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On the Use Of Polarimetry andInterferometry for SAR Image

Analysis

Jose Luis Gomez Dans

A Thesis presented for the degree of

Doctor of Philosophy

Department of Electrical and Electronic Engineering,Sheffield Centre for Earth Observation Science (SCEOS),

University of SheffieldFebruary 2004

Adicado o meu pai

AbstractThis Thesis focuses on the use of the indoor GB-SAR facility at Sheffield University for

interferometric and polarimetric SAR measurements. Differences between the coherence

and interferometric phase measured with the GB-SAR system and airborne and space-

borne sensors are reported, caused by the different geometry of GB-SAR, which results in

a change of angle of incidence within the resolution cell. The analysis was used to develop

an interferometric analysis procedure using three-dimensional volume reconstructions, as

well as an iterative interferometric phase to height conversion algorithm.

The use of polarimetric coherence optimisation techniques was studied. Since the

GB-SAR system is not affected by temporal decorrelation effects, a constrained proce-

dure (using identical polarisation states for both images in the interferometric pair) was

developed. A study of layered targets in the context of polarimetric interferometry was

undertaken, and it showed that the use of polarisation diversity allowed, under certain

conditions, the retrieval of interferometric information on individual layers. The use of

coherence optimisation (both constrained and unconstrained) was examined for single

layer information retrieval; it was found that unconstrained optimisation results are poor

unless a large number of independent looks were combined. Constrained optimisation

on the other hand was proved to be a more robust analysis technique.

Polarimetric interferometry analysis of an artificial layered target, comprising of two

layers made up of scatterers with known polarimetric properties, was carried out. Re-

trieved height was very accurate, and the coherence very high. The use of coherence op-

timisation algorithms backed the finding that constrained optimisation is recommended

for GB-SAR measurements.

A mature wheat canopy imaged at C band was examined. It was found that the hori-

zontal polarisation retrieved a height close to ground level, while the vertical polarisation

retrieved a height close to the flag leaves level for angles of incidence around 45◦. The

cross-polar return was located close to soil level. The interferometry results agreed well

with the conclusions from tomographic imaging of the same target. The study of coher-

ence as a function of polarisation suggested that using the left-hand circular polarisation

will result (for the angles of incidence considered) in a height very close to the soil level,

while a linear polarisation similar to VV will retrieve the top of the canopy for larger

angles of incidence.

Declaration

The work in this thesis is based on research carried out at the Department of

Electrical and Electronic Engineering, University of Sheffield, England. No part of

this thesis has been submitted elsewhere for any other degree or qualification and

is all my own work unless referenced in the text.

Copyright c© 2004 by Jose Luis Gomez Dans.

iv

Acknowledgements

Throughout the time I worked on this Thesis, a number of different people have

helped me in some way or other. First of all, I would like to thank both my

supervisors, Shaun Quegan and John Bennett, for introducing me into the area of

radar remote sensing, and for their guidance and encouragement. Their help was

critical to the development of this Thesis.

A number of people in Sheffield also helped with different aspects of this work,

either by giving advice, data or those little tricks that no one bothers to put on the

Internet. In particular, I am grateful to Keith Morrison, Sarah Brown and Geoff

Cookmartin for helping me understand how the GB-SAR system works. James

Matthews is also acknowledged for his help on dealing with computers, coffee and

generic social exchanges.

A number of other people have endured and supported me during this time.

Namely, my mother, Victoria Pastor Gonzalez, Miguel Fernandez Garrido and

a number of friends in A Coruna have tried by all means to lift my spirits. In

Sheffield, I am thankful to, inter alia, Francisco Cerezo, Antonio Feteira and

Miriam Rivas-Aguilar, for providing all sorts of credible excuses for disappearing

from work.

Financial support during the first two years of this PhD was granted by Fun-

dacion Caixa Galicia. I am grateful for both their support and interest in my

work.

A number of free software packages were used in the production of this Thesis:

Octave, GCC, Python and LATEX.

v

Contents

Abstract iii

Declaration iv

Acknowledgements v

1 Introduction 1

2 Polarimetric radar 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Wave polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Change of polarisation basis . . . . . . . . . . . . . . . . . . 8

2.3 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 The scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Change of polarisation basis . . . . . . . . . . . . . . . . . . 11

2.5 The scattering vector . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 The polarimetric covariance and coherence matrices . . . . . . . . . 14

2.7 Target decomposition theorems . . . . . . . . . . . . . . . . . . . . 15

2.8 Statistical properties of SAR data . . . . . . . . . . . . . . . . . . . 17

2.8.1 Properties of speckle fields . . . . . . . . . . . . . . . . . . . 17

2.8.2 Hermitian products of speckle patterns . . . . . . . . . . . . 19

2.8.2.1 Single look distributions . . . . . . . . . . . . . . . 21

2.8.2.2 Multi-look distributions . . . . . . . . . . . . . . . 22

vi

Contents vii

2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 SAR and SAR interferometry 26

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Synthetic Aperture Radar fundamentals . . . . . . . . . . . . . . . 28

3.2.1 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Range processing . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.3 Azimuth processing . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.4 System approach . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Interferometric SAR . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 The Interferometric Geometry . . . . . . . . . . . . . . . . . 33

3.3.2 The Interferometric Phase . . . . . . . . . . . . . . . . . . . 34

3.3.3 The Interferometric Coherence . . . . . . . . . . . . . . . . . 36

3.3.3.1 Thermal decorrelation . . . . . . . . . . . . . . . . 39

3.3.3.2 Temporal decorrelation . . . . . . . . . . . . . . . . 39

3.3.3.3 Geometric decorrelation . . . . . . . . . . . . . . . 40

3.4 Polarimetric Interferometry . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Interferometric Processing Using the Indoor GB-SAR Component 47

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 The GB-SAR Geometry . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 The GB-SAR Resolution Cell . . . . . . . . . . . . . . . . . 49

4.3 GB-SAR Coherence Analysis . . . . . . . . . . . . . . . . . . . . . . 51

4.3.1 Slant-Range Decorrelation . . . . . . . . . . . . . . . . . . . 52

4.3.2 Volumetric Decorrelation . . . . . . . . . . . . . . . . . . . . 54

4.4 Effective Height from the Interferometric Phase . . . . . . . . . . . 55

4.4.1 An Iterative Height Retrieval Algorithm . . . . . . . . . . . 57

4.4.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . 58

4.5 Coherence Analysis Using Three-Dimensional Data . . . . . . . . . 61

4.5.1 Application to the RADWHEAT data set . . . . . . . . . . 62

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Contents viii

5 Interferometric and Polarimetric Analysis of Layered Targets 72

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Coherence analysis of a layered target . . . . . . . . . . . . . . . . . 73

5.3 Polarimetric coherence optimisation applied to layered targets . . . 76

5.3.1 Orthogonal scattering vectors . . . . . . . . . . . . . . . . . 77

5.3.2 Linearly independent scattering vectors . . . . . . . . . . . . 80

5.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4.1 Orthogonal scattering vectors . . . . . . . . . . . . . . . . . 85

5.4.2 Linearly independent scattering vectors . . . . . . . . . . . . 88

5.5 Effect of finite number of independent samples in coherence optimi-

sation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.6 Variations in the scattering vectors within layers . . . . . . . . . . . 97

5.6.1 Random variations of scattering vectors . . . . . . . . . . . . 97

5.6.1.1 Numerical Simulations . . . . . . . . . . . . . . . . 100

5.6.2 Several scatterer types within individual layers . . . . . . . . 101

5.6.2.1 Numerical simulations . . . . . . . . . . . . . . . . 106

5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Experimental Verification of GB-SAR Polarimetric and Interferometric

Capabilities 116

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2 Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2.2 Imaging Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3 Initial Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.1 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.2 Three-Dimensional Reconstructions . . . . . . . . . . . . . . 119

6.3.3 Two-Dimensional Images . . . . . . . . . . . . . . . . . . . . 121

6.4 Interferometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . 125

6.5 Polarimetric Interferometry . . . . . . . . . . . . . . . . . . . . . . 128

6.6 Coherence Optimisation . . . . . . . . . . . . . . . . . . . . . . . . 129

6.6.1 Two Dimensional Coherence Optimisation . . . . . . . . . . 133

6.6.1.1 Unconstrained Optimisation . . . . . . . . . . . . . 133

Contents ix

6.6.1.2 Constrained Optimisation . . . . . . . . . . . . . . 135

6.6.2 Three-Dimensional Coherence Optimisation . . . . . . . . . 137

6.6.2.1 Unconstrained Optimisation . . . . . . . . . . . . . 137

6.6.2.2 Constrained Optimisation . . . . . . . . . . . . . . 139

6.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 141

7 Interferometric Studies of Wheat Canopies Using GB-SAR 146

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2 Description of the Experiment . . . . . . . . . . . . . . . . . . . . . 147

7.2.1 The Wheat Canopy . . . . . . . . . . . . . . . . . . . . . . . 147

7.2.2 Imaging Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.3 Three-dimensional Polarimetric Analysis . . . . . . . . . . . . . . . 150

7.4 Polarimetric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.5 Single Polarisation Interferometry . . . . . . . . . . . . . . . . . . . 153

7.6 Polarimetric coherence synthesis . . . . . . . . . . . . . . . . . . . 157

7.7 Pauli Basis Inteferometry Results . . . . . . . . . . . . . . . . . . . 161

7.8 Coherence Optimisation . . . . . . . . . . . . . . . . . . . . . . . . 161

7.8.1 Unconstrained Coherence Optimisation . . . . . . . . . . . . 161

7.8.2 Constrained Coherence Optimisation . . . . . . . . . . . . . 163

7.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8 Conclusions and Further Work 177

8.1 Polarimetry and Interferometry with GB-SAR . . . . . . . . . . . . 177

8.2 Polarimetric Interferometry of Layered Targets . . . . . . . . . . . . 179

8.3 Study of a Wheat Canopy . . . . . . . . . . . . . . . . . . . . . . . 180

8.4 Suggestions for Further Work . . . . . . . . . . . . . . . . . . . . . 182

Bibliography 184

Chapter 1Introduction

An imaging radar is an active device that generates reflectivity maps of

a scene by means of transmitting electromagnetic energy and analysing

the fields scattered by the scene. Synthetic Aperture Radars (SARs) are

a particular kind of imaging radar that have generated great interest due to their

ability to produce high resolution images. As SARs are operated at microwave

frequencies and are active sensors, they do not suffer from the limitations of optical

sensors, and can operate at night time and are not affected by cloud cover.

SAR research started in the 1950s, when Wiley [Wiley, 1954] suggested the use

of Doppler information to enhance the azimuth resolution of side looking aperture

radars. Within a decade, the first SAR images were produced [Cutrona et al.,

1961], and in the next 10 to 15 years, high resolution terrain maps were produced

using SAR optical processors [Cutrona et al., 1960]. Towards the end of the 1970s,

digital signal processors started to become available and were used to produce

off-line SAR images, opening the door to SARs mounted on orbiting satellites.

Since then SAR has continued to evolve, with new air and spaceborne sensors,

processing and analysis techniques, which have seen SAR images being used in a

wide variety of areas such as topographic mapping, agriculture and forestry, urban

planning, environmental monitoring, hydrology, oceanography and natural disaster

monitoring.

The use of microwaves provides a significant challenge for the correct inter-

pretation of SAR images. While optical frequencies are sensitive to processes at a

1

Chapter 1. Introduction 2

molecular scale (such as the absorption of visible light in photosynthetic processes),

microwaves are sensitive to geometric and dielectric properties of the imaged re-

gion. This makes SAR images sensitive to topography, surface roughness, plant

geometry and moisture content (soil moisture, plant moisture and snow/ice wet-

ness).

The aim of SAR remote sensing is to extract surface parameters from SAR im-

ages. The interaction of the electromagnetic wave with a natural scene is complex,

and depends on the properties of the scene, themselves functions of wave param-

eters such as polarisation and frequency [Ulaby et al., 1981, 1982]. Radar returns

are an amalgamation of all the interactions that happen within a resolution cell,

raising the question whether it is possible to separate these interactions from the

combined return. This problem is often called the inverse problem, and is often

investigated using either statistical models or simple physical models. Due to the

complexity of the problem, statistical approaches are often site and time of acqui-

sition dependent, and thus hard to generalise. On the other hand, model based

approaches are often too simplistic, and result in inaccurate estimations. More

complex models can be difficult to invert, but might be useful for understanding

the interaction of the wave with the natural scene.

In recent years, single polarisation, single frequency SAR systems have been

expanded to multiple polarisation (and fully polarimetric) systems, operating at

several frequencies to increase the amount of retrieved information.

Polarimetry studies the influence of wave polarisation in the scattering process.

It starts with the assumption that plane waves are used, and that the interaction

of the incident electric field with a fundamental scatterer in the scene is governed

by the properties of the fundamental scatterer, represented by its scattering matrix

[Ulaby and Elachi, 1990], an idea introduced by C. W. Sinclair in 1948. Polarisation

is sensitive to the shape of fundamental scatterers. For example, a vertically

polarised wave will be backscattered strongly by a vertical cylinder with a length

of the order of the wavelength or greater, whereas the backscattering from this

same cylinder would be negligible if a horizontally polarised wave were used.

A major benefit of polarimetry is the ability to separate different scattering

mechanisms to give a better appreciation of the scattering process. This results

in improvements over single-channel SAR imagery in a number of areas, such


as land-cover classification and segmentation [Van Zyl, 1989, Lee et al., 1994,

Ferrazzoli et al., 1999, Del Frate et al., 2003], hydrology through estimation of

soil roughness and moisture content [Hajnsek et al., 2003, Oh et al., 1992], sea-

ice monitoring [Drinkwater et al., 1991, Dierking et al., 2003] and forestry [Le

Toan et al., 1992, Dobson et al., 1992, Rignot et al., 1994, Kellendorfer et al.,

2003]. Another area where polarimetry is showing very encouraging results is in

agricultural applications [Ferrazzoli et al., 1999, Lemoine et al., 1994, Skriver et al.,

1999].

Another recent technique used to retrieve extra information from SAR images

is interferometric SAR (InSAR). This technique combines two SAR images of the

same scene, acquired with slightly different imaging geometries to extract infor-

mation on the vertical location of scatterers. Since SAR images are affected by

speckle (a noise-like phenomenon arising from the constructive and destructive in-

terference of the radar returns within the resolution cell that gives SAR images

itheir characteristic granular aspect), the phase of a single SAR image carries no

information. The phase difference between two images in an interferometric set-up

introduces a deterministic phase contribution due to the path length difference

arising from the antenna separation. Since the speckle patterns for each image are

nearly identical, the phase difference will tend to reduce this random contribution.

The phase difference can be related by triangulation to the vertical height of the

scattering centre. Two interferometer implementations are used: single and repeat

pass. In single pass interferometry, two different antennas are used to image the

scene simultaneously, while in repeat pass interferometry, the same antenna is used

twice to image the scene. The accuracy of the interferometric height estimate de-

pends on the correlation between the two images. Decorrelation between the two

images arises due to changes in the scene (in repeat-pass interferometry), thermal

noise effects and geometrical effects. The use of the magnitude of the correlation

between the two images has also been used in change detection applications.

The height retrieved from InSAR images relates to a phase centre which is de-

pendent on both the scene being imaged and on sensor parameters. Non-vegetated

areas imaged at high frequencies result in a phase centre that is coincident with the

top of the soil. On vegetated surfaces, a significant attenuation from the canopy

results in the effective height being somewhere between the top of the canopy


and the soil layer [Hagberg et al., 1995, Treuhaft et al., 1996]. This behaviour has

prompted the study of multiple parameter sensor configurations to retrieve vegeta-

tion height and underlying topography. The use of different frequencies is studied

in [Rosen et al., 2000], while the use of different wave polarisations is studied in

[Treuhaft and Siqueira, 2000, Papathanassiou and Cloude, 2001]. The use of data

gathered at different times to study changes occurring in the scene has also been

succesfully exploited [Strozzi et al., 2000, Wegmueller and Werner, 1995, Engdahl

et al., 2001, Askne et al., 2003, Engdahl and Hyyppa, 2003].

Differential interferometry (the combination of two or more interferograms that

enables the monitoring of vertical displacements) has been exploited to monitor

Earth crust dynamics, such as land subsidence [Ferretti et al., 2001, Strozzi et al.,

2003, Colesanti et al., 2003], earthquakes [Massonet et al., 1993], and vulcanol-

ogy [Massonnet et al., 1995]. Other areas where InSAR has had great impact

include glaciology and remote sensing of the polar regions [Goldstein et al., 1993,

Joughin et al., 1998]. InSAR has also been exploited for land cover classification

[Wegmueller and Werner, 1995, Strozzi et al., 2000, Engdahl and Hyyppa, 2003],

agricultural monitoring [Engdahl et al., 2001], and forestry [Askne et al., 2003,

Wagner et al., 2003], while an overview of the uses of interferometry for forestry

can be found in [Baltzer, 2001].

A recent addition to interferometry is polarimetric interferometry. The use of

different polarisations opens the possibility of separating different interferometric

phase centres within a resolution cell [Cloude and Papathanassiou, 1998, Treuhaft

and Siqueira, 2000]. Polarimetric InSAR is based on the ability to create inter-

ferograms in arbitrary polarisations, if fully polarimetric data are available. From

here, the polarisation state that maximises the correlation (and thus, minimises

the retrieved height uncertainty) can be calculated [Cloude and Papathanassiou,

1998]. The recent availability of polarimetric InSAR data from airborne sensors

such as E-SAR and the importance of monitoring forest biomass derived from the

Kyoto Protocol has led to a number studies of forest biomass estimation using

these techniques [Papathanassiou and Cloude, 2001]. The potential of PolInSAR

techniques for agricultural vegetation studies has also been reported in [Sagues

et al., 2000] using the facilities at the European Microwave Scattering Laboratory

(EMSL).


Scope and organisation of this Thesis

This Thesis aims to provide a characterisation of the indoor component of the GB-

SAR instrument [Brown et al., 2003] for polarimetric and interferometric applica-

tions. The availability of a tightly controlled, indoor facility is ideal for carrying

out experiments that enhance the understanding of the scattering process, prior to

the design of airborne and spaceborne campaigns, or to improve electromagnetic

scattering models.

Prior to any experimental attempts, the suitability of the GB-SAR indoor com-

ponent for interferometric measurements needs to be investigated, as the instru-

ment is significantly different from air and spaceborne sensors, commonly consid-

ered in the InSAR literature. A way to compare results between GB-SAR experi-

ments and air and spaceborne sensors is also needed to fully exploit the GB-SAR

data.

Processing techniques for GB-SAR experiments need to be addressed, so as to

recommend procedures that extract as much useful information from indoor experi-

ments as possible. These techniques and recommendations should be demonstrated

with both artificial and natural targets.

The Thesis starts with an introduction to polarimetry (Chapter 2), followed by

an overview of SAR imaging and interferometric SAR in Chapter 3. This sets the

basic theoretical framework for the rest of the Thesis. In Chapter 4, the GB-SAR

component is studied in the context of interferometric processing. Chapter 5 deals

with layered targets, an often used approximation in modelling environments. The

use of an artificial target to characterise both the system and processing algorithms

is presented in Chapter 6, and data from a wheat canopy from the RADWHEAT

experiment are presented in Chapter 7.

Chapter 2Polarimetric radar

2.1 Introduction

Radar Polarimetry studies the effect of wave polarisation on the radar re-

turn. Polarisation refers to the direction of orientation of the electric field

[Born and Wolf, 1999]. The scattering properties of a target are, in gen-

eral, dependent on the polarisation state of the wave with which it is illuminated.

Thus, extra information is gained by recording the backscatter for different polar-

isations. Polarimetry was first suggested in the 1940s, but only relatively recently

could technology produce the complex hardware required to exploit it.

This Chapter gives a brief introduction to polarisation in SAR imaging. This

includes a description of coordinate systems and polarisation parameters. The

scattering matrix and its basis transformations are also given, followed by the def-

inition of the scattering vector and the coherence and covariance matrices. Target

decomposition techniques are briefly described, and finally, a small treatment of the

statistics of SAR data (and its relevance to both interferometry and polarimetry)

is included.

2.2 Wave polarisation

In a monochromatic plane wave, the electric and magnetic field vectors are or-

thogonal to the direction of propagation. Polarisation describes the locus formed

6

2.2. Wave polarisation 7

Minor

Axis

Major

Axisη

ζ

h

v

χ

αaζ

aη

Ψ

ah

av

Figure 2.1: The polarisation ellipse. The direction of propagation is orthogonal to theplane of the paper.

by the tip of the electric field vector projected on a plane orthogonal to the di-

rection of propagation. In general, this locus is an ellipse [Born and Wolf, 1999]

(see Fig. 2.1). The polarisation ellipse is defined by the orientation angle, Ψ and

the ellipticity angle, χ. Additionally, the sense of rotation of the tip of the electric

field vector is determined by the handedness. In radar polarimetry, the handed-

ness is normally referred to the sense of rotation noted by an observer looking in

the direction of propagation. The handedness is right-handed (left-handed) if the

sense of rotation of the electric field is clockwise (counterclockwise), according to

the IEEE Antenna standard [IEEE, 1983].

The polarisation state of a plane wave can be written in terms of a right handed

orthogonal coordinate system{k, v, h

}[Ulaby and Elachi, 1990], where k is the

direction of propagation. The electric field lies in a plane normal to the direction


of propagation, so it can be written as a function of h and v:

� �

E = Ehh + Evv. (2.1)

Further, En (n = h, v) can be written as

En = an exp(jδn), (2.2)

It can be shown [Born and Wolf, 1999] that only three parameters are needed to

completely describe the polarisation state of the wave: av, ah and δ = δv − δh.

The last parameter describes the handedness. av, ah and δ can be related to the

ellipticity and orientation angles, χ and Ψ, as:

tan 2Ψ = cos δ tan

(2avah

)(2.3)

sin 2χ = sin δ sin

(2avah

). (2.4)

Table 2.1 shows the ellipticity and orientation angles for some commonly used

polarisation states.

Vertical Horizontal Linear at θ Left circular Right circular

Ψ (Orientation) 90 0 θ -90 to 90 -90 to 90χ (Ellipticity) 0 0 0 45 -45

Table 2.1: Ellipticity and orientation angles for some common polarisation states (an-gles are in degrees).

2.2.1 Change of polarisation basis

In the previous Section, the polarisation state was defined in terms of the H-V

linear polarisation basis. Any other orthonormal basis could have been used for

this purpose, i.e. Eq. 2.1 can be written as

� �

E = Ehh + Evv = Enn + Emm, (2.5)


where {n, m} is another basis. This change of basis can be performed by means

of a 2 × 2 complex unitary transformation matrix U :

� �

Ehv = U� �

Enm (2.6)

In Eq. 2.6,� �

Ehv is the electric field vector in the {h, v} basis, whereas� �

Enm is

the electric field vector in the {n, m} basis. The columns of U are given by the

coordinates of {n, m} expressed in the{h, v

}basis [Strang, 1988]. As an example,

consider a change of basis from the H-V basis into the right-left circular basis.

Circular polarisation states are characterised by av = ah, as the polarisation ellipse

is a circle. The sense of rotation is given by the phase δ, which will be π/2 (−π/2)

for the left (right) circular polarisation. In terms of the H-V basis, the polarisation

state of a left-handed circular field� �

Ebr can be written as

� �

Ebl= (−jh + v)

1√2

=1√2

[−j1

]. (2.7)

Similarly, for a right-handed circular polarised wave,

� �

Ebr = (h − jv)1√2

=1√2

[1

−j

]. (2.8)

The transformation matrix used to convert R-L circular polarisation states to the

H-V basis is given by

U =1√2

[1 −j−j 1

], (2.9)

and the conversion from polarisation states expressed in the H-V into the R-L

circular basis is given by� �

ERL = U−1� �

EHV , (2.10)

where

U−1 = U∗T . (2.11)

.

2.3. Coordinate systems 10

2.3 Coordinate systems

A radar transmits an electromagnetic wave and measures a scattered field. It is

common to have the transmit and receive antennas located at the same position,

indeed using a single antenna for both transmitting and receiving (monostatic

radar). A bistatic radar measurement occurs when the two antennas are posi-

tioned at different locations. For a monostatic radar, it makes sense to use a

local coordinate system centred in the transmit/receive antennas. In the case of

bistatic radar, it is more convenient to use the wave direction of propagation as a

basis for a local coordinate antenna system. In order to accommodate these two

situations, two conventions are commonly used: the Forward-Scatter Alignment

(FSA) convention and the Backscatter Alignment (BSA) convention. The BSA is

an antenna-based coordinate system, and hence preferred in backscattering mea-

surements. The FSA is a wave-based coordinate system, and as such, better suited

for bistatic measurements. In this thesis, the BSA convention (depicted in Fig.

2.2) will be used.

hi

φi

z

y

x

vi

φs

θs

θi

ki ks

vshs

Figure 2.2: The Backscatter Alignment (BSA) coordinate system

2.4. The scattering matrix 11

2.4 The scattering matrix

When an incident plane wave interacts with a scatterer, currents are induced in the

scatterer, which result in the scatterer acting as a source of radiation. Another way

of picturing the scattering phenomenon is to consider it as mapping an incident

vector into a scattered vector in a two-dimensional complex space.

Under the BSA convention, the polarisation state of the scattered field can be

related to that of the incident field by

[Esv

Esh

]=

exp (−jkr)r

[SV V SV H

SHV SHH

] [Eiv

Eih

], (2.12)

where the subscripts s and i refer to the scattered and incident fields, respectively.

r is the distance from the scatterer to the point where the field is measured, k is the

wavenumber, and the 2 × 2 matrix is usually called the scattering matrix, S. The

elements of the scattering matrix are called complex scattering amplitudes, and

they are a function of frequency and incidence direction. The scattering amplitude

is also dependent on the shape, dimensions, orientation and permittivity of the

scatterer.

Even though the scattering matrix shown in Eq. 2.12 has 4 complex terms, the

reciprocity theorem [Ulaby and Elachi, 1990] states that

SHV = SV H , (2.13)

so the scattering matrix is symmetrical.

2.4.1 Change of polarisation basis

The representation of the elements of S depends on the polarisation basis. The

use of an arbitrary polarisation basis will now be investigated. Eq. 2.12 is taken

as the starting point. If the propagation term is ignored, it can be written as

� �

Eshv = Shv

� �

E ihv ⇒ U∗

� �

Esnm = ShvU

� �

E inm, (2.14)

2.4. The scattering matrix 12

where use of Eq. 2.6 has been made, and the complex conjugate arises from the

use of the BSA convention. As before, an arbitrary basis {n, m} is considered.

Eq. 2.14, can be written as

� �

Esnm = UTShvU

� �

E inm. (2.15)

Essentially, Eq. 2.15 states the the scattered field in terms of the incident field in

the {n, m} basis (ignoring the propagation term), and by comparison to Eq. 2.12,

it can be seen that the scattering matrix in the {n, m} basis is given by

Snm = UTShv U . (2.16)

As an example, consider the change of basis transformation from the H-V basis

to the R/L circular basis. The basis vectors are given by

� �

Ebl=

1√2

[1

−j

](2.17)

� �

Ebr =1√2

[−j1

]. (2.18)

The transformation matrix is

U =1√2

[1 −j−j 1

], (2.19)

so the scattering matrix change of basis can be accomplished as in Eq. 2.15:

SRL =1

2

[1 −j−j 1

][SV V SV H

SV H SHH

] [1 −j−j 1

]

=1

2

[SV V − 2jSV H − SHH −jSV V − jSHH

−jSV V − jSHH −SV V − 2jSV H + SHH

]. (2.20)

Eq. 2.20 demonstrates three important properties of the change of basis trans-

formation:

2.5. The scattering vector 13

1. The total power (the sum of the squared magnitudes of all the elements)

has to be invariant under a basis transformation (in this example, the total

power is equal to |SV V |2 + 2|SV H |2 + |SHH |2),

2. The scattering matrix in the new basis is also symmetric,

3. As the transformation matrix is unitary, the determinant of the transformed

scattering matrix will not be changed by the transformation.

2.5 The scattering vector

In polarimetric applications, it is usually more convenient to deal with a 3 element

vector (in the BSA convention) representing the elements of the scattering matrix.

For a scattering matrix in the H-V basis, this can simply be written as

� �

k =[SV V ,

√2SHV , SHH

]T. (2.21)

In the previous equation, the scattering vector can be seen as a stacking of the

rows of the scattering matrix. The√

2 factor is needed to make the total power

represented by the 3-element vector identical to that of the scattering matrix. So,

for a generic scattering matrix, the scattering vector is written as

� �

k = [S1, S2, S3]T (2.22)

In the case of a change of basis transformation of the scattering matrix, the scatter-

ing vector will change accordingly1. As an example, and using the results from the

previous Section, the scattering vector for a scattering matrix in the RL circular

basis is given by

� �

k RL =1

2

[SV V − j2SV H − SHH , −j

√2(SV V + SHH) ,−SV V − j2SV H + SHH

]T.

(2.23)

In some applications, the scattering vector is defined in terms of a Pauli basis

[Cloude and Pottier, 1996]. This vector is useful, as its components can be directly

1For a more detailed description of the basis change in terms of a special unitary transformationof the scattering vector, see [Papathanassiou, 1999].

2.6. The polarimetric covariance and coherence matrices 14

related to ideal deterministic scattering mechanisms:

� �

k p =1√2

[SHH + SV V , SV V − SHH , 2SHV ]T . (2.24)

In Eq. 2.24, the first component of the vector can be seen as an “odd”-bounce

scattering mechanism, the second is an “even”-bounce scattering mechanism, while

the third is a version of the second rotated by 45◦.

2.6 The polarimetric covariance and coherence

matrices

The scattering matrix provides insight about point scatterers that can be char-

acterised by a single scattering matrix. In SAR systems, the imaging process

introduces the concept of resolution cell. A resolution cell can be seen as an

amalgamation of point scatterers. Hence, the SAR will measure the coherent su-

perposition of the returns from the scatterers within a cell. The recorded data will

be different from one resolution cell to another, due to different dispositions of the

scatterers in the resolution cell, and different samples from a homogeneous target

will only be statistically related.

The statistical analysis of polarimetric SAR images is helped by the use of the

polarimetric covariance and coherence matrices. The covariance matrix C uses the

scattering vectors defined in Eq. 2.22:

C =⟨

� �

k� �

k ∗T⟩

=

〈|S1|2〉〈S1S∗2〉〈S1S

∗3〉

〈S2S∗1〉〈|S2|2〉〈S2S

∗3〉

〈S3S∗1〉〈S3S

∗2〉〈|S3|2〉

, (2.25)

while the coherence matrix is written as

T =⟨

� �

k p� �

k ∗Tp

⟩=

1

2

〈|kp1|2〉⟨kp1k

∗p2

⟩ ⟨kp1k

∗p3

⟩⟨kp2k

∗p1

⟩〈|kp2|2〉

⟨kp2k

∗p3

⟩⟨kp3k

∗p1

⟩ ⟨kp3k

∗p2

⟩〈|kp3|2〉

, (2.26)

where� �

k p is defined in 2.24, and kpi is the ith component.

2.7. Target decomposition theorems 15

The coherence and covariance matrices are both Hermitian semi-definite and

share the same positive eigenvalues. This happens because the two matrices are

related by a unitary transformation:

C =1

2DT D∗T . (2.27)

D =1√2

1 0 1

1 0 −1

0√

2 0

and D−1 = D∗T =

1√2

1 1 0

0 0√

2

1 −1 0

(2.28)

In reality, the covariance and coherence matrices carry the same information, as

the only difference lies in the basis chosen to define each.

2.7 Target decomposition theorems

Target decomposition theorems aim to decompose polarimetric information from

random media into a combination of point scatterer-like contributions. For exam-

ple, in forestry applications, it would be useful if the return could be separated

into volumetric scattering from the crown, “single-bounce” scattering from the soil

and “double-bounce” scattering involving scattering by the soil and further scat-

tering by the tree trunks (or vice versa). However, each imaged resolution cell

is normally a combination of large numbers of different scatterers. Polarimetric

SAR offers the possibility of separating the three scattering mechanisms mentioned

above [Freeman and Durden, 1998].

The first attempts at target decomposition were carried out by Huynen in his

PhD thesis in 1970 [Huynen, 1970]. He proposed a technique to separate the

Mueller matrix (see [Ulaby and Elachi, 1990] for a definition of the Mueller matrix)

into a single, deterministic target plus a “noise” contribution.

Another simple yet useful target decomposition theorem has already been out-

lined in Eq. 2.21, where the scattering matrix is decomposed into four (three for

backscattering) components. Each element of the scattering vector can be seen as

a backscattering contribution from different scattering mechanisms: odd bounce,

even bounce and a π/4 radian rotation of the even bounce scattering mechanism

2.7. Target decomposition theorems 16

with respect to the horizontal. These considerations form the basis of the work

of Van Zyl [1989], Freeman and Durden [1998], who proposed a method in which

the radar returns are decomposed into three components: a volumetric component

(such as a tree canopy), a double bounce component (such as trunk-soil interac-

tions), and a single bounce component (direct returns from a rough surface, for

example). Several other decomposition theorems can be found in the literature

(there is a very complete review of these in [Cloude and Pottier, 1996]). A major

problem associated with many of these decompositions is that the results are not

invariant under basis transformations and therefore, the solutions are not unique

[Cloude and Pottier, 1996].

Other target decompositions are based on the analysis of the covariance and/or

coherence matrices, similar to the analysis which is carried out in Principal Compo-

nent Analysis (PCA) (for a detailed overview of PCA, see [Jolliffe, 1986, Kendall,

1980]). They were introduced by Cloude [Cloude, 1992], and are based on the

eigenvalue-eigenvector diagonalisation of the coherence matrix. These theorems

exploit the fact that the coherence matrix is Hermitian positive semidefinite, and

can be written as a sum of orthogonal matrices (made up from the eigenvectors)

weighted by the corresponding eigenvalue. If the set of eigenvalues is {λ1···3} and

the set of eigenvectors is { � �

e 1···3} , then the coherence matrix can be written as

T = λ1

(� �

e 1� �

e ∗T1

)+ λ2

(� �

e 2� �

e ∗T2

)+ λ3

(� �

e 3� �

e ∗T3

), λ1 > λ2 > λ3 ≥ 0. (2.29)

This algorithm will yield a three-component decomposition of the rank 3 coher-

ence matrix weighted by the significance of each rank 1 coherence matrix, which

can be seen as the coherence matrix of a point scatterer. A physical interpretation

[Cloude, 1997] of the eigenvalues suggests that these qualify the importance of

each scattering mechanism present in the resolution cell. It is useful to define the

entropy, given by

H :=3∑

i=1

−Pi log3 Pi (2.30)

Pi :=λi∑3j=1 λj

, so that

3∑

i=1

Pi = 1. (2.31)

2.8. Statistical properties of SAR data 17

A target characterised by only one significant scattering mechanism will exhibit a

low value of entropy. On the other hand, a random process would be characterised

by an entropy equal to unity.

The eigenvectors need further explanation. In general, they are not directly

related to scattering vectors present in the resolution cell (a resolution cell where

more than three scattering mechanisms are present will still be separated into three

scattering mechanisms). To understand how the eigenvector relates to a scattering

mechanism, [Cloude, 1997] introduces the α angle, which describes the type of

scattering mechanism. The value of α goes from 0 (sphere or flat plate) to 90◦

(dihedral). A value of α equal to 45◦ represents a dipole-like scattering mechanism.

2.8 Statistical properties of SAR data

Up to this point, no mention has been made of the statistical properties of polari-

metric SAR data. SAR images are affected by speckle, which is responsible for

their characteristic granular aspect. In this Section, a brief outline of the main

properties of SAR data is given, followed by treatment of the Hermitian product

of SAR channels, of vital importance in polarimetry and interferometry.

2.8.1 Properties of speckle fields

A SAR system records the scattered field defined in Eq. 2.12. Different channels

are used for different polarisations (polarimetric SAR) or for different imaging

geometries (interferometric SAR). The recorded data can be written as a vector

� �

R =

R1

R2

R3

=

N∑

p=1

� �

k (p) exp(−j2krp)rp

, (2.32)

where N scatterers have been assumed,� �

k (p) is the vectorised scattering matrix for

scatterer p, and rp is the distance from scatterer p to the antenna. If the region of

space where the scatterers are distributed is small compared to the distance to the

antennas, the rp term in the denominator can be treated as constant and it will be


ignored for the rest of the analysis. Assume further that the spatial distribution of

the scatterers is large compared to the wavelength (as it is often the case in imaging

radars where the resolution cell size is of the order of several wavelengths), so that

the phase in the exponential in Eq. 2.32 is uniformly distributed in [−π, π]. If

the amplitude and phase of each scatterer are independent, it can be shown (see

[Goodman, 1984]) that for largeN , the real and imaginary parts of each component

of� �

R, Ri, follow Gaussian distributions with zero mean and identical variances σi/2.

The variance is determined from the mean intensity as

σi = limN→∞

1

N

N∑

p=1

⟨∣∣∣S(p)i

∣∣∣2⟩. (2.33)

If Ri = zR + jzI , the distribution of real and imaginary parts can be written as

PzR,zI(zR, zI) =

1

πσiexp

(−z

2R + z2

I

σi

). (2.34)

The amplitude A for channel i is given by a Rayleigh distribution:

PA(A) =2A

σiexp

(−A

2

σi

). (2.35)

The intensity I = A2 follows an exponential distribution:

Pi(I) =1

σiexp

(− I

σi

). (2.36)

From Eqns. 2.34-2.36, it is clear that σi is the parameter that characterises all

the distributions (except the phase distribution, which is uniform). In practical

applications, σi needs to be estimated from the data, and its maximum likelihood

estimator (MLE) is given by [Oliver and Quegan, 1998]

σi = z2R + z2

I = I. (2.37)

Improved estimates of σi can be obtained by combining several independent

measurements or looks. In SAR imaging, this is often done by either splitting the

bandwidth or synthetic aperture and generating several uncorrelated sub-images


with decreased resolution. Since this is not always possible, several adjacent pixels

over a homogenous region can be averaged together (this is usually referred to

as spatial averaging). The averaging should be done in intensity, on maximum

likelihood grounds, and it results in the variance of the measurement being reduced

by a factor of 1/L for L looks. The multi-look intensity distribution is given by a

gamma distribution with order parameter L:

PI(I) =1

Γ(L)

(L

σ

)L

IL−1 exp

(−LIσ

). (2.38)

Often, L needs to be estimated from the data. This can be accomplished by using

the equivalent number of looks (ENL), defined as

ENL =mean2

variance, (2.39)

where the averages are carried out in intensity over a homogenous region. The

ENL is equivalent to the number of independent intensity values per pixel [Oliver

and Quegan, 1998].

2.8.2 Hermitian products of speckle patterns

In the previous Section, it was shown that individual speckle patterns follow circu-

lar Gaussian distributions. Linear combinations of these channels would also follow

a multivariate Gaussian distribution (see [Oliver and Quegan, 1998], for example).

A multivariate Gaussian distribution is completely defined by its covariance matrix

C, defined as the expected value of the Hermitian products between the considered

channels (as in Eq. 2.25 for polarimetric data). This motivates the interest in

understanding the statistics of the Hermitian products of SAR images.

The covariance between two channels, i and j can be written as

⟨RiR

∗j

⟩=

⟨Ni∑

p=1

S(p)i exp [−j2krp]

⟩ ⟨Nj∑

q=1

S(q)∗i exp [j2krq]

⟩. (2.40)

In Eq. 2.40, Ni and Nj are the scatterers in channels i and j. If the images

are assumed to be from the same scene, and have been taken simultanously and


co-registered, then it can be assumed that the number of scatterers in the two

channels is identical, Ni = Nj = N , and Eq. 2.40 can be re-written as

⟨RiR

∗j

⟩=

⟨N∑

p=1

S(p)i S

(p)∗j

⟩+

⟨∑∑

p6=q

S(p)i S

(q)∗j exp [−j2k(rp − rq)]

⟩.(2.41)

Eq. 2.41 states that there are two contributions to the covariance between two

channels: a term arising from the interaction between individual scatterers in the

two channels, and a second term arising from the interaction between pairs of

scatterers. If the scattering properties are independent of range, Eq. 2.41 can be

re-written as

⟨RiR

∗j

⟩=

⟨N∑

p=1

S(p)i S

(p)∗j

⟩+

∑ ∑

p6=q

⟨S

(p)i S

(q)∗j

⟩〈exp [−j2k(rp − rq)]〉 .(2.42)

The first term in Eq. 2.42 is only dependent on the properties of single scatterers

imaged by different channels. The second terms needs more attention. If the

resolution cell is large compared to the wavelength, the expected value of the

exponential will vanish, as its phase will be uniformly distributed. Otherwise, a

contribution from the second term will be present.

It has been shown that in a typical radar scenario, the Hermitian products

of SAR channels are dependent on the properties of scatterers. The distribu-

tions of these Hermitian products need to be addressed for single and multi-look

data, assuming an underlying multivariate Gaussian distribution. For the sake

of simplicity, two channels are assumed, but the extension to more channels is

straightforward.

The distribution of the two-element data vector� �

x is given by

P ��

x (� �

x ) =1

π2 |C| exp(− � �

x ∗TC−1 � �

x), (2.43)

where C is the covariance matrix, given by

C =

[σ1

√σ1σ2ρ√

σ1σ2ρ∗ σ2

], (2.44)


where σ1,2 are the backscattering coefficients for channels 1 and 2. ρ is the corre-

lation coefficient between the two channels; it is defined as

ρ =〈R1R

∗2〉√

σ1σ2= |ρ| exp(jφ0), (2.45)

and

σi =⟨|Ri|2

⟩. (2.46)

2.8.2.1 Single look distributions

The amplitude and phase distributions of the Hermitian product between two

channels can be obtained by letting Ri = ri exp(jφi) in Eqns. 2.45 and 2.46, and

substituting them back into Eq. 2.43. Eq. 2.43 can be marginalised to calculate

the distributions of interest (see [Tough et al., 1995] for details). The distributions

for single look amplitude and phase are given by

Pφ(φ) =(1 − |ρ|2)

2π

{β(π

2+ arcsin(β)

(1 − β2)3/2+

1

1 − β2

}(2.47)

PR(r) =4r

σ1σ2(1 − |ρ|2)I0(

2|ρ|r√σ1σ2(1 − |ρ|2)

)K0

(2r√

σ1σ2(1 − |ρ|2)

).(2.48)

In the previous two equations, β = |ρ| cos(φ − φ0), while I0(·) and K0(·) are

Bessel functions of the first and third kind, respectively.

Note that in Eq. 2.47, the only parameters controlling the distribution of the

phase are the magnitude and phase of the correlation coefficient. It is shown in

[Tough et al., 1995] that Eq. 2.47 is a delta function centred at φ0 if |ρ| = 1,

and tends towards a uniform distribution as |ρ| → 0, as indicated in Fig. 2.3.

In SAR interferometry, the phase difference between two channels is used to infer

vertical height. Accurate estimation of this height is critically dependent on the

two channels being highly correlated.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−4 −3 −2 −1 0 1 2 3 4

Rel

ativ

eFre

quen

cy

Phase [rad]

|ρ| = 0.9|ρ| = 0.5|ρ| = 0.1

Figure 2.3: Plots of normalised single look phase distributions for different values ofthe magnitude of the correlation coefficient. φ0 was set to 0 rad.

2.8.2.2 Multi-look distributions

Often, multi-channel data are multi-looked to improve the estimates of the sample

covariance matrix. The maximum likelihood estimator for C has been described in

e.g., [Goodman, 1984, Rodriguez and Martin, 1992] as

Cij =1

L

L∑

n=1

R(n)i R

(n)∗j , (2.49)

when L independent samples are averaged together. For i = j, the distribution is

given by the intensity distribution shown in Eq. 2.38, but for the other cases, the

analysis is more complicated. The distribution for the multi-look phase is given


by [Tough et al., 1995] as

PφL(φL) =

(1 − |ρ|2)L

2π

{(2L− 2)!

[(L− 1)!]2 22(L−1)

×[

(2L− 1)β

(1 − β2)L+ 12

(π2

+ arcsin β)

+1

(1 − β2)L

]

+1

2(L− 1)

L−2∑

r=0

Γ(L− 12)

Γ(L− 12− r)

Γ(L− 1 − r)

Γ(L− 1)

1 + (2r + 1)β2

(1 − β2)r+2

}.(2.50)

In Eq. 2.50, Γ(·) is the gamma function. If L = 1, the summation will have no

summands, and Eq. 2.50 will be identical to Eq. 2.47 (the single look case). Again,

it is noted that Eq. 2.50 is governed by the correlation coefficient and the number

of looks. Some distributions are shown in Fig. 2.4, where it is seen that increasing

the number of looks results in a narrower distribution. This is an efficient way of

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−4 −3 −2 −1 0 1 2 3 4

Rel

ativ

e F

requ

ency

Phase [rad]

2 looks8 looks

16 looks

Figure 2.4: Effect of number of looks on the distribution of the phase difference oftwo channels. The curves have been calculated with φ0 = 0 radians and|ρ| = 0.5. Note how the spread of the phase shrinks as the number of looksincreases.

2.9. Summary 24

reducing the uncertainty of the measured phase (and hence of measured vertical

height) in interferometric applications [Zebker and Villasenor, 1992, Rodriguez and

Martin, 1992].

Finally, it is important to consider the coherence and its estimation. The MLE

for the coherence has been given as [Oliver and Quegan, 1998]

ρ =

∑Lk=1R

(k)i R

(k)∗j√

∑Lk=1

∣∣∣R(k)i

∣∣∣2 ∑L

k=1

∣∣∣R(k)j

∣∣∣2. (2.51)

It has also been shown [Touzi et al., 1999] that the estimator provided by Eq.

2.51 is positively biased for a finite sample: for uncorrelated data (|ρ| = 0), the

estimation of the coherence for large L is equal to 12

√π/L [Oliver and Quegan,

1998].

2.9 Summary

This Chapter has given a brief description of polarimetry, emphasising the aspects

which are relevant to this thesis.

The polarisation state of a monochromatic electromagnetic wave describes the

locus of the tip of the field vector looking in the direction of propagation.

A fundamental concept is the scattering matrix, which links the incident and

scattered fields. The elements of the scattering matrix are a function of the fre-

quency of the incident field, as well as of the orientation, shape and dielectric

properties of the scatterer.

The scattering matrix representation depends on a polarisation basis. The

change from one polarisation basis to another can be carried out by means of

a congruence transformation.

It is often convenient to combine the elements of the scattering matrix into a

scattering vector, from where a definition of the covariance and coherence matrices

stems. The covariance matrix completely defines the statistics of polarimetric SAR

data if it follows a multivariate Gaussian distribution.

The pixels of a SAR image can be viewed as a superposition of different types

2.9. Summary 25

of scatterers, each characterised by its own scattering matrix. In a large number

of applications, it would be desirable to separate these contributions. A number

of approaches have been suggested, from simple physical models that separate the

contributions into some predetermined scattering mechanisms, to decompositions

of the covariance (or coherence) matrices that are able to separate the returns into

arbitrary scattering mechanisms.

Finally, the statistics of SAR data were considered. It was shown that SAR data

are characterised by a circular Gaussian distribution, and that to minimise speckle,

several independent samples are often combined (multi-looking). The statistics of

Hermitian products of channels were also examined, as these have a major impact

in interferometric and polarimetric applications. The distribution of the Hermitian

product phase was governed by the magnitude of the correlation coefficient between

the channels of interest, and by the number of looks in multi-look processing. A

narrow phase distribution, of critical importance in interferometry for example,

requires a high correlation between the two channels. If the correlation is poor,

the phase estimate can be improved by multi-looking.

Chapter 3SAR and SAR interferometry

3.1 Introduction

In this Chapter, the basic concepts needed to analyse interferometric SAR

images are introduced. A succinct overview of SAR imaging is given to support

some interferometric SAR concepts (a detailed overview of SAR and InSAR

in the context of the indoor GB-SAR facility is presented in Chapter 4).

The SAR system is shown to act as a measurement instrument that maps the

three-dimensional reflectivity function of a scene into a two dimensional image

space. InSAR uses the phase difference of two SAR images of a scene acquired using

nearly identical imaging geometries to accurately estimate vertical position. As

with other multichannel SAR measurements, the statistics of this phase difference

are governed by the complex correlation coefficient (or complex coherence). It

will be shown that a number of factors decrease the quality of the interferometric

phase (reduce the magnitude of the coherence): thermal noise, geometric effects

and the vertical distribution of scatterers within the scene. Finally, a combination

of polarimetric and interferometric SAR will be presented.

26

3.1. Introduction 27

θ

H

z

ys

y′

z′

x′

y

x

� �

r

Figure 3.1: SAR imaging geometry. The antenna is located at xs, ys,H and a sin-gle scatterer is depicted at x′, y′, z′. The range from the antenna to thescatterer is given by the vector −→r .

3.2. Synthetic Aperture Radar fundamentals 28

3.2 Synthetic Aperture Radar fundamentals

3.2.1 Resolution

Imaging radars measure the complex reflectivity of the imaged scene. Such systems

consist of a microwave transmitter, transmit and receive antennas and a receiver.

In monostatic radar, the same antenna (or set of antennas) is used both for transmit

and receive, whereas in bistatic radar, two different sets are used. The transmitter

generates a pulse of electromagnetic energy which is radiated by the transmit

antenna into free space. Targets scatter the incident wave, and the receiving

antenna records the returning echoes. If the radar is mounted on a moving platform

(a spacecraft, plane...), the direction in which the platform is moving is referred

to as azimuth or along-track. The range or across-track direction is the direction

orthogonal to the azimuth direction, and in this thesis, it is the direction of the

antenna boresight tilted by the angle of incidence. The standard SAR imaging

geometry is shown in Fig. 3.1.

Discrimination between targets located at different ranges is carried out by mea-

suring the time differences in their radar echoes. As such, the range resolution of

the radar depends on the transmitter pulse length τ , or its signal bandwidth, W

[Skolnik, 1967], according to

∆R =cτ

2=

c

2W, (3.1)

where c is the velocity of propagation. From this relationship, it is clear that a

radar with fine range resolution will need a wide bandwidth. Achieving a wide

bandwidth by using short pulses is problematic for system design, as high powers

are usually needed, resulting in very inefficient duty cycles. A way around this is

to use chirp pulses, where the pulse is frequency modulated to produce a longer

pulse with the required bandwidth [Skolnik, 1967].

The resolution in azimuth is controlled by the angular resolution of the antenna

beam, approximately given by λ/D, where D is the physical dimension of the

antenna. The azimuth spatial resolution at some range R is then given by

∆x ≈λ

DR. (3.2)


��

��

��

��

L

Figure 3.2: The aperture length is defined by the spatial interval in which a scattereris illuminated by the radar.

Eq. 3.2 states that the azimuth resolution is range dependent, implying that, for

example, high resolution Earth surface observation using spaceborne sensors would

require the use of impractically large antennas. Aperture synthesis overcomes this

problem by using the idea of an antenna array to coherently combine the returns

from each scatterer. This results in very high resolution, even when using relatively

small antennas. In SAR systems, a single antenna is displaced over the aperture

region, effectively using the motion of the platform to simulate an antenna array,

Similar to Eq. 3.2, the azimuth resolution of a synthetic aperture is given by

∆x =λ

2LR, (3.3)

where L is the total length of the synthetic aperture. The factor of two in the

denominator arises due to the two-way transmit-receive operation of active sys-

tems. The length of the synthetic aperture is given by the spatial interval within

which a stationary scatterer lies in the beam, i.e. L = Rλ/D, as illustrated in

Fig. 3.2. Hence, the maximum achievable resolution in azimuth for a synthetic

aperture system is

∆x =D

2, (3.4)


a remarkable result, which states that the maximum resolution is independent of

both wavelength and range, and that the resolution improves as the real antenna

length gets shorter.

3.2.2 Range processing

In order to obtain range resolution, pulse compression techniques are used. Range

processing is based on the resolution being a function of the bandwidth of the

transmitted pulse. In order to use long pulses for efficient radar operation, the

transmitted pulse is multiplied by a short-time signal (usually, a chirp with a

linear frequency shift [Skolnik, 1967]), which equips the pulse with the required

bandwidth to obtain the desired range resolution.

The returned echoes, a scaled and time shifted version of the transmitted signal,

are detected by a process of correlation with a local copy of the transmitted signal,

in a process called matched filtering [Papoulis, 1977]. If the time-bandwidth prod-

uct of the radar, Wτ , is large, then the spatial range resolution is approximately

given by Eq. 3.1 [Oliver and Quegan, 1998].

3.2.3 Azimuth processing

Azimuth processing can also be seen as a pulse compression process. The pulse

duration τ in the previous Section has an equivalent in azimuth: the synthetic

aperture length. This is the distance illuminated by a beam in the azimuth direc-

tion, and can be written as Rα (where R is range, and α is the antenna angular

resolution in azimuth). It can be shown that the maximum azimuth resolution

possible with a SAR is approximately given by Eq. 3.3 [Oliver and Quegan, 1998].

3.2.4 System approach

The imaging process can also be examined from a systems point of view, as the SAR

can be described as a system that produces some output based on the reflectivity

of the imaged scene. The aim of this Section is to explore this more general

viewpoint, with the aim of better understanding the information content of the

final SAR image.


y

a( ��r ′)

��r

ys

θ

��r ′

H

z

Figure 3.3: The SAR integrates the ground reflectivity along circular segments (thedashed curved segments). Within each of these resolution cells, all scattererreturns are combined in the final image. In the far field, these circularsegments can be approximated by straight lines.

To accommodate natural scenes, the scatterer density function a(� �

r ′) =

a(x′, y′, z′) can be used to quantify the three-dimensional distribution of scatterers

within the scene. The radar can be characterised by its impulse response function,

h(x, r), ideally approximated as a product of sinc functions for the slant range and

azimuth directions [Zebker and Villasenor, 1992, Rodriguez and Martin, 1992],

h(x, r) = sinc (πx/∆x) sinc (πr/∆R) , (3.5)

where ∆x and ∆R are the azimuth and range resolutions, assuming that ∆x is

constant.

Throughout the following, two assumptions are made:

1. The scene is stationary during the imaging process,

2. Only first order interactions are of any significance (Born approximation).

The imaging process can be seen as a projection of the scatterer density function

on to the azimuth-range plane, followed by a (two-dimensional) convolution with

3.3. Interferometric SAR 32

s(x, r)a( ��r ′)

exp(−j2krs)

h(x, r)

n(x, r)

Figure 3.4: System diagram of a SAR system. The scattering object is multiplied bya phase factor, and is then convolved with the SAR’s transfer function,h(x, r). Finally, the effect of additive noise is also included.

the impulse response function of the radar [Jakowatz et al., 1996, Marechal, 1995].

The projection occurs along a radial path of constant radius r′ (see Fig. 3.3). In

the far-field, this path can be approximated by a straight line, so that the SAR

image can be written as [Bamler and Hartl, 1998]

s(x, r) ≈ exp(−j2krs)∫a(

� �

r ′) exp(−j2

� �

k · � �

r ′)h(x−x′, r−r′)dV ′+n(x, r), (3.6)

where rs =√H2 + y2

s , and a noise contribution n(x, r) has also been included. As

a block diagram, the imaging process is shown in Fig. 3.4. An interpretation of

Eq. 3.6 is that the final SAR image contains a band-limited representation of the

scatterer density function; this transduced band can be shown to be a function of

the impulse response of the radar and of the angle of incidence [Prati and Rocca,

1993].

3.3 Interferometric SAR

A SAR image is a projection of a three-dimensional distribution of scatterers onto

the range-azimuth plane. Elements at different heights may be mapped in different

range bins. Hence part of the information is lost due to the projection on the image

plane. A way to estimate the height location of the imaged scatterers is to use

an interferometric set-up. This relies on the acquisition of two SAR images of


90◦

n(x, r1)

a(� �

r ′)

exp(−j2kr1)

h(x, r1)

exp(−j2kr2)

h(x, r2)

n(x, r2)

s1s∗

2

Figure 3.5: A system view of interferogram formation. The radar reflectivity is mul-tiplied by a propagation factor and is convolved with two different radartransfer functions, one for each image. Different noise contributions areadded, and the complex conjugate product of the two images is formed toproduce the interferogram.

the same scene using slightly different imaging geometries. The two images can be

combined to form the phase difference between them (an interferogram), which can

be used to infer vertical position. The interferogram processing chain is depicted

in Fig. 3.5.

Interferometry has a broader meaning than that outlined above. Throughout

the literature, SAR interferometry refers to the combination of two complex SAR

images to study their phase differences. In this thesis, across-track interferome-

try (two images acquired using different geometries combined for vertical location

retrieval) will be considered.

The aim of this Section is to present the basic concepts of interferometric SAR:

the InSAR geometry, the nature of the interferometric phase, the interferometric

coherence and the factors that govern both the phase and the coherence.

3.3.1 The Interferometric Geometry

The standard InSAR geometry is depicted in Fig. 3.6: two antennas A1 and A2,

separated by a baseline B, image a scene. It is useful to decompose the baseline into


z

yx

r1

∆r

θ1

θ2

B

αA1

A2

∆θ

r2

Figure 3.6: Standard InSAR geometry. Two antennas image a single scatterer, locatedat distances r1 and r2 from each antenna. Each antenna has a differentlook angle (θ1, θ2), and the path length difference is given by ∆r = r2 − r1.The two antennas are separated by a baseline B. The baseline maintainsan angle α with respect to the horizontal.

two components, normal and parallel to the line of sight, B⊥ and B‖, respectively.

The baseline is tilted by an angle α with respect to the horizontal. The angle of

incidence is given by θ1 (antenna A1) and θ2 (antenna A2).

Two configurations are possible:

• using two antennas to simultaneously image the scene (single-pass interfer-

ometry),

• using one antenna, flown twice over the scene (repeat-pass interferometry).

Repeat pass is often used in satellite systems, where it is difficult to mount two

antennas on the same platform. The time interval between passes raises the possi-

bility of changes in the scene, conditioning the information that can be extracted

from an interferogram.

3.3.2 The Interferometric Phase

The phase of a SAR image is uniformly distributed over distributed targets due to

speckle (see Chapter 2), and carries no information. InSAR extracts information


from the phase difference between two images. This is equivalent to estimating

the phase difference between each scatterer in the two images, and carrying out

an average of this phase difference weighted by the RCS of each scatterer within

a resolution cell.

For the geometry shown in Fig. 3.6, the interferometric phase is calculated as

the argument of the Hermitian product of the signals acquired from both antennas.

The recorded signals are

s1(r) = a(r) exp(−j2kr) (3.7)

s2(r + ∆r) = a(r + ∆r) exp [−j2k(r + ∆r)] . (3.8)

For simplicity, assume that the scene is made up of a single scatterer, and that

a(r) = a(r + ∆r). The interferometric phase ∆φ can be written as

∆φ = 2k∆r =4π

λ∆r. (3.9)

The interferometric phase can be related to the geometry as follows,

∆φ ≈[4π

λ

B⊥

r tan θ∆y +

4π

λ

B⊥

r sin θh

]

2π

, (3.10)

where the average look angle θ = (θ1 + θ2)/2 has been introduced, ∆y is the

difference in slant range from one pixel to the next and h is the height of the

scatterer over a reference plane. Eq. 3.10 states that the interferometric phase

is the sum of two components: a term arising from the range position (usually

subtracted in a process called flat earth removal) and a term due to the vertical

position of the scatterer. The phase is recorded modulo 2π, which introduces an

ambiguity. In this case, the topography is related to an unwrapped phase,

∆φunwrapped = ∆φ+ 2πN, (3.11)

where N is an integer that needs to be estimated using phase unwrapping tech-

niques (see [Zebker and Lu, 1998], for example).

Eq. 3.10 states that after flat earth removal and phase unwrapping, there is a


direct relation between the interferometric phase and the vertical position of the

scatterer.

Natural targets are made up of large numbers of scatterers, so that the Hermitian

product where the interferometric phase is derived from can be written as

〈s1s∗2〉 =

⟨N∑

p=1

ap exp [−j2krp]⟩ ⟨

N∑

q=1

a∗q exp [j2k(rq + ∆rp)]

⟩, (3.12)

following the rationale outlined in Section 2.8.2 (Eq. 2.40). In contrast with Eq.

2.40, it has been assumed that the two images only contain the same scatterers N ,

and that the path length difference for a scatterer i due to the different imaging

geometries is ∆ri. Assuming that the scattering properties are independent of

range, and that the resolution cell is large compared to the wavelength, Eq. 3.12

can be written as

〈s1s∗2〉 =

N∑

p=1

⟨|ap|2 exp [j2k∆rp]

⟩, (3.13)

where similar arguments as those used in Section 2.8.2 have been used. Eq. 3.13

states the nature of the interferometric phase for a distributed target. In these

circumstances, the interferometric phase is the measure of the expected change

of the scattering phase for each of the scatterers in the scene due to the different

imaging geometries. The contribution of each scatterer is weighted by its backscat-

ter intensity. The interferometric phase can be related to the vertical location of

the scattering phase centre for the imaged target, using the same arguments that

were previously put forward.

3.3.3 The Interferometric Coherence

The complex coherence ρ between two signals s1 and s2 is defined as

ρ =〈s1s

∗2〉√

〈|s1| 2〉⟨|s2|2

⟩ (3.14)


(see Eq. 2.45). Making use of Eq. 3.6, s1 and s2 can be written as

s1(x1, r1) = exp(−jk1r1)

∫

V

a(� �

r ′) exp[−j

� �

k 1 ·� �

r ′]h(x1 − x′1, r1 − r′1)dV

′

+ n1(x1, r1) (3.15)

s2(x2, r2) = exp(−jk2r2)

∫

V

a(� �

r ′) exp[−j

� �

k 2 ·� �

r ′]h(x2 − x′2, r2 − r′2)dV

′

+ n2(x2, r2), (3.16)

where ri is the slant range distance from the i-th antenna to the centre of the

resolution cell. It has been assumed that the two images have uncorrelated noise

contributions n1(x1, r1) and n2(x2, r2), and the different wave-vectors account for

the fact that each image will have a different look direction due to the differ-

ent imaging geometry. It is also useful to allow for the centre frequency used

for each image to vary. The second image s2 needs to be co-registered with re-

spect to the first image, to ensure that the transfer functions overlap [Fornaro and

Franceschetti, 1995].

The intensities of the individual images and the cross-correlation of the pair are

(a factor of 4π has been omitted)

〈s1(x, r1)s∗1(x, r1)〉 =

∫

V

σ1(� �

r ′) |h(x− x′1, r1 − r′1)|2dV ′ +N1 (3.17)

〈s2(x, r2)s∗2(x, r2)〉 =

∫

V

σ2(� �

r ′) |h(x− x′2, r2 − r′2)|2dV ′ +N2 (3.18)

〈s1(x, r1)s∗2(x, r2)〉 = ejϕ

∫

V

σeff (� �

r ′) exp[−j2(

� �

k 1 −� �

k 2) ·� �

r ′]

× |h(x− x′, r − r′)|2 dV ′. (3.19)

A phase term ϕ = 2 [k1r1 − k2r2] has been introduced in the cross-correlation, and

an effective backscatter coefficient σeff is also present. This effective coefficient

accounts for the correlation between the reflectivity functions in the two images.

In the previous Equations, two assumptions have been made:

1. The autocorrelation of the reflectivity function can be written as [Rodriguez


and Martin, 1992, Hagberg et al., 1995]

〈a( � �

r )a∗(� �

r ′)〉 = σ(� �

r )δ(� �

r − � �

r ′). (3.20)

2. The noise is uncorrelated with the signal.

By putting together Eqns. 3.17-3.19, and assuming that the individual backscat-

tering coefficients in both images are identical (σ1 = σ2 = σ) and N1 = N2 = N

is the same noise power in both images, the complex correlation coefficient can be

written as

ρ = ejϕ

∫

V

σeff (� �

r ′) exp[−j2(

� �

k 1 −� �

k 2) ·� �

r ′]|h(x− x′, r − r′)|2 dV ′

∫

V

σ(� �

r ′) |h(x− x′, r − r′)|2 dV ′ +N

(3.21)

The phase of Eq. 3.21 is the phase retrieved from an interferogram over a

distributed scene. The interferometric phase will no longer be a single discrete

value over the imaged region, but will be statistically distributed. The phase of

Eq. 3.21 is essentially the expected value of the phase of the hermitian product

of two circular Gaussian distributions, already been introduced in Section 2.8.2.

The distribution of the interferometric phase will have a mode given by the phase

of Eq. 3.21, and its shape will be governed by the magnitude of Eq. 3.21. Since

the interferometric phase is converted to height, the shape of the phase difference

distribution will directly impact on the accuracy of retrieved height. For small

height uncertainty, it is critical to have a narrow phase distribution, and hence the

magnitude of Eq. 3.21 should be close to unity. Note that the coherence can be

increased by increasing the number of looks used to estimate it, as described in

Section 2.8.2.

Given that the quality of the height estimation is governed by the magnitude

of the correlation coefficient between the two interferometric channels, shown in

Eq. 3.21. This expression has contributions both from sensor parameters and

from the imaged target, so it is of interest to examine the different effects of each

contribution. The magnitude of Eq. 3.21 can be broken down into the product of


several contributions [Zebker and Villasenor, 1992, Rodriguez and Martin, 1992]

|ρ| = ρn · ρt · ρh. (3.22)

where

ρn is the decorrelation accounting for finite signal-to-noise ratio (SNR) in the

imaging process,

ρt accounts for changes in the scene between the two passes. This contribution

is unity in single-pass (simultaneous acquisition) InSAR systems,

ρh describes the effect that the different imaging geometries and spatial disposi-

tion of the scatterers within the resolution cell has on the value of the coherence.

3.3.3.1 Thermal decorrelation

Different noise contributions in each channel (see Eqns. 3.17 and 3.18) will decor-

relate the interferometric image pair. If the same SNR is assumed for both images,

this contribution follows readily from Eq. 3.21 and is given by [Zebker and Vil-

lasenor, 1992]

ρn =1

1 + SNR−1. (3.23)

3.3.3.2 Temporal decorrelation

If sub-resolution changes occur in the imaged scene between the two passes, the

arrangement of the imaged scatterers will be different, leading to a loss of corre-

lation between the two images. Examples of these changes are the movement of

branches due to wind, changes in land cover type, etc.

While the drop in coherence due to temporal changes is a nuisance for height

retrieval, there are a number of situations where temporal decorrelation can be

useful. For example, if one image in the pair is produced over temperate forests in

summer, and the other in winter, the coherence can be used to distinguish decid-

uous and coniferous forests [Wegmueller and Werner, 1995]. Similarly, changes in

land-cover type (from a developed crop to a bare field, from forest to clear-cut...)

will show low coherence values [Askne and Smith, 1996, Wegmueller and Werner,

1997, Strozzi et al., 2000].


3.3.3.3 Geometric decorrelation

Geometric decorrelation occurs due to the different imaging geometries and scat-

terer positioning. Referring back to Eq. 3.21, the different geometries are encom-

passed by the different wave-vectors� �

k 1 and� �

k 2. The two images have effectively

been acquired with different angles of incidence, so the sum of contributions from

each point scatterer in the final image will be different. To understand geometric

decorrelation effects, a study of the wave-vector difference projection along the

slant range direction is needed. The projection exponential in Eq. 3.21 can be

written as

(� �

k 1 −� �

k 2

)· � �

r ′ ≈[kB⊥ cos θ

R− ∆k sin θ

]y′ +

[kB⊥ sin θ

R+ ∆k cos θ

]z′, (3.24)

where ∆k = k1−k2 = 2π(f1−f2)/c, B⊥ is the component of the baseline orthogonal

to the line of sight, and θ is the mean look angle (see Fig. 3.6). The wave-vectors

are given by

� �

k 1 = [0, sin θ1,− cos θ1]T (3.25)

� �

k 2 = [0, sin θ2,− cos θ2]T . (3.26)

ρh can now be written as a product of a slant range and a height (volume) com-

ponent contribution,

ρh = ρrange · ρvol,

where

ρrange =

∣∣∣∣∫

exp

[−j2

(kB⊥ cos θ

r− ∆k sin θ

)y′

]|h(x− x′, r − r′)|2 dx′dy′

∣∣∣∣∫

|h(x− x′, r − r′)|2dx′dy′

(3.27)

ρvol =

∣∣∣∣∫σeff (z

′) exp

[−j2

(kB⊥ sin θ

r+ ∆k cos θ

)z′

]dz′

∣∣∣∣∫σeff(z′)dz′

. (3.28)

The slant range component can be seen as a Fourier transform of the magnitude

3.4. Polarimetric Interferometry 41

of the impulse response of the radar. Depending on the impulse response and

imaging geometry, a critical baseline [Zebker and Villasenor, 1992] can be calcu-

lated from Eq. 3.27. If the baseline is larger than this critical value, the image pair

will be completely uncorrelated. Another way of picturing this total decorrelation

is to think that the two images represent non-overlapping bands of the ground

reflectivity spectrum. This decorrelation contribution can be eliminated by choos-

ing ∆k, either by means of a tunable radar (using different frequency bands to

acquire each image), or by filtering out the ground reflectivity bands which are

not common to both images. This procedure is called wavenumber shift filtering

[Gatelli et al., 1994].

If the vertical distribution of scatterers in the resolution cell is non-zero, the

volumetric effect will result in a loss of coherence, referred to as ρvol in Eq. 3.28

[Rodriguez and Martin, 1992, Gatelli et al., 1994]. Vegetated areas will show a

lower coherence than non-vegetated areas, which could again be used to map bare

soils and vegetated regions [Hagberg et al., 1995, Wegmueller and Werner, 1995].

3.4 Polarimetric Interferometry

So far, a single polarisation has been assumed throughout. However, polarimetric

data provide new possibilities for improving the coherence (and hence increasing

the retrieved height accuracy) estimation, as well as giving other potentially valu-

able information. Recently, a procedure to combine interferometry and polarimetry

has been presented [Cloude and Papathanassiou, 1998], where a formalism for gen-

erating interferograms using any two polarisation states (one for each image in the

interferometric pair) is presented.

Let the scattering vectors (in an arbitrary basis) associated with each image be� �

v 1 and� �

v 2 (see Section 2.5). The choice of a given polarisation for each image is

achieved by a linear combination of the elements of the scattering vector. This can

be achieved by transforming each vector by the same 3× 3 special unitary matrix

[Sagues et al., 2000]. Rather than use a transformation matrix, the scattering

vector can be projected on to vectors� �

w1 and� �

w2, that define the polarisations for

images 1 and 2 [Cloude and Papathanassiou, 1998]. The results of these projections


are two scalars, µ1 and µ2:

µ1 =� �

w∗T1

� �

v 1 (3.29)

µ2 =� �

w∗T2

� �

v 2. (3.30)

The resulting interferogram is obtained as µ1µ∗2. The interferometric phase is

given by

φ = arg {〈µ1µ∗2〉} = arg

{⟨(� �

w∗T1

� �

v 1

) (� �

w∗T2

� �

v 2

)∗⟩}, (3.31)

It is useful to define the covariance matrices for each image (Eqns. 3.32 and

3.33), and a third matrix which contains the interferometric phase (Eq. 3.34):

T =⟨

� �

v 1� �

v ∗T1

⟩(3.32)

P =⟨ � �

v 2� �

v ∗T2

⟩(3.33)

Q =⟨

� �

v 1

� �

v ∗T2

⟩. (3.34)

Making use of Eq. 3.34, Eq. 3.31 can be written as

φ = arg{

� �

w∗T1 Q � �

w2

}. (3.35)

The magnitude of the coherence can now be written as

|ρ| =|〈µ1µ

∗2〉|√⟨

|µ1|2⟩ ⟨

|µ2|2⟩ =

∣∣ � �

w∗T1 Q � �

w2

∣∣√

� �

w∗T1 T � �

w1� �

w∗T2 P � �

w2

. (3.36)

Eq. 3.36 suggests that the coherence can be maximised by choosing two suitable

polarisation states. [Cloude and Papathanassiou, 1998] approach the optimisation

problem defining a complex Lagrangian function L:

L =� �

w∗T1 Q � �

w2 + λ1

[ � �

w∗T1 T � �

w1 − C1

]+ λ2

[ � �

w∗T2 P � �

w2 − C2

], (3.37)

where λ1,2 are the Lagrange multipliers and C1,2 are constants. Since L is complex,

the maximal coherence will be obtained optimising LL∗, but since the two terms in

the right of Eq. 3.37 are always real (as T and P are hermitian), the optimisation

of LL∗ can be simplified to the maximisation of L [Cloude and Papathanassiou,


1998]:

∂L

∂� �

w∗T1

= Q � �

w2 + λ1T� �

w1 = 0 (3.38)

∂L∗

∂� �

w∗T2

= Q∗T � �

w1 + λ∗2P� �

w2 = 0. (3.39)

Letting ν = λ1λ∗2, and solving Eqns. 3.38 and 3.39 leads to two coupled eigenvector

problems:

T −1QP−1Q∗T � �

w1 = ν� �

w1 (3.40)

P−1Q∗TT −1Q � �

w2 = ν� �

w2. (3.41)

In [Cloude and Papathanassiou, 1998], the matrices in Eqns. 3.40 and 3.41 are

shown to share the same real eigenvalues, but to have different eigenvectors. The

highest eigenvalue results in the square of the optimal coherence, obtained using

the associated eigenvectors to generate the interferogram (as in Eqns. 3.29-3.30).

Another proof can be found in [Colin et al., 2003]. Note that the projection

vectors that define the polarisations are complex, and will induce a phase shift in

the interferogram if arg{

� �

w1� �

w∗T2

}6= 0. This phase shift needs to be taken into

account and discarded in order to interpret the phase of the interferogram as a

height (as shown in Section 3.3.2).

A further interpretation of the eigenvector pairs in Eqns. 3.40 and 3.41 is that,

if they are orthogonal, a set of interferograms can be generated using them:

φi = µ1iµ∗

2i=

� �

p(i)1 Q � �

p(i)2 , (3.42)

where i = 1 . . . 3 represents the eigenvector pair. The phase differences between

the 3 interferograms produced using Eq. 3.42 would represent the height differ-

ence between scattering phase centres related to the retrieved polarisations. As

an idealised example, a forest canopy might be separated into three backscatter-

ing components: a ground component, a trunk-ground component and a canopy

component. If the phase centres of these three components are associated with

the three extrema retrieved from coherence optimisation, the interferograms will

result in the height differences between the phase centres of the three components


in the forest canopy.

The previous discussion indicates that the retrieved polarisation states may be

different for both antennas (in general,� �

w1 6= � �

w2); the difference in polarisations

would arise from changes taking place in the scene between data acquisitions.

However, this thesis is mainly concerned with investigating the GB-SAR indoor

component, in which case there should be no changes in the scene between acqui-

sitions, so the polarisation states for both images in the pair should be identical

when a small baseline is used. Moreover, this argument also implies that the co-

variance matrices for the individual images will be identical, so that the Lagrange

multipliers introduced in Eq. 3.37 can be assumed to be nearly identical, as they

are used to maximise the numerator of Eq. 3.36 while keeping the denominator

constant. Rewriting Eqns. 3.38 and 3.39 using� �

w1 =� �

w2 =� �

w and λ1 ≈ λ2 = λ

yields

Q � �

w = −λT � �

w (3.43)

Q∗T � �

w = −λP � �

w. (3.44)

Adding these two equations results in a single eigenvalue problem:

[T + P]−1 [Q + Q∗T

]� �

w = −λ � �

w. (3.45)

As in the unconstrained case outlined in Eqns. 3.40 and 3.44, the eigenvectors are

associated with the polarisation state used for both acquisitions. The associated

coherence value can be calculated by substituting the eigenvectors into Eq. 3.36.

This same result is obtained (using another procedure) by Colin et al. [2003].

Constraining the coherence optimisation algorithm to retrieve the same polari-

sations for both images makes physical sense for the GB-SAR instrument. Other

benefits of constrained optimisation over unconstrained optimation will be explored

in Chapter 5.

3.5. Summary 45

3.5 Summary

In this Chapter, the building blocks of InSAR have been introduced. The standard

InSAR geometry is shown, where two slightly displaced antennas acquire an image

of the same scene. The phase difference of these two images (the interferometric

phase) can be used to infer the vertical position of the scatterers if the imaging

geometry is known. The interferometric phase has a distribution controlled by

the coherence. In particular, the magnitude of the coherence controls the shape

of the phase difference distribution, directly affecting the quality of the height

measurements that can be carried out with InSAR.

A number of effects that decorrelate the images in the interferometric pair have

been introduced: thermal decorrelation, temporal decorrelation and geometrical

decorrelation. Thermal decorrelation occurs due to different noise contributions

in the two images in the pair. Temporal changes in the scene also result in decor-

relation. The geometric contribution can be split further into a baseline and a

volumetric component. The former represents the decorrelation due to the dif-

ferent imaging geometry, and can be mitigated by either using a tunable SAR,

or wavenumber shift filtering prior to interferogram formation. The volumetric

decorrelation effect arises from the vertical distribution of scatterers in the scene,

with large vertical distributions resulting in decorrelation.

The Chapter closes with an overview of polarimetric interferometry. Polarimet-

ric interferometry combines polarimetric information (related to the shape of the

scatterers) with the height distribution information available from interferome-

try. This technique is based on the generation of interferograms using arbitrary

polarisation states, and opens the possibility for improving the coherence using

a particular polarisation. This results in an optimisation problem, which finds

the polarisation states that maximise the coherence, and which also results in the

possibility of calculating the height difference between different scattering phase

centres within the resolution cell. A constrained version of this algorithm, where

the polarisation states for both images in the pair are constrained to be identical,

has also been presented.

Some implications for the GB-SAR instrument and its interferometric capabil-

ities can be discussed here (a more systematic treatment is deferred to Chapter

3.5. Summary 46

4). Interferometric processing using the GB-SAR indoor system is carried out

in a repeat-pass configuration; however, the scene is not disturbed between the

passes and temporal decorrelation should be negligible. The nature of the system

results in high SNR, and thus relatively little decorrelation arising from thermal

noise. The geometrical contribution might be significant, due to the geometry of

the system: large changes of angle of incidence over the swath, fine resolution and

compact geometry might play a substantial role. Finally, volume effects will be

a major issue for the analysis of crop targets, which have a significant volumet-

ric scattering contribution. In terms of polarimetric interferometry, and the use

of constrained optimisation is recommended for GB-SAR data processing will be

described in detail in Chapter 5.

Chapter 4Interferometric Processing Using the

Indoor GB-SAR Component

4.1 Introduction

The indoor component of the GB-SAR facility consists of an anechoic cham-

ber, in which targets are imaged using antennas mounted on either a wall

or a roof scanner. Typical resolution cell sizes are of the order of two

to three wavelengths, and the system is fully polarimetric. Three-dimensional

imagery is also possible using two-dimensional scans.

InSAR processing in GB-SAR can be carried out by using two parallel azimuth

scans, providing a horizontal baseline which can be very accurately estimated using

the scanner controls.

The geometry of the GB-SAR system restricts the use of standard InSAR pro-

cessing techniques. The aim of this Chapter is to study the consequences of the

GB-SAR configuration for InSAR measurements. As shown before, the main prod-

ucts of InSAR processing are the interferometric phase and coherence. The values

of these are governed by the geometry of the system, the point spread function of

the radar and the geometry of the resolution cell.

This Chapter starts with a discussion of the GB-SAR geometry, followed by

consideration of the factors that affect the coherence. The analysis results in

an iterative height estimation algorithm for GB-SAR data, and leads naturally

47

4.2. The GB-SAR Geometry 48

to an analysis procedure that makes use of three-dimensional data to estimate

the InSAR performance of the system. The iterative phase to height conversion

procedure is demonstrated with simulated data, whereas the three-dimensional

data analysis technique is demonstrated with real data from the RADWHEAT

experiment. Finally, some concluding remarks are presented.

4.2 The GB-SAR Geometry

The GB-SAR system in an interferometric configuration is a repeat-pass system.

An antenna set is attached to the roof scanner, and moved in the azimuth direction

to synthesise an aperture. The scanner then shifts the antenna set in the range

direction, and performs another azimuth scan, generating a horizontal baseline.

Throughout this Chapter, it will be assumed that the antennas are located 2.54

m above the reference processing plane (this plane is parallel to the floor of the

chamber and has been used for the experiments reported in Chapter 5), the target

region is a 2× 2 m area, and the average angle of incidence at the reference plane

at the centre of this region is 45◦, unless otherwise stated. The horizontal baseline

B is defined as the distance that separates the two antennas. It is also useful to

define the component of the baseline orthogonal to the line of sight, B⊥ = B cos θ,

where θ is the average angle of incidence for the two antennas (see Section 3.3.3.3).

The angle of incidence (at reference layer level) varies from around 31◦ at near

range to about 55◦ at far range. The system azimuth resolution is range-dependent,

and is a function of the synthetic aperture length. For X band imaging centred

at 10 GHz, and using a 0.91 m aperture (a rather typical configuration), the

theoretical azimuth resolution varies from 1.6 to 2.4 wavelengths from near to far

range in the target region. In range, the equivalent resolution in a horizontal plane

varies with position and using a 4 GHz bandwidth at X band, yields a theoretical

slant range resolution around 4 cm.

During the SAR image generation process, the data are windowed using a Han-

ning or raised cosine window. This reduces sidelobes in the PSF, but broadens it

by about a factor of 2 [Oppenheim and Schaeffer, 1989] both in range and azimuth,

so the nominal resolution of the system is around 4 wavelengths in azimuth, and

2.5 wavelengths in range.


4.2.1 The GB-SAR Resolution Cell

The GB-SAR resolution cell is dependent on the geometry, imaging parameters and

processing. Two crucial effects occur in GB-SAR imaging which do not usually

happen with spaceborne systems, although the first is usual in airborne systems:

• the angle of incidence varies significantly across the swath

• the angle of incidence can experience significant changes over the vertical

spread of the scatterers within a resolution cell

The change of angle of incidence across the swath can result in changes in the

backscattering behaviour of the target. The geometry considered for GB-SAR

also produces a change in the angle of incidence within the resolution cell, an

effect which is particular to the GB-SAR system. An example for a crop canopy

is shown in Fig. 4.1: the ground and the top of the canopy would be imaged with

different angles of incidence, whereas in spaceborne and airborne sensors, the angle

of incidence can be taken as constant within the resolution cell.

The value recorded by GB-SAR in a given range bin is an integration of the

contributions of all scatterers within the volume of the resolution cell (see Eq. 3.6).

The interferometric phase from this bin can be converted into an effective height,

which depends on the distribution of backscattering within the resolution cell.

For example, if the ground backscatters strongly and there is little attenuation,

this effective height will have a value close to the ground. Since backscattering

properties are often a function of both height and angle of incidence, the analysis

of GB-SAR data needs to take this into account.

The change in angle of incidence within the resolution cell is a function of the

target’s depth. To find an approximate estimate of the change of angle of incidence

within a resolution cell, it is useful to define the angle of incidence at an arbitrary

plane under the antennas, θ0. The distance between a scatterer located at this

reference plane (typically, the SAR processing plane) and a scatterer located at

the top of the target (height z, angle of incidence θ) can be written as r(θ − θ0).

r is the distance between the scatterer and the antenna. If the distance between

these two points can be approximated by a straight line (i.e., the scene is assumed


H

B

θ0

θt

Figure 4.1: GB-SAR resolution cell geometry. The antennas are separated by a hori-zontal baseline B and are at a height H above a reference plane. The twosets of coloured dashed lines represent the range resolution cell for each an-tenna. The scene is assumed to be in the far field, so the range resolutionlines can be approximated by straight lines. The two angles shown are theincidence angles at the reference layer level, and at the highest point in thevertical spread of scatterers (for example, the top of a canopy).

to be in the far field region), then

r(θ − θ0) ≈z

sin θ0, (4.1)

from where an approximation to the value of the incidence angle at the top of the

layer can be written as

θ = θ0 +z

H tan θ0. (4.2)

A more accurate expression which does not approximate the wavefront as a

straight line yields

θ = arccos[(

1 − z

H

)cos θ0

],

which can be approximated by a Taylor series:

θ =π

2−

(1 − z

H

)cos θ0 −

(1 − z

H

)3

cos3 θ0

6+ · · · . (4.3)

4.3. GB-SAR Coherence Analysis 51

Eq. 4.3 is a better approximation to θ than Eq. 4.2 only if the higher order terms

are taken into account. However, using a third order polynomial (or higher) will

unnecessarily complicate the analysis, and given that the maximum error between

Eq. 4.2 and the exact angle is small (around 3.5◦ for a layer depth of 0.5 m and

θ0 = 30◦). Eq. 4.2 will be used throughout this work.

For a layer of depth 0.5 m, the difference in angle of incidence between the

reference plane (with associated angle of incidence θ0) and the top of the target

can be quite large, as shown in Table 4.1. The variation decreases as θ0 increases.

As an example, for a mature wheat canopy imaged with a ground angle of incidence

of 30◦, the top of the canopy would be imaged with an angle of incidence around

50◦. In this case, the ground return would suffer little attenuation, and the canopy

return would increase as the angle of incidence increased. The integration over

the resolution cell and further interferometric processing might return an effective

height somewhere in the middle of the canopy, or slightly biased towards ground

level.

θ0 [deg] Top-level angle of incidence [deg] Difference [deg]

30 49.5 19.545 56.3 11.360 66.5 6.5

Table 4.1: Angle of incidence variation over a resolution cell for a vertical scatteringdistribution of 0.5 m as a function of the ground-layer angle of incidence,θ0. The antenna was located at a height H of 2.54 m.

4.3 GB-SAR Coherence Analysis

Coherence in InSAR systems can be broken down into a product of three factors:

a SNR contribution, a temporal contribution and a geometrical contribution, itself

made up of a baseline and a volumetric contribution (see Chapter 3). In this

Section, these contributions are considered with respect to the particularities of

the GB-SAR system, particularly its geometry. The treatment leads to an iterative

algorithm for height retrieval.


The use of an anechoic chamber ensures high SNR. However, the SNR figure

needs to be estimated for each imaging configuration, as it is dependent on band-

width, number of samples along the aperture, etc. Based on a worst-case scenario

figure of 20 dB SNR, the thermal decorrelation factor will be 0.99, so that the

thermal decorrelation contribution can be ignored for the purposes of the analysis.

In the stable environment of the anechoic chamber, target changes between

scans will be very small, so that the temporal decorrelation contribution can also

be ignored. The controlling effects for decorrelation are all geometrical.

The different imaging geometries result in decorrelation. In Section 3.3.3.3, two

contributions could be separated: a slant-range, ρrange, and a volumetric decorre-

lation contribution, ρvol.

4.3.1 Slant-Range Decorrelation

The expression for ρrange was given in Eq. 3.27, and is repeated here for conve-

nience:

ρrange =

∣∣∣∣∫

exp

[−j2

(kB⊥ cos θ

r− ∆k sin θ

)y′

]|h(x− x′, r − r′)|2 dx′dy′

∣∣∣∣∫

|h(x− x′, r − r′)|2dx′dy′

(4.4)

The slant range decorrelation contribution can be mitigated by choosing ∆k so

that the exponential in the numerator in Eq. 4.4 vanishes, in a process called

common band filtering:

∆k =kB⊥

r tan θ. (4.5)

Common band filtering can be carried out either by changing the centre fre-

quencies for each scan for each acquisition, or after acquisition by filtering out

the non-common bands of the ground reflectivity spectrum. The disadvantage of

this latter procedure is that part of the bandwidth is discarded, and thus range

resolution broadens.

Eq. 4.5 shows that ∆k depends on angle of incidence, hence is variable over

the GB-SAR swath. Two approaches are possible to implement common-band

filtering:


1. Process each range bin with a different value of ∆k

2. Take the smallest value of θ, and use this shift throughout the image

While the first option makes better use of the bandwidth, the second option is

substantially easier to implement, and it retains a constant slant range resolution

cell size with range. For a typical GB-SAR configuration, with a baseline B of

0.06 m, and a minimum angle of incidence of θe = 30◦, the frequency shift in MHz

is

∆f =B cos2 θeλH tan θe

c = 306.8594 MHz, (4.6)

using a centre frequency of 10 GHz. The frequency shift accounts for about 8% of

the available bandwidth (if using 4 GHz), and, with an angle of incidence of 45◦,

the range resolution would increase from 0.038 m to 0.041 m, a small difference.

Using a frequency shift calculated for the smallest angle of incidence results in

the exponential in Eq. 4.4 not vanishing, leading to a small amount of residual

decorrelation. Assume that the angle of incidence does not change over a resolution

cell, and that the width of the point spread function in the range direction is 3.5

wavelengths. The range decorrelation contribution for the shift calculated in Eq.

4.6 will be 0.86 at 60◦ (for lower angles of incidence, this contribution increases).

This small loss at large angles of incidence is a small price to pay for some of the

simplifications that can be achieved, and that will be used in the following sections.

In detail, the common band filtering procedure is implemented as follows:

1. The bandwidth of the first image of the pair is reduced by ∆f at the begin-

ning of the bandwidth. A new centre frequency is calculated.

2. The bandwidth of the second image of the pair is reduced by ∆f at the end

of the bandwidth. A new centre frequency is calculated.

3. The two images are processed with the new centre frequencies

After carrying out this procedure, the only remaining source of decorrelation is

due to the vertical distribution of the target.


4.3.2 Volumetric Decorrelation

The volumetric decorrelation contribution is given by Eq. 3.28 (repeated here for

convenience):

ρvol =

∣∣∣∣∫f(z′) exp

[−j2

(kB⊥ sin θ

r+ ∆k cos θ

)z′

]dz′

∣∣∣∣∫f(z′)dz′

. (4.7)

In spaceborne systems, Eq. 4.7 states that the volumetric decorrelation contri-

bution is related to the (normalised) Fourier Transform of the vertical distribution

of scatterers f(z′), as the exponential in the numerator is linear in z′. If common

band filtering is applied, then ∆k → kB⊥/(r tan θe). θe is the minimum incidence

angle along the swath, as discussed in Section 4.3.1. The exponential term in Eq.

4.7 can then be written as

exp

[−j 2kB⊥ cos θ0

H tan θe(sin θ tan θe + cos θ) z′

](4.8)

In the case of GB-SAR, the exponential cannot be interpreted as a Fourier kernel,

because θ is a function of z′ (cf. Eq. 4.2). Eq. 4.7 can be written as (using Eqns.

4.2 and 4.8):

ρvol =

∫ θt

θ0

f (z′(θ)) exp

[−j2kB sin θ0 cos θ

sin θ tan θe + cos θ

tan θe(θ − θ0)

]dθ

∫ θt

θ0

f(θ)dθ

. (4.9)

Eq. 4.9 expresses the value of the volumetric decorrelation in terms of the

vertical distribution of the scatterers (and hence as a function of angle of incidence).

θt is the angle of incidence at the top of the scatterer layer.

Eq. 4.9 is illustrated in Fig. 4.2, where the coherence calculated from Eq.

4.9 is shown compared to full-system numerical simulation coherence estimations

(relevant parameters for the simulations are shown in Table 4.2). The numerical

simulations randomly position scatterers within the target region. A set of two

transmit-receive antennas are shifted along a linear path to simulate the synthetic

4.4. Effective Height from the Interferometric Phase 55

aperture. The path lengths from each antenna to every scatterer (and back to the

antennas) are calculated for each frequency and each antenna position along the

aperture. The data are then windowed both in azimuth and range, and processed

using the standard GB-SAR backpropagation algorithm [Bennett and Morrison,

1996], with resulting pixel spacing of 0.01 m (azimuth or range). The resulting

images from each antenna are combined in an interferogram, and subsampled by

a factor of 10 in both azimuth and range. The coherence is estimated over a 5× 3

averaging window, which results in an estimated number of looks close to 9.

The presented numerical simulations agree well with the expected results for the

45◦ case (Fig. 4.2). The 60◦ case overestimates the coherence with respect to the

simulations, due to the extra slant-range decorrelation arising from common-band

filtering, as outlined in the previous Section. For the 30◦ incidence case, two effects

result in the estimation not following the simulations. First, the estimation of the

variation of the angle of incidence becomes less accurate as the angle of incidence

decreases (cf. Eq. 4.2), and secondly, the estimated coherence in the simulations

will be positively biased in areas of low coherence due to the small number of looks

used. Using 9 looks and Eq. 11.53 in [Oliver and Quegan, 1998], the bias for the

estimated coherence on an area where the true coherence is zero is found to be

around 0.3.

Parameter Value

Centre Frequency 10.0 GHzBandwidth 4.0 GHz

Approx. Resolution (after filtering) 0.09 × 0.08 m (Range,Azimuth)Scatterers per Cell 20

Baseline (horizontal) 0.06 mApprox. Number of Looks 9

Table 4.2: Parameters used for the full system simulations.

4.4 Effective Height from the Interferometric Phase

If slant range decorrelation contributions are negligible, the phase of ρvol in Eq. 4.9

is the value of the interferometric phase due to volume scattering in the scene. The


0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5

Coh

eren

ce

Vertical Spread [m]

Coherence of a uniform distribution of scatterers as a function of vertical spread

30 deg45 deg60 deg

Figure 4.2: Coherence estimation for standard GB-SAR geometry using Eq. 4.9 (red,green and blue curves, for θ0 equal to 30, 45 and 60◦, respectively). Thecircles show the values calculated from full system simulations for the sameangles of incidence. The coherence was estimated over 9 looks.


interferometric phase can be converted to an effective height, h, which will depend

on the properties of the imaged volume. The interferometric phase can thus be

interpreted as coming from a single scatterer located at the effective height h. The

vertical scattering distribution of the equivalent single-scatterer scene is given by

a Dirac δ function, f(z) = δ(z − h), which is substituted in Eq. 4.9, so that the

coherence can be written as

ρvol = exp

[−j2kB sin θ0 cos θf

tan θe sin θf + cos θftan θe

(θf − θ0)

], (4.10)

where θf is the angle of incidence associated with height h (see Eq. 4.2). The

phase of ρvol in Eq. 4.10 will be the phase shift induced by the imaged volume. In

the equivalent single scatterer model, the phase of ρvol can be written as

arg{ρvol} = ∆ϕ = −2kB

rcos θf

tan θe sin θf + cos θftan θe

h, (4.11)

where the angle variation approximation introduced in Eq. 4.2 has been used.

4.4.1 An Iterative Height Retrieval Algorithm

Eq. 4.11 can be used to retrieve the effective height, provided that θf can be

estimated, which itself requires knowledge of the effective height. However, the

effective height can be retrieved using an iterative procedure: θf can initially be

assumed to be equal to θ0, and some approximate effective height can be calculated.

This new height can be used to refine the estimate of θf , providing an improved

estimate of the effective height, and so on. This approach is shown in Algorithm

1.

Algorithm 1 stops when the difference in the estimation of θf (or h) between

two iterations is smaller than some predetermined value Dθ (Dh). Algorithm 1 will

converge if it is a contraction mapping. To prove that the algorithm is indeed a

contraction mapping, start from the equation for h(i) based on θ(i)f (Step 2), and

substitute h(i) for H tan θ0(θf − θ0) to provide a mapping from θ(i)f to θ

(i+1)f :

θ(i+1)f = f(θi). (4.12)


Algorithm 1 Iterative Height Retrieval Algorithm

1. θ(0)f = θ0, i = 0

2. h(i) = −∆ϕr tan θe2kB

1

cos θ(i−1)f (tan θe sin θ

(i−1)f + cos θ

(i−1)f )

3. θ(i)f = θ0 +

h(i)

H tan θ0

4. If |θ(i)f − θ

(i−1)f | < Dθ (or |h(i) − h(i−1)| < Dh), then stop. Otherwise, let

i = i + 1 go back to 2.

Eq. 4.12 converges if

|f ′(θ(0)f )| < 1, (4.13)

where f ′(x) = ∂f/∂x. Eq. 4.13 translates to

tan θe cos 2θ(0)f − sin 2θ

(0)f + 1

2[1 + cos 2θ

(0)f + sin 2θ

(0)f tan θe

]2 <8kB

∆ϕ tan θe tan θ0. (4.14)

For practical GB-SAR configurations (see Table 4.2), and assuming the initial

guess in Step 1 above, the algorithm will converge.

In areas where there is uncertainty in the estimation of ∆ϕ, convergence is not

guaranteed unless the uncertainty is relatively small and Eq. 4.14 still holds. This

uncertainty in the estimate of ∆ϕ can be seen as a fluctuation of the right hand

side of Eq. 4.14, resulting in inaccurate estimates of θf . This problem would be

greatest in areas of low coherence.

4.4.2 Numerical Simulations

The previous ideas can be illustrated using numerical simulations and comparing

them with estimates of Eq. 4.11. The simulations are based on a scene composed

of scatterers uniformly distributed in a volume in space. For this arrangement, the

effective height lies at the centre of the vertical spread (see [Hagberg et al., 1995],

for example). The proposed scenario can be analysed using Eq. 4.11 to estimate the


interferometric phase (the vertical height distribution f(z(θ)) is just a constant),

and from there, the effective height can be retrieved either using the iterative

procedure outlined in Algorithm 1, or a phase to height conversion approximate

procedure often used in the InSAR literature (see [Zebker and Villasenor, 1992]

for example):

∆ϕ = − 2kB

r tan θ0h. (4.15)

The results of estimating the effective height from Eq. 4.11 using these two phase-

to-height conversion approaches are shown in Fig. 4.3; the results from Algorithm

1 form the red curve, those for the approximate height estimation (Eq. 4.15) are in

blue, and the expected value is shown in green. The angle of incidence at ground

level, θ0, was set to 45◦.

The results presented in Fig. 4.3 in general show a good agreement between

expected retrieved heights and estimated heights using either phase to height con-

version approach. The iterative procedure is closer to the expected value than the

approximation. This is due to the small angle approximation used to derive Eq.

4.15 being increasingly unsuitable with increasing layer depth. A similar argument

explains the departure of the iterative approach from the expected result: as the

depth of the layer increases, Eq. 4.2 becomes less accurate.

Fig. 4.3 also shows the results from full system simulations at 45◦ angle of

incidence. The retrieved interferometric height has been obtained using both the

iterative (red circles) and approximate height conversions (blue squares). The

simulation parameters are identical to those used in the simulations presented in

Section 4.3.2. While the iterative estimation closely follows the estimation using

that same algorithm (red curve), the approximate approach departs from the value

expected for this algorithm and follows the expected height curve more closely than

predicted, offering a better height estimate, particularly for large vertical spreads.

The results presented in Fig. 4.3 suggest that the proposed iterative height

retrieval procedure will result in an accurate estimate of the height. The approxi-

mate estimation results in an overestimate of the retrieved height at larger heights.

Eq. 4.15 assumes that the change in angle of incidence between the effective height

location and the processing plane is small. As the effective height grows, the small


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6

Ret

rieve

d H

eigh

t [m

]

Vertical Spread [m]

Retrieved Height from a Uniform Distribution of Scatterers

IterativeApproximate

Expected

Figure 4.3: Retrieved height for a uniform distribution of scatterers with different ver-tical spreads. The retrieved heights were calculated using Eq. 4.9 andapplying the iterative height retrieval algorithm (Algorithm 1) (red curve)and the approximate height retrieval algorithm (Eq. 4.15, blue curve). Thegreen curve shows the expected height. The red circles show the retrievedheight from full system numerical simulations at 45◦ angle of incidence,where the retrieved phase has been processed using the iterative heightretrieval algorithms, while the blue squares show the results of processingthe full system simulation with the approximate phase to height approach.

angle approximation becomes less accurate.

Also shown in Fig. 4.3 are results from the full system simulations used in Sec-

tion 4.3.2. The phase of the correlation coefficient (whose magnitude was shown

in Fig. 4.2) was converted to height using both the iterative (red circles) and

the approximate geometric approach (blue squares). Either of these procedures

results in an accurate estimate of the effective height. As discussed in the previous

paragraph, the iterative procedure slightly underestimates the height, whereas the

4.5. Coherence Analysis Using Three-Dimensional Data 61

approximate approach overestimates it slightly, in line with the expected behaviour

for each algorithm. Both solutions are lower than the expected lines, which sug-

gests that the estimated interferometric phase corresponds to an effective height

slightly under the expected height. This variation is due to the poor coherence

(and small number of looks) and to the small number of scatterers in the resolution

cell.

The simulations presented in this Section confirm that the iterative height esti-

mation algorithm is suitable for height retrieval with GB-SAR data. The approx-

imate formulation (Eq. 4.15) also offers a suitable alternative. For small vertical

displacements, either method produces accurate height estimates.

4.5 Coherence Analysis Using Three-Dimensional

Data

For the GB-SAR instrument, after common-band filtering, and neglecting other

sources of decorrelation, the coherence depends only on ρvol, where value is given by

Eq. 4.9. The numerator can be thought of as a line integral over some height-angle

of incidence two-dimensional space, multiplied by some complex function of height

and angle of incidence. Example paths are shown in Fig. 4.4. The denominator is

just the line integral over the same path without the complex function factor. The

volumetric decorrelation work presented in this Chapter can be used for analysis

purposes if a model of the backscattering distribution as a function of height and

angle of incidence is available. Alternatively, the use of three-dimensional data

can be made. The three-dimensional GB-SAR datasets reconstruct the RCS over

a volume, and can be averaged in azimuth to produce an averaged height-angle

of incidence graph of the imaged target. The azimuth averaged and angle of

incidence corrected image can then be used as an estimate of f(θ), and Eq. 4.9

can be evaluated.

The benefit of using three dimensional data to estimate the coherence is that it

does away with the need for a model to analyse the interferometric data. While

simple models can be applied to analyse the data (for example, a random volume

plus a ground return), this might be of little advantage, since the imaged scenes


IntegrationPaths

Hei

ght [

m]

Angle of Incidence [deg]

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

20 30 40 50 60 70 80

f(θ)

Figure 4.4: Integration paths for Eq. 4.9 for ground angles of incidence of 30◦, 45◦ and60◦.

might not relate to the proposed models at all. On the other hand, a number of

models could be studied with this approach and compared to the real data in an

effort to see which model best predicts the measured data.

4.5.1 Application to the RADWHEAT data set

During the RADWHEAT campaign [Bennett and Morrison, 1996, Brown et al.,

2003], indoor three-dimensional imaging of wheat canopies was carried out, and

these data can be used to estimate the interferometric behaviour of wheat canopies

using the techniques discussed in Section 4.5, provided a height-angle of incidence

representation of the data is possible.

Three-dimensional imaging results in a reconstruction of the RCS of the scene at

the imaging frequency. To estimate the height-angle of incidence plane represen-

tation, azimuth averaging is carried out, assuming that the imaged scene does not

vary significantly across the azimuth direction. This process reduces the 3D data

to a two-dimensional one, a height-range plane representation. Unfortunately, the

size of the resolution cell and of the target region limit the number of azimuth


samples that can be averaged to around 15-20 [Brown et al., 2003]. A simple

transformation converts the range axis to angle of incidence, which results in a

depiction of the backscattering distribution in terms of vertical position and angle

of incidence, similar to that shown in Fig. 4.4.

As an example, data from the RADWHEAT experiment [Brown et al., 2003] will

be used to demonstrate the analysis suggested above. A wheat canopy measuring

1.7 × 1.6 m with an average height of 58 cm was imaged on 14/06/1999. At the

time, the ears were just emerging, and the moisture content was less than 10%.

Crop density was 442 shoots per square meter. The results of three-dimensional

imaging of this canopy at C band, converted into a height-angle of incidence image,

are shown in Figs. 4.5-4.7 for the VV, HH and VH channels.

The HH channel in Fig. 4.6 shows a substantial contribution from the soil, with

little contribution from the wheat canopy itself. The soil contribution is larger

at low angles of incidence, due to the the increasing attenuation with increasing

angle of incidence. At about 40◦ some backscattering from the top of the canopy is

present, but its magnitude is small compared with that of the soil return. The VV

channel also shows significant backscattering from the soil level at small angles of

incidence, but the returns from this level diminish substantially as the attenuation

through the canopy increases with increasing angle of incidence. At around 40◦,

backscattering from the top of the canopy is present, comparable to that of the

ground return for smaller angles of incidence. This contribution is due to the

alignment of the flag leaves with respect to the incident field. The cross-polar VH

channel shows low returns in comparison with the co-polar channels, with the soil

contribution being dominant, and a small return from the top of the canopy at

around 40◦.

The images in Figs. 4.5-4.7 suggest that in interferometric processing, the HH

response will be mostly influenced by the soil layer and that the VV response

will have a significant contribution from the flag leaves at incidence angles larger

than 40◦. It is thus expected that the effective height derived from the HH-HH

interferogram will be close to the soil level for all angles of incidence, whereas the

effective height derived from the VV-VV interferogram will increase towards the

top of the canopy with increasing angle of incidence. The GB-SAR interferomet-

ric picture is more complex than this. There is a significant change in angle of


Figure 4.5: RADWHEAT (18/06/1999) height-angle of incidence reconstructions at Cband for VV polarisations.


Figure 4.6: RADWHEAT (18/06/1999) height-angle of incidence reconstructions at Cband for HH polarisations.


Figure 4.7: RADWHEAT (18/06/1999) height-angle of incidence reconstructions at Cband for VH polarisations.


incidence within the resolution cell which is dependent on the range position. The

change is greatest for areas that have a small angle of incidence at ground level, as

discussed in Sect. 4.2.1). This effect can be seen in Fig. 4.8, where the variation

in angle of incidence for different areas of the scene has been superimposed on the

VV image. For low angles of incidence at ground level, the resolution cell includes

a significant contribution from elements at the top of the canopy level imaged with

a relatively large angle of incidence, apart from the strong contribution from the

soil; the interferometric effective height in this case will lie somewhere between the

two layers. For areas where the ground layer is imaged with a large angle of inci-

dence, the top of the canopy will have a larger impact, and the VV interferometric

effective height will be closer to the top of the canopy.

Figure 4.8: Integration paths superimposed on the VV image.

The results of analysing the images with the procedure outlined in Section 4.3,

using the integration paths shown in Fig. 4.8, are shown in Figs. 4.9 and 4.10

for the coherence and effective height, respectively. The coherence results are very

high for any angle of incidence and polarisation combination: the coherence is

always larger than 0.93. This is due to the use of a small baseline. The three


0.93

0.935

0.94

0.945

0.95

0.955

0.96

0.965

0.97

0.975

0.98

0.985

10 15 20 25 30 35 40 45 50

VVHHVH

Cooher

ence

Incidence Angle [deg]

RADWHEAT Delivery 4 (19990614), C band, 0.01 m Baseline

Figure 4.9: Estimated coherence for the images shown in Fig. 4.7 using a 0.01 mhorizontal baseline.

polarimetric channels show a similar trend, with higher coherence at small angles

of incidence, a broad dip in coherence at around 20◦ and high values again at

around 40◦. The reason for this behaviour lies in the relative influence of the

ground and top of canopy contributions: at small angles, the ground return is very

strong, and dominates the return (especially in the HH channel), whereas around

40◦, the ground return is feeble and the top of canopy return is quite strong,

especially in the VV channel. In both cases, there is a principal contribution from

a relatively thin layer (ground in the former, top of the canopy in the latter),

with some vestigial contributions from the stalks and either the ground (large

angles) or the top of the canopy (small angles). By contrast, the middle-angle

region will have similar contributions from both the ground and top of the canopy,

resulting in a larger volumetric contribution, and hence decorrelation. In terms

of polarimetric channels, the HH channel shows the largest coherence values over

most of the angular range. This is due to the strong backscattering from the

soil, and relatively minor contributions from the wheat canopy itself. The VV


0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

10 15 20 25 30 35 40 45 50 55

VVHHVH

Effec

tive

Hei

ght

[m]

Incidence Angle [deg]

RADWHEAT Delivery 4 (19990614), C band, 0.01 m Baseline

Figure 4.10: Estimated height for the images shown in Fig. 4.7 using a 0.01 m hori-zontal baseline.

channel shows lower coherence over most of the angular range because it has a

combination of strong backscattering from the ground level as well as from the top

of the canopy. At larger angles of incidence, the ground level contribution vanishes,

and the coherence rises as only the top of the canopy is present. The cross-polar

channel shows a similar behaviour to the HH channel, as both channels seem to

have relatively small returns from the top of the canopy.

In general terms, the effective height (Fig. 4.10) increases with increasing angle

of incidence, as the backscattering from the top of the canopy increases and that

from the ground decreases. This is more evident in the VV channel, which shows

a higher effective height due to the strong contribution from the top of the canopy.

The HH channel shows lower values than the VV channel, as this channel shows

an important soil return over the whole range of angles of incidence. The cross-

polar effective height is larger than either of the co-polar effective heights for

smaller angles of incidence because the resolution cell does not have such a strong

contribution from the soil. In this region, the returns from the VH channel are

4.6. Conclusions 70

of similar value throughout the canopy. As the angle of incidence increases, the

signal has an increasing contribution from the flag leaves level, which increases the

effective height slightly.

4.6 Conclusions

In this Chapter, the relevant aspects of the GB-SAR indoor component related

to interferometric SAR processing have been discussed. It has been shown that

the geometry of the GB-SAR indoor component is significantly different to the

geometry of air- and space-borne SAR systems, and that the processing approaches

used for these cannot be used directly in GB-SAR data, but need to be adapted.

Three issues which need treating carefully are:

1. The large change in angle of incidence across the GB-SAR swath.

2. The change in angle of incidence within a resolution cell.

3. The effect of the change of angle of incidence across the swath on wavenumber

shift filtering.

The large change in angle of incidence over the swath results in a limitation on how

the GB-SAR data have to be analysed, as different areas in the image will have

different backscattering behaviour. In terms of processing, the wavenumber shift

filter needs to take the angular variation into account: the discarded bandwidth is

angle of incidence dependent. It is suggested that the smallest angle of incidence of

any significance should be used to calculate the discarded bandwidth, as this will

automatically cater for areas of larger angles of incidence (at the cost of discarding

useful correlated signal in these latter areas).

The change of angle of incidence within a resolution cell presents a challenge

particular to interpreting the GB-SAR data, because in conventional SAR systems

the angle of incidence is constant within the resolution cell. The implication is that

a resolution cell might contain contributions from two completely different parts

of the scene, each imaged with different angles of incidence. For example, for a

wheat canopy, the soil contributions would be imaged at small angles of incidence,

4.6. Conclusions 71

and in the same cell, there would be a contribution from the top of the canopy

imaged with a large angle of incidence.

The change of angle of incidence also impacts on the system model of the GB-

SAR instrument, as the expressions for the coherence discussed in the literature

need to be adapted. This results in a new iterative algorithm to estimate the

effective height from the interferometric phase, and in a procedure to analyse the

data from three-dimensional backscattering reconstructions.

The iterative height retrieval algorithm has been tested using system simulations,

and shown to work as expected. The data analysis was based on data from the

RADWHEAT indoor measurement campaign and resulted in a prediction of the

interferometric behaviour of the imaged wheat canopy at C band. These results

will be compared with true InSAR processing in Chapter 7.

Chapter 5Interferometric and Polarimetric Analysis

of Layered Targets

5.1 Introduction

Layered targets are often used to model natural scenes in remote sensing.

As an example, a forest may be modelled using 3 layers: a crown layer, a

trunk layer and a soil layer. Similarly, a crop may be modelled by a canopy

layer, a stem layer and a soil layer. Each of these layers are located at different

heights, have different depths and the scatterers making up each of the layers often

belong to a few scatterer types. The interferometric SAR signal is sensitive to the

vertical distribution of the scatterers, whereas the polarimetric signal is sensitive

to the polarimetric properties of the scatterers. Combining interferometry and

polarimetry potentially provides a powerful way of analysing layered targets.

The Chapter opens with a discussion of the coherence properties of layered tar-

gets. A theoretical study of coherence optimisation procedures applied to layered

targets where all the scatterers belonging to a layer are characterised by the same

scattering matrix is then presented. The behaviour of these targets is investigated

in terms of theory and of numerical simulations. The effect of estimation inac-

curacies in coherence optimisation is then reviewed, followed by a study of more

realistic scenarios where the scattering matrices of the scatterers in a layer either

fluctuate, or belong to different types. Finally, some conclusions are presented.

72

5.2. Coherence analysis of a layered target 73

5.2 Coherence analysis of a layered target

In Section 3.3.3, the coherence of a target was related to the vertical spread of

the scatterers making up the target by a Fourier Transform relationship, ignoring

temporal, baseline and thermal decorrelation.

If the vertical distribution of scatterers is g(z), the coherence is given by its

normalised Fourier Transform:

ρ =

∫g(z′) exp(−j2πfz′)dz′

∫g(z′)dz′

, (5.1)

where f is a function of the imaging geometry, sensor wavelength and baseline:

f =kB⊥

πR sin θ, (5.2)

where k is the wavenumber, R is the range, B⊥ is the component of the baseline

perpendicular to the look direction and θ is the angle of incidence (see Section

3.3.3.3 for more details). For the GB-SAR system operating at X band with a

baseline of 0.06 m, the value of f is close to unity. Fig. 5.1 shows the vertical

scattering distribution for a target made up of homogeneous scattering layers. For

an M -layer target, g(z) is written as

g(z) =M∑

i=1

κigi(z − zi

∆i

), (5.3)

where gi(z), ∆i and zi are respectively the distribution of scatterers in layer i, the

depth of the layer and the height of the centre of the layer. κi accounts for different

backscattering intensities for each layer. The Fourier Transform of a single layer j

is given by

FT{κjgj(z − zj

∆j)} = κj∆jGj(2πf∆j) exp(−j2πfzj), (5.4)

where the Fourier Transform of gi(z) is defined as Gi(2πf). The magnitude of the


κ1

∆1

z1

κ2

∆2

∆3

κ3

z2

z3

g(z)

z

Figure 5.1: The vertical scattering distribution of a three layered target. The scatterersthat make up each layer are homogeneously distributed within the layervolume.


coherence for layer j taken individually is given by

|ρj| =

∣∣∣FT{κjgj( z−zj

∆j)}

∣∣∣

κj

∫gj(

z′ − zj∆j

)dz′= |Gj(2πf∆j)| , (5.5)

where it has been assumed that∫gi(z

′)dz′ = 1. The coherence for the ensemble

target is given by

ρ =

∑

i

Gi(2πf∆i) exp(−j2πfzi)κi∆i

∑

i

κi∆i

. (5.6)

An important results is that the magnitude of the coherence of the layer with the

highest coherence will be larger than that of the ensemble of layers. To see this,

Eqns. 5.5 and 5.6 can be used with the triangle inequality to write:

∣∣∣∣∣∑

i

Gi(2πf∆i) exp(−j2πfzi)κi∆i

∣∣∣∣∣ ≤∑

i

|Gi(2πf∆i)κi∆i| . (5.7)

If layer M has the largest coherence, then using Eq. 5.5

∑

i

|Gi(2πf∆i)κi∆i| ≤ |GM(2πf∆M)|∑

i

κi∆i. (5.8)

Combining Eqns. 5.8 and 5.7,

|∑iGi(2πf∆i) exp(−j2πfzi)κi∆i|∑

i

κi∆i

≤ |GM(2πf∆M)|

⇒ |ρ| ≤ |ρM |. (5.9)

The inequality expressed in Eq. 5.9 is independent of the backscattering inten-

sity of each of the layers and of the distribution of the scatterers within the layers,

gi(z). The only assumption is that one layer has a larger individual coherence than

the other layers. In fact, similar expressions to Eq. 5.9 can be written for other

5.3. Polarimetric coherence optimisation applied to layered targets 76

layers j where the condition expressed by Eq. 5.8 holds:

∑

i

|Gi(2πf∆i)κi∆i| ≤ |Gj(2πf∆j)|∑

i

κi∆i

⇒ |ρ| ≤ |ρj| (5.10)

The implication from both Eqns. 5.9 and 5.10 is that coherence maxima will be

associated with single layers, and not combinations of layers.

5.3 Polarimetric coherence optimisation applied to

layered targets

The previous Section demonstrated that in a layered target, the coherence maxima

were associated with single layers. The combination of polarimetry and interfer-

ometry allows the expression of the coherence as a function of polarisation. If the

polarisation properties of the layers are such that layers can be separated using

their polarimetric properties, then the coherence maxima can be associated with

individual layers. The coherence maxima could be estimated either by an exhaus-

tive search, or by using a coherence optimisation algorithm such as those described

in Section 3.4. This Section studies the use of coherence optimisation techniques

to retrieve interferometric information (height) about individual layers in layered

targets.

The separation of layers based on their polarimetric properties depends on find-

ing polarisation states that maximise the returns from one layer while blocking

or minimising the returns from other layers. Here we first assume that layers are

composed of identical scatterers. The return from a scatterer for a given incident

polarisation can be calculated [Kostinski and Boerner, 1986] as the scalar product

of the scattering vector of the scatterer and the vector that defines the incident

polarisation; in order to separate layers, the polarisation state needs to be aligned

with the scattering vector of one of the layers while being orthogonal to the scat-

tering vectors of other layers. In this Section, attention is given to targets made

up of orthogonal and linearly independent scattering vectors. The reason behind

studying orthogonal layered targets stems from an experiment carried out using


GB-SAR and which will be presented in Chapter 6.

5.3.1 Orthogonal scattering vectors

Consider a three-layered target, where each layer is made up of a homogeneous

spatial distribution of scatterers characterised by the same orthogonal scattering

vectors� �

S 1...3. In a suitably chosen basis, the scattering vectors for each layer can

be written as

� �

S 1 = [1, 0, 0]T

� �

S 2 = [0, 1, 0]T

� �

S 3 = [0, 0, 1]T . (5.11)

The recorded data vectors at the receiving antennas are

� �

v =

∑N1

p=1 exp(−j2krp)∑N2

p=1 exp(−j2krp)∑N3

p=1 exp(−j2krp)

and

� �

v′ =

∑N1

p=1 exp(−j2kr′p)∑N2

p=1 exp(−j2kr′p)∑N3

p=1 exp(−j2kr′p)

, (5.12)

where the primed terms refer to the second antenna, rp(r′p) is the distance from

scatterer p to the first (second) antenna, and there are Ni scatterers in layer i. The

covariance and cross-covariance matrices can be written as (see Section 2.8.2)

T =P =

N1 0 0

0 N2 0

0 0 N3

(5.13)

Q =

N1e

jφ1 0 0

0 N2ejφ2 0

0 0 N3ejφ3

, (5.14)

where φ1...3 are the interferometric phases associated with each of the layers (see

Section 3.3.2). The unconstrained optimisation procedure combines the covariance


and cross-covariance matrices into matrices

M1 = T −1Q (5.15)

M2 = T −1Q∗T . (5.16)

Both M1 and M2 turn out to be identity matrices. Coherence optimisation re-

sults in three pairs of polarisation states defined by the eigenvectors of M1M2

and M2M1. In this case, these are undefined, as both matrices are identity ma-

trices, with a degenerate eigenvalue spectrum {1, 1, 1}. In these circumstances,

unconstrained coherence optimisation does not yield any useful information. The

numerical implementation will result in an ill-conditioned eigenvector estimation

problem [Quarteroni et al., 2000, Anderson et al., 1999], as stability in the es-

timation of the eigenvectors requires separation in the eigenvalues. The small

fluctuations arising from inaccuracies in parameter estimation will not affect the

eigenvalues substantially, but will have a major impact on the stability of the

eigenvectors. A further effect of these fluctuations is to make M1M2 different to

M2M1, and while the eigenvalues are guaranteed to be identical, the retrieved

eigenvectors will differ (this assertion is proved in Section 5.5). This means that

the retrieved polarisation states from coherence optimisation would be different for

each of the images in the interferometric pair. For a target that does not undergo

any changes between passes or that is imaged using a single pass, the polarisation

states should be the same for both images.

The scenario above causes the optimisation procedure to fail as a means to

separate layers since all layers are characterised by the same coherence. In practice,

differences in coherence between layers will arise due to

• Different scatterer distributions in each layer, leading to different volumetric

decorrelation effects,

• Temporal decorrelation effects selectively affecting different layers,

• Different signal levels in different layers, resulting in different thermal decor-

relation contributions.

Such different coherence values result in different eigenvalues and thus allow the


optimisation algorithm to accurately separate the layers in the polarisation space.

Numerically, the differences in coherence result in differences in the eigenvalues,

which stabilise the estimation of the eigenvectors. Under the assumption of no

changes in the scene between acquisitions, the differences in coherence are due to

volume effects and do not affect the covariance matrices for the individual images:

the coherence changes only affect the cross-covariance matrix Q. A way to model

the changes in coherence on a per layer basis is to multiply the interferometric

phase by the magnitude of the coherence associated with that layer (as done in

[Jakowatz et al., 1996]). If the coherences associated with the three layers are α, β

and γ for (respectively) layers one, two and three, then the cross covariance matrix

can be written as

Q =

αN1e

jφ1 0 0

0 βN2ejφ2 0

0 0 γN3ejφ3

. (5.17)

The matrices where the eigenvectors are extracted, M1M2 and M2M1, using the

covariance matrices shown in Eq. 5.13 and the cross-covariance matrix in Eq. 5.17

are still diagonal, but not identity matrices. The diagonal elements are α2, β2

and γ2. The eigenvectors of these matrices are identical to� �

S 1...3 (Eq. 5.11), with

eigenvalues identical to the coherence of each layer squared. The stability of the

retrieved eigenvectors will improve as the difference between eigenvalues increases.

The use of constrained optimisation avoids one of these problems, as it forces

the use of the same polarisation state for each of the two passes. The constrained

optimisation procedure described in Section 3.4 results in the extraction of eigen-

vectors from a matrix M defined as [T + P]−1[Q + Q∗T ]. From Eqns. 5.13 and

5.14,

M =

cos φ1 0 0

0 cosφ2 0

0 0 cosφ3

, (5.18)


for identical coherence in all the layers, and using Eq. 5.17

M =

α cosφ1 0 0

0 β cos φ2 0

0 0 γ cosφ3

(5.19)

for the case where each layer has a different coherence associated with it. In either

case, the constrained algorithm will retrieve the polarisation states� �

S 1...3 in Eq.

5.11. The stability problem is less serious than in the unconstrained case, as the

eigenvalues are related to both the coherence and the interferometric phase, hence

for layers located at different heights, they will be different, even if the coherence

is the same or very similar.

5.3.2 Linearly independent scattering vectors

Another interesting scenario is a layered target where each of the layers is populated

by scatterers with scattering vectors which are linearly independent from layer to

layer for three layers. The scattering vectors associated with each layer can be

expressed in a suitable basis as

� �

S 1 = [1, 0, 0]T� �

S 2 = [a, b, 0]T� �

S 3 = [c, d, e]T . (5.20)

The polarisation states that retrieve the individual layers are given by a set of

three vectors aligned with one of the layers, but orthogonal to the other two, such

as

� �

u 1 =

[1, −a

b,

−bc + ad

be

]T

� �

u 2 =

[0,

1

b,

−dbe

]T

� �

u 3 =

[0, 0,

1

e

]T, (5.21)


where� �

u i retrieves the layer made up of scatterers characterised by� �

S i. The tech-

nique used in the previous Section to estimate the optimal polarisation states is

of little benefit in this case, as the matrix equations and subsequent eigenvector

problems become complicated.

We wish to prove that the unconstrained coherence optimisation algorithm will

indeed recover single layers using the polarisation states defined in Eq. 5.21. To

do this, it is useful to go back to the definition of the Lagrangian function where

the optimisation procedure stems from [Cloude and Papathanassiou, 1998]. Let

the Lagrangian function L to be maximised be defined as

L =� �

w∗T1 Q � �

w2 + λ1

( � �

w∗T1 T � �

w1 + C1

)+ λ2

( � �

w∗T2 P � �

w2 + C2

), (5.22)

where λ1,2 are Lagrange multipliers and C1,2 are constants. The optimisation of L

results in a pair of equations:

∂L

∂� �

w∗T1

= 0 → Q � �

w2 = −λ1T� �

w1 (5.23)

∂L∗

∂� �

w∗T2

= 0 → Q∗T � �

w1 = −λ∗2P� �

w2. (5.24)

In the absence of changes in the target between acquisitions, it can be assumed

that the same polarisation states will be recovered from the two images, so that� �

w1 =� �

w2 =� �

w. The covariance and cross covariance matrices are defined as

T = P =

3∑

i=1

Ni

⟨� �

S i

� �

S ∗Ti

⟩(5.25)

Q =3∑

i=1

ρi exp(jφi)Ni

⟨� �

S i

� �

S ∗Ti

⟩(5.26)

where each layer has been characterised as in the previous Section by a coherence

ρi, Ni scatterers and the interferometric phase for that layer is given by φi.

Combining Eqns. 5.25 and 5.26 with Eqns. 5.23 and 5.24, and multiplying both


sides of Eqns. 5.23 and 5.24 by� �

w∗T ,

� �

w∗T

[3∑

i=1

ρiejφiNi

⟨� �

S i

� �

S ∗Ti

⟩]� �

w

= −λ1

� �

w∗T

[3∑

i=1

Ni

⟨� �

S i

� �

S ∗Ti

⟩]� �

w (5.27)

� �

w∗T

[3∑

i=1

ρie−jφiNi

⟨� �

S i

� �

S ∗Ti

⟩∗T]

� �

w

= −λ∗2� �

w∗T

[3∑

i=1

Ni

⟨� �

S i

� �

S ∗Ti

⟩]� �

w.(5.28)

To check whether the polarisations outlined in Eq. 5.21 are indeed solutions, set

the solution polarisations,� �

w1...3 to be� �

u 1...3 in Eqns. 5.27 and 5.28. As pointed

out above, each of these polarisation states will be orthogonal to the scattering

vectors of two of the layers, and aligned with the third one, so that� �

u ∗Ti

� �

S j = 0 if

i 6= j. The substitution results in the following two equations:

� �

u ∗Ti ρie

jφiNi

⟨� �

S i

� �

S ∗Ti

⟩� �

u i = −λ1

� �

u ∗Ti Ni

⟨� �

S i

� �

S ∗Ti

⟩� �

u i (5.29)

� �

u ∗Ti ρie

−jφiN1

⟨� �

S i

� �

S ∗Ti

⟩∗T� �

u i = −λ∗2� �

u ∗Ti Ni

⟨� �

S i

� �

S ∗Ti

⟩� �

u i, (5.30)

where due to the orthogonality condition put on the test solutions most of the

terms in Eqns. 5.29 and 5.30 vanish. By inspection, the two resulting equations

are solved by setting

λ1 = ρi exp(jφi) (5.31)

λ2 = ρi exp(jφi). (5.32)

Thus, it has been proved that� �

u 1...3 are indeed solutions to the optimisation prob-

lem. The only assumptions made are that scatterers in each layer are homoge-

neously distributed and there are no changes in the target between acquisitions,

so that the polarisation states for the two acquisitions should be identical.

Note that the same comments made in the Section 5.3.1 on stability of the

retrieved polarisation states for targets with identical coherences for each layer

5.4. Numerical Simulations 83

apply equally to the case considered in this Section.

The proof can be extended to constrained optimisation by noting that the con-

strained optimisation problem is defined as

Q � �

w = −λ1T� �

w (5.33)

Q∗T � �

w = −λ∗2P� �

w, (5.34)

where Eqns. 5.24 and 5.23 have been rewritten constraining the optimal polari-

sation states to be identical for the two acquisitions. Substituting the covariance

and cross-covariance matrices from Eqns. 5.25-5.26 and multiplying both pairs of

equations by� �

w∗T leads to Eqns. 5.29-5.30, demonstrating that the same polarisa-

tion states� �

u 1...3 that were a solution to the Eqns. 5.29-5.30 are a solution for the

constrained optimisation case.

This Section has demonstrated that:

• both unconstrained and constrained coherence optimisation algorithms

should retrieve individual layers when applied to targets consisting of 3 in-

dependent layers,

• problems will arise in unconstrained optimisation if the coherence values of

the layers are similar.

5.4 Numerical Simulations

This Section illustrates the theoretical findings of the previous Section using nu-

merical simulations. In the case of orthogonal scattering mechanisms, the problem

has been reduced to a two-dimensional rather than three-dimensional problem, as

a similar scenario has been imaged in the GB-SAR chamber.

The simulations start from a homogeneous distribution of scatterers within each

of the layers. Each of the scatterers within a layer is characterised by a scattering

matrix, which is identical for all scatterers in the layer (more complex scenarios are

deferred until Section 5.6). The imaging geometry emulates the GB-SAR system

as it was used for the experimental validation shown in Chapter 6; the details

common to all the simulations presented in this Chapter are shown in Table 5.1.


There were ten scatterers per resolution cell, as this number was sufficient to give

rise to an approximately fully-developed speckle pattern (i.e., the recorded data

were approximately characterised by circular Gaussian distributions).

Parameter Value

Angle of Incidence 45◦

Baseline 0.06 m

Range to Target 2.54√

2 mNo. Scatterers in Res. Cell 10

Frequency 10.0 GHzResolution Cell Size 0.08 × 0.08m

Table 5.1: Common parameters for all the simulations presented in this Chapter. Thegeometry has been chosen to emulate that of the GB-SAR indoor system.

The first step in the simulations after defining the position of all the scatterers

is to calculate the recorded scattering vectors at the two ends of the interfero-

metric baseline, as described by Eq. 5.12. Several scatterer configurations are

generated to simulate independent samples, and the covariance matrices for each

of the two antennas and the cross-covariance matrix are estimated. These matrices

are combined according to the theory given in Chapter 3 to estimate the polari-

sation states that maximise the coherence from the relevant eigenvector problem

(depending on whether the optimisation is constrained or unconstrained). This

process is repeated a thousand times, and the eigenvectors, retrieved height and

coherence are recorded.

One of the aims of these simulations is to examine the implications for a system

like GB-SAR. As discussed in Chapter 4, the GB-SAR indoor system is charac-

terised by a large change in angle of incidence over a relatively small target region.

This means that there will be few independent samples to estimate polarimet-

ric or interferometric parameters. In the simulations, no more than 40 looks are

considered, and a minimum of 8 is initially considered.


5.4.1 Orthogonal scattering vectors

A two layered target with identical layer geometries (0.08 × 0.08 × 0.01 m range-

cross range-depth) were populated with scatterers characterised by scattering vec-

tors [0, 1]T and [1, 0]T in an arbitrary basis. 8 independent samples (looks) were

combined to estimate the covariance matrices. The results are shown in Fig. 5.2

for unconstrained optimisation. The retrieved height using the polarisation states

[0, 1]T and [1, 0]T which retrieve the individual layers are also shown for compar-

ison. The height distributions from unconstrained optimisation are distributed

between the heights of the two layers. These height distributions arise from the

identical geometries for the two layers, which result in identical coherence values.

0

0.2

0.4

0.6

0.8

1

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

1st Evector2nd Evector

Fre

quen

cy

Retrieved Height [m]

[1, 0]T

[1, 0]T

Figure 5.2: Retrieved height distributions for the numerical simulations of a two-layered target populated by orthogonal scattering mechanisms. Forthe [1, 0]T layer, the mean retrieved height was -0.098 and for the[0, 1]T interferogram, 0.11m. Both distributions had a standard deviationof 0.008 m. The first solution from unconstrained optimisation (red line)had a mean value of -0.039 (SD: 0.19); the second solution (green line) hada mean value of 0.012 (SD: 0.19).

An improvement in the localisation of the two layers is obtained if different


coherence values are associated with each layer. This was achieved by giving the

layers different depths. The results for 8 looks are shown in Fig. 5.3(a) (the

descriptive statistics are in Table 5.2), where the distributions for each of the two

solutions are unimodal and centred roughly around the layer heights obtained using

the polarisation states defined by [0, 1]T and [1, 0]T . In comparison with the latter,

the results from coherence optimisation are broadly distributed. The reason for

this is the poor estimation of the covariance matrices due to the small number of

looks employed. In Fig. 5.3(b), the simulations are repeated, but 40 looks are

used. The retrieved height distributions from optimisation are now narrower than

in the 8 look case, as inaccuracies in the estimation are reduced. The distributions

are still broader than the ones obtained using the polarisation states expected to

retrieve the individual layers.

8 looks 40 looks

Mean SD Mean SD[0, 1]T - interferogram 0.1 0.004 0.1 0.002

Solution 1 0.056 0.06 0.09 0.02[1, 0]T - interferogram -0.098 0.006 -0.098 0.002

Solution 2 -0.04 0.06 -0.075 0.03

Table 5.2: Descriptive statistics for the height distributions shown in Fig. 5.3. Unitsare metres.


0

0.2

0.4

0.6

0.8

1

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25



Fre

quen

cy

[0, 1]T[1, 0]T

(a) 8 looks

0

0.2

0.4

0.6

0.8

1

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25



Fre

quen

cy

[1, 0]T[0, 1]T

(b) 40 looks

Figure 5.3: Unconstrained optimisation. Simulation results for a layered target madeup of orthogonal scattering mechanisms. Each layer has been given a differ-ent vertical spread to simulate different values of coherence for each layer.All other parameters are as in Fig. 5.2. Descriptive statistics for thesesimulations in Table 5.2.


The simulations for unconstrained optimisation presented in this Section suggest

that this optimisation procedure will struggle to separate layers when the geometry

of these layers is identical (or in other situations where the coherence of the two

layers is very similar). The situation is worsened if only a limited number of

looks are available for the estimation of the matrices governing the second order

statistics.

Constrained optimisation is not affected by layers having identical depths. The

results for constrained optimisation and identical layer geometries (coherence) and

8 looks are shown in Fig. 5.4(a) The results from constrained optimisation and

using the [0, 1]T and [1, 0]T polarisation states are nearly identical. For 40 looks

(Fig. 5.4(b)), the optimisation height distributions overlap those obtained using

the expected polarisation states. These results suggest that the constrained optimi-

sation algorithm will perform very well in circumstances where the unconstrained

algorithm completely fails to separate the layers. The reasons for this improved

behaviour will be covered in greater detail in Section 5.5.

5.4.2 Linearly independent scattering vectors

For these simulations, a target made up of three layers each populated with linearly

independent scattering vectors. In order to avoid the limitations of unconstrained

optimisation when layers are characterised by identical coherences, the layers have

been given three different depths. The scattering vectors that have been used for

these simulations were

� �

S 1 =

1

0

0

� �

S 2 =1√2

1

−1

0

� �

S 3 =1√2

1

0

1

. (5.35)


0

0.2

0.4

0.6

0.8

1

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

Fre

quen

cy



[1, 0]T [0, 1]T

(a) 8 looks

0

0.2

0.4

0.6

0.8

1

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

Fre

quen

cy



[1, 0]T[0, 1]T

(b) 40 looks

Figure 5.4: Constrained optimisation. Simulation results for a layered target made upof orthogonal scattering mechanisms. Each layer has been given a differentvertical spread to simulate different values of coherence for each layer. Allother parameters are as for the simulations shown in Fig. 5.2.


Each layer will be retrieved using the polarisation states characterised by (cf. Eq.

5.21)

� �

u 1 =1√3

1

1

−1

� �

u 2 =1√2

0

−1

0

� �

u 3 =

0

0

1

. (5.36)

The results from unconstrained optimisation and 8 looks are presented in Fig.

5.5 (see also Table 5.4). The geometry of the three layers is outlined in Table

5.3. The height distributions from coherence optimisation are clearly unimodal,

with modes centred about the vertical position of each of the layers. However, the

distributions are broad compared to the distributions using the polarisation states� �

u 1...3. This is due to the poor estimation of the covariance matrices due to the

small number of looks. Increasing the number of looks to 40 results in a much

better layer separation, as shown by the results in Table 5.4.

Layer Scattering Type Retrieved with Layer centre [m] Layer depth[m]

1 ��

S 1 ��u 1 -0.4 0.001

2 ��

S 2 ��u 2 0 0.1

3 ��

S 1 ��u 2 0.3 0.4

Table 5.3: Layer structure for three layers populated with linearly independent scat-terer types.

The results from constrained optimisation for 8 looks are presented in Fig. 5.6,

with descriptive statistics shown in Table 5.4. As in the orthogonal case, the results

from constrained optimisation are nearly identical to using� �

u 1...3. The distributions

are significantly narrower than with unconstrained optimisation. Results for 40

looks are reported in Table 5.4; as expected, the increase in number of looks


0

0.2

0.4

0.6

0.8

1

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

� �

u 1

� �

u 2

� �

u 3

Fre

quen

cy


1st Evector

3rd Evector

2nd Evector

Figure 5.5: Unconstrained optimisation. Three-layered target made up of linearly in-dependent scattering vectors. Results shown for 8 looks.

decreases the standard deviation of the retrieved height distribution.

The estimated coherence distributions for the three cases outlined previously

using 8 looks are shown in Fig. 5.7. From these distributions, it is clear that

constrained optimisation does not come up with the highest possible coherence.

This is as expected, as this technique is suboptimal. By contrast, the coherence

distribution of unconstrained optimisation is nearly identical to that obtained using

the polarisations expected to retrieve the individual layers.

To summarize the results of this Section, it was found that

• constrained optimisation showed a clear separation of layers under all the

considered circumstances,

• Unconstrained optimisation required a larger number of looks to come up

with results comparable to those of constrained optimisation,

• The poor results obtained by unconstrained optimisation are due to estima-


0

0.2

0.4

0.6

0.8

1

−0.6 −0.4 −0.2 0 0.2 0.4 0.6Retrieved Height [m]

1st Evector

3rd Evector

2nd Evector

� �

u 1� �

u 2� �

u 3

Fre

quen

cy

Figure 5.6: Constrained optimisation. Three-layered target made up of linearly inde-pendent scattering vectors. Results shown for 8 looks.

5.5. Effect of finite number of independent samples in coherence

optimisation 93

Unconstrained Constrained

Sol. 8 looks 40 looks 8 looks 40 looks

1 -0.236 (0.277) -0.382 (0.08) -395 (0.077) -0.38 (0.079)

2 -0.034 (0.205) -0.0043 (0.067) -0.0155 (0.082) -0.017 (0.081)

3 0.260 (0.132) 0.294 (0.019) 0.307 (0.033) 0.298 (0.009)

Expected Pols.

Pol. 8 looks 40 looks� �

u 1 -0.392 (0.003) -0.392 (0.001)� �

u 2 0 (0.006) 0 (0.004)� �

u 3 0.296 (0.013) 0.296 (0.007)

Table 5.4: Descriptive statistics illustrating the results of different coherence optimisa-tion techniques on a 3-layered target (see Table 5.3). The results are givenin terms of the mean retrieved height; the standard deviation is given inbrackets for each case.

tion errors arising in the calculation of the covariance and cross-covariance

matrices.

While the data used are identical for both optimisation procedures, the constrained

technique shows a greater resistance to fluctuations. In the next Section, the

reasons for this behaviour are explored further.

5.5 Effect of finite number of independent samples

in coherence optimisation

As it was demonstrated in the previous Section, the use of a small number of

independent looks resulted in poor results for unconstrained optimisation. By

contrast, constrained optimisation resulted in acceptable results even for a small

number of samples. This Section examines the behaviour of both algorithms and

the effect of the number of looks on the optimisation procedure. While it is well-

established that using a large number of looks is beneficial, in the case of the GB-

SAR indoor component this is not an option, as the small size of the target region

in terms of resolution cells limits the number of available independent samples.


optimisation 94

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.5

1

1st vector2nd vector3rd vector

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Coherence

Fre

quen

cy

Constrained Optimisation


0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.5

1

Coherence

Fre

quen

cyF

requ

ency

Expected Projection Vectors


Unconstrained Optimisation

Figure 5.7: Histograms of the calculated coherence from different optimisation tech-niques and coherence using the projection vectors that retrieve individuallayers (8 looks).


optimisation 95

Covariance and cross-covariance matrices need to be estimated from the data.

The elements of the sample matrices arise from the maximum likelihood estimator

of the conjugate product of two polarimetric channels (covariance matrices) or two

polarimetric channels imaged with slightly different geometries (cross-covariance

matrix) over a finite number of looks. The accuracy with which these values can

be estimated is limited by the Cramer-Rao bound, which in the case of unbiased

estimators guarantees the lowest possible variance of the estimate. The asymptotic

variance for the estimated amplitude of the complex product Z of two channels 1

and 2, (L→ ∞) is given by [Tough et al., 1995]

var(Z(L)) =

√σ1σ2

L

(1 + |ρ|2

2

), (5.37)

where σi is the power associated with channel i and ρ is the magnitude of the

correlation coefficient between the two channels. The effect of these inaccuracies

in the estimation of the covariance matrix elements can be seen as an additive

process: the sample covariance matrices for each image, T and P are equivalent

to the sum of true covariance matrix T plus some perturbation matrices:

T = T + E (5.38)

P = T + E ′, (5.39)

where it has been assumed that there are no changes in the target between acqui-

sitions. In short, the estimated covariance matrices for each image are different,

and these differences will tend to be larger the smaller the number of looks. The

coherence optimisation algorithms are affected in different ways by this effect. In

the case of unconstrained optimisation, the differences in the covariance matrices

force the estimated polarisation states for each image to be different. The con-

strained optimisation overcomes the problem by combining both sample covariance

matrices, and arriving at a better estimate.

In the preceding Sections, the analysis of unconstrained optimisation usually

assumed that the polarisation states for the two images in the interferometric pair

should be identical. To prove this assertion and to point out situations where this

assumption does not hold, start by defining the two coupled eigenvector problems


optimisation 96

as

T −1QP−1Q∗T � �

w i = νi� �

wi (5.40)

P−1Q∗TT −1Q� �

w′i = νi

� �

w′i, (5.41)

where νi are the eigenvalues of the combination of matrices on the left hand side,

and� �

w i and� �

w′i are the optimal polarisation states associated with the first and

second images. The eigenvectors will be identical when the matrix combinations

on the left hand side of Eqns. 5.40-5.41 are identical:

T −1QP−1Q∗T = P−1Q∗TT −1Q. (5.42)

While these two matrices are not hermitian positive semidefinite, they can still be

written in terms of an eigenvector-eigenvalue expansion [Papathanassiou, 1999]:

T −1QP−1Q∗T =3∑

i=1

νiT 1/2[

� �

w i� �

w∗Ti

]T −1/2 (5.43)

P−1Q∗TT −1Q =

3∑

i=1

νiP1/2[ � �

w′i

� �

w′∗Ti

]P−1/2. (5.44)

The substitution of Eqns. 5.43-5.44 into Eq. 5.42, forcing the eigenvectors to be

identical (and hence dropping the primed vector), yields

3∑

i=1

νiT 1/2[

� �

w i� �

w∗Ti

]T −1/2 =

3∑

i=1

νiP1/2[

� �

wi� �

w∗Ti

]P−1/2, (5.45)

which is satisfied if T and P are identical. In other words, if there are no changes

in the target between acquisitions, the retrieved polarisation states are identical

for both images. If the covariance matrices are different, Eq. 5.45 will not hold

as written, but will hold if the eigenvectors are not forced to be identical. In

practice, the estimated covariance matrices will be different, and so the retrieved

polarisation states will be different for both images in the pair.

Constrained optimisation has so far proved to be a robust algorithm for scenarios

similar to the GB-SAR indoor component. The estimation of the polarisation

5.6. Variations in the scattering vectors within layers 97

states that maximise the coherence is given by

[T + P]−1[Q + Q∗T ]� �

w = λ� �

w. (5.46)

Assuming that the differences between the covariance matrices are small, then the

addition of these two matrices can be pictured as the use of an average covariance

matrix, generated from two sets of independent samples.

The simulations shown in Section 5.4.2 for three layers populated with scatterers

characterised by linearly independent scattering vectors have been run for different

numbers of looks; the results are shown in Figs. 5.8 and 5.9. From the plots,

unconstrained optimisation needs a larger number of looks to arrive at an accurate

layer height estimate. By contrast, constrained optimisation shows accurate height

estimates irrespective of the number of looks: with only ten looks, the retrieved

height shows an accurate mean height with small standard deviation.

5.6 Variations in the scattering vectors within layers

Up to this point, an idealised scenario of all scatterers belonging to a layer having

the same polarimetric properties has been considered. In practice, this is rarely the

case. On the one hand, the scatterers in a layer might all share similar polarimet-

ric characteristics (e.g. dihedral-like scattering mechanisms), but the individual

scattering matrices will fluctuate due to differences in orientation, scatterer size,

etc. On the other hand, it is common for scatterers of several types to be present

in the same layer. Usually, one type will be more common than the other (or oth-

ers), or its contribution to the backscattered signal will be larger than the other

types. This Section aims to study coherence optimisation in the presence of these

deviations from the ideal cases studied up to now, using a theoretical discussion

illustrated by numerical simulations.

5.6.1 Random variations of scattering vectors

In this Section, we assume all scatterers in an individual layer to be characterised

by a single scattering matrix plus some random fluctuations that change from


−0.4

−0.2

0

0.2

0.4

5 10 15 20 25 30 35 40 45 50

Mea

n R

etrie

ved

Hei

ght [

m]

Number of Looks


Solution 1Solution 2Solution 3

(a) Mean

0

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20 25 30 35 40 45 50

Ret

rieve

d H

eigh

t Std

. Dev

. [m

]

Number of Looks



(b) Standard Deviation

Figure 5.8: Unconstrained optimisation. Evolution of the retrieved height with numberof looks. The simulation parameters are identical to the results shown inFig. 5.5.

scatterer to scatterer. In other words, the scattering matrix for scatterer p is

S(p) = Si + S(p)ε , (5.47)


−0.4

−0.2

0

0.2

0.4

5 10 15 20 25 30 35 40 45 50

Mea

n R

etrie

ved

Hei

ght [

m]

Number of Looks



(a) Mean

0

0.05

0.1

0.15

0.2

5 10 15 20 25 30 35 40 45 50

Ret

rieve

d H

eigh

t Std

. Dev

. [m

]

Number of Looks



(b) Standard Deviation

Figure 5.9: Constrained optimisation. Evolution of the retrieved height with numberof looks. The simulation parameters are identical to the results shown inFig. 5.6.

where Si is the ideal scattering matrix for layer i, and S (p)ε is the perturbation

contribution for this particular scatterer. The perturbations are assumed to be


small, so that ‖Si‖2 �∥∥∥S(p)

ε

∥∥∥2, where ‖·‖2 is the l2 norm of the matrix, and the

elements of the perturbation matrix are assumed to uniformly distributed. The

scattering vector for the scatterer characterised by Eq. 5.47 can be written as

� �

S (p) =� �

S i +� �

S (p)ε , (5.48)

The covariance matrix for an L-layer target is defined as

C =

L∑

i=1

[Ni

⟨� �

S i

� �

S ∗Ti

⟩+

⟨N∑

p=1

� �

S (p)ε

� �

S (p)∗Tε

⟩], (5.49)

under the assumptions outlined in Section 2.8, and assuming Ni scatterers per

layer. The first term in the summation is identical to the ideal case shown in

Eq. 5.25 (for L = 3), perturbed by a second term. Under the assumption of

a uniformly distributed perturbation of the scattering matrix, the perturbation

contribution (i.e., the second term in Eq. 5.49) will vanish in the mean. However,

the perturbations will have a role to play in the estimation of the sample covariance

matrix. In Section 5.5, the effect of a finite number of looks was found to add some

variation to the first term of Eq. 5.49. The estimation over a number of looks also

means that the second term (the random contribution) will not vanish, but will

act as a source of fluctuations, on top of the estimation variations in the ideal

case. Again, increasing the number of looks will decrease the contribution of this

second term, and therefore, the sample covariance matrix will approach the true

covariance matrix.

The effect of these fluctuations in coherence optimisation can be argued in ex-

actly the same terms that were used in the ideal case in Section 5.5, but adding

an extra source of fluctuations due to the variations in the scattering matrices.

5.6.1.1 Numerical Simulations

To illustrate the previous discussion, the numerical simulations presented in Sec-

tion 5.4.2 have been modified to account for random variations in the scattering

matrices. This was achieved by applying random rotations to the scattering vec-


tors:� �

P = RφRθRψ

� �

S , (5.50)

where� �

P is the perturbed scattering vector,� �

S is the unperturbed scattering vec-

tor, and Rφ, Rθ and Rψ are plane rotation matrices. For these simulations, the

scattering vectors were rotated by a uniformly distributed angle between ±5◦, in-

dependently in each direction. The rest of the simulation parameters were left as

in Section 5.4.2.

The results are shown in Figs. 5.10 (a-b) (unconstrained optimisation, 8 and

40 looks) and in Figs. 5.11 (a-b) (constrained optimisation, 8 and 40 looks).

The retrieved height distributions for unconstrained optimisation are similar to

those found in Section 5.4.2, each centred at the position where each of the layers

was positioned. However, distributions are broader than those reported in the

preceding Sections. The situation improves when increasing the number of looks,

as this tends to minimise the random fluctuations in the scattering matrices and

also increases the accuracy of the unperturbed contribution depicted in Eq. 5.49.

The results from constrained optimisation are in line with the results obtained

using the polarisation states that were expected to separate the layers, with nar-

rower distributions as the number of looks is increased.

The numerical simulations confirm the behaviour outlined in the previous Sec-

tion: the effect of scattering matrix perturbations results in a broadening of the

height distribution curves, either using the polarisation states that are expected

to separate the layers, or any of the optimisation techniques. The broadening of

the retrieved height curves can be particularly severe for unconstrained optimisa-

tion, while results from constrained optimisation are not particularly different to

the results obtained using the polarisation states that theoretically separate the

layers. The results improve for all distributions increasing the number of looks, as

the random contributions decrease.

5.6.2 Several scatterer types within individual layers

Another departure from the ideal model outlined in Section 5.3 is to assume several

types of scatterers present in individual layers. Further, consider that each layer

has a strong contribution from one type of scatterer, plus some minority contribu-


0

0.2

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0

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(b) 40 looks

Figure 5.10: Unconstrained optimisation. Simulation results for a layered target madeup of linearly independent scattering mechanisms (cf. Table 5.3 for layergeometry) with scattering matrix variations. Black curves represent theheight distributions obtained using the polarisations expected to separatethe layers, and the red, green and blue curves are respectively the retrievedheight with the first, second and third projection vectors estimated fromunconstrained optimisation.


0

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Fre

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(b) 40 looks

Figure 5.11: As Fig. 5.10, using constrained coherence optimisation (black, green andblue for first, second and third set of projection vectors).


0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Fre

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Layer 1Layer 2Layer 3

0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Fre

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0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Coherence

Fre

quen

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Figure 5.12: Coherence distributions for perturbed scattering matrices scenario and40 looks: expected projection vectors (top), unconstrained optimisation(middle) and constrained optimisation (bottom).




1 -0.10 (0.185)- -0.19 (0.33) -0.298 (0.195) -0.36 (0.10)

2 -0.057 (0.227) -0.053 (0.14) -0.11 (0.184) -0.03 (0.011)

3 0.231 (0.185) 0.295 (0.047) 0.30241 (0.046) 0.298 (0.007)

Expected Pols.


u 1 -0.392 (0.006) -0.395 (0.002)� �

u 2 0 (0.01) 0 (0.004)� �

u 3 0.296 (0.014) 0.296 (0.006)

Table 5.5: Descriptive statistics illustrating the results of different coherence optimi-sation techniques on a 3-layered target (see Table 5.3). Each scatterer ischaracterised by a perturbed scattering matrix, as outlined in the main text.The results are given in terms of the mean retrieved height; the standarddeviation is given in brackets for each case.

tions from other scatterer types. While random scatterering matrix fluctuations

can be reduced by increasing the number of looks because these fluctuations do

not have a structure, in the present case the minority scatterers have a well-defined

polarimetric structure, and cannot be minimised by similar means.

In the context of this Section, a dominant scatterer type is defined as the scat-

terer type that is responsible for the largest part of the radar signal, be it because

its numbers in a layer are large relative to other scatterer types, or because its

backscattering amplitude is substantially larger than that of the other scatterer

types. A minority scatterer type is any type of scatterer present in a layer that

only contributes a small fraction of the backscattered signal. In this Section, it will

be assumed that layers are populated by two types of scatterers, one dominating

the other.

The effect of the minority scatterers in interferometric processing is that while

a layer might be selected using a suitable polarisation state that minimises the

returns from the dominant scatterers in the other layers, this polarisation state

might not be able to block the contributions of the minority scatterers present

in one or more layers. This results in the combination of returns from several


layers rather than from a single layer. If the minority scatterer contribution in

these layers is small compared to the dominant scatterer contribution, then the

effects will be a drop in the coherence (as outlined in Section 5.2), and a bigger

uncertainty in the interferometric phase. The worse case scenario occurs when the

minority scatterers in one layer are characterised by the same scattering matrix as

the dominant scatterers in another layers, as this guarantees the minority scatterers

will have the greatest possible contribution to the backscattered signal.

Coherence optimisation will be affected by minority scatterer contributions, as

these will effectively change the nature of the covariance and cross covariance

matrices, and thus, the coherence as a function of polarisation. In situations

where estimation inaccuracies or large random variations of the scattering type

are present, the minority contribution could be buried under these effects, ap-

pearing as a source of fluctuations. In less adverse conditions, the optimisation

procedures will either try to optimise a function where, in comparison with the

ideal non-perturbed case, the maxima are lower but located in the same areas in

the polarisation space if the minority scatterer types are dominant in other layers.

In these untoward conditions, the unconstrained optimisation will fare worse than

the constrained optimisation, as the minority contributions will be an effective

source of fluctuations.

In scenarios where the estimation errors are small, the nature of the optimisation

problem is different, as the covariance and cross-covariance matrices take a different

shape. Assuming a small minority contribution, the coherence can be seen as a

smeared version of the ideal case. In these circumstances, it is expected that the

optimisation procedures will retrieve polarisation states that are similar to those

retrieved in the ideal case.

5.6.2.1 Numerical simulations

The simulations presented in Section 5.4.2 have been modified to simulate a sit-

uation in which each of the layers is populated with a mixture of two scatterer

types. The dominant scatterer types are distributed on the layers as in the ideal

case, and the minority scatterers, accounting for a 5% of the scatterers in the layer,

are characterised by the scatterer type of one of the other layers. The number of

5.7. Discussion 107

scatterers per footprint was increased from 10 to 20 so as to accommodate the 5%

of minority scatterers. The results for unconstrained and constrained optimisation

are shown in Figs. 5.13 and 5.14 (8 and 40 looks). The descriptive statistics can

be found in Table 5.6.

From Fig. 5.13, it is immediately clear that unconstrained optimisation is not

able to separate layers when using only eight looks. When the number of looks is

increased to 40, the distributions become unimodal, but are still quite broad. This

behaviour arises because in the eight look simulations, the minority scatterers are

buried in the estimation inaccuracies. Increasing the number of looks improves the

situation to some extent.

The constrained optimisation results (Fig. 5.14) show three distributions centred

at the layer heights. There is some confusion in assigning samples between the first

and second layers (at -0.4 and 0.0 m, respectively). This confusion arises in samples

where the coherence values for the two layers are very similar, so that samples in

the second layer (expected to have a lower coherence due to larger vertical spread)

result in a slightly larger coherence value, and are associated with the first solution

(highest coherence). As the number of looks increases, this behaviour tends to

disappear.

The coherence distributions are shown in Fig. 5.15. The coherence distributions

for the second and third layer (using the polarisation states that would be expected

to recover each of these layers) tend to overlap. This overlap diminishes slightly for

the constrained case, while the unconstrained case shows a very clear separation

of the coherence obtained from the three solutions.

5.7 Discussion

This Chapter has examined polarimetric interferometry applied to layered targets.

The aim of this Chapter was to determine the suitability of polarimetric interfer-

ometry to retrieve information about individual layers, such as the height of the

layer, an indication of its depth (through the coherence of this layer taken individ-

ually), and an idea of the scatterer types (or most important scatterer type) in the

layer. While the treatment is general, care has been taken to adapt the findings

to the GB-SAR indoor component.

5.7. Discussion 108

0

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(b) 40 looks

Figure 5.13: Simulation results for a layered target made up of linearly independentscattering mechanisms (cf. Table 5.3 for layer geometry). Each layerhas a dominant and secondary scattering mechanism. Thick black curvesare height distributions for expected scattering vectors, and the red, greenand blue curves are respectively the retrieved height with the first, secondand third projection vectors estimated from unconstrained optimisation

5.7. Discussion 109

0

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(b) 40 looks

Figure 5.14: As Fig. 5.13, using constrained coherence optimisation (black, green andblue for first, second and third set of projection vectors).

5.7. Discussion 110

0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Fre

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0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Fre

quen

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0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Coherence

Fre

quen

cy



Figure 5.15: Coherence distributions for dominant scattering vector scenario and 40looks: expected projection vectors (top), unconstrained optimisation(middle) and constrained optimisation (bottom).

5.7. Discussion 111



1 0.070 (0.3) -0.29 (0.23) -0.348 (0.15) -0.40 (0.03)

2 -0.018 (0.28) 0.07 (0.18) -0.06 (0.16) 0.005 (0.07)

3 0.133 (0.12) 0.127 (0.208) 0.287 (0.06) 0.279 (0.06)

Expected Pols.


u 1 -0.40 (0.013) -0.40 (0.005)� �

u 2 0 (0.02) 0 (0.007)� �

u 3 0.288(0.02) 0.296 (0.009)

Table 5.6: Descriptive statistics illustrating the results of different coherence optimisa-tion techniques on a 3-layered target (see Table 5.3). All the scatterers inevery layer have the same reflectctivity, but 95% are of a “dominant” type,and 5% are from a minority type, as outlined in the main text. The resultsare given in terms of the mean retrieved height; the standard deviation isgiven in brackets for each case.

To achieve these objectives, the coherence of a layered target was examined,

and it was proved that the coherence of the layer that individually had the largest

coherence was larger than that of the ensemble, or in other words, no combination

of layers obtains a larger coherence than that of the individual layer with the

highest coherence.

In PolInSAR processing, the coherence is a function of the polarisation states

used to generate the interferogram. In some circumstances (e.g., when the scat-

tering vectors characterising the scatterers in the different layers are independent),

a polarisation state can be found that recovers the returns from one of the layers

while strongly attenuating the returns from the other layers. The choice of po-

larisation state requires knowledge of the scattering matrix of each layer. If the

scattering matrices for each layer are not known, and they are assumed indepen-

dent with no more than three layers are present, a single layer can be identified

by locating the coherence maximum in the polarisation space. This can be carried

out either by means of an exhaustive search, or by using a coherence optimisation

algorithm. Coherence optimisation retrieves the optimal polarisation states, which

can then be used to infer details of the scattering behaviour of a particular layer.

5.7. Discussion 112

The resulting (optimal) interferogram can also be used to estimate the height of

the layer.

Coherence optimisation has been analytically applied to a target made up of

three orthogonal mechanisms. It was found that under these conditions, the un-

constrained coherence optimisation algorithm will be unstable if the layers have

similar coherences. Physically, this means that there are several maxima (with

nearly identical values) in different regions of the polarisation space, and the al-

gorithm fails to separate these. This happens because under these conditions, the

estimation of the eigenvectors that characterise the optimal polarisation states is

numerically unstable. The situation improves if the coherence for the individual

layers differs. Under the same scenario, constrained optimisation suffers no limita-

tions due to similar coherence of individual layers, because of the different nature

of the eigenvector problem.

A layer made up of three layers populated by linearly independent scattering

types was also investigated. The layer separation can be obtained using polar-

isation states that are aligned with the scattering vector associated with one of

the layers and orthogonal to the scattering vectors characterising the other lay-

ers. For a generic target, the polarisation states that enable layer separation were

calculated based on these premises, and were then substituted in the coherence

optimisation to prove that these polarisation states are indeed solutions to the

problem. Identical results are obtained using constrained optimisation.

The previous findings have been illustrated using numerical simulations. For the

orthogonal case, a two layered target with identical layers was considered, as this

is a model of an artificial target that was measured in the GB-SAR indoor cham-

ber (see Chapter 6). The use of unconstrained optimisation produced very poor

results, with no appreciable layer separation in terms of the retrieved height. The

reasons for this were the instability of the polarisation state estimation problem, as

explained above. Increasing the difference in coherence between different layers by

giving each layer a different depth improved things. However, the results for a small

number of looks were poor, with broad retrieved height distributions. Increasing

the number of looks resulted in improvements, with narrower distributions, but

the results were worse than those obtained using the expected polarisation states.

Using constrained optimisation resulted in retrieved heights that were in good

5.7. Discussion 113

agreement with those found using the polarisation states expected to separate the

layers.

A more general scenario, a three layered target made up of three linearly in-

dependent scattering mechanisms, each of the layers characterised by a different

coherence was also considered. The results for all optimisation methods show uni-

modal distributions. The results from constrained optimisation are better (in the

sense that the retrieved height distributions are significantly narrower) than the

unconstrained optimisation for the same number of looks.

The effect of the number of looks in the optimisation problems was also consid-

ered. Using a small number of looks results in inaccuracies in the estimation of the

covariance and cross-covariance matrices. In terms of unconstrained optimisation,

the different covariance matrices result in different polarisation states for each of

the images in the interferometric pair, a situation that does not have physical

grounds in the absence of temporal decorrelation effects and when the baseline is

very small. This difference results in different combinations of the polarimetric

channels in the final interferogram, and thus, in height uncertainty. Constrained

optimisation overcomes this problem by constraining the polarisation state in both

images to be the same. This constraint is seen as a way of averaging the covari-

ance matrices from both images. Simulations of the three layered target populated

with linearly independent scattering vectors were carried out, varying the number

of looks. These confirmed that constrained optimisation produces accurate results

with around 10 looks, while unconstrained optimisation needed more than 30 looks

to produce similar results.

The above conclusions are for highly idealised targets. In practice, scatterers in

a layer will suffer from scattering matrix fluctuations, and more than one type of

scatterer will often be present in the same layer. The random scattering matrix

fluctuations can be seen as a source of additive perturbations. Given the random

nature of these perturbations, they can be minimised by choosing a large number

of looks. In terms of coherence optimisation, they have the same effect that poor

covariance matrix estimation, and as such, unconstrained optimisation will be more

vulnerable to them.

The presence of different types of scatterers in each layer leads to a fundamen-

tally different optimisation problem. In the context of this Thesis, it is helpful

5.7. Discussion 114

to assume that each layer is characterised by a “dominant” and one or more “mi-

nority” scatterer types. The former can be thought of as the scatterer type that

is more frequent in a layer, or the one that contributes to the largest part of the

backscattered signal. The contribution of the minority scatterer type (or types)

to the radar return is small in comparison with the dominant type. If only a few

independent samples are available, and the minority contribution is small, the lat-

ter will be buried under the estimation inaccuracies, and will effectively act as a

source of perturbations, with similar consequences to the problems arising from

estimation errors. If the number of looks is sufficiently large, effectively a different

optimisation problem has to be solved. The worse case scenario is a situation were

the minority scatterers in one layer are identical to the dominant scatterers in an-

other layer, as it is impossible to separate the dominant scatterers using different

polarisations. This results in the reduction of the coherence maxima associated

with the individual layers, but does not significantly alter their position, so coher-

ence optimisation will retrieve the same polarisation states as in the ideal case. If

the minority scatterers are not identical to the dominant scatterers in other layers,

the shape of the coherence function will change, but if the minority contribution

is small, the estimated polarisation states will not change drastically from the

ideal case. Clearly, as the minority contribution increases, the results will start

departing from the ideal case.

To summarize the findings presented in this Chapter, and in the context of

the indoor GB-SAR component, it is clear that if the polarimetric properties of

different layers allow layer separation by choosing the polarisation states of the

imaging radar, coherence optimisation will provide useful interferometric informa-

tion on each of the layers. On the other hand, if the polarimetric properties are

not known a priori, the use of coherence optimisation techniques appears very

promising for the geometrical localisation of the layer, as well as being a useful

tool for examining the polarimetric properties without any prior knowledge of the

scatterer types. Throughout this Chapter, the use of constrained algorithms in

single-pass scenarios has proven to be substantially more robust than the use of

unconstrained optimisation. In the light of this, and bearing in mind that all the

numerical simulations use geometries which are typical of a system like GB-SAR,

this Chapter strongly recommends the use of a constrained algorithm with this

5.7. Discussion 115

system.

Chapter 6Experimental Verification of GB-SAR

Polarimetric and Interferometric

Capabilities

6.1 Introduction

This Chapter provides experimental validation of the interferometric ca-

pabilities of the GB-SAR instrument, both in single polarisation InSAR

processing and in polarimetric InSAR processing. The validation was

carried out by imaging an artificial target, with both a well-defined geometry and

well-defined polarimetric behaviour.

The Chapter starts with a description of the experimental set-up, including the

geometry, imaging parameters and a succinct analysis of the processed data. This

is followed by the results from an interferometric analysis and from the applica-

tion of polarimetric coherence optimisation algorithms to the data. Finally, some

conclusions are presented.

6.2 Experimental Set-Up

The experiment consisted of imaging two layers of 20 mm aluminium nails embed-

ded in expanded polystyrene tiles. The layers were located at different heights, and

116

6.2. Experimental Set-Up 117

H

r21

A1

r22

r12

A2

2.54

0.06

r11

Figure 6.1: Experimental Geometry.

in each layer the nails were all either vertically or horizontally aligned. The aim of

this set-up was to construct a target that would exhibit different polarimetric char-

acteristics at different heights: the vertical nails would backscatter strongly with a

vertically polarised incident wave, whereas the horizontal nails would backscatter

strongly when illuminated by a horizontally polarised wave.

6.2.1 Geometry

The experimental geometry is shown in Fig. 6.1. It shows the two layers, separated

by a distance D (in the experiment, this was equal to 0.15 m). The top and bottom

layers are populated with vertically and horizontally oriented nails, respectively.

Each layer measures 0.90×0.90 m, and consists of 100 nails embedded in each 0.30×0.30 m polystyrene tile, producing 900 nails in each layer. The VV channel will

retrieve the vertical layer, whereas the HH channel would be expected to retrieve

the horizontal layer. The return in the cross-polar channel should be negligible,

as neither layer will depolarise the incident field. However, small perturbations,

particularly in the orientation of the vertical nails might result in a small signal

6.3. Initial Data Analysis 118

associated with this layer appearing in the VH channel.

Due to the GB-SAR chamber geometry, there will be a change in angle of in-

cidence across the swath. The angle of incidence on a reference plane coincident

with the bottom layer varied between 38◦ and 50◦ over the target region. This

variation results in a variation of the ground range resolution across the swath,

and it also influences the backscattering behaviour of the vertical nails.

6.2.2 Imaging Set-Up

The geometric set-up described in Sect. 6.2.1 was imaged at X band, with centre

frequency 10 GHz and bandwidth 4 GHz. The data were calibrated using the

standard GB-SAR procedure [Sarabandi et al., 1990]. As an additional check

to test the quality of the calibration, the RCS of a sphere was measured and

compared to theoretical predictions. The good agreement between experimental

and measured data over the whole bandwidth indicated that the whole bandwidth

could be used.

The synthetic aperture measured 0.91 m, and was sampled at 0.01 m intervals.

Two parallel scans, separated by a 0.06 m baseline were made. A full three-

dimensional reconstruction [Bennett and Morrison, 1996] of the same scene at X

band is also available for comparison.

6.3 Initial Data Analysis

The aim of this Section is to introduce the processed data and to point out some

relevant imaging features which affect the interferometric analysis.

6.3.1 Resolution

System resolution affects the experiment in a number of different ways, for example

the number of scatterers per resolution cell or the number of independent samples

available in multi-look processing. The nominal resolution (i.e., with no windowing

applied to the data) on a reference plane 2.54 m below the antennas with an

incidence angle of 45◦ was 0.053 × 0.059 m (ground range-azimuth).


A resolution cell size of around 0.1 × 0.1 m can be assumed after windowing

and common band filtering (estimated from full-system numerical simulations),

giving around 10 nails per resolution cell. Simulations show that this number of

scatterers is enough for the recorded data to be approximately characterised by a

circular Gaussian distribution.

The relatively small target region limits the accuracy of the estimated covariance

matrix through multi-looking because there were only a few uncorrelated samples

over the whole region of interest. If the backscattering behaviour has a significant

variation over the swath, due to the change in angle of incidence, the scene could

not be assumed homogeneous in range, further limiting the available number of

independent samples.

6.3.2 Three-Dimensional Reconstructions

Additionally to the InSAR pair scans, a full three-dimensional reconstruction [Mor-

rison et al., 2001] of the target was carried out. These measurements help to

interpret the interferometric results, and are briefly outlined here.

The azimuth-averaged backscatter in the three-dimensional reconstruction can

be seen in Fig. 6.2 for the HH, VH and VV channels (the SAR aperture was

located at the top right hand side of these images). As expected, the HH channel

shows the lower layer of nails (horizontally-oriented), with a small return from the

top layer. The VV channel shows strong backscattering from the top layer, and

a small signal from the horizontal nails (the relatively strong signal at the the

front of the bottom layer comes from radar absorber used to cover metallic railings

in the chamber). The VH channel shows residual contributions from both layers,

which arise from small perturbations in the position of the nails and from the finite

cross-polar isolation of the antennas.

The top layer imaged in the VV channel shows an increasing returns as the

incident wave becomes closer to the broadside direction of the nails. The bottom

layer, imaged in the HH channel shows less variation in its return across the swath,

as the horizontal nails will not significantly modify their backscattering behaviour

with increasing angle of incidence.

From Fig. 6.2 and the theory presented in the Chapter 2, it seems reasonable to


0.32

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t [m]

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−30−75 −70 −65 −60 −55 −50 −45 −40 −35 −25 −20

(a) HH

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(b) VH

��

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t [m

]

Range [m]

85

dbm2

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0.36

−40 −35−45−50−55−60−65−70−75−80

(c) VV

Figure 6.2: Azimuth-averaged three dimensional reconstruction.


expect that the VV-VV and HH-HH interferograms will respectively retrieve the

top and bottom layers. The coherence associated with each of these interferograms

will be large, due to the small volumetric decorrelation. The height recovered from

the VH-VH interferogram will lie at some point between the two layers, as returns

from both layers are present in this channel and are of a similar value. It is expected

that the coherence of the VH-VH interferogram will be lower, mainly due to the

lower signal to noise ratio, which will result in thermal decorrelation.

6.3.3 Two-Dimensional Images

The intensity of the images reconstructed from single scans is shown in Figs. 6.3.

These images have been processed using the standard GB-SAR plane wave back-

propagation algorithm [Bennett and Morrison, 1996], and each pixel represents

0.01 m in ground range and azimuth.

As part of the processing, the data need to be corrected to account for antenna

patterns. For two-dimensional reconstructions, the antenna pattern is corrected

for a given height plane (i.e., parallel to the ground of the chamber). The VV

and VH images were both horn pattern corrected with respect to a plane 0.15 m

above the reference plane, whereas the HH image was corrected with respect to

the reference plane itself, based on the evidence from the previous Section, which

suggested that the VV and VH channels mainly consist of returns from the vertical

nails (located 0.15 m above the reference plane), whereas the HH channel consist

of returns from the horizontal nails. As a further test, all channels were antenna

pattern corrected with respect to the reference plane. This resulted in a decrease

of the magnitude of the VV and VH channels of around 1.5 dB. This difference

outlines the problem of using a particular plane for targets that show some vertical

distribution.

In Fig. 6.3, the target area is clearly visible in the three channels, but the

VH return is lower than that of either the HH or VV channels. The cross-polar

channel return could arise from two sources: finite polarisation isolation between

the transmit and receive antennas and depolarising effects in both layers. Since

the antennas have polarisation isolation better than 30 dB, the return associated

with finite polarisation isolation is negligible. The vertical nails will depolarise the


(a) HH (b) VH

(c) VV

Figure 6.3: Images for the HH, VH and VV channels.


incoming field, if they are not perfectly vertical. The horizontal nails will show a

smaller depolarising contribution as their positions depart less from the horizontal.

The return from the cross-polar channel should therefore mainly consist of a return

from the vertical nails layer. This is confirmed by the magnitude of the correlation

coefficient between each of the co-polar channels and the cross-polar channel: for

the VV-VH case, it was found to be 0.56, whereas for the HH-VH, it was 0.24,

indicating that the vertical nails layer is contributing strongly to the cross-polar

return.

. Outside the main target region, and towards the top and bottom on the right of

the image are areas of relatively large returns. These areas coincided with metallic

railings in the chamber floor. Since they are outside the region of interest, they

can be neglected.

Fig. 6.3 also shows some residual signal down-range from the target region.

The origin of this signal is not immediately obvious, as system simulations do

not suffer from it. Previous investigation of this phenomenon [Ghinelli, 1997]

suggested that backscattering from a group of nails cannot be viewed simply as a

first order problem. Indeed, the experimental arrangement of a low permittivity

binder (polystyrene) populated with a random arrangement of high permittivity

particles (nails) results in a frequency-dependent phase shift [Gauss, 1982] (in other

words, the medium is dispersive), which would affect the point spread function,

and result in the observed range trickle effect. A detailed study of the statistical

properties of the speckle within the region of interest indicates that if the effect is

present within the main target region, it does not significantly affect the nature of

the data (the data are still Gaussian), and shall thus be neglected from the rest of

the analysis.

In interferometry, the signal to noise ratio (SNR) is important in determining the

amount of thermal decorrelation. For the nails set-up, a theoretical calculation is

complex, as their backscattering intensity will change with both angle of incidence

and frequency. The SNR can however be estimated by calculating the ratio between

the intensity inside and outside the region of interest, leading to an SNR for the

HH channel of 36 dB, 33 dB for the VV channel and 19 dB for the VH channel.

The images shown in Fig. 6.3 are highly oversampled, which results in highly cor-

related pixels. While this might not be critical for visual amplitude image analysis,


it poses problems for interferometric and polarimetric processing, as the theory of

multi-look processing is based on uncorrelated samples. The autocorrelation func-

tion of the images shown in Fig. 6.3 is shown in Fig. 6.4. The broad curves (similar

0

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0.3

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0.7

0.8

0.9

1

1.1

−40 −30 −20 −10 0 10 20 30 40

VVHHVH

ACF

Lag

Aucorrelation function in azimuth

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−40 −30 −20 −10 0 10 20 30 40

VVHHVH

ACF

Lag

Autocorrelation function in range

(a) Azimuth (b) Range

Figure 6.4: Autocorrelation function of the images shown in Fig. 6.3

for all polarisations) indicate high correlation between adjacent pixels: at least 15

lags are needed for decorrelated samples. Averaging over relatively large sliding

windows in the original images will result in approximately single look data, as

shown in Fig. 6.5, where the multi-look intensity histogram using a 5 × 5 pixel

sliding window filter is plotted together with the expected distribution using the

calculated ENL for the HH and VV channels. The ENL for these two channels was

close to one (1.2 and 1.5 looks, for HH and VV respectively), suggesting that the

combined 25 adjacent pixels mostly had the same information content. A number

of techniques for obtaining uncorrelated samples can be found in the literature (see

[Oliver and Quegan, 1998], for example). Some require re-processing of the raw

data or complex interpolation. A simple way of obtaining uncorrelated pixels is

to sub-sample the images by a suitable factor. The effect of sub-sampling can be

seen in Fig. 6.6, where the autocorrelation functions for the oversampled and sub-

sampled images are shown for the VV channel. The plots show that sub-sampling

by a factor of 8 results in a substantial drop of the autocorrelation function of the

sub-sampled image. In the remaining data analysis, sub-sampled data will be used

for all multi-look processing.

6.4. Interferometric Analysis 125

0

0.2

0.4

0.6

0.8

1

0 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

Rel

ativ

e F

requ

ency

Intensity of HH channel 5x5 mask, ENL=1.2

Expected

Intensity [RCS,m2 ]

0

0.2

0.4

0.6

0.8

1

0 0.0005 0.001 0.0015 0.002 0.0025

Rel

ativ

e F

requ

ency

Intensity of VV channel 5x5 mask, ENL=1.5

Expected

Intensity [RCS,m2 ]

(a) HH (b) VV

Figure 6.5: Histograms of multi-look intensity for the HH and VV channels using a5 × 5 sliding window average and the theoretical predictions based on theequivalent number of looks.

6.4 Interferometric Analysis

The interferometric analysis of the data studies the phase and amplitude of the

correlation coefficient. In this Section, the phase will be translated into height,

following the discussion presented in Chapter 4, as height can be directly related

to the geometry of the scene (layer separation, in this case).

The images that make up the interferometric pair have been sub-sampled by a

factor of 8, resulting in each pixel corresponding to an area of 0.08×0.08 m (ground

range, azimuth), with the autocorrelation function dropping to a value of around

0.25 at lag 1 (in both range and azimuth) suggesting that adjacent pixels can be

combined for multi-look processing. The interferometric coherence was estimated

using a 9×5 (azimuth×range) sliding window averaging filter, resulting in an ENL

of around 25 for the co-polar channels, and 29 for the cross-polar channel. Only

areas inside the target region unaffected by edge effects were considered, resulting

in a total of 36 samples.

The results from single polarisation interferometry are shown as histograms in

Figs. 6.7, 6.8 and 6.9. The descriptive statistics are shown in Table 6.1.


0

0.2

0.4

0.6

0.8

1

1.2

−10 −5 0 5 10

AC

F

Azimuth (Sub−sampled)Azimuth

Range (Sub−sampled)Range

Lag

ACF of VV channel for original and sub-sampled data

Figure 6.6: Autocorrelation function for original data and data sub-sampled by a factorof 8 in both ground range and azimuth. The presented data is for the VVchannel.

0

5

10

15

20

25

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ple

s


HH-HH

Figure 6.7: Retrieved height in the HH-HH interferogram.


0

5

10

15

20

25

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ple

s


VV-VV

Figure 6.8: Retrieved height in the VV-VV interferogram.

0

5

10

15

20

25

0 0.05 0.1 0.15

Sam

ple

s


VH-VH

Figure 6.9: Retrieved height in the VH-VH interferogram.

The two co-polar interferograms are characterised by nearly identical mean co-

herence values, close to 0.96. The mean coherence of the cross-polar channel is


HH-HH VV-VV VH-VH

Height [m] 0.008 (0.003) 0.149 (0.003) 0.116 (0.02)Coherence 0.966 (0.004) 0.958 (0.008) 0.876 (0.038)

Table 6.1: Mean values from single polarisation interferometry of the nails data. Thevalue in brackets is the standard deviation.

0.88, due to the lower signal to noise ratio in this channel. The mean height for the

VV interferogram is very close to 0.15 m, agreeing very well with the real height of

the vertical nails layer. For the HH-HH interferogram, the mean retrieved height

is 0.01 m, in good agreement with the position of the horizontal nails layer. The

height of the cross-polar channel is close to 0.12 m, a value between the two lay-

ers, but closer to the upper one, due to the correlation between the VV and VH

channels (see Sect. 6.3.3).

6.5 Polarimetric Interferometry

For a fully polarimetric data-set, a polarimetric coherence matrix Γ can be defined

with respect to the VH basis as

Γ =

ρhh−hh ρhh−vv ρhh−vh

ρvv−hh ρvv−vv ρvv−vh

ρvh−hh ρvh−vv ρvh−vh

, (6.1)

where ρxx−yy is the coherence resulting from the xx − yy interferogram. It is

possible to express Γ in any other polarisation basis using a special unitary 3 × 3

transformation matrix (Mattia et al. [1997], Sagues et al. [2000]). This procedure

is similar to polarisation synthesis used in polarimetry [Evans et al., 1988], but,

in the case of interferometry, care must be taken to define the phase consistently

for different polarisation bases [Cloude and Papathanassiou, 1998, Sagues et al.,

2000]. This technique allows a representation of the coherence as a function of the

ellipticity and orientation angles of the transmit and receive antennas, or, if the

expression for coherence for polarimetric data presented in Chapter is used, for

all possible projection vectors (described in terms of the orientation and ellipticity

6.6. Coherence Optimisation 129

angles).

The polarimetric coherence matrix has been calculated for all possible combi-

nation of orientation and ellipticity angles for a region in the centre of the nails

area, where the angle of incidence was 45◦ (at the reference plane). The coherence

for the co-polar and cross-polar solutions are shown in Figs. 6.10 and 6.11. The

contour levels represent retrieved height. Vertical polarisation is represented by

an orientation of 90◦ and an ellipticity of 0◦(shown in the centre of the image),

and horizontal polarisation is described by an ellipticity of 0◦ and an orientation

of either 0 or 180◦ (centre top and bottom of the image).

The co-polar plot has an absolute coherence maximum with value 0.977, and

retrieved height of 0.01 m. It is located at (ellipticity = 10◦, orientation = 3◦),

i.e., close to the HH polarisation. A second maximum is located at the centre

of the plot, (ellipticity = 11, orientation = 86◦), i.e., very close to VV, with a

coherence of 0.965, and a retrieved height value of 0.143 m. This is what we would

expect given the VV-VV and HH-HH interferograms. The cross-polar channel

shows maxima at the right and left sides of the image, areas which characterise

circular polarisations. The cross-polar coherence exhibits symmetry [Sagues et al.,

2000], so these areas are identical. There are also local maxima at the centre of

the image (small values of ellipticity), in the VH region.

The co-polar coherence maxima are broad, and within these coherence maxima,

the retrieved height varies slowly. This can be seen for polarisation states charac-

terised by the ellipticity equal to zero; Figs. 6.12 and 6.13 respectively show the

value of the coherence and height as a function of orientation.

The images in Figs. 6.10 and 6.11 show some interesting features. It can be

seen that coherence varies slowly with choice of polarisation; the maxima are rather

broad, and no quickly changing regions are present within them. Also, the retrieved

height changes slowly around the maxima, but then decays rapidly outside them.

6.6 Coherence Optimisation

The nails data can be used to test coherence optimisation of layered structures, as

discussed in the previous Chapter. However, this presents a number of challenges

for coherence optimisation:


0.12 m

0.14 m

0.10 m

Figure 6.10: Coherence as function of transmit and receive polarisations. Co-polarvalues. Contour levels indicate retrieved height.


0.12 m

0.14 m

0.10 m

Figure 6.11: Coherence as function of transmit and receive polarisations. Cross-polarvalues.Contour levels indicate retrieved height.


0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

0 20 40 60 80 100 120 140 160 180

Coh

eren

ce

Orientation Angle [deg]

Co−Polar

Figure 6.12: Coherence for different polarisation orientations and 0◦-ellipticity for thenails data at 45◦ angle of incidence.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 20 40 60 80 100 120 140 160 180

Ret

rieve

d H

eigh

t [m

]

Orientation Angle [deg]

Co−Polar

Figure 6.13: Height for different polarisation orientations and 0◦-ellipticity for the nailsdata at 45◦ angle of incidence.

1. The two layers have similar geometries and, therefore, similar coherence val-


ues.

2. While the return from the co-polar channels is dominant, a cross-polar return

is still present, as seen in Section 6.3.3.

3. The coherence is a slowly varying function over large areas of the polarisation

space.

The first point will be a problem for unconstrained coherence optimisation, as

shown in Chapter 5. The nearly identical coherence values result in an unstable

estimation of the projection vectors, and in inaccurate height estimation. Finally,

the slowly varying coherence implies that optimisation algorithms may mistake

small rapidly varying perturbations as maxima, and result in estimated projection

vectors far from the expected maxima. This problem will be particularly acute for

the unconstrained optimisation algorithm, which has an extra degree of freedom

to come up with maximal coherence values.

The two and three dimensional reconstructions clearly show a signal present in

the VH channel, indicating that the covariance matrices will be full rank. How-

ever, from the experimental set-up, it can be assumed that the signal in the VH

channel will be due to experimental imperfections. This leads to the possibility

of discarding the VH channel, and carrying out coherence optimisation using a

two dimensional covariance matrix, as well as the three-dimensional solution. The

two-dimensional results allow for a clearer assessment of the problems of coher-

ence optimisation noted in Chapter 5 in comparison with the three-dimensional

approach. The use of both approaches can also be used to test the robustness of

the optimisation procedures: the algorithm will be robust if the results from two-

and three-dimensional optimisation do not change significantly.

6.6.1 Two Dimensional Coherence Optimisation

6.6.1.1 Unconstrained Optimisation

The results for two-dimensional unconstrained coherence optimisation (see Table

6.2 for descriptive statistics, Fig. 6.14 for retrieved height histograms) show val-

ues of coherence near 1 for both solutions. The retrieved height distributions



Height [m] Coherence Height [m] CoherenceSolution 1 0.0587 (0.043) 0.972 (0.006) 0.008 (0.003) 0.971 (0.008)Solution 2 0.120 (0.032) 0.960 (0.004) 0.1495 (0.004) 0.930 (0.014)

Table 6.2: Retrieved height and coherence from 2D coherence optimisation applied tothe nails data. The mean and standard deviation (in brackets) are shown.

depart from the narrow distributions presented for the VV-VV and HH-HH in-

terferograms, being broader distributions and having significant tails, similar to

those shown for simulated data (Fig. 5.3). A typical set of projection vector pairs

(chosen for a random sample) is1

[� �

p 1� �

p 2] =

[0.86 −0.17

0.40 + j0.31 0.98

][

� �

p ′1

� �

p ′2] =

[0.89 −0.104 + j0.22

j0.46 0.97

],

(6.2)

where the projection vectors are the columns of the two matrices, the first column

being the eigenvector associated with the larger eigenvalue, and the second column

showing the eigenvector associated with the lower eigenvalue. The first matrix

corresponds to the first image, and the second to the second image.

The retrieved projection vectors show two important features:

1. The estimated projection vectors for the two images are very different

2. The estimated projection vectors depart significantly from the [1, 0]T and

[0, 1]T vectors expected results to separate the horizontal and the vertical

layers.

The variations in the eigenvectors arise from the coherence being virtually identical

for both solutions. Equivalently, the matrices have nearly identical eigenvalues: the

main diagonal elements are both very close to unity, and the off-diagonal terms

dominate the definition of the eigenvectors.

1Note that the eigenvectors are undefined up to a constant of value ejϕ. Throughout this Thesis,the eigenvectors are chosen so that the largest component is real. Where a phase constraintis needed, it will be applied.


A physical interpretation of the results is that the minimal increases in coherence

obtained by the optimisation algorithm with respect to the interferograms using

prior knowledge of the projection vectors is used result from small perturbations

in the data, due to the imperfect estimation of the covariance and cross-covariance

matrices. In the vicinity of the maxima, the coherence does not change significantly

as a function of the projection vectors; the optimal value will be the highest possible

value within this region, and will be heavily influenced by fluctuations in the data.

6.6.1.2 Constrained Optimisation

Results from two-dimensional constrained optimisation are shown in Table 6.2,

and retrieved height distributions are shown in Fig. 6.15. The coherence is similar

(though slightly lower) to that obtained in the unconstrained optimisation results.

The retrieved height is comparable with that from the VV-VV and HH-HH in-

terferograms, showing a very clear layer structure with no tails. The retrieved

eigenvector matrix is very close to an identity matrix (with the magnitude of the

main diagonal elements always larger than 0.98), suggesting that, in this case,

the optimisation procedure effectively results in the VV-VV and HH-HH interfer-

ograms.

The constrained procedure does not suffer as a result of the eigenvalues being

very close. As was shown in Section 5.3.1, the final matrix will be diagonal, but

with the main diagonal elements related to the real parts of the HH-HH and VV-

VV interferogram. Given the relatively large spacing between the layers (which

results in an interferometric phase difference of around 1 radian), the main diago-

nal elements have a large separation compared with the value of the off-diagonal

elements, significantly reducing the impact of off-diagonal terms in the final result.

The constraint to find an optimal coherence value using the same projection

vector for each image reduces the degrees of freedom of the procedure, and results

in a search on a reduced problem space, where small fluctuations cannot be used

for minor coherence enhancements.

The results depart slightly from those obtained with prior knowledge (in partic-

ular, the coherence for the second solution, associated with the VV-VV interfer-

ogram, is lower than the actual VV-VV interferogram) due to small variations of


0

1

2

3

4

5

6

7

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(a) Solution 1

0

2

4

6

8

10

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(b) Solution 2

Figure 6.14: Two-dimensional unconstrained coherence optimisation. Retrieved heighthistograms for 36 samples within the main target region. The first solutionis associated with the highest eigenvalue, the second with the lowest.


the projection vectors, which are produced in turn by variations in the values of

the main diagonal terms of the final matrix from which the projection vectors are

extracted. These variations are the result of the interferometric phase distribu-

tion, but physically they arise from the fact that the constrained algorithm is an

approximation that assumes that the coherence is unity. In this case, the approx-

imation implies Dirac δ-functions for phase distributions, and hence no variations

in the main diagonal terms of the final matrix. The small variations in the esti-

mated projection vectors arise from the imperfect experimental conditions, where

the coherence for either layer is close to, but not quite, 1.

6.6.2 Three-Dimensional Coherence Optimisation


Height [m] Coherence Height [m] CoherenceSolution 1 0.097 (0.025) 0.977 (0.007) 0.008 (0.003) 0.973 (0.009)Solution 2 0.118 (0.035) 0.965 (0.004) 0.144 (0.01) 0.915 (0.022)Solution 3 0.109 (0.029) 0.848 (0.037) 0.117 (0.036) 0.857 (0.038)

Table 6.3: Retrieved height and coherence from 3D coherence optimisation applied tothe nails data. The mean and standard deviation (in brackets) are shown.

6.6.2.1 Unconstrained Optimisation

Results from three-dimensional unconstrained optimisation are shown in Table

6.3 and in Fig. 6.16. While the coherence is in line with that of the single-

polarisation interferograms, the retrieved height is not. The first solution (highest

coherence) shows a mean height between the two layers, with a broad distribution,

while the second solution is biased towards the vertical (upper) nail layer, but

with a significant tail extending towards lower heights. These distributions are

similar to the height distribution of the 2D unconstrained optimisation, but with

a lower mean height. The third solution is similar to the VH-VH interferogram.

These results consistent with the optimisation procedure resulting in three pairs

of projection vectors that are similar to HH, VV and VH, i.e., the eigenvectors

form an identity matrix. The first two solutions are identical to those presented in


0

5

10

15

20

25

30

35

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(a) Solution 1

0

5

10

15

20

25

30

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(b) Solution 2

Figure 6.15: Two-dimensional constrained coherence optimisation. Retrieved heighthistograms for 36 samples within the main target region. The first solutionis associated with the highest eigenvalue, the second with the lowesr one.


the two-dimensional case; the third solution would arise from a maximum in the

cross-polar interferogram for the VH polarisation.

Typical eigenvectors are

[� �

p 1� �

p 2� �

p 3] =

0.82 + j0.20 0.20 − j0.13 0

−0.33 − j0.14 0.81 j0.12

−0.16 − j0.36 0.49 + j0.22 0.99

(6.3)

[� �

p ′1

� �

p ′2

� �

p ′3] =

0.81 −j0.21 0

−0.17 − j0.29 0.79 j0.17

−j0.47 −0.51 + j0.26 0.98

. (6.4)

These two matrices have their largest elements along the main diagonal, suggesting

that the retrieved projection vectors are perturbed versions of the HH, VV and

VH projection vectors. The perturbation of the VV and HH projection vectors

(associated with the first and second eigenvalues) are larger than that of the third

projection vector (associated with the lowest eigenvalue). This is due to the simi-

larity of the coherence between the vertical and horizontal layers, which affects the

value of the eigenvectors for these two solutions in the same way as it did in the

2D case. Since the difference between the second and third eigenvalue is relatively

large, the third eigenvector is not badly affected by this effect. Note also that the

second eigenvalue has a significant contribution from the VH channel. This arises

from the correlation between the VV and VH channels, as outlined in Section 6.3.3.

The relatively large conditioning number of the problem (around 60) results in a

fairly ill-conditioned problem, and in significant differences between the retrieved

polarisation states for each image. Ill-conditioning also results in large variations

in the eigenvectors from sample to sample.

6.6.2.2 Constrained Optimisation

Results from three-dimensional constrained optimisation are shown in Table 6.3

with height distributions presented in Fig. 6.17. The height distributions show

that the first solution is associated with the horizontal layer, the second with

the vertical layer, and the third is similar to the VH-VH interferogram. These

distributions are very different for those for the 3D unconstrained case (Fig. 6.16),


0

1

2

3

4

5

6

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(a) Solution 1

0

2

4

6

8

10

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18Retrieved Height [m]

Sam

ples

(b) Solution 2

0

2

4

6

8

10

12

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(c) Solution 3

Figure 6.16: Unconstrained Optimisation

6.7. Summary and Discussion 141

which do not show such a clear layer separation. The coherence for each of the

solutions is high. In the case of the first solution, it is similar to that of the HH-

HH interferogram and to the first solution of the 2D constrained optimisation; for

the second solution, the value is slightly lower than the second solution of the 2D

constrained optimisation, and the third solution is similar to that of the VH-VH

interferogram.

Typical eigenvectors are

[� �

p 1,� �

p 2,� �

p 3] =

0.98 0 0

0 0.80 j0.13

−0.13 − j0.15 0.37 − j0.28 0.99

. (6.5)

The first eigenvector is well-aligned with an HH projection vector. The second

eigenvector has contributions from the VH and VV channels, while the third is

nearly coincident with a VH projection vector. The first and third eigenvectors

thus result in interferograms similar to the HH-HH and VH-VH interferograms.

The second eigenvector results in a mixture of the VV and VH channels due to

the correlation between these two channels, with a predominant VV contribution.

Physically, the algorithm is using a combination of two correlated channels to

improve the coherence. The third eigenvector is forced to be orthogonal to the

other two, as the final matrix is Hermitian [Colin et al., 2003]. This results in a

value of the coherence that is a local maximum.

6.7 Summary and Discussion

An artificial target, made up of two layers of nails separated by 0.15 m, vertically

oriented in the top layer, and horizontally oriented in the bottom layer, has been

used to test the interferometric and polarimetric capabilities of the GB-SAR sys-

tem. The scene was imaged using a two-dimensional aperture at X band, which

allows both the generation of three-dimensional scene reconstructions and interfer-

ometric SAR. For the two-dimensional image reconstructions, the resolution cell

size was around 3-4 wavelengths.

The three-dimensional reconstruction shows that the VV channel mainly con-


0

5

10

15

20

25

30

35

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(a) Solution 1

0

5

10

15

20

25

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(b) Solution 2

0

1

2

3

4

5

6

7

8

9

−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Sam

ples


(c) Solution 3

Figure 6.17: Constrained Optimisation.

tains the returns from the vertical nails, whereas the HH channel mainly contains

the returns from the horizontal nails. The VH channel contains returns from both

layers, but the intensity of the returns is much lower.

In order to produce the two-dimensional images needed for interferometry, a

choice needs to be made as to where the scattering centre is. The antenna beam


pattern correction for each polarisation needs to be calculated for a single horizon-

tal plane. In this experiment, the scattering centres of the VV and HH channels

are clearly located in different planes (with the VH channel being located close to

the upper nails layer). It is thus sensible to use different planes to calculate the

beam pattern corrections for different polarisations. In this case, the correction

factors for VV and VH were calculated using a plane coincident with the upper

nails layer, whereas the HH corrections were calculated using a plane coincident

with the lower nails layer.

Two dimensional reconstructions clearly show the shape of the target region,

with a strong signal in the co-polar channels, and a weaker signal in the cross-

polar channel. The cross-polar channel was found to be correlated with the VV

channel, due to the imperfect alignment of the vertical nails resulting in a larger

depolarising contribution than that from the horizontal nails.

After processing, the two-dimensional images are highly oversampled. In order

to perform multi-look processing, the data were subsampled by a factor of 8. Using

an averaging window of 9× 5 subsampled pixels, the ENL was estimated between

25 and 30 (depending on the channel considered).

Single polarisation interferometry resulted in the VV-VV and HH-HH interfero-

grams accurately estimating the heights of the vertical and horizontal nail layers.

The coherence exceeded 0.95 (nearly identical for both interferograms), and the

standard deviation of the retrieved height was very low (around 0.003 m) for both

co-polar channels. The VH-VH interferogram showed a lower coherence, due to the

weaker signal level, and a height of 0.12 m, close to the top nail layer, confirming

the hypothesis that the VH signal would mostly come from the vertical nails layer.

The polarimetric coherence matrix Γ was calculated for all possible combina-

tions of ellipticity and orientation angles. The co-polar solution showed two broad

maxima, broadly coincident with the VV-VV and HH-HH interferograms. The

height for the maximum close to the VV-VV interferogram was close to that of the

vertical nails layer, whereas the height for the maximum close to the HH-HH inter-

ferogram was close to the horizontal nails layer. The cross-polar solution showed

a number of maxima, one close the VH-VH interferogram, and another one close

to the circular polarisation regions.

The data were analysed using coherence optimisation techniques. While prior


knowledge clearly suggests a two-dimensional problem (the experiment is designed

so that signal in the VH channel should be negligible), the signal in the VH channel

is well above the noise floor, justifying a three-dimensional approach. Both three-

and two-dimensional height retrievals were carried out, as this would test the

robustness of the coherence optimisation algorithms.

For 2D coherence optimisation, the unconstrained optimisation results were dis-

appointing. While the retrieved height distributions show some sort of two layer

structure, the tails in the distributions are important, resulting in a poor estimate

of the height of each layer. This poor performance arises from the nearly identical

coherence values for the VV and HH channels (see Section 6.4). As was pointed

out in Chapter 5, this causes the matrices from where where the projection vectors

are extracted being close to identity matrices. The eigenvectors will change sub-

stantially between different samples, and the projection vectors for the two images

in the pair will be different.

The 2D constrained optimisation procedure worked remarkably well on the nails

data. The retrieved height distributions were narrow, unimodal and with modes

located at the heights of the two layers. The coherence was slightly lower than for

the unconstrained optimisation, but the projection vectors were nearly identical

to the expected VV and HH projection vectors. This algorithm gives a greater

differentiation of the coherence maxima than unconstrained optimisation.

The 3D unconstrained algorithm results were poor. To the problems outlined in

the 2D case, the addition of the VH channel leads to the inversion of two matrices

with a conditioning number of approximately 60. This relative ill-conditioning

causes relatively large differences in the projection vectors estimated for each image

in the pair. All the retrieved projection vectors have significant contributions from

the VH channel, particularly in the case of the second solution (closer to the vertical

layer), as the VV and VH channels are correlated.

The 3D constrained optimisation results were slightly worse than the 2D con-

strained case, but still compared favourably with the single polarisation interfer-

ograms. The addition of the VH channel slightly decreases the coherence for the

first two solutions with respect to the 2D approach, and the vertical layer is again

retrieved using a combination of the VV and VH channels.

This particular experiment confirms some of the points addressed in Chapter 5


and the analysis of layered scenes using polarimetric interferometry. It has been

shown that unconstrained coherence optimisation is not well-suited for systems

such as GB-SAR, where the scene is unlikely to suffer a change in polarimetric

behaviour between passes. The problems identified in Chapter 5 regarding similar

coherence values and the inability of the unconstrained approach to separate lay-

ers were experimentally confirmed. Constrained optimisation provided very good

results, and demonstrated the ability to blindly separate a layered target using

polarimetric interferometry. The addition of a channel correlated with one of the

other channels (but with a low signal level) did not degrade the quality of the

separation substantially, asserting the robustness of the algorithm.

Chapter 7Interferometric Studies of Wheat

Canopies Using GB-SAR

7.1 Introduction

This Chapter presents an application of the GB-SAR indoor component

to the study of wheat canopies using interferometric and polarimetric

techniques, as a demonstration of the techniques and issues which have

been discussed in this Thesis. To this end, data gathered during the RADWHEAT

experiment in 1999 will be analysed. During the RADWHEAT campaign, wheat

was grown under outdoor conditions, and was then transported to Sheffield, where

a canopy was assembled inside the GB-SAR anechoic chamber. Several deliveries

were made during different stages of crop development. The reconstructed canopies

were imaged to produce three-dimensional backscatter reconstructions. A detailed

description of this campaign is available in [Brown et al., 2003].

This Chapter uses wheat measurements from June 18, 1999, which have al-

ready been considered in Chapter 4. It is structured as follows: first, the wheat

canopy and experimental conditions are described, followed by some comment

on the three-dimensional reconstructions. Next, single-polarisation interferometry

results are presented, followed by polarimetric coherence synthesis analysis. Inter-

ferograms using the Pauli basis are also presented, and using both constrained and

unconstrained coherence optimisation procedures. These results are presented in

146

7.2. Description of the Experiment 147

the light of recent developments in electromagnetic modelling of wheat canopies.

7.2 Description of the Experiment

7.2.1 The Wheat Canopy

The canopy is also described in [Brown et al., 2003]. The imaged spring wheat

(Triticum aestivum, “Chablis” variety) was handsown in containers in March 1999,

and grown under normal field conditions. Batches of containers were then delivered

at different growth stages to the University of Sheffield, where they were assembled

together to reconstruct a canopy. This was achieved by packing the contents of

the containers in the GB-SAR trolley. Any empty spaces were covered with spare

soil, so as not to have any gaps. The canopy size was 1.56× 1.74m. The soil used

was Kettering loam (41% sand, 37% silt and 22% clay), and had a depth of 0.25

m. The rms height was around 0.01m, while soil moisture was less than 10%.

In this Chapter, the fourth wheat delivery (imaged on June 18, 1999) will be

studied; at this stage, the ears were just emerging (stage 51 on the BBCH scale).

The crop was green, with gravimetric moisture between 71 and 80%. The mean

height of the crop was 0.58 m (with a standard deviation of 0.09 m) and the shoot

density was 441 shoots m−2. The Green Area Index (GAI) was around 2.9.

7.2.2 Imaging Set-Up

The canopy described in Sect. 7.2.1 was imaged at C band in order to produce

three-dimensional reconstructions. The imaging geometry is shown in Fig. 7.1.

The canopy was grown on 0.25m of soil, which was located on top of the trolley,

itself 0.55m from the floor of the GB-SAR chamber. The two-dimensional aperture

measured 0.78×1.86m (azimuth×range), and was situated on a plane 3.20m above

the chamber floor. The distance from the antenna cluster phase centre to the top

of the soil was 2.13 m. In this Chapter, the top of the soil level is taken as the

reference height plane; positive and negative heights indicate height above and

below the soil, respectively. Although the front of the trolley was covered with

radar absorbing material (RAM), it produced significant returns.


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2.129m

3.2m

0.78m

0.25m

0.55m

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1.86n

Figure 7.1: RADWHEAT imaging geometry. Average crop height was 0.58m.

The two-dimensional aperture was sampled every 0.02m. InSAR processing will

be carried out by selecting any two azimuth scans and processing them individually

to create the interferometric pair. For InSAR analysis, the azimuth scans which are

further from the region of interest are of greatest use, as the angle of incidence will

be larger, resulting in a smaller angle of incidence variation within the resolution

cell. To minimise the change in angle of incidence, scans 4 and 5 were used (where

scan 1 is the scan farthest from the target). Scans 1 and 2 were also considered,

but these showed glitches in the reconstructed images.

The data were recorded over a 1.43 GHz bandwidth centred at 5.44 GHz, with an

aperture measuring 0.78m. The angle of incidence over the region of interest varied

from 33◦ to 55◦. The baseline was 0.02 m, and the theoretical range resolution

varied between 0.19m (near range) and 0.13m (far range), whereas the theoretical

azimuth resolution varied between 0.10m (near range) and 0.13m (far range). The

data were windowed both in range and azimuth with a raised cosine (or Hanning)


window, which broadens the point spread function by around a factor of two. The

relatively small size of the canopy and the large change in angle of incidence over

the target region and within the resolution cell results in different regions of the

image containing very different contributions from the canopy. Three regions can

be readiliy identified (see also Fig. 7.2):

Area I: The contribution from the soil suffers no attenuation as it does not traverse

a full canopy. The angle of incidence at the top of the canopy is around 50◦,

and around 35◦ at ground level.

Area II: The signal from the soil traverses the canopy and is thus attenuated. The

angle of incidence at the top of the canopy is larger than 55◦. At ground

level, the angle of incidence goes up to 46◦.

Area III: The returned signal only contains returns from soil level (attenuated),

and no returns from the top of the canopy, as the resolution cell no longer

includes this part of the target. This region starts at around 47◦ incidence

at ground level.

��

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Soil

Resolution Cell

Canopy

35◦ 45◦40◦

Sensor

Figure 7.2: Diagram showing the angle of incidence variation of the resolution cell overthe target region.

7.3. Three-dimensional Polarimetric Analysis 150

The images were initially processed to achieve a pixel size of 0.02 × 0.02 m, and

were subsequently subsampled by a factor of 8, resulting in the range and azimuth

autocorrelation functions dropping to around 0.2-0.3 in the first lag, so that adja-

cent pixels could be used for multi-look processing. Due to the change in look angle

between consecutive range bins (around 3◦ at close range and 1◦ at far range), mul-

tilooking has been carried out using a rectangular 3× 13 (range× azimuth) mask,

which resulted in an ENL of between 31 and 34 (depending on the polarisation).

The interferometry results are displayed as a function of angle of incidence at

soil level. To reduce contributions from non-target regions, 13 azimuth samples

are combined at each range bin to produce a single mean height at each range bin.

7.3 Three-dimensional Polarimetric Analysis

The azimuth-averaged reconstructions for the present canopy have already been

presented in Chapter 4 (see Figs. 4.6, 4.5 and 4.7). The soil return is clearly

dominant for HH polarisation and for all angles of incidence. For VV, it is only

dominant for smaller angles of incidence: as the angle of incidence increases, there

is strong two-way attenuation of the vertically polarised wave arising from the

strong coupling between the incident field and the stems of the crop. The VH

polarisation shows a small return. If Bragg scattering is assumed, no return is

expected in this channel, so it has been postulated [Brown et al., 2003] that the

returns in this channel might arise from canopy-soil interactions, rather than direct

soil returns. This is an interesting conclusion, as the cross-polar channel is often

associated with volumetric scattering. Another important feature is the increase of

the top of the canopy return with increasing angle of incidence for all polarisations.

This top layer mainly comprises flag leaves and the emerging ears.

In the previous discussion, the existence of double bounce scattering was not

clear. The phase of the co-polar correlation coefficient can be examined to assess

this contribution. A phase close to 0◦ can be associated with an odd bounce con-

tribution (direct backscattering from canopy and soil), whereas a value close to

180◦ would indicate even bounce contributions (canopy-soil interactions) Van Zyl

[1989], Freeman and Durden [1998]. The phase of the co-polar correlation coef-

ficient is shown in Fig. 7.3 (a). As noted in Section 2.8, the distribution of this

7.3

.T

hree-d

imensio

nalPola

rim

etr

icA

naly

sis151

0

20

40

60

80

100

120

140

160

180


Ver

tical

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ght [

m]

Phase of the co−polar complex correlation coefficient

10 20 30 40 50 60

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

(a) Phase

0.1

0.2

0.3

0.4

0.5

0.6

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0.8

0.9

1


Ver

tical

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ght [

m]

Magnitude of the co−polar correlation coefficient

10 20 30 40 50 60

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

(b) Magnitude

Figure 7.3: Phase and Magnitude of the co-polar correlation coefficient for RADWHEAT three-dimensional data.

7.4. Polarimetric analysis 152

phase is conditional on the number of independent samples and of the magnitude

of the co-polar correlation coefficient. The number of looks used to estimate the

the correlation coefficient varies with position, but was always greater than 15.

The magnitude of the correlation coefficient is shown in Fig. 7.3(b). The phase

image shows relatively low values for small angles of incidence at soil level, and for

large angles of incidence at flag-leaf level. This suggests that in these areas, the

dominating scattering mechanism is a direct return. At soil (and lower canopy)

level, the phase difference increases substantially after 30◦, but at the same time,

the magnitude of the correlation drops significantly, resulting in a large uncertainty

in the estimation of the phase difference which explains its variability. In these

circumstances, no firm conclusions can be made, although the trend in this region

is for the phase difference to be large, indicating an important contribution from

canopy-soil interactions in this region.

In summary, it is expected that polarisation states close to HH will result in

a significant soil return, while polarisation states close to VV will show a strong

soil return for small angles of incidence and a strong top of the canopy return for

large angles of incidence. The existence of canopy-soil interactions appear to be

important at large angles of incidence.

7.4 Polarimetric analysis

It is enlightening to examine the coherence matrices for the data considered in this

Chapter, and their characterisation in terms of parameters derived from their eigen-

decomposition, which can be summarised in terms of entropy and average α angle

(see Section 2.7). The coherence matrices were estimated from the reconstructed

SAR image for one of the scans (scan 4) using a 13 × 3 averaging filter, and the

entropy and average α angle were calculated for each bin and averaged. The results

are presented in Figs. 7.4(a)-(c). The entropy rises rapidly from around 0.4 to

around 0.75 at around 45◦ incidence at ground level. This can be interpreted as a

fast transition from a region characterised by one or two scattering mechanisms, to

an area increasingly characterised by a complex mixture of scattering mechanisms

(e.g., a random volume). After 45◦, the entropy drops back to around 0.4 again,

suggesting that again only one or two scattering mechanisms are significant. This

7.5. Single Polarisation Interferometry 153

region coincides with the resolution cell no longer having any returns from the top

of the canopy, as discussed in Section 7.2.2.

The anisotropy is a useful way to analyse the relative importance of the second

and third eigenvalues. The anisotropy experiences a very rapid rise, indicating the

fast transition from a two scattering mechanism area to a more complex area, with

at least three scattering mechanisms, due to the important contribution from the

top of the canopy. At around 45◦, the anisotropy drops again, indicating a drop in

the value of the third eigenvalue, explained by the resolution cell no longer having

a contribution from the canopy in this region.

The average α angle increases up to about 45◦ incidence, where it reaches values

very close to α = 45◦, indicating the presence of dipole-like scattering mechanisms,

associated in this case with flag leaves. The transitions in both the entropy and

the α angle are consistent with the three regions present in the scene (as described

in Section 7.2.2). The first region is characterised by one or two scattering mech-

anisms, with a low α angle, suggesting an isotropic surface (the soil). As the

angle increases, α increases to coincide with the flag leaves located at the top of

the canopy. After that, the entropy drops again to account for the loss of one

scattering contribution (the top of the canopy).

7.5 Single Polarisation Interferometry

The two scans which have been considered sustain a 0.02 m horizontal baseline.

This provides an unambiguous height range of ±2m, suitable for the wheat canopy.

Note that the images used to produce the results presented in the following Sec-

tions have been wavenumber-shift filtered (see Chapter 4). The minimum angle

of incidence was taken to be 30◦, which resulted in a discarded bandwidth of 67.8

MHz. Coherence and effective height as a function of angle of incidence for VV,

HH and VH polarisations are shown in Figs. 7.5(a) and (b).

The effective height shows two distinct trends with respect to angle of incidence

and polarisation: for the VV channel, there is an increasing trend with a dip after

46◦, whereas for the HH and VH channels, the retrieved height is mostly flat.

The VV channel shows an effective height close to soil level at near range. The

height increases nearly linearly with angle of incidence up to around 45◦, where it


0.45

0.5

0.55

0.6

0.65

0.7

30 35 40 45 50 55

Ent

ropy


(a) Entropy

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

30 35 40 45 50 55

Ani

sotr

opy


(b) Anisotropy

30

32

34

36

38

40

42

44

46

48

30 35 40 45 50 55

Avg

. Alp

ha [d

eg]


(c) Average α

Figure 7.4: Two-dimensional polarimetric analysis


−0.1

0

0.1

0.2

0.3

0.4

0.5

30 35 40 45 50 55

Ret

rieve

d H

eigh

t [m

]


Wheat 19990618 C band

VVHHVH

(a) Effective Height

0.6

0.7

0.8

0.9

1

30 35 40 45 50 55

Coh

eren

ce



VVHHVH

(b) Coherence

Figure 7.5: Effective height and coherence from single polarisation InSAR processingfor C band data.

reaches a value of 0.43 m. It then decreases again back to soil level. The HH-HH

interferogram results in a low effective height around -0.07 m at near range, which

climbs up to around 0 m throughout most of the target, with small fluctuations.


The VH-VH interferogram follows a similar trend.

The coherence for the HH-HH and VH-VH interferograms show little variation

with angle of incidence. The coherence is very high, typically above 0.9. The

coherence of the VV-VV interferogram is slightly lower than that of the other two

interferograms, and after 46◦ shows a marked dip.

The HH-HH and VH-VH interferograms retrieve heights associated with the soil

level. There is little variation in height with increasing angle of incidence, and the

coherence is very high, suggesting a very small volume decorrelation contribution.

In other words, most of the signal in these channels is coming from a very thin

layer located at the soil level for all angles of incidence. This behaviour suggests

that there is very little interaction between HH signal and the canopy, in line with

the behaviour expected from the 3D reconstructions. It is interesting to note that

even for large angles of incidence, when there is a significant return from the flag

leaves level in the 3D reconstruction of the HH channel, the retrieved height is

still close to the ground, suggesting that not even this contribution is enough to

compete with the strong ground return. In the case of the VH polarisation, the

scattering seems to be coming from the soil level. This can either be due to direct

soil returns, or due to second order interactions within the canopy. The direct

returns should be small, as the soil can be seen as a Bragg surface, with small

cross-polar contributions. If the second order canopy interactions were responsible

for the VH signal, these would occur towards the top of the canopy, as if they

occurred towards the bottom of the canopy, the effective height would be under

the soil level.

The VV-VV interferogram shows a marked angle of incidence dependence. The

effective height close to soil level at the front of the canopy occurs in the area

where there is no canopy attenuation, and the soil return is very strong. As the

angle of incidence increases, the ground return suffers larger attenuation due to the

coupling between incident field and stems, while at the same time, the return from

the flag leaves becomes more intense. The combined effect is to raise the effective

height, up to around 45◦, where the maximum height of 0.43 m is obtained. At

around 46◦, the effective height rapidly drops back to soil level, as the resolution

cell only contains returns from the soil level, and not from the top of the canopy, as

discussed in Section 7.2.2. The coherence is slightly lower than that of the HH-HH

7.6. Polarimetric coherence synthesis 157

and VH-VH interferograms because of the strong attenuation of the VV signal.

This lower signal level results in a reduced SNR, and thus, higher decorrelation

(lower coherence) due to thermal effects.

7.6 Polarimetric coherence synthesis

As in the previous Chapter, the coherence can be calculated for any polarisation

state, and plotted as a function of the ellipticity and orientation angles of the

used polarisation state. A sample at 43.6◦ incidence angle was selected, as it

was an area in the middle of the canopy, where full attenuation effects would be

visible. The coherence for all possible polarisation states was calculated for this

sample. The retrieved height has also been calculated, and the results are shown

in Figs. 7.6 (co-polar) and 7.7 (cross-polar). From the co-polar solution, it can

be seen that the maximum height is located towards the centre of the plot, not

far from the VV-VV interferogram (which would be located exactly at the center

of the plot). The minimum height is located close to the RR-RR interferogram

(i.e., the interferogram generated using circular right handed polarisations, top

left corner), whereas the maximal coherence is close to the LL-LL interferogram

(i.e., the interferogram generated using circular left handed polarisations, top right

corner). The lowest coherence is again found close to the centre of the plot.

The polarimetric variation of coherence and retrieved height suggests that ver-

tical polarisation is an effective way of retrieving the top of the canopy at around

45◦ incidence. On the other hand, the use of the RR-RR (or LL-LL) interfer-

ogram effectively retrieves the soil layer. This can be explained by the strong

attenuation of the nearly vertical wave by the canopy in the first case, and by the

weak attenuation that the circularly polarised waves suffer travelling through the

canopy.

The cross-polar coherence shows a maximum for the RL/LR interferograms, with

a retrieved height close to 0. The coherence minimum is located at the HV/VH

interferogram, with a maximum height of around 0.36 m. These results indicate

that the circular polarisations mainly contain returns located at soil level. The

VH/HV coherence minima, with a height close to the soil could point to second

order canopy interactions, characterised by a small return and a relatively large


variation in path length.

A study of the variation of the co-polar coherence (and associated retrieved

height; see Table 7.1) with angle of incidence at ground level shows that the max-

imal heights between 38 and 45◦ rise from 0.34 to 0.53 m, with an associated

coherence value varying from 0.9 at 38◦ incidence to 0.72 at 45◦. The polarisation

states used to retrieve these maximum heights are characterised by small values

of ellipticity angle, and orientation values in the vicinity of 70◦, suggesting that

the maximum height occurs for linear polarisation states, equivalent to the polar-

isation state obtained by a vertical dipole rotated by around 20◦. Note that the

ellipticity angle of these regions varies with angle of incidence, from around −17◦

at 36◦incidence to 5◦ at 46◦.

The maximum coherence is typically found at the top and bottom corners of the

polarisation space, in the RR-RR (χ = −45◦) and LL-LL (χ = 45◦) regions. The

value of coherence is slightly higher in the LL-LL region, where the retrieved height

is located within 0.03 m from the soil level. However, the results from RR-RR are

very similar to LL-LL.

These findings suggest that an estimate of the crop height can be found by using

the height difference between the RR-RR interferogram and an interferogram using

the polarisation state characterised by χ = 00;ψ = 70◦(ellipticity; orientation).

The best crop height estimate would be obtained at around 45◦ incidence, where

the second polarisation state seems to indicate an important contribution from the

top of the canopy, and the estimated height would be within a standard deviation

of the mean crop height. The crop height estimate would underestimate the mean

height of the crop for smaller angles of incidence, due to the weaker contribution

from the top of the canopy and to the strong soil return. These findings are

backed by the tomographic data in Section 7.3, where it was apparent that the

contribution from the top of the canopy is significant at around 45◦ incidence, and

that horizontal polarisations have a strong ground return for all angles of incidence.


0.0 m

0.1 m

0.3 m

0.4 m

0.2 m

Figure 7.6: Coherence and retrieved height as a function of antenna polarisationstates.Contours represent height.


0.0 m

0.1 m

0.3 m

0.35 m

0.2 m

Figure 7.7: Coherence and retrieved height as a function of antenna polarisation states.Cross-polar solution. Contours represent height.

7.7. Pauli Basis Inteferometry Results 161

7.7 Pauli Basis Inteferometry Results

The availability of fully polarimetric data can be used to produce interferograms

using other polarisation combinations. The Pauli basis (see Section 2.5) is useful for

data analysis, as the polarisations that make up the basis represent odd and even

bounce and diffuse scattering mechanisms. The new polarisations are obtained as

combinations of the elements of the recorded scattering matrix: Svv + Shh (odd

bounce), Svv − Shh (even bounce) and Svh(diffuse scattering).

In terms of a wheat canopy, the odd-bounce contribution would be made up

of the direct returns from the canopy and the soil, the even-bounce contribution

would include some second order interactions, such as stem-ground (located near

to or at soil level). The diffuse scattering contribution would characterise crop

elements such as leaves and emerging ears and second order canopy interactions

(located above the soil level for interactions happening towards the top of the

canopy, and below soil level for interactions occurring close to the soil level due

to the long path lengths). The results are shown in Fig. 7.8 (effective height and

coherence).

The odd and even bounce contributions only make sense for small angles of

incidence, where the backscattering recorded by the VV and HH channels is located

at similar locations. At larger angles of incidence, the strong attenuation of the

vertical polarisation results in the scattering centres being different, so no useful

information will be found by using the sum or difference of these two channels,

which are essentially imaging different targets. The diffuse scattering results are

of course identical to the VH-VH interferogram presented in Section 7.5.

7.8 Coherence Optimisation

7.8.1 Unconstrained Coherence Optimisation

The results from unconstrained coherence optimation in Fig. 7.9 show that the

effective height retrieved using the optimal polarisation states is usually located

slightly below ground level. The second solution rises to around 0.1 m between 38◦

and 44◦, then dropping back to below soil level at larger angles of incidence. The


−0.2

−0.15

−0.1

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0

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rieve

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]



OddEven

Diffuse


0.8

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0.9

0.92

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30 35 40 45 50 55

Ret

rieve

d H

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]



OddEvenDiffuse

(b) Coherence

Figure 7.8: Effective height and coherence from InSAR processing using odd bounce,even bounce and diffuse scattering projection vectors at C band.

third solution closely tracks the top of the soil up to 44◦, and then rises rapidly.

The coherence of the optimal solution is very close to unity and is flat over all

angles of incidence, whereas the second solution shows a slight variation with angle


of incidence. This variation is small, and the coherence for this solution is over 0.9

for all points except that at 49◦. The third solution is very similar to the coherence

of the VV-VV interferogram. shown in Fig. 7.5(a).

The retrieved polarisation states are shown in Tables 7.2-7.4 for samples located

at the centre of the target (different range bins located at the same cross-range

position). These states change substantially with azimuth (for the same range bin),

suggesting instabilities. The optimal solution is characterised by a very strong

contribution from the HH channel for all angles of incidence. At the front of

the canopy, there are contributions from the VV and VH channels. As range

increases, the contributions from the VV channel diminish, whereas those from

the VH channel rise. The polarisation state associated with the second solution

is nearly identical to the VH channel, whereas the third one is a mixture of a

strong VV component and smaller VH contributions. In all three solutions, there

is a substantial difference in the optimal polarisation states for the two images,

even when the small baseline (0.02 m) suggests that these should be identical

if they correspond to real physical scattering properties of the target. The fact

that different polarisations are found, and that there is a relatively large variation

of the retrieved polarisation states within samples at similar incidence angle is

probably due to the inadequate estimation of the covariance matrices, as discussed

in Chapter 5.

The optimal solution shows an effective height close to the soil level, which is

obtained either by a very strong contribution from the HH channel at near range,

or by a combination of the HH and VH channels at far range. The second solution

is basically the VH channel, while the third solution is made up of a significant

VV contribution.

7.8.2 Constrained Coherence Optimisation

The arguments outlined in Chapter 5 suggest that the results from constrained

coherence optimisation are expected to be similar to those of the unconstrained

case, with better stability in the retrieved height and slightly lower coherence

values.

The coherence plots shown in Fig. 7.10(b) are virtually identical to those shown


AOI[deg] Max. Height [m] Ellipticity [deg] Orientation [deg] Coherence

36 0.067 -19 90 0.8038.7 0.341 –17 79 0.9141.3 0.400 -3 72 0.8943.7 0.450 0 71 0.8845.8 0.526 5 69 0.7247.8 0.125 -22 98 0.68

Table 7.1: Maximum retrieved height and polarisation states needed to obtain it as afunction of angle of incidence (AOI) for samples located along the centre ofthe target region (co-polar solution).

Angle [deg] Image 1

HH VV VH

33.14 0.97 (-0.15,0.11) (0.15,0.08)

36.06 0.90 (-0.13,-0.04) (0.23,0.35)

38.77 (0.04,-0.44) (-0.02,0.19) 0.88

41.29 (-0.29,0.53) (0.10,-0.22) 0.76

43.64 0.69 (-0.22,0.01) (-0.34,-0.58)

45.81 (-0.26,0.58) (0.06,-0.20) 0.74

47.82 0.95 (-0.18,0.10) (-0.21,-0.04)

49.70 0.94 (-0.24,0.11) (0.02,0.23)

Angle [deg] Image 2

HH VV VH

33.14 0.93 -0.16 (0.32,0.07)

36.06 0.85 (-0.10,-0.04) (0.40,0.33)

38.77 (0.03,-0.46) (0.01,0.21) 0.86

41.29 (-0.38,0.50) (0.18,-0.19) 0.73

43.64 0.72 (-0.29,-0.04) (-0.38,-0.50)

45.81 0.72 -0.30 (-0.16,-0.61)

47.82 0.95 (-0.30,0.04) -0.13

49.70 0.96 (-0.16,0.05) (0.10,0.22)

Table 7.2: Polarisation states retrieved from the first solution to unconstrained opti-misation. Numbers in brackets are complex (real, imaginary).


Angle [deg] Image 1

HH VV VH

33.14 (-0.08,0.14) (0.03,-0.16) 0.97

36.06 (-0.03,0.20) (-0.10,-0.06) 0.97

38.77 (-0.08,0.20) (-0.08,-0.06) 0.97

41.29 (0.07,-0.16) (-0.02,0.09) 0.98

43.64 (0.06,-0.13) (-0.03,0.07) 0.99

45.81 (0.04,-0.15) (0.00,0.06) 0.99

47.82 (0.04,0.01) (0.03,-0.04) 0.99

49.70 (-0.07,0.07) (0.03,-0.09) 0.99

Angle [deg] Image 2

HH VV VH

33.14 (-0.14,0.11) (0.07,-0.15) 0.97

36.06 (-0.09,0.20) (-0.02,-0.02) 0.97

38.77 (0.03,0.15) (0.04,0.09) 0.99

41.29 (0.07,-0.18) (-0.02,0.09) 0.98

43.64 (0.07,-0.15) (-0.03,0.09) 0.98

45.81 (0.05,-0.16) (0.09,0.10) 0.98

47.82 (0.04,0.02) (0.12,-0.02) 0.99

49.70 (-0.10,0.08) (0.10,-0.10) 0.98

Table 7.3: Polarisation states retrieved from the second solution to unconstrained op-timisation. Numbers in brackets are complex (real, imaginary).


Angle [deg] Image 1

HH VV VH

33.14 (0.00,-0.21) (-0.22,0.63) 0.71

36.06 (-0.30,0.03) 0.80 (-0.16,-0.48)

38.77 (0.02,0.03) 0.78 (0.46,0.42)

41.29 (0.14,0.09) 0.75 (0.13,0.63)

43.64 (0.15,0.08) 0.71 (0.20,0.65)

45.81 (0.23,0.15) 0.77 (-0.04,0.57)

47.82 (0.08,0.14) 0.96 (0.06,-0.22)

49.70 (-0.13,0.14) 0.86 (0.46,-0.13)

Angle [deg] Image 2

HH VV VH

33.14 (0.14,-0.09) (-0.18,0.61) 0.75

36.06 (-0.26,-0.15) 0.76 (-0.21,-0.53)

38.77 (-0.04,0.06) 0.93 (0.15,0.33)

41.29 (0.02,0.17) 0.75 (-0.19,0.19)

43.64 (0.06,0.17) 0.93 (-0.15,0.28)

45.81 (0.08,0.25) 0.88 (-0.40,0.02)

47.82 (0.03,0.20) 0.91 (0.10,-0.35)

49.70 (-0.04,0.20) 0.89 (0.19,0.37)

Table 7.4: Polarisation states retrieved from the third solution to unconstrained opti-misation. Numbers in brackets are complex (real, imaginary).


−0.1

0

0.1

0.2

0.3

0.4

0.5

30 35 40 45 50 55

Ret

rieve

d H

eigh

t [m

]



Solution 1Solu1ion 2Solution 3


0.5

0.6

0.7

0.8

0.9

1

30 35 40 45 50 55

Coh

eren

ce




(b) Coherence

Figure 7.9: Effective height and coherence from unconstrained coherence optimisation.

in Fig 7.9(b), except that the constrained optimisation coherence is slightly lower

due to the loss of a degree of freedom in the optimisation procedure. The effec-

tive height shown in Fig. 7.10(a) for the highest coherence solution is close to

ground level for all angles of incidence, as in the unconstrained optimisation case.


−0.1

0

0.1

0.2

0.3

0.4

0.5

30 35 40 45 50 55

Ret

rieve

d H

eigh

t [m

]




(b) Effective Height

0.5

0.6

0.7

0.8

0.9

1

30 35 40 45 50 55

Coh

eren

ce




(b) Coherence

Figure 7.10: Effective height and coherence for constrained coherence optimisation.

However, the result from constrained optimisation shows less height variation over

all angles of incidence than the unconstrained case. The second solution is again

similar to that of unconstrained optimisation, being very close to soil level. In the

7.9. Conclusions 169

constrained case, the effective height remains closer to the soil level than in the

unconstrained case. Finally, the third solution shows a very similar pattern to the

VV-VV interferogram. The retrieved maximum height occurs at around 45◦, wth

a value of 0.48 m, closer to the mean crop height than the largest retrieved height

from the VV-VV interferogram.

The retrieved polarisation states are more stable than in the unconstrained case.

Samples adjacent in cross-range result in nearly identical retrieved polarisation

states and effective heights. In this respect, the constrained procedure is more

robust, as expected.

The interpretation of the polarisation states (see Tables 7.5-7.7) is similar to

that outlined in the previous Section, but some trends become more obvious. The

highest coherence retrieved from constrained optimisation is obtained by a mixture

of all polarimetric channels. While not inmediately clear, it can be shown that

these polarisation states are very similar to co-polar circular left-handed (LL)

polarisations. The second solution essentially consists of the cross-polar channel

signal, whereas the third solution mostly consists of the VV channel.

The results described in this Section are revealing. On the one hand, they show

that an interferogram generated from a polarisation similar to LL would be com-

posed of returns originating from the soil layer, irrespective of angle of incidence,

in line with the results of Section 7.6. The second solution recovers the cross-polar

channel, also located close to the soil, and a relatively large coherence. The third

solution is similar to the VV-VV interferogram, but results in a higher effective

height for large angles of incidence. This ties in with the results from Section 7.6,

which indicated that the maximum height is found with a linear polarisation close

to VV. The largest effective height occurs at around 45◦ with the third solution,

where the difference between the first and third solutions is of 0.53 m. This is very

close to the average crop height of 0.58 m.

7.9 Conclusions

This Chapter reports the application of the techniques presented in the preced-

ing chapters to a wheat canopy from the indoor RADWHEAT experiment when

the ears were beginning to emerge. This canopy was imaged inside the GB-SAR


Angle [deg] HH VV VH

33.14 0.94 (0.14,-0.07) (0.22,0.21)36.06 0.76 (0.13,-0.08) (0.27,0.57)38.77 0.87 (-0.37,-0.03) (0.30,-0.11)41.29 0.74 (-0.33,-0.01) (0.02,-0.58)43.64 0.79 -0.34 (0.03,-0.51)45.81 0.78 (-0.26,0.05) (0.04,-0.56)47.82 0.97 (-0.22,0.09) (0.08,0.04)49.70 0.94 (-0.19,0.09) (0.05,0.25)

Table 7.5: Polarisation states retrieved by constrained optimisation, first solution.Numbers in brackets are complex (real,imaginary).


33.14 (-0.15,0.14) (0.08,-0.01) 0.9836.06 (-0.09,0.22) (0.01,0.04) 0.9738.77 (0.01,-0.06) (0.01,0.04) 0.9941.29 (0.04,-0.16) (0.01,0.01) 0.9943.64 (0.04,-0.14) (0.01,0.01) 0.9945.81 (0.03,-0.15) (0.05,0.04) 0.9947.82 (0.02,0.01) (0.06,-0.04) 0.9949.70 (-0.08,0.07) (0.05,-0.09) 0.99

Table 7.6: Polarisation states retrieved by constrained optimisation, second solution.Numbers in brackets are complex (real,imaginary).


33.14 (-0.26,-0.08) 0.70 (-0.30,-0.59)36.06 (-0.36,-0.10) 0.78 (-0.26,-0.42)38.77 (-0.04,0.01) 0.88 (0.17,-0.45)41.29 (0.05,0.08) 0.96 (0.17,-0.19)43.64 (0.08,0.11) 0.98 (0.13,-0.12)45.81 (0.16,0.23) 0.91 (-0.26,0.18)47.82 (0.06,0.17) 0.94 (0.00,0.29)49.70 (-0.09,0.17) 0.89 (0.34,-0.24)

Table 7.7: Polarisation states retrieved by constrained optimisation, third solution.Numbers in brackets are complex (real,imaginary).


Single Polarisation Unconstrained Opt. Constrained Opt.

AOI [deg] VV-VV HH-HH VH-VH Sol. 1 Sol. 2 Sol. 3 Sol. 1 Sol. 2 Sol.3

36.1 0.041 -0.075 -0.086 -0.092 -0.116 -0.030 -0.048 -0.104 -0.066

38.8 0.332 0.009 0.052 0.017 0.065 0.014 -0.011 0.004 0.349

41.3 0.387 -0.001 0.051 -0.022 0.102 0.009 -0.018 0.020 0.406

43.6 0.433 -0.007 0.050 -0.032 0.102 -0.006 -0.020 0.017 0.455

45.8 0.429 -0.047 -0.033 -0.044 -0.019 0.173 -0.032 -0.040 0.484

47.8 0.079 -0.044 -0.065 -0.027 -0.033 0.212 -0.028 -0.068 0.143

49.7 -0.082 0.026 -0.012 0.026 -0.041 0.253 0.032 -0.058 0.118

51.4 0.012 0.088 -0.076 -0.040 -0.071 0.285 0.059 -0.086 0.212

Table 7.8: Mean retrieved heights (in meters) using single polarisation interferometryand coherence optimisation.



36.1 0.858 0.967 0.922 0.984 0.920 0.785 0.975 0.895 0.705

38.8 0.906 0.932 0.927 0.983 0.964 0.910 0.979 0.942 0.891

41.3 0.892 0.943 0.930 0.984 0.957 0.894 0.982 0.930 0.877

43.6 0.885 0.948 0.924 0.985 0.950 0.882 0.983 0.921 0.866

45.8 0.767 0.975 0.938 0.989 0.933 0.722 0.987 0.916 0.645

47.8 0.677 0.981 0.902 0.987 0.915 0.633 0.986 0.904 0.595

49.7 0.644 0.963 0.882 0.973 0.892 0.595 0.969 0.886 0.535

51.4 0.679 0.940 0.877 0.972 0.902 0.656 0.958 0.878 0.621

Table 7.9: Mean coherence using single polarisation interferometry and coherence op-timisation.



36.1 0.016 0.008 0.008 0.040 0.030 0.062 0.004 0.015 0.057

38.8 0.036 0.026 0.009 0.023 0.026 0.106 0.010 0.031 0.039

41.3 0.033 0.031 0.013 0.023 0.048 0.121 0.005 0.030 0.050

43.6 0.036 0.030 0.016 0.020 0.046 0.134 0.003 0.032 0.060

45.8 0.040 0.018 0.025 0.032 0.015 0.125 0.007 0.033 0.055

47.8 0.076 0.006 0.018 0.016 0.018 0.237 0.006 0.018 0.188

49.7 0.208 0.023 0.017 0.064 0.039 0.254 0.017 0.039 0.301

51.4 0.200 0.016 0.041 0.055 0.068 0.162 0.016 0.034 0.218

Table 7.10: Standard deviation of the retrieved heights (in meters) using single polar-isation interferometry and coherence optimisation.


anechoic chamber using a two-dimensional synthetic aperture at C band. The

gathered data were fully polarimetric, and interferometric processing was carried

out by selecting two pairs of azimuth scans.

In order to help the interpretation of the interferometric results, an analysis of

the 3D dataset has been carried out, in addition to that presented in [Brown et al.,

2003]. The aim of this extra analysis was to examine the polarimetric nature of

the scattering within the canopy. In particular, it tried to ascertain under which

conditions single and double bounce contributions had a significant effect. At the

front of the canopy at ground level, and at the top of the canopy for large angles of

incidence, the returns were clearly classified as a single bounce. Elsewhere in the

target region, the low magnitude of the correlation coefficient prevented any firm

conclusions on the nature of the phase difference, even though the distribution of

the phase difference in this region was biased towards large phases, indicative of

even bounces.

The use of single polarisation interferometry using the linear H/V basis resulted

in the HH and VH channels retrieving a height close to the top of the soil layer,

whereas the retrieved height from the VV channel increased with the angle of

incidence, reaching a maximum value of 0.43 m at around 45◦. The coherence for

the HH and VH channels was very high, while that of the VV channel was slightly

lower, due to the larger volumetric contribution.

The variation of coherence with polarisation state was investigated. The plots

for coherence (and the related retrieved height) showed the following significant

features:

1. The largest values of coherence occurred in the region associated with circu-

larly polarised states,

2. The highest coherence (with an associated height located very close to the

soil layer) was obtained in the vicinity of the LL-LL region,

3. The lowest coherence values were located close to the vertical polarisation,

4. The largest values of retrieved height were found to range from 0.34 to 0.53 m

(the largest values for larger angles of incidence). These values were obtained


using polarisation states characterised by an orientation angle around 70◦ and

an ellipticity angle close to 0◦, i.e. a linear polarisation.

These findings suggest that a good estimate of the mean crop height could be

retrieved at around 45◦ incidence, by subtracting the the LL-LL retrieved height

from the top of the canopy contribution described by the previously indicated

linear polarisation state.

Interferograms using a Pauli polarisation basis were also presented. They pro-

vided little or no insight into the interpretation of the data, as the combinations

of the co-polar channels resulted in combination of two different phase centres

(especially at large angles of incidence).

Results from both constrained and unconstrained coherence optimisation have

also been reported in this Chapter. It was found that unconstrained optimisa-

tion results were unstable, probably due to poor estimation of the covariance and

cross-covariance matrices. The retrieved height varied significantly between neigh-

bouring pixels using this method, and the algorithm also resulted in different po-

larisation states for the two images, which would not be expected for the small

baseline used for the interferometric pair generation.

Constrained optimisation provided stable results between neighbouring pixels,

and the retrieved polarisation states that agreed with some of the findings outlined

in the polarisation synthesis analysis. The optimal polarisation states retrieved

were very similar to the LL-LL interferogram (highest coherence, retrieved height

very close to soil level), the VH-VH interferogram (very high coherence, very close

to the soil), and the third solution showed a significant VV contribution, with

similarly valued HH and VH contributions.

The results presented in this Section agree well with some of the results obtained

using electromagnetic models by Stiles and Sarabandi [Stiles et al., 2000] and Pi-

card [Picard et al., 2003] for the VV-VV interferogram, where it is shown that there

is a significant contribution from the flag leaves at the top of the canopy with a

highly attenuated ground return, which is explained by the vertical orientation of

the stems. The HH-HH signal according to these two models mostly consists of

a significant ground return and a stem-ground component. These contributions

suffer little attenuation when travelling through the canopy as there is little wave


interaction with the stems. The results from the work of [Marliani et al., 2002]

are not confirmed from the experimental data presented in this Section. [Marliani

et al., 2002] suggest an equivalent height for all the single polarisation interfero-

grams very close to the soil, whereas the results presented in this Section show

a significant deviation from this behaviour for the VV-VV interferogram. This

discrepancy arises from the assumption in [Marliani et al., 2002] that the main

scattering mechanism in a wheat canopy will be a stem-ground interaction (irre-

spective of polarisation).

In the light of these results, the following points can be made

• The soil level can be retrieved with a circular polarisation state, LL.

• The top of the canopy can be retrieved with a linear polarisation state close

to VV for large angles of incidence.

• The cross-polar interferogram consistently results in a signal close to the soil

level.

In terms of the insight gained into the scattering behaviour of the crop, note that

• The horizontal polarisation suffers little attenuation going through the

canopy, irrespective of angle of incidence.

• The vertical polarisation is strongly attenuated. This arises from the align-

ment of the field with vertical structures in the canopy (stems). The atten-

uation increases with angle of incidence.

• The cross-polar return presents problems in its interpretation. The retrieved

height is very close to the soil level, suggesting that either the backscattering

is located there, or that second order canopy interactions result in path

lengths that are equivalent to a direct ground return. The relatively small

soil roughness and the retrieved height being a few centimetres above the

soil seem to suggest that this return could arise from interactions occurring

within the top of the canopy.

Some interesting methodological conclusions can also be drawn:


• The use of unconstrained optimisation for the GB-SAR system is ill-advised,

as the results are unstable due to the small number of samples available,

• Constrained optimisation provides a robust solution that agrees well with

the analysis carried out using coherence synthesis,

• Interferograms created using a Pauli basis seem to be of little use for natural

targets such as that presented here, as the scattering centres in different

channels are not coincident.

The results presented in this Chapter show great promise for the use of polari-

metric interferometry to retrieve crop height. It would be of interest to examine

this behaviour throughout the growing season, with different crop varieties, soil

conditions and crop densities, so as to try to generalise these findings.




36.1 0.018 0.003 0.011 0.004 0.008 0.038 0.006 0.011 0.029

38.8 0.023 0.013 0.006 0.008 0.008 0.015 0.008 0.006 0.025

41.3 0.033 0.011 0.005 0.006 0.008 0.020 0.006 0.009 0.020

43.6 0.032 0.010 0.005 0.005 0.009 0.018 0.006 0.006 0.019

45.8 0.051 0.010 0.005 0.003 0.005 0.055 0.004 0.008 0.047

47.8 0.023 0.002 0.015 0.003 0.019 0.036 0.002 0.018 0.022

49.7 0.030 0.003 0.027 0.004 0.019 0.087 0.004 0.018 0.103

51.4 0.053 0.005 0.018 0.003 0.019 0.096 0.003 0.019 0.101

Table 7.11: Standard deviation of the coherence using single polarisation interferome-try and coherence optimisation.

Chapter 8Conclusions and Further Work

This Chapter sums up the main findings of this Thesis and suggests avenues

for further work in the area of polarimetric interferometry. The Chapter

is structured into four Sections: the use of the indoor GB-SAR instrument

for polarimetry and interferometry, basic science results for polarimetric interfer-

ometry of layered targets, polarimetric interferometry studies of a wheat canopy

and suggestions for further work.

8.1 Polarimetry and Interferometry with GB-SAR

GB-SAR provides a highly controlled environment to carry out interferometric

experiments that can be used to understand the interaction of electromagnetic

waves with targets, prior to the design of airborne and spaceborne campaigns.

However, the geometry of GB-SAR is different to that of conventional sensors.

This difference needs to be understood to allow meaningful comparisons between

results obtained using GB-SAR and conventional air and spaceborne sensors.

The main difference between GB-SAR and other SARs is the change in angle

of incidence across the resolution cell, which results in returns located at different

heights and imaged with a different angle of incidence being combined in the

resulting resolution element. This is not the case in air and spaceborne sensors,

where the angle of incidence is approximately constant within the resolution cell.

In GB-SAR, the change of angle of incidence is dependent on the position of the

177

8.1. Polarimetry and Interferometry with GB-SAR 178

swath and on the vertical spread of the target. The largest angle changes occur at

near range and for targets that exhibit a large vertical spread.

To reconcile results obtained from GB-SAR and from conventional systems, it

is recommended that the target should be imaged with a large angle of incidence,

and that the vertical spread should be as small as possible. In Chapter 4, the

change in angle of incidence at 45◦ for a target with a vertical spread of 0.5 m is

calculated to be around 11◦, for a typical GB-SAR set-up.

Another property of GB-SAR is that the angle of incidence changes across the

swath. While on the one hand this is beneficial, as it allows the study of angle of

incidence effects, it also leads to a decreased number of independent samples, as

the target can only be considered homogeneous in azimuth, not in range. This will

present a problem in polarimetric and interferometric applications, where multi-

look processing is often used. To overcome this limitation, two approaches are

suggested:

1. the bandwidth can be split to generate several reduced resolution indepen-

dent looks,

2. the target can be rotated to generate independent samples, if the target is

symmetric in azimuth.

The study of interferometric coherence in GB-SAR interferometry resulted in two

new procedures for data analysis:

1. A way to predict the interferometric height and coherence based on azimuth-

averaged three-dimensional reconstructions,

2. An iterative interferometric phase to height conversion.

The ability to produce interferometric measurements with GB-SAR has been tested

with an artificial target that confirmed that the GB-SAR system is capable of

retrieving height information from an interferometric set-up. The measurements

show very good height estimation accuracy, very high coherence, and have also

been used for polarimetric interferometry validation.

8.2. Polarimetric Interferometry of Layered Targets 179

8.2 Polarimetric Interferometry of Layered Targets

In this Thesis, layered targets play an important role, since natural targets are often

modelled as a group of layers located at different heights, each layer characterised

by aspecific geometric and scattering properties. In Chapter 5, the usefulness of

the combination of polarimetry and interferometry to retrieve information about

individual layers was investigated. Two important properties of layered targets

were demonstrated:

1. The coherence of the layer with the highest coherence is always larger than

that of the ensemble.

2. Layers can be separated depending on their polarimetric properties, if the

scattering vectors characterising the scatterers in the different layers are in-

dependent (i.e., a polarisation state that masks returns from all layers but

one can be found)

The second property allows layer separation, provided the scattering properties of

the target are known beforehand, which is not usually the case. However, the com-

bined use of polarimetry and interferometry allows, under some conditions (such as

layers being characterised by linearly independent scattering mechanisms) the sep-

aration of layers by locating coherence maxima as a function of polarisation. This

can be achieved either by an exhaustive search or by using coherence optimisation

techniques.

The use of coherence optimisation techniques was studied. The published op-

timisation technique assumes that two different polarisation states will be needed

(one for each pass), but in situations where temporal decorrelation is not an issue,

optimisation can be constrained to use a common polarisation state for both im-

ages in the pair. This constrained algorithm was developed, and both algorithms

were used to analyse layered targets. It was found that:

1. Unconstrained optimisation fails to separate layers when these are charac-

terised by similar coherences;

2. Constrained optimisation results in greater retrieved height accuracy for the

same number of looks, compared to unconstrained optimisation.

8.3. Study of a Wheat Canopy 180

The previous results were demonstrated using an analytical approach, numerical

simulations, and experiments using GB-SAR with an artificial target made up of

two layers, each populated wtih orthogonal scattering mechanisms. The exper-

imental results confirmed the superior results from constrained optimisation for

layer separation, whereas unconstrained optimisation failed to separate the lay-

ers, as the coherence associated with each layer was nearly identical. On the

basis of this evidence, constrained coherence optimisation provides a useful tool

for analysing polarimetric interferometry data with the GB-SAR system.

Another useful procedure for analysing polarimetric interferometry data is the

study of the co-polar and cross-polar coherence as a function of polarisation. These

plots provide a visual means of interpreting the effect of polarimetry on the in-

terferometric coherence, and are easier to interpret than the optimal scattering

mechanisms from coherence optimisation.

8.3 Study of a Wheat Canopy

As a demonstration of the techniques and issues discussed in this Thesis, C band

images of a wheat canopy from the RADWHEAT experiment were used to gain

insight into the interactions that give rise to the interferometric and polarimetric

responses. Full 3D reconstructions of the canopy are also available. The chosen

canopy was imaged on June 18, 1999. At this stage, the ears were just emerging

and the mean crop height was 0.58 m.

The magnitude and phase of the co-polar complex correlation coefficient were

studied in the three-dimensional reconstructions. The three-dimensional data were

averaged in azimuth, taking care of averaging only samples with the same angle of

incidence. The results separate three distinct areas:

1. Small phase difference (i.e., single bounce) and large correlation magnitude

in areas of small angle of incidence at ground level where the waves have

traversed a small amount of canopy;

2. Small phase difference (i.e., single bounce) and large correlation magnitude

in areas of large angle of incidence, and towards the top of the canopy,

indicating significant backscattering from the flag leaves;

8.3. Study of a Wheat Canopy 181

3. A large area of low correlation magnitude and relatively large phase differ-

ences (i.e., double bounce), located at soil level in zones where the waves

had traversed a full canopy. The low magnitude of the correlation coefficient

results in a large phase difference spread.

Results from single polarisation interferometry (using the linear and Pauli bases),

coherence synthesis and coherence optimisation (both constrained and uncon-

strained) are analysed and the following observations were made:

1. In GB-SAR experiments with targets similar to wheat, due to the shape of

the resolution cell, several regions can be identified: an area at the front

of the canopy where the incident wave suffers little attenuation as it only

traverses an incomplete canopy; an area where the incident wave suffers full

canopy attenuation; an area where only attenuated soil returns are present;

2. The height retrieved from the HH-HH interferogram is located at soil layer

for all angles of incidence, indicating that interactions with the canopy are

of small consequence;

3. The VH-VH interferogram retrieves a height close to the soil layer, which

could arise from higher order interactions occurring within the canopy;

4. The retrieved height of the VV-VV interferogram shows a marked depen-

dence on angle of incidence, with height increasing up to the flag leaves level

with increasing angle of incidence. This is due to the increased coupling of

the flag leaves with the incident field, and to the large attenuation suffered

by the wave as it traverses the canopy;

5. The use of the Pauli basis did not result in useful results, since the combina-

tions of channels resulted in the combination of different scattering centres

(i.e., scattering mechanisms not located at the same level);

6. Coherence synthesis indicates that interferograms made using circular polar-

isations (LL-LL in particular) will accurately retrieve the soil layer;

7. The top of the canopy can be estimated using linear polarisations close to

VV-VV. Best estimations of canopy height are obtained using larger angles

8.4. Suggestions for Further Work 182

of incidence (see point 3 above), and a linear polarisation characterised by

an orientation angle of around 70◦;

8. Unconstrained coherence optimisation results are not particularly stable, due

to the small number of independent samples available.

9. Constrained optimisation results in three solutions:

a) The height retrieved from the optimal coherence solution is very close

to the top of the soil, and is characterised by a large coherence. The

polarisation states used is close to LL;

b) The height retrieved from the second solution is close to the soil level,

and the polarisation state used is very similar to VH;

c) The retrieved height from the third solution increases with angle of

incidence, up to 0.48 m at around 45◦. The polarisation state used to

retrieve this height has a large VV component

10. The results from constrained optimisation broadly agree with the results from

coherence synthesis, indicating that this technique provides a useful means

of analysing GB-SAR data.

8.4 Suggestions for Further Work

Clearly, the GB-SAR instrument has shown great potential for vegetation studies.

In particular, the use of the whole RADWHEAT dataset could help understand

temporal variations of the polarimetric and interferometric measurements, which

could be used as a basis to define strategies for wheat monitoring using microwave

radar. Clearly investigation of other varieties of wheat, crop densities and the

influence of soil moisture could also be carried out in future experiments.

The study of other crop types could result in interesting and useful results, while

the testing and development of soil moisture estimation algorithms could also be

explored by using GB-SAR polarimetric capabilities.

Finally, while this Thesis has only been concerned with the use of the indoor

component of GB-SAR, a relatively simple modification of the outdoor component

8.4. Suggestions for Further Work 183

(ading an extra stepper motor to lower the boom and thus be able to provide

a vertical baseline) would result in a portable, high resolution, fully polarimetric

interferometric SAR system, which could be deployed in fields for measurements

similar to those outlined in this Thesis. The outdoor system would have a number

of advantages over the indoor system: realistic canopies, real imaging conditions,

larger frequency coverage, a larger number of samples and angles of incidence.

Results from the experiments suggested above should be contrasted with data

obtained from airborne and spaceborne sensors and from electromagnetic models.

This would allow for the generalisation of the measurements, and could pave the

way for the design of new missions.

Finally, a careful study of polarimetric interferometry and its use for biomass

monitoring applied to forests should be undertaken. This study should be formu-

lated around a study of sound physical models, both at the tree as well as at the

forest level. These studies could be combined with VHF measurements, so as to

come up with a sensor configuration for forest biomass measurements.

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