposition spike and negativepolarization observed in the reflectance and polarization phase curves of partic-ulate surfaces. We concentrate our studies on particle-size separates of alumina(bright powders), boron carbide (absorbing powders), and glass powders withintermediate albedo. The results suggest that the negative polarization has twodominant mechanisms, (1) the coherent-backscatter enhancement and (2) single-particle scatter, and that the contributions of the mechanisms are a function ofparticle size.

We also present results of photometric and polarimetric laboratory measure-ments of particulate surfaces undergoing different degrees of compression andaerosol particles at a phase angle range from 7◦ to 150◦. Several samples ofnatural materials with different albedo and particle sizes were measured in red(λ = 0.63 μm) and blue (λ = 0.44 μm) light using two different instruments:Kharkov’s large-phase-angle photopolarimeter for measurements of surfaces andAmsterdam’s polar nephelometer for particles in air. We compare phase curvesfor particulate surfaces with the corresponding measurements for single particles.Our measurements suggest that the maximum of positive polarization, which isobserved for particulate surfaces at large phase angles is mainly due to the con-tribution of single-particle scattering. We observe an increase in the negativepolarization at slanting view (70◦), which is explained via double scattering.

We carried out goniometric photometry of different samples at extremelysmall phase angles using a new laboratory setup with a laser light source (λ =0.658 μm). No opposition effect of carbon soot is found in the range 0.008–1.5◦;whereas MgO deposits and very disperse porous SiO2 powder demonstrate anarrow brightness spike of significant amplitude. We find a strong dependenceof phase-angle slope in this range on albedo when studying mixtures of whitechalk and carbon soot.

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10 Laboratory measurements of reflected light intensity 401

References

Belskaya, I.N., Ortiz, J.L., Rousselot, P., Ivanova, V., Borisov, G., Shevchenko, V.G.,and Peixinho, N., 2006: Low phase angle effects in photometry of trans-neptunianobjects: 20000 Varuna and 1996 TO66, Icarus 184, 277–284.

Bondarenko, S., Ovcharenko, A., Shkuratov, Y., Videen, G., Eversol, J., and Hart, M.,2006: Backscatter by surfaces composed of small spherical particles, Applied Optics,45, 3871–3877.

Lyot, B., 1929: Recherches sur la polarisation de la lumiere des planetes et de quelquessubstances terrestres, Ann. Obs. Meudon, 8, 1–161.

Muinonen, K., 1989: Electromagnetic scattering by two interacting dipoles, In Proc.1989 URSI Electromagnetic Theory Sympos. Stockholm: Sweden, 428–430.

Munoz, O., Volten, H., de Haan, J., Vassen, W., and Hovenier, J., 2000: Experimentaldetermination of scattering matrices of olivine and Allende meteorite particles,Astron. Astrophys., 360, 777–788.

Nelson, R., Hapke, B., Smyth, W., Shkuratov, Yu., Ovcharenko, A., and Stankevich,D., 1999: The reflectance phase curves at very small phase angle: a comparativestudy of two goniometers, Lunar and Planet. Sci. 30-th. LPI Houston. Abstract#2068.

Oetking, P. 1966: Photometric studies of diffusely reflecting surfaces with applicationsto the brightness of the Moon, J. Geophys. Res., 71, 2505–2513.

Ovcharenko, A.A., Bondarenko, S.Yu., Zubko, E.S., Shkuratov, Yu.G., Videen, G., andNelson R., 2006: Particle size effect on the opposition spike and negative polariza-tion, J. Quant. Spectrosc. Rad. Transfer, 101, 394–403.

Psarev, V., Ovcharenko, A., Shkuratov, Yu., Belskaya, I., and Videen G., 2006: Pho-topolarimetry of surfaces with complicated structure at extremely small phase an-gles, 9th Conference on Electromagnetic and Light scattering by Nonspherical Par-ticles. June 5–9, Russia, St. Petersburg. 235–238.

Shkuratov, Yu.G., 1985: On the origin of the opposition effect and negative polarizationfor cosmic bodies with solid surface, Astronomicheskii. Circular (Sternberg StateAstron. Inst., Moscow), no.1400, 3–6 (in Russian).

Shkuratov, Yu.G., 1988: The diffraction mechanism of opposition effect of surface withcomplex structure, Kinematika i Fisika Nebesnykh Tel, 4, 33–39 (in Russian).

Shkuratov, Yu., Ovcharenko, A., Zubko, E., Miloslavskaya, O., Nelson, R., Smythe, W.,Muinonen, K., Piironen, J., Rosenbush, V., and Helfenstein, P., 2002: The opposi-tion effect and negative polarization of structurally simulated planetary regoliths,Icarus, 159, 396–416.

Shkuratov, Yu., Videen, G., Kreslavsky, M., Belskaya, I., Ovcharenko, A., Kaydash,V., Omelchenko, V., Opanasenko, N., and Zubko, E., 2004a: Scattering propertiesof planetary regoliths near opposition, in: Photopolarimetry in Remote Sensing,G. Videen, Ya. Yatskiv, and M. Mishchenko (Eds). NATO Science Series. KluwerAcademic Publishers, London, 191–208.

Shkuratov, Y., Ovcharenko, A., Zubko, E., Volten, H., Munos, O., and Videen G.,2004b: The negative polarization of light scattered from particulate surfaces and ofindependently scattering particles, J. Quant. Spectrosc. Rad. Transfer, 88, 267–284.

Shkuratov, Y., Bondarenko, S., Ovcharenko, A., Pieters, C., Hiroi, T., Volten, H.,Munoz, O., and Videen, G., 2006: Comparative studies of the reflectance and degreeof linear polarization of particulate surfaces and independently scattering particles,J. Quant. Spectrosc. Rad. Transfer, 100, 340–358.

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Shkuratov, Yu., Bondarenko, S., Kaydash, V., Videen, G., Munoz, O., Volten, H.,2007a: Photometry and polarimetry of particulate surfaces and aerosol particlesover a wide range of phase angles, J. Quant. Spectrosc. Rad. Transfer, 106, 487–508.

Shkuratov, Yu., Opanasenko, N., Zubko, E., Grynko, Ye., Korokhin, V., Pieters, C.,Videen, G., Mall, U., and Opanasenko, A., 2007b: Multispectral polarimetry as atool to investigate texture and chemistry of lunar regolith particles, Icarus, 187,406–416.

Volten, H., Munoz, O., Rol, E., de Haan J., Vassen, W., Hovenier, J., Muinonen, K.,and Nousianen, T., 2001: Scattering matrices of mineral aerosol particles at 441.6nm and 632.8 nm, J. Geophys. Res., 106, 17,375–17,401.

Weidner, V. and Hsia, J., 1981: Reflection properties of pressed polytetrafluoroethylenepowder, J. Opt. Soc. Am., 71, 856–861.

Zhang, H. and K. J. Voss, 2005: Comparisons of bi-directional reflectance distributionmeasurements on particulate surfaces and radiative transfer models, Applied Optics,44, 597–610.

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Index

absorbing medium, 273, 343absorption, 3, 16, 29–32, 47–50, 52, 53,

55, 58, 59, 61, 70, 87, 134–137,140–147, 151–157, 162, 166, 167,171, 174, 187, 267, 297, 321,323–325, 327, 333, 336, 340, 343,345, 347, 356, 362, 365, 369–371,373, 375, 377, 379

absorption coefficient, 137, 138, 312, 318,320–325, 327, 356, 379

adjoint, 72, 73, 143, 179–181, 188, 189,230, 242, 262, 272, 275

aerosol, 13, 18, 19, 70, 105, 136, 137, 140,188, 227, 231, 237, 263, 266–269,271–274, 381, 395–397, 400, 402

albedo, 3, 4, 22, 24, 134, 136, 137, 151,152, 154, 157, 160, 166, 180, 181,232, 236, 237, 240, 258, 260–262,266–268, 281, 283, 284, 286,294, 297, 312, 314, 316–318, 320,323–325, 327, 332, 334, 335, 345,346, 349, 351–354, 366, 370, 376,378, 380, 383, 385, 387, 389–391,393, 398–400

alumina, 389, 392, 393, 400anomalous diffraction theory (ADT), 27,

28asymmetry parameter, 3, 5, 6, 13, 21, 23,

284, 286, 294

Bacillus subtilis, 28, 67bacteria detection, 52backscattered signal, 192backscattering, 5, 7, 9, 11, 12, 192, 194,

195, 204, 211, 218–220, 222–224,226–228, 287, 295, 297, 300, 301,307, 308, 314, 322, 324, 331, 347,351, 352, 356, 361, 365, 377, 380,383, 397, 398, 400

backscattering matrix, 194, 204, 219, 220benthic floor, 301bi-directional reflectance, 279–281, 327,

402biosphere, 133boron carbide, 389, 392, 393, 400boundary conditions, 72–74, 77–79, 81,

83, 84, 120, 180, 236, 238, 240, 247,264, 265

BRDF, 231, 237, 257–261, 266, 269,279–282, 285, 287, 288, 301, 308,310, 322, 324, 325

calibration, 279controlled measurement, 279, 281, 292,

301, 312instrumentation, 286models, 279, 282

calibration, 109, 279, 288–291, 294, 325,384–386

carbon soot, 391, 398–400CBH, 134, 136, 147–149, 151, 153, 154,

160–162, 165–167, 171, 175, 179,182

circular polarization, 193, 194, 218, 225,356, 360

cirrus cloud, 3, 5–7, 9–11, 13–17, 19,21–25, 28, 29, 46, 226, 228

cloud aerosol, 262cloud optical thickness, 134, 138, 146,

148–151, 153, 158, 166, 167, 169,174, 181, 182, 216

cloud parameters, 134–136, 138, 141,146–153, 155–160, 162–165, 179,186, 191

cloud retrieval algorithm, 186–188cloud top height, 134, 136, 138, 139, 141,

146, 147, 150, 151, 154, 160, 163,165, 166, 169, 171, 174, 178, 187

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404 Index

cloudiness, 133, 135, 136, 140, 142, 151,158, 159, 187

coupled-dipole, 109, 130CTH, 134, 136, 147–149, 151, 153, 154,

160–163, 165–179, 182

depolarization, 194, 195derivative, 142, 143, 147, 182, 183, 189,

230, 237, 240, 242, 243, 245, 246,249, 251, 252, 257–261, 267, 268,275

differential absorption, 142–145differential scattering cross-section,

125–128diffraction, 6, 20, 27–30, 32, 47, 212, 286,

300, 301, 333, 347discrete sources method, 124–127double scattering, 397, 400

Earth, 3, 5, 21, 133, 186, 187, 253, 255,325, 326, 329, 330, 380

effective radius, 4, 11–15, 21, 25, 26, 139,161, 224, 269

evanescent scattering, 117expansion coefficients, 111, 113, 117, 118,

123, 197, 234–236, 256, 283extinction coefficient, 6, 13, 17, 23, 137,

138, 181–184, 214, 221, 255, 257,268, 334

extinction efficiency, 32, 48, 50, 51

filling factor, 284, 286, 294, 297, 324film, 110, 119, 121, 127–129, 307Fresnel reflection, 111, 112, 114, 119, 120,

257, 259, 357, 365, 366

Gaussian random field, 338Gaussian ray approximation (GRA), 30geometric optics, 3, 6, 25, 326, 329, 347,

357, 375, 378, 379, 382geometrical path statistics of rays

cylinder, 36, 38, 40, 42–45speroid, 42–45

GOME, 133, 135, 136, 154, 157, 185–189,230, 231, 253, 270–272

Green’s functiondefinition, 72free-space, 78, 80, 83, 84matrix, 196, 197, 199, 205surface, 71, 79, 80, 82, 84, 87, 92

volume, 78, 80, 84, 85, 94

Helmholtz equation, 72, 74, 78, 82, 84,104

ice clouds, 6, 22–26, 137, 165, 191, 192ice crystal, 3–16, 18–25, 29, 69, 133, 134,

137, 138, 151, 169, 181, 187, 226,227, 330, 334, 378, 379, 381, 382

interaction operator, 79–82, 84–87, 92,94, 104

inverse problem, 27, 51, 66, 134, 135, 147,151, 155, 179, 188, 194, 230, 272

Lambertian, 9, 151, 154, 187, 237, 247,258, 260, 265, 281, 287, 288, 290,291, 301, 302, 307, 308, 314, 320,324, 325, 349, 351, 352

Lambertian surface, 183, 257, 261, 281,346, 350, 352–354, 376

laser, 6, 16, 25, 67, 109, 191–195, 199,211, 212, 214, 223, 225–228, 287,289, 290, 384, 387–389, 391, 400

laser beam scattering, 263LER, 134–136, 146, 151, 152, 154, 155,

160, 162–178lidar sounding, 192–195, 203, 213, 218,

223, 224light absorption, 29, 32, 51, 52light scattering, 3, 5, 8, 24, 25, 27–30, 46,

50–53, 59, 63–67, 69, 71, 107, 109,130, 225, 280, 325, 329, 331–336,345, 351, 374, 377–379, 382, 383,386, 387, 393

linear polarization, 192, 193, 212, 336,346, 355, 356, 365, 366, 383, 385,389, 392, 401

linearization, 143, 145, 152–157, 230, 231,237, 238, 240, 242–244, 246, 248,249, 252, 253, 256, 257, 263, 270

linearized radiative transfer, 272Lorenz–Mie theory, 400

magnesia, 399mean geometrical path, 44–46, 50mean-squared-root geometrical path, 30,

42, 45, 47, 50, 63measurement, 5–7, 9, 11–25, 28, 50, 51,

53–55, 58, 59, 63, 64, 66, 133–135,141, 151, 154, 157, 178, 185–189,194, 200, 203, 224–228, 230,

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Index 405

271–274, 279, 281–283, 285–301,303, 305, 307–319, 321, 323–327,329, 332–334, 337, 346, 347, 365,369, 377, 380, 383–385, 387–389,391, 393–395, 397–402

Mie, 27, 29, 63, 70, 105, 109, 137, 138,268, 269, 271, 272, 286, 292–295,300, 301, 325, 334, 336, 399, 400

Mie scattering, 271multi-layered cloudiness, 133multiple scattering, 11, 53, 55, 124, 137,

140, 142, 180, 187, 191, 192, 194,195, 210, 211, 218, 219, 223–230,271, 274, 330, 332–336, 347, 351,363, 369, 373, 375–377, 380, 393,395, 396

negative polarization, 327, 332, 355,360, 362, 363, 368, 376, 380, 384,392–395, 397, 398, 400, 401

nonspherical particles, 27, 29, 64,105–107, 194, 226, 336, 379, 382

opposition effect, 283, 300, 326, 327,368, 376, 380, 383, 387, 389, 391,399–401

oxygen, 66, 137, 186–188, 274ozone, 133, 178, 185–187, 270, 272, 273,

275

packed surface, 292, 297packing density, 335, 342, 343, 347, 348,

352, 366, 369, 376particle size distribution, 28, 54, 193,

269, 292particle sizing, 28, 50, 51, 64particles deposited upon a surface, 109particulate layer, 263, 279, 325–327particulate surface, 279, 281, 283, 285,

287, 289, 291, 293, 295, 297,299, 301, 303, 305, 307, 309, 311,313, 315, 317–319, 321, 323, 325,327, 329–331, 333–335, 337, 339,341, 343, 345, 347, 349, 351, 353,355, 357, 359, 361, 363, 365, 367,369, 371, 373, 375, 377–381, 383,393–402

phase angle, 280, 286, 290, 293–298, 300,301, 308, 314, 319, 321, 322, 324,329–332, 336, 346, 347, 351–353,

355–363, 369–377, 380, 383–387,389, 393–402

phase curve, 332, 348, 354, 363, 368, 378,384, 387, 392–395, 397–401

phase function, 3, 6, 11, 13, 22–24, 104,137, 169, 180, 181, 197, 205, 208,209, 213–216, 223, 234, 236, 252,255–257, 261, 263, 265, 267, 275,284–286, 292–295, 300, 301, 326,327, 332, 334, 336, 350, 351, 365,376, 379, 389

phase matrix, 107, 196, 197, 199–201,206–208, 232, 233, 235, 265, 274,275, 379, 381

photopolarimetry, 363, 384plane surface, 109–111, 113, 115–119,

121–125, 127–130polarization, 192–194, 212, 218, 336, 346,

355, 356, 360, 365, 366, 383, 385,389, 392

polarization lidar, 226, 228polarization maximum, 332, 366, 384,

393–395polarized radiation transfer, 207, 224, 228polarized return, 193powdery sample, 385

ray tracing, 32, 333–337, 343, 345, 347,349, 350, 354, 363, 369, 376, 378

reflectance, 25, 188, 231, 237, 257–260,273, 275, 279–281, 283, 285–291,293–295, 297, 299–301, 303, 305,307–319, 321, 323–327, 333, 335,346, 347, 350, 353, 369–373,375–379, 381, 387, 400–402

reflectance spectroscopy, 292, 326, 333,363

reflection, 371reflection function, 136, 140–144,

147–150, 152–158, 258, 281reflection matrix, 110, 113–117, 119, 123,

236refractive index, 27–30, 47, 48, 50–54, 59,

60, 62–64, 66, 112, 125, 138, 179,257, 259, 269, 279, 292, 294, 312,316, 324, 327, 333, 336, 343, 355,356, 369, 370

regolith, 325–327, 330, 331, 347, 348, 363,369, 375, 377–381, 383, 401, 402

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406 Index

satellite, 19, 26, 133–135, 154, 171,186–189, 253, 272, 279, 326, 331,398

scattering coefficient, 124, 146, 147, 166,196, 214, 215, 220, 228, 255–257

scattering efficiency, 46scattering matrix, 105, 106, 196, 202,

204, 234, 253, 273, 334, 336, 339,345, 354–362, 367, 368

scattering media, 194, 195, 223, 224, 233SCIAMACHY, 133, 135, 136, 154, 179,

185–187, 230, 231, 271SCIATRAN, 136, 140, 143, 148, 179, 188,

189sediment, 279, 292, 301, 302, 307, 308,

314, 315, 318, 324, 325, 327semi-analytical solution, 195, 213, 223shadow-hiding effect, 330, 337, 339, 347,

348, 351, 363, 365, 366, 376, 393,395, 398, 400

shape, 3, 5, 6, 9–12, 15, 21, 22, 25, 28,30, 32, 40, 41, 46, 47, 50–53, 55, 56,58, 63–67, 70, 104, 105, 109, 138,139, 151, 152, 161, 169, 179, 192,227, 273, 279, 308, 309, 318, 326,330, 333, 334, 336–343, 353–355,357–363, 366, 367, 376–378, 380,381, 389–391, 393

silica sphere, 399single scattering, 25, 53–55, 180, 195–197,

201, 203, 206, 208, 210, 211, 214,218, 220, 232, 233, 260, 330, 334,335, 350, 373, 394, 396, 397

single scattering albedo, 180, 233Snell’s law, 112sounding of clouds, 195, 223sphere, 34, 35, 43–45, 59, 70, 104,

113, 114, 121, 130, 204, 208, 290,294–298, 324, 339, 355, 357

spheroid, 32, 33, 39, 41–45, 48–50

spore germination, 47, 63Spore refractive index, 59, 62Stokes, 191, 197, 211, 266, 344substrate, 66, 112, 117–119, 124, 130,

326, 342, 365, 366, 377, 378, 389,393

symmetries, 69–71, 73, 75, 77–79, 81–83,85, 87, 89, 91, 93, 95, 97, 99, 101,103–107

T-matrix, 20, 27, 29, 70, 71, 78, 80, 82,85–87, 92–95, 103, 104, 107, 268,273, 336, 379, 382

Taylor series, 142, 143, 147, 152, 160,198, 230, 251

transition matrix, 110transmission, 28, 248transneptunian object, 384

Umov effect, 332, 363, 366, 377

vector radiative transfer, 186, 228, 229,235, 272

vector spherical harmonics, 116vertical column, 133–137, 149, 150,

155–159, 164–174, 176–179VHC, 134–136, 151–153, 160–179, 181VIC, 134–136, 151–153, 160, 162–164,

170, 172–179, 182

wafer inspection, 109water clouds, 17, 65, 137, 170, 187, 192,

195, 203, 212, 215, 224–226weighting function, 81, 82, 135, 136,

142–144, 147–149, 154, 157, 158,160, 179, 189, 230, 231, 237,258–261, 265, 269, 270, 273–275

wetting, 279, 312, 314, 316, 318, 320,322–325

WF, 142, 143, 148, 150, 154, 155, 157,179, 181–185