375

A vector radiative-transfer model forthe Odin/OSIRIS project

C.A. McLinden, J.C. McConnell, E. Griffioen, andC.T. McElroy

Abstract: A vector radiative-transfer code has been developed that is able to accuratelyand efficiently calculate radiance and polarization scattered from Earth’s limb. A primaryapplication of this code will be towards generating weighting functions, based on calculatedlimb radiances, for the retrieval of trace gases (O3, NO2, BrO, OClO, and O4) from theoptical spectrograph and infrared imaging system (OSIRIS). OSIRIS is a UV–visibleinstrument on board the Odin satellite that measures limb-scattered light. This modelsolves the vector radiative-transfer equation using an iterative technique simultaneously inboth plane-parallel and spherical-shell atmospheres. OSIRIS simulated limb radiance andpolarization and OSIRIS weighting functions are presented along with a discussion of thenumerical solution parameters, model intercomparisons and timings, and necessary modelimprovements. Overall agreement with other models was found to be very good and modelspeed is comparable to a fast finite-difference code. A set of OSIRIS reference atmosphereshave been compiled for use with radiative-transfer models. Each of the 216 atmospheres (18latitudes× 12 months) include profiles of air, pressure, temperature, ozone, NO2, BrO, andstratospheric aerosols.

PACS Nos.: 42.68-w, 94.10Gb

Résumé: Nous avons développé un programme pour calculer le transfert radiatif vectorielet pouvant évaluer avec précision et efficacité la radiance et la polarisation de la lumièrediffusée par le limbe terrestre. Une application importante de ce programme est la générationde fonctions de poids basées sur le calcul des radiances du limbe, afin d’identifier lescomposés à l’état de trace (O3, NO2, BrO, OClO et O4) à partir des mesures obtenues àl’aide du spectrographe optique et du système d’imagerie infrarouge (OSIRIS). OSIRIS estun instrument du satellite Odin dont le rôle est de mesurer la diffusion de la lumière parle limbe terrestre dans le domaine UV–visible. Notre modèle numérique résoud l’équationde transfert radiatif vectoriel en utilisant une méthode itérative simultanément pour desatmosphères à plans parallèles et à couches sphériques. Nous présentons des résultatsOSIRIS simulés pour la radiance du limbe et la polarisation, ainsi que les fonctions de poidssimulés OSIRIS, accompagnés de comparaisons avec d’autres modèles et de suggestionspour améliorer le modèle. Nous trouvons un très bon accord global avec d’autres modèlesnotre vitesse de calcul est comparable à celle de modèles rapides utilisant les différencesfinies. Nous avons complété un ensemble de coupes d’atmosphères OSIRIS de référencepour utilisation avec des modèles de calcul de transfert radiatif. Chacune des 216 coupes

Received 13 May 1999. Accepted 23 November 2001. Published on the NRC Research Press Web site athttp://cjp.nrc.ca/ on 21 March, 2002.

C.A. McLinden.1,2 Department of Physics and Astronomy, York University, Toronto, ON M3J 1P3, Canada.J.C. McConnell and E. Griffioen. Department of Earth and Atmospheric Science, York University, Toronto,ON M3J 1P3, Canada.C.T. McElroy. Meteorological Service of Canada, Downsview, ON M3H 5T4, Canada.

1Present Address: Meteorological Service of Canada, Downsview, ON M3H 5T4, Canada.2Corresponding author (email: chris.mclinden@ec.gc.ca).

Can. J. Phys.80: 375–393 (2002) DOI: 10.1139/P01-156 © 2002 NRC Canada

376 Can. J. Phys. Vol. 80, 2002

(18 latitudes× 12 mois) inclut des profils pour l’air, la pression, la température, l’ozone, lebioxyde d’azote, l’oxyde de bore et des aérosols stratosphériques.

[Traduit par la Rédaction]

1. Introduction

Satellite measurements of near-UV to near-IR solar radiation have been used for many years toprovide global information on the trace composition of the atmosphere. These instruments generallymeasure either scattered sunlight in the nadir direction [1] or direct sunlight via solar occultation [2].Until recently, the use of sunlight scattered from the limb of the Earth was generally avoided due to thecomplexity of the inversion process. The optical spectrograph and infrared imaging system (OSIRIS)[3,4] will be the first satellite instrument to make measurements of limb-scattered sunlight with others,such as SCIAMACHY (scanning imaging absorption spectrometer for atmospheric chartography) [5],to follow.

OSIRIS, a Canadian instrument, was launched on the Swedish Odin satellite in February 2001. Odinwas inserted into a circular, Sun-synchronous, near-terminator orbit at an altitude of approximately600 km [4]. The optical sensor of OSIRIS measures scattered sunlight from Earth’s limb over a range oftangent heights from 280–450 nm at 1 nm resolution and from 450–800 nm at 2 nm resolution. Inversionof OSIRIS radiances will yield vertical profiles of important atmospheric constituents between 10 and70 km including aerosols, O3, O4, NO2, BrO, and OClO. These measurements will enable studies ofpolar and midlatitude ozone loss, stratospheric–tropospheric exchange, as well as other processes in themiddle atmosphere that relate to anthropogenic impacts on the Earth’s atmosphere [4].

Retrieval of these constituents will require a forward radiative-transfer model capable of accuratelycalculating the radiation field. The forward model is used to generate the weighting functions used in theinversions and also to simulate OSIRIS measurements to test the retrieval algorithms [6,7]. There are agreat many radiative-transfer codes described in the literature; however, the vast majority of them eitherignore polarization and (or) obtain a solution appropriate only for a plane-parallel (PP) atmosphere. Formany applications, calculations using PP geometry is adequate. However, for solar zenith angles (SZAs)larger than about 75◦ the attenuation of the direct solar beam is overestimated in a PP atmosphere. Similarproblems arise in the calculation of scattered light within about 10◦ of the limb. Further, the conceptof tangent height is not applicable as any down-looking angle must intercept the entire atmospherebelow it (as well as the surface). Given OSIRIS’ limb-viewing nature, it is critical to account for thecurvature of the Earth in forward modelling. The Odin orbit is such that the angle between the OSIRISline-of-sight and the Sun is usually close to 90◦. This implies the measured radiances will be largelypolarized and hence polarization effects such as the grating efficiency are potentially important [8]. Fewmodels currently exist that calculate polarization while allowing for the sphericity of the Earth, andmost of these use the computationally expensive Monte Carlo solution method, see, for example, refs.9 and 10. One exception is that of Herman et al. [11].

This paper describes a vector radiative-transfer model that incorporates these key features withoutincurring the computational overhead required by most other models. A new source-function scalingmethod designed to further speed up the model by as much as a factor of 20 is also outlined.

2. Vector radiative-transfer model

It is convenient to describe the atmospheric radiation field, including polarization, using the Stokesvector

I =

I

Q

U

V

= [

I Q U V]T (1)

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McLinden et al. 377

whereI is the Stokes vector, a 4× 1 column matrix, often written as the transpose of a row matrixfor convenience ([ ]T denotes transpose). Each component has units of radiant intensity, or radiance(e.g.,µW cm−2 nm−1 sr−1). The first component,I , describes the total (polarized and unpolarized)radiance. If polarization is neglected, the Stokes vector reduces to a single element:I (the so-calledscalar approximation). The second component,Q, is the radiance linearly polarized in the directionparallel or perpendicular to a reference plane. The third,U , is the radiance linearly polarized in thedirection 45◦ to a reference plane, and the fourth,V , is the radiance circularly polarized. This referenceplane is taken to be the meridian plane, or the plane defined by the direction of propagation and thevertical. Previous studies have determined that the exclusion of polarization can lead to errors in radianceas large as 10% for pure-Rayleigh atmospheres [12].

The basic quantities that are used to describe the optical properties of the atmospheric medium are

• σe, σs, andσa: The extinction, scattering, and absorption cross sections, expressed in units of areaand are related byσe = σs + σa. These are the macroscopic optical properties of the atmosphericmedium.

• ke(z), ks(z), andka(z): The extinction, scattering, and absorption coefficient profiles expressedas functions of altitude,z, and having units of inverse length. Each is related to its cross-sectionanalogue byk(e,s,a) = n(z)σ(e,s,a) (so thatke = ks + ka) and wheren(z) is the number-densityprofile. For an atmosphere that consists ofNl different types of molecules and aerosols, the totalextinction coefficient is

ke(z) =Nl∑l

nl(z)σe,l =Nl∑l

nl(z)(σs,l + σa,l) (2)

• ω̃: the single-scattering albedo defined as

ω̃(z) = ks(z)

ke(z)= ks(z)

ks(z) + ka(z)(3)

that represents the fraction of radiation scattered to the total removed. For conservative scattering,ω̃ = 1, and for pure absorption,̃ω = 0.

• τ : vertical optical depth. It is defined as

dτ

dz= −ke(z), τ (z) =

∫ ∞

z

ke(z′) dz′ (4)

so that at the top of the atmosphere,τ = 0, and it increases with decreasing altitude with the totaloptical depth of the atmosphere defined asτ1. It is convenient to useτ in lieu of z as the verticalco-ordinate.

2.1. Solution in a plane-parallel atmosphere

The equation that governs the transfer of solar radiation in an absorbing and scattering atmosphereis called the vector radiative-transfer equation (VRTE). It can be derived based solely on heuristicarguments [13] or from the Boltzmann transport equation for photons [14] and its solution results in adescription of the radiation field, or Stokes vector, at each point in the atmosphere.

The VRTE is initially solved in a one-dimensional planar, or plane-parallel (PP), atmosphere inwhich all atmospheric quantities, including the radiation field, are invariant in thex- andy-directions.It is natural to describe the directional dependence of the Stokes vector in terms of spherical polar

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378 Can. J. Phys. Vol. 80, 2002

co-ordinates: a zenith angle,θ , and an azimuthal angle,φ. In a PP atmosphere, the monochromaticVRTE has the form [15]

µdI (τ ; µ, φ)

dτ= I (τ ; µ, φ) − J (τ ; µ, φ) (5)

whereµ = cosθ (by convention+µ represents upward propagating radiance streams and−µdownwardpropagating) andJ (τ ; µ, φ) is the source-function vector for a scattering atmosphere. Physically, thesource-function vector represents scattering into the radiance stream propagating in direction(µ, φ)

from all other directions and can be expressed as,

J (τ ; µ, φ) = ω̃(τ )

4π

∫ 2π

0

∫ 1

−1I (τ ; µ′, φ′)Z(τ ; µ, φ; µ′, φ′) dµ′ dφ′

+ ω̃(τ )

4ππFoZ(τ ; µ, φ; −µo, φo) e−τs(θo) (6)

whereπFo = [πFo 0 0 0]T is the unpolarized extra-terrestrial solar flux vector incident in the directiondefined by(−µo, φo) andZ is the phase matrix. The phase matrix is a 4×4 linear transformation matrixthat relates the scattered Stokes vector to the incident Stokes vector (see Appendix A). The first term onthe right in (6) represents the multiple-scattered radiation and the second term on the right representsthe single-scattered radiation. The only allowance made for the sphericity of the Earth at this point isin the calculation of the attenuated solar flux andτs is used to represent the slant optical depth alongthe direction of the direct solar beam. Note thatI in (5) and (6) describes the diffuse, or scattered, fieldonly. The direct, or unscattered, component must be added separately.

There are many techniques available to solve (5), each with its own advantages and drawbacks.A survey of some of the more common ones can be found in Hansen and Travis [15]. The methodused herein is the successive orders of scattering technique. This is an iterative technique in which theStokes vector is calculated for photons scattered once, twice, three times, etc., with the total Stokesvector obtained as the sum over all orders [15]. This method offers the advantage of being physicallyintuitive and easily adaptable to a spherical-shell (SS) geometry. A main drawback of this method isthat for optically-thick atmospheres the number of scattering orders required for convergence becomesprohibitive. However, as only optically thin, cloud-free atmospheres are considered, this presents noproblems.

It is convenient to first expand out the azimuthal dependence of the Stokes vector and phase matrixusing a Fourier series, such that,

I (τ ; µ, φ) = I c0(τ ; µ) + 2M∑

m=1

[I cm(τ ; µ) cosmφ + I sm(τ ; µ) sinmφ] (7)

Z(τ ; µ, φ; µ′, φ′) = Zc0(τ ; µ; µ′) + 2M∑

m=1

[Zcm(τ ; µ; µ′) cosm(φ − φ′)

+ Zsm(τ ; µ; µ′) sinm(φ − φ′)] (8)

wherem is the harmonic index andM is the highest harmonic used in the expansion. Substituting (7)and (8) into (5) and making use of the orthogonality property of sines and cosines, a series of 2M + 1azimuthally-independent VRTEs are arrived at

µdI c0(τ ; µ)

dτ= I c0(τ ; µ) − ω̃(τ )

2

∫ 1

−1Zc0(τ ; µ; µ′)I c0(τ ; µ′) dµ′

= I c0(τ ; µ) − J c0(τ ; µ) (9)

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McLinden et al. 379

µdI cm(τ ; µ)

dτ= I cm(τ ; µ) − ω̃(τ )

2

∫ 1

−1[Zcm(τ ; µ; µ′)I cm(τ ; µ′) − Zsm(τ ; µ; µ′)I sm(τ ; µ′)] dµ′

= I cm(τ ; µ) − J cm(τ ; µ) (10)

µdI sm(τ ; µ)

dτ= I sm(τ ; µ) − ω̃(τ )

2

∫ 1

−1[Zsm(τ ; µ; µ′)I cm(τ ; µ′) + Zcm(τ ; µ; µ′)I sm(τ ; µ′)] dµ′

= I sm(τ ; µ) − J sm(τ ; µ) (11)

wherem = 1, 2, . . . , M. The second equality in each equation simply defines the even- and odd-source-function vectors that will be used below.

Each of the 2M + 1 expansion coefficients are solved using the successive orders of the scatteringsolution technique [15]. The formal solution to the VRTE is obtained by integrating (9)–(11) fromτ ′to τ . The upward and downward Stokes vector streams of scattering ordern for Fourier coefficientmare given by

I (c,s)mn (τ ; µ) = I (c,s)m

n (τ ′; µ) e−(τ ′−τ)/µ +∫ τ ′

τ

J (c,s)mn (τ ′′; µ) e−(τ ′′−τ)/µ dτ ′′

µ(12)

I (c,s)mn (τ ; −µ) = I (c,s)m

n (τ ′; −µ) e−(τ−τ ′)/µ +∫ τ

τ ′J (c,s)m

n (τ ′′; −µ) e−(τ−τ ′′)/µ dτ ′′

µ(13)

wheren ≥ 1. The superscript(c, s)m is a shorthand used to denote either the even,cm, or odd,sm,coefficients. Note thatµ ≥ 0 in the PP solution (likewise,−µ ≤ 0). The source-function vector ofordern + 1 is calculated based on the Stokes vectors of ordern

J cmn+1(τ ; µ) = ω̃

2

∫ 1

−1[Zcm(µ; µ′)I cm

n (τ ; µ′) − Zsm(µ; µ′)I smn (τ ; µ′)] dµ′ (14)

J smn+1(τ ; µ) = ω̃

2

∫ 1

−1[Zsm(µ; µ′)I cm

n (τ ; µ′) + Zcm(µ; µ′)I smn (τ ; µ′)] dµ′ (15)

The zero-order Stokes vector is

I cm0 (τ ; −µ) = πFo e−τs(θo)δ(µ + µo) (16)

that gives rise to the single-scattering source-function vector when equation (16) is substituted into (14)and (15). The total Stokes vector is the sum over all orders

I (c,s)m(τ ; ±µ) =∞∑

n=1

I (c,s)mn (τ ; ±µ) (17)

In practice, the summation in (17) is truncated once some convergence criterion is met, as discussedbelow. Once all the required Fourier coefficients are known, the azimuthally-dependent Stokes vectoris reconstructed using (7).

Integration overµ is performed using Gaussian quadrature where theµj ’s are determined by thezeros of an even Legendre polynomial [16]. The vertical grid is divided up intoNi layers in altitudespace where the lowest layer is bounded by levelsi = 1 andi = 2 and the top layer byi = Ni andi = Ni +1. The source-function vector elements are taken as varying linearly through a layer that gives,based on (12) and (13),

I (c,s)mn (τi+1, µj ) = I (c,s)m

n (τi, µj ) e−1τi,i+1 + αi+1J(c,s)mn (τi, µj ) + βi+1J

(c,s)mn (τi+1, µj ) (18)

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380 Can. J. Phys. Vol. 80, 2002

I (c,s)mn (τi, −µj ) = I (c,s)m

n (τi+1, −µj ) e−1τi,i+1 + αiJ(c,s)mn (τi+1, −µj ) + βiJ

(c,s)mn (τi, −µj ) (19)

where

αi = 1 − e−1τi,i+1

1τi,i+1− e−1τi,i+1 (20)

βi = 1 − 1 − e−1τi,i+1

1τi,i+1(21)

are linear interpolating coefficients and

1τi,i+1 =(

ke,i + ke,i+1

2

) (zi+1 − zi

µj

)(22)

represents the slant optical depth through a layer between levelsi andi + 1 andke,i ≡ ke(zi).For each scattering order, the solution begins at the top of the atmosphere using (19) to calculate

the downward Stokes vector ati = Ni . The solution is propagated down to the surface where thesurface-reflected Stokes vector is calculated (see below). The upward Stokes vectors are then calculatedusing (18) in an analogous manner.

2.2. Boundary conditionsThe two boundary conditions that must be specified are the inward-source Stokes vectors at the

bottom and top of the atmosphere. From the near-UV to the near-IR, there is no diffuse source at thetop of the atmosphere soI (0; −µ, φ) = [0 0 0 0]T.

At the bottom, a diffuse source arises from reflection by the surface. The reflected Stokes vector isa function of the downward Stokes vector incident upon the surface

I (τ1; µ, φ) =∫ 2π

0

∫ 1

0R(µ, φ; −µ′, φ′)I (τ1; −µ′, φ′)µ′ dµ′ dφ′ (23)

whereR is a 4× 4 reflection matrix relating each reflected Stokes vector component to each incidentStokes vector component. Also,Q R Q (whereQ is a diagonal matrix of elements[1, 1, −1, 1]), mustbe used instead of simplyR in (23) to account for the change in symmetry when the atmosphereis illuminated from the bottom [17]. In practice, however, it is very difficult to determine all matrixelements ofR though either theoretical or experimental methods. The situation is further complicatedas most real surfaces are heterogeneous and may even be a mixture of several completely differentsurface types over the spatial scale of interest. To make the solution tractable it is usually assumed thatthe surface is depolarizing so that all matrix elements other thanR11 are zero.A further approximation isto take the surface-reflected Stokes vector as isotropic and independent of the downward Stokes vector.In this case, the surface is said to be Lambertian and it is convenient to express the reflectivity of thesurface in terms of a surface albedo,3. The surface albedo represents the fraction of energy incidenton a plane surface that is reflected back into the atmosphere

3 = E↑(τ1)

E↓(τ1)(24)

whereE↑(τ1) andE↓(τ1) represent the up-welling and down-welling irradiance at the surface, respec-tively. Albedo may vary with wavelength or other parameters such as solar zenith angle or wind speed,depending on the type of surface it represents. For a Lambertian surface, the azimuthally-independentboundary conditions can be summarized by

I (c,s)mn (τN+1; −µj ) = [0 0 0 0]T (25)

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McLinden et al. 381

Fig. 1. Illustration of a spherical-shell atmosphere illuminated by an extra-terrestrial source,πFo, at a solar zenithangle ofθo,t at the tangent point, TP. OSIRIS (or the observation point) is located atA. Note the variation in localzenith angle and solar zenith angle along the line of sight and thatBC = 1s`,`+1.

OSIRISline-of-sight

θo,t

Re

πFoπFo

ATP BCD

∆φo,t

θo,l∆φo,l

θl

I cmn (τ1; µj ) =

[3E↓(τ1)

π0 0 0

]Tn = 1, m = 0[

3E↓dif (τ1)

π0 0 0

]T

n > 1, m = 0

[0 0 0 0]T m > 0

(26)

I smn (τ1; µj ) = [0 0 0 0]T (27)

so that only them = 0 term is nonzero (which necessarily means that all odd coefficients are zero). NotethatE↓(τ1) represents both the diffuse and the direct irradiance components whileE

↓dif (τ1) is simply

the diffuse component. As the direct component of irradiance is counted in the single-scattered Stokesvector, higher scattering orders possess only the diffuse component (of that particular scattering order).

2.3. Solution in a spherical shell atmosphere

While use of the PP solution for limb radiances will lead to large errors, a spherical-shell (SS)solution throughout the entire atmosphere will require a large increase in computer time. It is sufficientto solve for the internal radiation field in a PP atmosphere while simultaneously solving for the top ofatmosphere limb radiances in a SS atmosphere.

A SS atmosphere is depicted in Fig. 1 and it can be seen that it introduces a number of complicationsover its PP counterpart [18]. The sphericity causes the local zenith angle, SZA, and change in azimuthalangle to vary along the line of sight (LOS). This means parallel paths between the same levels willhave different (slant) optical depths. The SZA variation may be particularly important if the observationpoint and the tangent point are on opposite sides of the terminator. The specified SZA and change inazimuthal angle are that at the tangent point, now denoted asθo,t and1φo,t. The local zenith angle isreferenced to the observation point and is determined by the altitude of the model atmosphere top andthe tangent height.

The first step in solving for the SS-limb Stokes vector is to reconstruct the azimuthal-dependentPP source-function vector at each point along the LOS at the appropriate local SZA and change inazimuthal angle,θo,` and1φo,`. The subscript̀ identifies a point along the LOS that intersects withthe SS grid and ranges from̀= 1 at the observation point toN` at the tangent point to 2N` − 1 on thefar side of the tangent point. The source-function vector,J` = J (τ`; θo,`, φo,`; θ`, φ`), is reconstructedby summing over the source-function-vector Fourier-expansion coefficients defined in (9)–(11) (where

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382 Can. J. Phys. Vol. 80, 2002

θ` is the the local zenith angle and1φ` is the local change in azimuthal angle; see Fig. 1). To allow forthe LOS variation in SZA, multiple PP calculations are performed to obtain the source function at theproper SZA. The local SZA and change in azimuthal angle are calculated using

cosθo,` = − cosθ` sinθo,t cos1φo,t + sinθ` cosθo,t (28)

cos1φo,` = sinθ` sinθo,t cos1φo,t + cosθ` cosθo,t

sinθo,`

(29)

The Fourier expansion allows the LOS variation in1φ` to be handled analytically. Recalculationof the source-function vector at all 2N` − 1 SZAs along the LOS would involve a large number of PPcalculations. Instead, only a subset of allθo,`’s are used concentrating on those near 90◦ and (or) thosenear the tangent point with quadratic interpolation in cosθo,` used to obtainJ` at intermediate values.Likewise, the local zenith angle varies along the LOS from a maximum at the observation point, typically100◦ or so in an atmosphere which extents up to 100 km, to 90◦ at the tangent point (by definition) to aminimum of about 80◦ at the top of the atmosphere on the far side of the tangent point. A standard gridof cosθ` = −0.18 to cosθ` = 0.18 in increments of 0.01 is used with quadratic interpolation employedto obtain intermediate values.

The limb Stokes vector for a particular tangent height is found by simply integrating over thesource-function vectors along the LOS using

I` = I`+1 e−1τ`,`+1 + α`J`+1 + β`J` (30)

1τ`,`+1 =(

ke,` + ke,`+1

2

)1s`,`+1 (31)

where1s`,`+1 is the physical pathlength between` and` + 1 (see Fig. 1). The definitions ofα` andβ`

are analogous to (20) and (21). The solution is initiated on the far side of the tangent point at` = 2N`

using a boundary condition ofI2N`−1 = [0 0 0 0]T. It proceeds down to the tangent point and then upto the observation point. The approximation involved in using a PP source has been assessed throughcomparisons using a source obtained for a SS atmosphere. For many different viewing geometries andwavelengths, differences never exceeded 0.8% and were more typically 0.1–0.3%.

Regenerating the PP source-function vectors, even on the reduced SZA grid, may lead to significantincreases in CPU requirements. The SZA variation can be accounted for in an approximate manner byinitially calculating the source-function vector at a reference SZA and then scaling it for all points alongthe LOS by the ratio of their single-scattered source-function vectors. This scaling factor,γ , is appliedto theα andβ interpolating coefficients in (30), where

γ` = exp[τs(θo,t) − τs(θo,`)] (32)

Generally,θo,t is taken as the reference SZA except when it is larger than 90◦, in which case 90◦ isused. This approximation performs quite well with errors not exceeding 10%. For OSIRIS geometry(1φo,t = 60–120◦) it is typically within 0–5%, depending on viewing geometry, tangent height, andwavelength. The only exception to this is for large optical depths along the LOS (e.g., tangent heightsbelow 20 km and wavelengths smaller than 315 nm) when the tangent point is very close to the terminator(θo,t = 89.5–90.5◦). The choice of regenerating the source function or the single-scattered scalingapproximation will depend primarily on the accuracy required.

3. OSIRIS reference atmospheres and input data

Comprehensive testing of the OSIRIS retrieval algorithms requires atmospheres representative ofdifferent latitudes and seasons. Towards that end, a set of reference atmospheres have been compiled for

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McLinden et al. 383

use in the forward RT models. In the current version, an individual atmosphere consists of a temperatureprofile and air, ozone, NO2, BrO, and stratospheric sulphate-aerosol number-density profiles. Each is ona 0.5, 1.0, or 2.0 km grid, as required, from 0–100 km. As this version does not include any atmospheresrepresentative of perturbed or “ozone-hole” conditions, OClO is not included.Atmospheres are availableat 18 latitudes (85◦S, 75◦S, ..., 85◦N) and for each month.

The ozone and temperature profiles are zonal, monthly-mean climatologies derived from sonde andsatellite measurements [19,20].The profiles of NO2 and BrO have been calculated using a photochemicalbox model [21,22]. For these calculations, ozone, temperature, families (NOy , Cly , and Bry), andimportant long-lived species are specified according to either observed climatologies or tracer–tracercorrelations. The box model is integrated to a near-steady state with the diurnal cycle fixed at midmonth. Given the diurnal nature of NO2 and BrO, profiles have been tabulated for high-sun, twilight,and diurnal-mean.

Background stratospheric sulphate-aerosol profiles are generated based on seasonal 1µm extinctionprofiles [23] converted to number density using the Mie-scattering cross section for a log-normal sizedistribution with an effective radius (reff ), or area-weighted mean radius, of 0.2µm and an effectivevariance (veff ) of 0.17. (The aerosol size distribution and parameters are defined in Hansen and Travis[15].) Future versions will include atmospheres representative of perturbed (ozone-hole) conditions andvolcanic aerosol conditions.

All input spectroscopic data, including an extra-terrestrial solar spectrum [24] and absorption crosssections of ozone [25], NO2 [26], BrO [27], OClO [28], and O4 (a short-lived O2–O2 collision complex)[29] are mapped onto a user-defined wavelength grid. The Rayleigh-scattering cross section and thedepolarization factor (used to account for the anisotropy of the molecular scatterers) are taken fromChance and Spurr [30].

Values for the ozone and NO2 cross section at three temperatures (202, 241, and 273 K) [25,26]allow height-dependent cross sections to be used assuming a quadratic temperature dependence [31] ofthe form

σ(λ, T ) = σTo(λ)[1 + a(λ)(T − To) + b(λ)(T − To)2] (33)

whereTo = 241 K anda andb are interpolation coefficients.

4. Simulation results

4.1. Limb radiance and polarizationThis section presents calculations of OSIRIS simulated total radiance and linear polarization. Total

radiance is simply the first element of the Stokes vector,I , and linear polarization is

LP =√

Q2 + U2

I(34)

Simulations are carried out using the OSIRIS reference atmosphere for May, 45◦N (using the twilightprofile option),θo,t = 80◦, 1φo,t = 90◦, a surface albedo of 0.3, and the reduced[I Q U ]T Stokesbasis. NeglectingV is equivalent to assuming the circularly polarized radiance is zero, which is exactfor pure-Rayleigh atmospheres and introduces errors less than 0.1% for the level of aerosols present inEarth’s atmosphere.

Radiances are shown in Fig. 2 as a function of wavelength (panela) and tangent height (panelsbandc). The most prominent feature of the radiances in Fig. 2a is the increase in radiance with decreasingtangent height. This is the result of the near-exponential increase in atmospheric density with decreasingaltitude. This is not the case for spectral regions in which significant absorption by ozone occurs: below320 nm in the Huggins bands and to a lesser extent between 550 and 650 nm in the Chappius bandsfor lower tangent heights. At each tangent height, the radiance falls off exponentially with increasing

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384 Can. J. Phys. Vol. 80, 2002

Fig. 2. OSIRIS limb radiance as (a) a function of wavelength at selected tangent heights and (b) and (c) as afunction of tangent height at selected wavelengths (whereA = 280 nm,B = 290 nm,C = 300 nm,D = 310 nm,E = 320 nm,F = 330 nm,G = 350 nm,H = 400 nm,I = 500 nm,J = 600 nm,K = 700 nm, andL = 800 nm). Based on the May, 45◦N reference atmosphere forθo,t = 80◦, 1φo,t = 90◦, and3 = 0.3.

0.001 0.01 0.1 1 1010

20

30

40

50

60

70

Tang

ent H

eigh

t (km

)

Radiance (µW cm-2 nm-1 sr-1 )

G H

IJKL

(c)

0.001 0.01 0.1 1 1010

20

30

40

50

60

70

Tang

ent H

eigh

t (km

)

Radiance (µW cm-2 nm-1 sr-1 )

A B C D E F G

(b)

300 400 500 600 700 80010

-3

10-2

10-1

100

101

Wavelength (nm)

Rad

ianc

e (µ

W c

m-2

nm-1

sr-1

)

10 km

20 km

30 km

40 km

50 km

60 km

70 km

(a)

wavelength in the visible to near-IR due to theλ−4 dependence of the Rayleigh cross section. Examiningradiance as a function of tangent height in the near-UV, from Fig. 2b, the tangent height at which themaximum radiance occurs decreases with increasing wavelength due to a rapidly decreasing ozoneabsorption cross section. In the visible, from Fig. 2c, radiance is observed to increase with decreasingtangent height at all wavelengths although at 350 and 400 nm there is sufficient scattering that theradiance peaks near 20 km.

Linear polarization is shown in Fig. 3 as a function of wavelength (panela) and tangent height(panelb). From Fig. 3a, polarization rapidly decreases with wavelength between 280 and 350 nm dueto an increasing importance of multiple scattering. In this region, as well as in the Chappius region,the structure follows that of the ozone absorption cross section and is driven by the variation in therelative importance of multiple scattering with absorption. Longward of 350 nm, the slow decreasein polarization with wavelength results from an increase in Mie scattering relative to Rayleigh. In apure-Rayleigh atmosphere, the opposite would be observed: a gradual increase in polarization withwavelength due to a smaller contribution from multiple scattering. As a function of tangent height, fromFig. 3b, the polarization varies little between tangent heights of 30 to 70 km as the entire tangent pathis above the bulk of the ozone and aerosols and all have roughly the same relative contributions fromsingle and multiple scattering. Below this, multiple scattering and scattering by aerosols become more

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McLinden et al. 385

Fig. 3. OSIRIS limb linear polarization as (a) a function of wavelength at selected tangent heights and (b) afunction of tangent height at selected wavelengths (whereA = 280 nm,C = 300 nm,D = 310 nm,E = 320 nm,F = 330 nm,H = 400 nm,J = 600 nm, andL = 800 nm). Based on the May, 45◦N reference atmosphere forθo,t = 80◦, 1φo,t = 90◦, and3 = 0.3.

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.9510

20

30

40

50

60

70

Tang

ent H

eigh

t (km

)

Linear Polarization

A

C

DEFHJL

(b)

300 400 500 600 700 8000.6

0.65

0.70

0.75

0.80

0.85

0.90

0.95

Wavelength (nm)

Line

ar P

olar

izat

ion

10 km

20 km

30 km

70 km

(a)

important as they act to depolarize the light. The polarization minimum near 20 km at 600 and 800 nmis an effect of the slant optical depth approaching unity. Below 20 km, a larger fraction of the lightreaching the observation point originates above the tangent point where there are fewer aerosols andhence reduced depolarization.

4.2. Limb weighting functionsAn example of limb weighting functions are shown in Fig. 4 for NO2 at 450 nm and ozone at 550 nm.

These weighting functions are dimensionless and relate how an absorber vertical column density (cm−2)in a 2 kmlayer contribute to the overall slant column abundance (cm−2) for a given tangent height.A single-wavelength version is presented for simplicity; actual OSIRIS weighting functions will becalculated based on a range of wavelengths, typically that used for the spectral fitting [6,7].

Weighting functions are calculated by perturbing the absorber profile at a particular height and recal-culating the limb radiances. The peak for a given tangent height typically occurs at the tangent altitude

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386 Can. J. Phys. Vol. 80, 2002

Fig. 4. OSIRIS limb weighting functions for NO2 at 450 nm (continuous line) and O3 at 550 nm (dotted line) shownfor five tangent heights (as indicated). Based on the May, 45◦N reference atmosphere forθo,t = 80◦, 1φo,t = 90◦,and3 = 0.3.

0 20 40 60 80 100 12010

20

30

40

50

60

Weighting Function

Alti

tude

(km

)

10 km

20 km

30 km

40 km

50 km

as this is the location of the largest path-length enhancement. The weighting functions are nonzerobelow the tangent point (unlike for occultation) owing to multiple scattering and surface reflection. Themagnitude of the peaks are seen to decrease with decreasing tangent height. This is a result of a near-exponential increase in slant optical depth with decreasing tangent height, which makes it increasinglyimprobable for light scattered into the LOS at the tangent point to reach the observation point. For thisreason, the 10 km tangent height NO2 weighting functions peak near an altitude of 20 km.

A subsequent study will estimate the errors introduced by using the scalar approximation on limbradiance, weighting functions, and trace-gas retrievals.

4.3. Solution parametersThere exists some uncertainty as to the choice of the numerical parameters that arise in the discrete

solution. Care must be taken in the selection of these parameters as they significantly influence theaccuracy and computational efficiency of the code. These parameters include: the number of zenithangles, the number of Fourier-expansion coefficients, the number of scattering orders, and the numberand spacing of the vertical levels. The criteria used for their selection are a maximum “error” of 0.01in limb radiance and linear polarization relative to benchmark values. The maximum is taken over arange of wavelengths (300–800 nm), SZAs (0–93◦), changes in azimuth (0–180◦), and tangent heights(10–70 km). Errors in radiance and linear polarization are defined as,

εr = |I − Io|Io

, εp = |LP − LPo| (35)

whereIo andLPo are benchmark values calculated using very tight numerical constraints.The number of zenith angle streams is determined by the order of the Legendre polynomial used to

numerically integrate the the source-function vector coefficients (i.e., (14)–(15)). The number requiredfor accurate integration is a function of the degree of anisotropy of the radiation field, which in turn islargely controlled by the phase matrix (and in particular theP11 element). The RayleighP11 elementvaries slowly with scattering angle and so few angles are required.The variation of the MieP11element isheavily dependent on the aerosol size to wavelength ratio. The maximum and average errors in radianceand linear polarization were determined for 4 to 16 zenith angle Stokes vector streams per hemisphere,

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McLinden et al. 387

Fig. 5. Maximum error (+) and average error (◦) for total radiance (continuous line) and linear polarization (brokenline) as a function of (a) number of zenith angles (per hemisphere) and (b) highest Fourier-expansion coefficient.Errors assessed using the May, 45◦N reference atmosphere over a range of tangent heights (10–70 km), wavelengths(300–800 nm), SZAs (0–93◦), and changes in azimuth (0–180◦). Errors are based on comparisons with benchmarks(see text).

4 6 8 10 12 14 1610

-4

10-3

10-2

10-1

Number of Zenith Angles

Err

or(a)

3 4 5 6 7 8 9 1010

-4

10-3

10-2

10-1

100

Fourier Coefficient

Err

or

(b)

relative to a benchmark of 20, with the results plotted in Fig. 5a. These runs were performed using thebackground stratospheric sulphate-aerosol profile, as discussed above. Based on the 0.01 criteria, 14zenith angle streams are required. The polarization is observed to be considerably less sensitive andmaximum errors are about three times larger than the average. Larger aerosols would require morestreams. For pure-Rayleigh scattering, 5 zenith angle streams are sufficient.

The number of Fourier-expansion terms required is controlled by the nature of the scatterer. Forisotropic scatterers, which have only azimuthally-independent phase-matrix elements, only them = 0term is required for an exact series representation. The Rayleigh-scattering phase matrix contains termsup to 2(φ − φ′) and so only terms up tom = 2 are required. The number of terms necessary whenincluding Mie scatterers is again dependent upon the nature of the scatterers and the size to wavelengthratio. Maximum values ofm from 3 to 10 have been used and compared against a benchmark of 20 withthe results shown in Fig. 5b. A 0.01 maximum error in radiance is reached atm = 8.

The number of scattering orders is handled by recognizing that at some point during the iterationsthe ratio of successive radiance orders becomes a constant. In this case all remaining orders can beadded using a geometric series. Testing has revealed that an almost exact solution is obtained when

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388 Can. J. Phys. Vol. 80, 2002

Table 1. Comparison of model calculations with various sources of tabulated modeloutput.

Solution Atmosphere and Quantity PercentNo. Ref. method geometry compared difference

1 32 Doubling Rayleigh Reflected Avg = 0.04and adding τ1 = 1, ω̃ = 1.0 I, Q, U Max = 0.08

3 = 0.25,µo = 0.8φ − φo = 90◦

2 32 Doubling Mie (reff = 0.2 µm, Reflected Avg = 0.03and adding veff = 0.07) I, Q, U, V Max = 0.06

τ1 = 1, ω̃ = 0.99 expansion3 = 0.1, µo = 0.2 coefficients

3 33 Spherical Mie (reff = 0.2 µm, Internal Avg = 0.03harmonics veff = 0.07) I, Q, U, V Max = 0.07

τ1 = 1, ω̃ = 0.993 = 0.1, µo = 0.2φ − φo = 0, 90, 180◦

4 34 Doubling Rayleigh Internal Avg = 0.1and adding τ1 = 10, ω̃ = 1.0 I Max = 0.4

3 = 0.1, µo = 0.6φ − φo = 0, 90, 180◦

5 34a Doubling Mie (reff = 1.05µm, Internal Avg = 0.08and adding veff = 0.07) I Max = 0.35

τ1 = 10, ω̃ = 1.03 = 0.1, µo = 0.6φ − φo =0, 90, 180◦

aSimulates Venusian atmosphere.

performing calculations for four scattering orders. That is

I = I1 + I2 + I3 + I4

(1

1 − I4/I3

)(36)

and applies equally forQ, U , andV .Model levels are selected by specifying altitudes. Testing has determined that a grid spacing of

1 km near the tangent point is necessary for an accurate calculation. This is a consequence of the longpath-length between the tangent grid point and the next layer, about 113 km for 1 km thick layers. Asany simulation usually encompasses a range of tangent heights, a 1 kmgrid spacing is used throughout.

5. Model evaluation

5.1. Model accuracy

Comparisons were made with tabulated results of three models [32–34] for PP, homogeneous at-mospheres, a summary of which is given in Table 1. The atmospheres considered consisted entirelyof either Rayleigh or Mie scatterers and the the total optical depth and single-scattering albedo wereprescribed. The overall agreement was found to be very good with maximum errors of less than 0.5% fora simulation of the thick Venusian atmosphere and less than 0.1% for optical depths not exceeding unity.In general, the Mie atmospheres seem to have slightly better agreement than the Rayleigh atmospheres.This is believed to be because in the Mie atmospheres, the energy is distributed over a greater number

©2002 NRC Canada

McLinden et al. 389

Fig. 6. Distribution of percent difference in limb radiance between the current model and LIMBTRAN [35] for allcombinations of: wavelengths = 300, 350, 400, and 600 nm;θo,t = 0, 30, 75, 90, and 93◦; 1φo,t = 90◦; 3 = 0.0and 1.0; and tangent height = 10 and 40 km. The comparison was performed with aerosols (open bars) and withoutaerosols (solid bars).

-4 -3 -2 -1 0 1 2 3 40

5

10

15

20

25

Percent Difference

Num

ber

of Fourier harmonic frequencies. As a result, the Rayleighm = 0 term does not converge as fast as theMie m = 0 term.

A comparison with the scalar model LIMBTRAN [35] was also carried out. Using identical atmo-spheres, limb radiances were calculated for all combinations of: wavelengths of 300, 350, 400, and600 nm; SZAs of 0, 30, 75, 90, and 93◦; change in azimuthal angle of 90◦; surface albedo of 0.0 and1.0; tangent heights of 10 and 40 km; and with and without stratospheric sulphate aerosols (opticaldepth of 0.2 at 400 nm). The distribution of percentage difference in limb radiance with LIMBTRANfor these combinations is shown in Fig. 6. Agreement is satisfactory with an average difference of about0.5% without aerosols and about 1.3% with aerosols. Given this, together with a favourable comparisonbetween LIMBTRAN and a three-dimensional backward Monte Carlo code [35,36], the present modelis deemed able to accurately calculate limb radiances (for a horizontally homogeneous atmosphere).

5.2. Model timingA summary of a model-timing study is presented in Table 2 in which the code was run on a Silicon

Graphics SGI O200 4x MIPS R10000 processor. Using two OSIRIS-like geometries it was run fora pure-Rayleigh atmosphere at 51 wavelengths (300–800 nm) and calculated radiances at 31 tangentheights (10–70 km). A 1 km vertical grid (0–100 km), 5 zenith angles (per hemisphere), and Fouriercoefficients up tom = 2 were used. For the first geometry (θo,t = 80◦, 1φo,t = 90◦), the variation ofthe SZA along the LOS is small (about 0.1◦) and so only a single PP calculation is required which took26 s. For the second geometry (θo,t = 90◦, 1φo,t = 120◦), the variation in SZA, about 10◦, is roughlythe largest possible given the Odin orbit. Using 21 SZAs required 209 s, or 8 times that for the singleSZA calculation. The source-function scaling technique is slightly slower than no SZA variation. Theaddition of polarization slowed the code down by a factor of 6. As polarization will be used only whenexamining polarization-related issues, this is not expected to pose a large problem.

Also shown in Table 2 are the timings for LIMBTRAN, which solves the RTE using a fast finite-difference scheme running on the same machine and for the same scenarios. While comparable, LIMB-TRAN is faster: 30% for one SZA and 60% for 21 SZAs.

More relevant are the timings for weighting-function calculations. Unfortunately, CPU time in-creases almost linearly with the number of perturbations, where one perturbation run is performed for

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390 Can. J. Phys. Vol. 80, 2002

Table 2. Computer time on an SGI O2000 (in seconds) required to calculate limb radianceat 31 tangent heights (10–70 km) at 51 wavelengths (300–800 nm). The atmosphere is pureRayleigh for May 45◦N. Quantities in brackets represent the error introduced into the limbradiances at tangent heights of 10 km and 40 km and 400 nm using the two source-functionapproximations: no SZA variation and single-scattering scaling. Vector calculations aremade using the reduced[I Q U ]T Stokes basis.

Model geometry

Run type θo,t = 80◦, 1φo,t = 90◦ θo,t = 90◦, 1φo,t = 120◦

Present modelScalar 26 209Scalar, no SZA variation 26 27 (32%,1%)Scalar, scaled-source function 26 38 (1%,1%)Vector 148 1377

LIMBTRANScalar 20 132Scalar, no SZA variation 20 20

each tangent height in a limb scan. Thus, using 51 wavelengths and 31 tangent heights, roughly 10 min(for geometry that requires one PP calculation) to almost 2 h (for 20 PP calculations) of CPU timeare required to generate one set of weighting functions. The benefits of using a single PP calculation(e.g., using the source-function scaling technique described above) become evident here. The additionof aerosols will further increase the CPU burden by a factor of 5–10. Reduction of the CPU overheadis a priority of future work.

6. Summary

This paper has presented a vector radiative-transfer model that is able to accurately calculate thepolarized radiation field, including limb radiance and polarization, in a one-dimensional, inhomogeneousatmosphere. It employs the successive orders of scattering solution technique simultaneously in plane-parallel and spherical-shell atmospheres. Comparisons carried out against other models indicated goodoverall agreement and the model timing is comparable (within 50%) of a fast finite-difference code.

Numerical parameters required for accurate calculations are as follows. With background strato-spheric sulphates 14 zenith angles (per hemisphere), Fourier coefficients up tom = 8, and four ordersof scattering (with remaining orders added through a geometric series) are required. In a pure-Rayleighatmosphere, these requirements are reduced to 5 streams andm = 2.

A straightforward extension of the current version is planned that will allow for a varying atmospherealong the line of sight analogous to the SZA variation. This will permit a better assessment of howhorizontal inhomogeneities may impact weighting functions and retrievals, for example, when OSIRISis looking through the polar vortex. Raman scattering, via the Ring effect [37], can introduce small-scalestructure into scattered-light spectra, which is comparable in magnitude to the absorption signatures ofNO2 and BrO. While an important effect, including a multiple-scattered Raman component is beyondthe capability of current radiative-transfer models. Refraction of the direct solar beam and light travelingalong the tangent path is currently being tested. Reducing the computational overhead of the weighting-function calculations is a priority of future work.

Acknowledgements

The authors thank two anonymous reviewers for helpful comments. J.McC. wishes to thank the Nat-ural Sciences and Engineering Research Council of Canada (NSERC) and the Meteorological Serviceof Canada for continuing support.

©2002 NRC Canada

McLinden et al. 391

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Appendix A: The phase matrix

Scattering by isotropic, homogeneous spheres can be described with a scattering matrix of the form

P (2) =

P11(2) P12(2) 0 0P12(2) P11(2) 0 0

0 0 P33(2) −P34(2)

0 0 P34(2) P33(2)

(A.1)

where2 is the scattering angle. This type of matrix is valid for Rayleigh scattering (whereP11(2) =(1 + cos2 2)/2; P12(2) = −(1 − cos2 2)/2; P33(2) = cos2; P34(2) = 0) and Mie scattering. Miescattering is the application of Maxwell’s equations to an isotropic, homogeneous, dielectric sphere; itis equally applicable to spheres of all sizes and refractive indices, and for radiation at all wavelengths[38] and it is used to describe scattering by aerosols. The molecules that comprise air (N2 and O2) arediatomic and hence slightly anisotropic. This anisotropy can be accounted for by considering molecularscattering to be a combination of true-Rayleigh and isotropic scattering using a depolarization factor[13,30]. The scattering angle,2, can be expressed in terms of the direction of the incident radiation(µ′, φ′) and direction of scattered radiation(µ, φ) through

cos2 = µµ′ +√

1 − µ2√

1 − µ′2 cos(φ − φ′) (A.2)

The scattering matrix is defined relative to the plane of scattering, or the plane containing both theincident and scattered directions. However, the Stokes vector is defined with respect to the meridianplane, or the plane containing the direction of propagation and the vertical. Thus,I must first betransformed from the incident meridian plane to the plane of scattering so that the scattering calculationscan be carried out, and then from the plane of scattering to the scattered meridian plane.

The Stokes vector can be rotated through an anglei (≥0) in the anticlockwise direction, whenlooking into the direction of propagation, by the rotation matrix [39]

L(i) =

1 0 0 00 cos 2i sin 2i 00 − sin 2i cos 2i 00 0 0 1

(A.3)

such that the rotated Stokes vector isI ′ = L(i)I . The elementsI andV are invariant under rotation.The phase matrix is arrived at upon applying the required rotations

Z(τ ; µ, φ; µ′, φ′) = L(π − i2)P (τ ; µ, φ; µ′, φ′)L(i1) (A.4)

where the anglesi1 andi2 are shown in Fig.A.1. Carrying out the matrix multiplication gives the explicitform for the phase matrix

Z(τ ; µ, φ; µ′, φ′) = (A.5)

P11(2) P12(2)c1 P12(2)s1 0P12(2)c2 c2P11(2)c1 − s2P33(2)s1 c2P11(2)s1 + s2P33(2)c1 −P34(2)s2

−P12(2)s2 −s2P11(2)c1 − c2P33(2)s1 −s2P11(2)s1 + c2P33(2)c1 −P34(2)c20 −P34(2)s1 P34(2)c1 P33(2)

©2002 NRC Canada

McLinden et al. 393

Fig. A.1. Rotation of the Stokes vector for a scattering event at O from direction OP1 into direction OP2. Theincident meridian plane is defined by ONP1, the scattering plane by OP1P2, and the scattered meridian plane byONP2. The rotation angles, i1 and i2, are also shown.

�

'

�

�����'

�',�'( )

���( )

x

z

y

i1i2

O

N

P1

P2

�

where

c1 = cos(−2i1) c2 = cos[2(π − i2)]; s1 = sin(−2i1) s2 = sin[2(π − i2)] (A.6)

The rotation angles i1 and i2 can be related to the incident and scattered directions, (µ′, φ′) and (µ, φ),respectively, using spherical trigonometry,

cos i1 = −µ + µ′ cos

±(1 − cos2 )1/2(1 − µ′2)1/2 (A.7)

cos i2 = −µ′ + µ cos

±(1 − cos2 )1/2(1 − µ2)1/2 (A.8)

The denominators are positive when π < (φ − φ′) < 2π and negative when 0 < (φ − φ′) < π .The model atmosphere will, in general, contain both Rayleigh and Mie scatterers, which means

the effective-phase matrix is the weighted average of each scatterer’s phase matrix with its scatteringcoefficient. Also, in an inhomogeneous atmosphere the effective-phase matrix is height dependent. ForNl different types of scatterers (Rayleigh or Mie)

Z(τ ; µ, φ; µ′, φ′) =∑Nl

l=1 ks,l(z)Zl (τ ; µ, φ; µ′, φ′)∑Nl

l=1 ks,l(z)σs,l

(A.9)

where the sum is performed over all scatterers and Zl is the phase matrix for scatterer l.When expanding the elements of Z, given in (A.5), in a Fourier series, many of the coefficients will

be zero. Elements of Z can be classified as

e e o −e e o oo o e e− o e e

where e denotes an even function (Zsm = 0) and o an odd function (Zcm = 0).

©2002 NRC Canada