Rotational Raman scattering of polarized light in the...

Journal of Quantitative Spectroscopy &Radiative Transfer 87 (2004) 399–433

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Rotational Raman scattering of polarized light in the Earthatmosphere: a vector radiative transfer model using the

radiative transfer perturbation theory approach

J. Landgraf∗, O.P. Hasekamp, R. van Deelen, I. AbenSRON, National Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands

Received 26 August 2003; accepted 17 March 2004

Abstract

A plane parallel vector radiative transfer model is presented to simulate the e:ect of rotational Ramanscattering on radiance and polarization properties of sunlight re;ected by the Earth atmosphere in the ultravioletand visible part of the solar spectrum. The model employs the radiative transfer perturbation theory, whichtreats inelastic rotational Raman scattering as a perturbation to elastic Rayleigh scattering. The approachprovides a perturbation series expansion for a simulated radiation quantity, where each term describes thee:ect of one additional order of Raman scattering. The model is worked out in detail to =rst order. Here, theadjoint formulation of radiative transfer reduces signi=cantly the numerical e:ort of computational applications.Numerical simulations are presented for the ultraviolet part of the solar spectrum and the e:ect of Ramanscattering on the Stokes parameters I; Q and U of the re;ected sunlight is studied. Furthermore, the accuracyof both the single scattering approximation and the scalar radiative transfer approach is considered for thesimulation of Ring structures. The use of these approximation techniques is investigated for the simulation ofRing structures in polarization sensitive GOME measurements.? 2004 Elsevier Ltd. All rights reserved.

Keywords: Rotational Raman scattering; Ring e:ect; Polarization; Perturbation theory; Green’s function; Adjointformulation; GOME

1. Introduction

Satellite measurements of backscattered sunlight in the ultraviolet and visible part of the so-lar spectrum contain essential information about the composition of the atmosphere. In the years1978–2002 a series of solar backscattered ultraviolet (SBUV) instruments were launched on

∗ Corresponding author. Tel.: +31-30-253-5942; fax: +31-30-254-0860.E-mail address: [email protected] (J. Landgraf).

0022-4073/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.jqsrt.2004.03.013

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400 J. Landgraf et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 399–433

NASA’s NOAA satellites, which measure the backscattered ultraviolet radiances at 12 wavelengthsbetween 250 and 340 nm with a spectral resolution of approximately 1.1 nm. The global ozone mon-itoring experiment (GOME), launched in 1995 on board of the ESA’s ERS-2 satellite, representsthe =rst space borne spectrometer of a series of European instruments for remote sensing of atmo-spheric trace gas distributions. It performs measurements in the spectral range 240–790 nm in a nadirviewing geometry with a spectral resolution of approximately 0.2 nm. In 2002 the scanning imag-ing absorption spectrometer for atmospheric cartography (SCIAMACHY) was launched on ESA’sENVISAT satellite, which has an extended spectral range of 240–2880 nm. SCIAMACHY measuresin limb and occultation geometry as well as in nadir and its spectral resolution in the ultraviolet andvisible is comparable to GOME. In 2005 the =rst of three GOME-2 instruments will be launchedon the METOP-1 platform of ESA and EUMETSAT with the same spectral coverage and spec-tral resolution as its precursor GOME. In 2004 NASA will launch the ozone monitoring instrument(OMI), also a GOME-type spectrometer on the EOS-AURA satellite. OMI covers a spectral range of270–500 nm with a spectral resolution of 0.4 nm, which is slightly worse than that of GOME.

The retrieval of atmospheric constituents from these types of measurements needs a forwardmodel, which simulates the radiation re;ected by the Earth atmosphere in the viewing direction ofthe instrument. It requires a proper simulation of radiative transfer in the atmosphere including theinteraction of light with di:erent atmospheric constituents and with the ground surface. A commonapproach in radiative transfer theory is to assume a model atmosphere, where scattering by airmolecules is described by elastic Rayleigh scattering. However, due to the anisotropic polarizability ofair molecules the formulation of Rayleigh scattering provides only an e:ective description. Rayleighscattering actually consists of three spectral components: (a) an elastic scattering component, whichis called the Gross line, (b) the Brillouin lines, describing inelastic translational Raman scattering,and (c) inelastic rotational and rotational–vibrational Raman scattering [1]. For the simulation of bothSBUV and GOME-type measurements the Gross and Brillouin lines may be combined into one elasticscattering component, the so-called Cabannes line [2]. Furthermore, rotational Raman scattering isthe most prominent inelastic scattering process in the atmosphere [3]. If we describe atmosphericRayleigh scattering in the ultraviolet in terms of Cabannes and rotational Raman scattering, about96% of all scattering events can be described by Cabannes scattering and only 4% by Ramanscattering. In the ultraviolet the change in wavelength of Raman scattered light can be signi=cant,e.g. up to 2 nm at 320 nm. This causes spectral signatures on the backscattered sunlight, measuredby space borne spectrometers. For example for pronounced Fraunhofer lines in the solar spectrum butalso for narrow band absorption lines of atmospheric constituents light e:ectively is scattered fromthe line wings to the line center. This =lling-in of spectral lines is present clearly in the re;ectancespectra measured by SBUV- and GOME-type instruments [2,4,5].

Furthermore, GOME, GOME-2, and SCIAMACHY are polarization sensitive instruments, whichmeans that the detected radiance signal depends on the polarization properties of the observed light.For these instruments the oOcial data processor corrects for the polarization sensitivity of the mea-surements. Here, the state of polarization of the backscattered light has to be known at the spec-tral resolution of the radiance measurement. For this purpose broadband polarization measurementsare taken, from which spectra at the required resolution are reconstructed. However, the state ofpolarization can vary signi=cantly with wavelength [6–8]. Here, not only atmospheric absorption(e.g. by ozone and molecular oxygen) but also atmospheric Raman scattering is of relevance. Soany attempt to reconstruct the polarization information at the required spectral resolution of the

J. Landgraf et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 399–433 401

instrument using broadband polarization measurements introduces errors in the polarization correctedspectrum and thus can bias signi=cantly the interpretation of the measurements. To overcome thisproblem, Hasekamp et al. [9] suggested to simulate directly the polarization sensitive measurementtaken by the instrument. This requires a vector radiative transfer model, which simulates the Stokesvectors of the re;ected sunlight in the viewing direction of the instrument. For radiative transfersimulations at =ne spectral resolution this model has to account for inelastic Raman scattering in theatmosphere.

The simulation of radiative transfer including inelastic rotational Raman scattering becomes com-plicated due to the coupling of radiation at di:erent wavelengths. A =rst numerical approach wasproposed by Kattawar et al. [10]. Their computational technique is based on a separation of sin-gle and multiple scattering using a scalar Monte Carlo model for a Rayleigh scattering atmosphere(by which we mean a model atmosphere where atmospheric scattering is described by Rayleighscattering only). For both the single and multiple scattering component a correction is proposedfor polarization e:ects and inelastic Raman scattering contributions. A more rigorous approach ispresented by Joiner et al. [2]. They utilize a successive order of scattering model of scalar radiativetransfer for a Rayleigh scattering atmosphere. Here, each scattering contribution is corrected for oneorder of Raman scattering, where the spectral dependence of Raman scattering is taken fully intoaccount. The approach contains an exact solution for singly scattering light. To ease the correctionfor multiple scattering it is assumed that the incoming radiation =eld of a Raman scattering processis isotropic. The work of Vountas et al. [5] represents an alternative approach to treat inelasticRaman scattering in scalar radiative transfer. The radiative transfer equation, which describes Ca-bannes and Raman atmospheric scattering, is divided into two parts, the radiative transfer equationfor a Rayleigh scattering atmosphere and a perturbation term, which describes the additional inelasticscattering contribution. The e:ect of Raman scattering on the radiation =eld is described by a Picarditeration series that uses the radiance =eld for a Rayleigh scattering atmosphere as a =rst guess. Thistechnique represents a sound treatment of the problem in its scalar approximation, because it iteratestoward the correct solution. Vountas et al. [5] presents numerical results for a =rst iteration of thisapproach.

For the =rst time Humphreys et al. [11] presented a vector radiative transfer model, which simulatesthe e:ect of inelastic Raman scattering on the full Stokes vector. Here the frequency variation ofindividual rotational Raman lines is described in a simpli=ed form. This e:ect is described fully in themodel of Aben et al. [12] and Stam et al. [13]. They demonstrated that di:erential =ne structuresin polarization measurements could be simulated already by a second order of scattering model.However, for a proper simulation of the continuum of the radiation signal higher orders of scatteringmust be included. Recently, Sioris and Evans [14] have improved on the earlier e:ort of Joiner etal. with a successive order of scattering model including polarization. The underlying assumption oftheir approach is that for scattering events of second order and higher only the scattering phase matrixdetermines the angular distribution of the outgoing radiation =eld. So the angle distribution of theincoming radiation is neglected. This is not entirely true for scattering contributions of lower order.

In this work, we present a vector radiative transfer model, where inelastic Raman scattering istaken into account using the radiative transfer perturbation theory approach. Similar to the work ofVountas et al. [5], the vector radiative transfer equation is separated in an unperturbed radiativetransfer problem describing radiative transfer in a Rayleigh scattering atmosphere, and a perturba-tion term for additional Raman scattering (Section 2). In Section 3, the e:ect of this perturbation


on the measurement simulation is described by a classical perturbation series using the Green’sfunction formalism. In this approach the nth order perturbation term can be interpreted as a correc-tion to the simulated measurement for n orders of Raman scattering. To =rst order the numericalimplementation of the perturbation can be eased signi=cantly by the adjoint formulation of radiativetransfer, which is also discussed in Section 3. Section 4 shows numerical simulations for the Stokesparameters I , Q and U of the re;ected sunlight. Here the e:ect of one order of Raman scatteringon the polarization components is considered in detail. Furthermore, the accuracy of both the singlescattering approximation and the scalar approach of radiative transfer is studied for the simulationof Ring structures in the re;ectance spectra. Finally, in Section 5 we investigate the suitability ofthe single scattering and scalar radiative transfer approximation for the simulation of Ring structuresin polarization sensitive measurements of GOME within the error margins of the instrument. Here,we focus on both the spectral range 290–313 nm, which contains information on the vertical ozonedistribution in the probed atmosphere, and the =lling-in of the Ca II Fraunhofer lines, which can beused to retrieve cloud properties.

2. The radiative transfer problem

2.1. The radiative transfer equation including both elastic Cabannes scattering and inelasticRaman scattering

The radiance and state of polarization of light can be described by an intensity vector I, whichhas the Stokes parameters I; Q; U , and V as its components (see e.g. [15,16])

I = [I; Q; U; V ]T: (1)

Here, the superscript T indicates the transposed vector. The intensity vector is de=ned with respectto a certain reference plane, which in this paper is given by the local meridian plane. In general,the intensity vector is a function of altitude z, of wavelength and of direction �= (; ’), where’ is the azimuthal angle measured clockwise when looking downward and is the cosine of thezenith angle (¡ 0 for downward directions and ¿ 0 for upward directions). The domain of thesevariables de=ne the so-called phase space of the radiative transfer problem.

In the following, we consider a plane-parallel, macroscopically isotropic atmosphere boundedfrom below by a Lambertian re;ecting surface. For such an atmosphere the equation of transfer forpolarized light is given in its forward formulation by

LI = S; (2)

with the transport operator [17–20]

L=∫ ∞

0d

∫4�

d�

[�(− )

{[@@z

+ �e(z; )]�(�− �)E

− A()�

�(z)�()||�(−)||}

− J(z; �; | z;�; )]

◦ : (3)


Here d� = d d’, E is the 4 × 4 unity matrix, � represents the Heaviside step function, and � isthe Dirac-delta with �(�− �) = �( − )�(’− ’). �e is the extinction coeOcient and the symbol‘◦’ indicates that L is an integral operator acting on a function to its right. The =rst term of theradiative transfer equation describes the extinction of light for a wavelength in a direction � andthe second term represents the isotropic re;ection on a Lambertian surface, where the matrix A()is given by

A() = diag[A(); 0; 0; 0] (4)

with Lambertian albedo A(). Finally, J(z; �; | z;�; ) de=nes the source function at height z forscattered light from direction � to � with a change in wavelength from to . To simplify matterswe assume a model atmosphere consisting of N2, O2, and Ar molecules only. For Cabannes androtational Raman scattering by an ensemble of randomly oriented molecules J is described by

J(z; �; | z;�; ) =∑N=

N2 ;O2 ;Ar

{�(− )

�cabs;N (z; )4�

ZcabN (; �;�) − �rrs;N (z; ; )

4�Zrr(�;�)

}: (5)

Here �cabs;N and �rrs;N represent the scattering coeOcients of O2, N2 and Ar for Cabannes and rotationalRaman scattering, respectively, with corresponding scattering phase matrices Zcab

N and Zrr. Noticethat Ar as an inert gas has an isotropic polarizability and thus rotational Raman scattering on Ardoes not exist, �rrs;Ar = 0. The elastic feature of Cabannes scattering is represented by the Dirac-delta�(−), whereas the change in wavelength from to due to inelastic Raman scattering is describedby the scattering coeOcient �rrs;N (z; ; ). The scattering source function can be written in a morecompact form, viz.

J(z; �; | z;�; ) = �(− )�cabs (z; )

4�Zcab(; �;�) − �rrs (z; ; )

4�Zrr(�;�); (6)

if we use the e:ective scattering phase matrix

Zcab(; �;�) =∑N=

N2 ;O2 ;Ar

�cabs;N (z; )

�cabs (z; )ZcabN (; �;�) (7)

with the total scattering coeOcients

�cabs (z; ) =∑N=

N2 ;O2 ;Ar

�cabs;N (z; ) (8)

and an analogous expression for the Raman scattering contribution. The explicit form of the scatteringphase matrices as well as the scattering coeOcients are given in Appendix A. The right-hand side ofEq. (2) provides the source of light and can either be a volume source inside the atmosphere or asurface source chosen to reproduce the incident ;ux conditions at the boundaries of the atmosphere,or some combination of the two. In the ultraviolet and visible part of the spectrum the radiationsource S is determined by the unpolarized sunlight that illuminates the top of the Earth atmosphere:

S(z;�; ) = 0�(z − ztop)�(�−�0)F0(): (9)


Here, ztop is the height of the model atmosphere, �0 = (−0; ’0) describes the geometry of theincoming solar beam (we de=ne 0¿ 0), and F0 is given by

F0() = [F0(); 0; 0; 0]T; (10)

where F0 is the solar ;ux per unit area perpendicular to the direction of the solar beam.Because the re;ection of light at the ground surface is already included in the radiative transfer

equation (2) and the incoming solar beam is represented by the radiation source S in Eq. (9), theintensity vector I is subject to homogeneous boundary conditions

I(ztop;�; ) = [0; 0; 0; 0]T for ¡ 0;

I(0;�; ) = [0; 0; 0; 0]T for ¿ 0: (11)

In combination with these boundary conditions, the radiation source S can be interpreted as locateda vanishingly small distance below the upper boundary. Similarly, the surface re;ection takes placea vanishingly small distance above the lower boundary (see e.g. [21]).

The solution of Eq. (2) gives the internal intensity vector =eld inside the model atmosphere. Inthe context of measurement simulations, one is generally interested in a certain radiative e:ect Eof this =eld. This scalar quantity can be derived from the vector intensity =eld with an appropriateresponse vector function R through the inner product [17,18,22]

E = 〈R; I〉: (12)

Here, the inner product is de=ned by the phase space integration

〈I1; I2〉 =∫ ∞

0d

∫ ztop

0dz

∫4�

d�I1T(z;�; )I2(z;�; ) (13)

for two arbitrary vector functions I1 and I2.For retrieval purposes of atmospheric constituents from space borne measurements one is typically

interested in a radiative e:ect E given by one of the four Stokes parameters (or a linear combinationof those) at a wavelength v, at the top of the model atmosphere pointed toward the viewing directionof an instrument. In this particular case the response function for the ith Stokes parameter is givenby

Ri(z;�; ) = �(z − ztop)�(�−�v)�(− v)ei; (14)

where ei is the unity vector in the direction of the ith component of the intensity vector, and�v = (v; ’v) denotes the viewing direction of the instrument.

2.2. Decomposition of the radiative transfer problem

The spectral coupling of the intensity vector =eld due to inelastic Raman scattering renders anysolution approach of the integro-di:erential equation (2) diOcult. However, in the case of rotationalRaman scattering the coupling is weak. One solution technique is to solve =rst the radiative transferequation, which takes only elastic scattering processes into account and to treat the wavelengthcoupling due to inelastic Raman scattering as a perturbation e:ect. In other words, we decomposethe transport operator L in two parts,

L= L0 + UL: (15)


Here L0 describes a monochromatic radiative transfer problem

L0I0 = S; (16)

which can be solved with standard techniques (see e.g. [23]), and UL represents a correspondingperturbation in transport given by inelastic scattering.

For example, we can choose a radiative transfer operator L0, which describes monochromaticradiative transfer including pure Rayleigh scattering. So L0 is de=ned by Eq. (3) but with a scatteringsource

J0(z; �; | z;�; ) = �(− )�rays (z; )

4�Zray(; �;�): (17)

Here, �rays is the total Rayleigh scattering coeOcient for an ensemble of O2, N2 and Ar molecules,which is de=ned analogous to Eq. (8), and Zray represents the corresponding scattering phase matrix.This in turn results in the perturbation operator

UL=∫ ∞

0d

∫4�

d�UJ(z; �; | z;�; ) ◦ (18)

with the integral kernel

UJ(z; �; | z;�; ) = �(− )

{�cabs (z; )

4�Zcab(; �;�) − �rays (z; )

4�Zray(; �;�)

}

+�rrs (z; ; )

4�Zrr(�;�): (19)

Thus the perturbation in Eq. (18) represents a replacement of Rayleigh scattering processes bypartly Cabannes and partly inelastic Raman scattering processes. The e:ect of this perturbation onthe radiative transfer can be described by a perturbation series approach, which is the subject of thenext section.

3. Perturbation theory

3.1. The perturbation series approach

A general formalism to calculate the e:ect of a perturbation UL on the solution of a radiativetransfer problem is given by the perturbation theory using the Green’s function technique. Green’sfunctions provide a concept of broad and universal applicability to solve inhomogeneous di:erentialequations. Several applications of the formalism have been discussed in radiative transfer theory (seee.g. [24–28]). Here we apply the Green’s function method to show the theoretical concept of theradiative transfer perturbation theory.

The formal solution of the radiative transfer equation (2) means to =nd the inverse of the transportoperator L, i.e., an operator G with

LG = GL= 1; (20)

where 1 is the identity operator. So for any source S the solution is given by

I = GS: (21)


In other words, G represents the propagation of light emitted at a source point in the phase spaceto a target point, where the intensity vector I is de=ned. Here, S describes the source strength atthe source point. With the knowledge of G the radiative transfer problem is completely solved andany radiation e:ect may be calculated by

E = 〈R; GS〉 (22)

for any radiation source S and any response function R.The operator G can be given in the form of an integral, viz.

G =∫4�

d�′∫ ztop

0dz′

∫ ∞

0d′G(z; �; | z′; �′; ′) ◦; (23)

where the kernel G is a function with a 4 × 4 matrix structure. Because of Eq. (20) the kernel Ghas to obey the relation

LG(z;�; | z′;�′; ′) = �(z − z′)�(�−�′)�(− ′)1: (24)

This equation can be written also as

Lgi(z;�; | z′;�′; ′) = �(z − z′)�(�−�′)�(− ′)ei ; (25)

where gi is the ith column vector of G. Here, gi represents the intensity vector =eld at the targetpoint (z; �; ) for a given unity light source in the ith Stokes parameter at the source point (z′; �′; ′).The matrix G is commonly called the Green’s function of the corresponding di:erential equationand accordingly we call the inverse operator G = L−1 the Green’s operator.

If the calculation of the Green’s function intends to solve the actual problem for a particularsource S, this formulation does not provide any simpli=cation. However, the concept is neededfor the perturbation theory approach. Suppose that a simpli=ed problem, which in our case is themonochromatic radiative transfer problem in Eq. (16), is already solved with a corresponding Green’soperator G0. Then the operator G and L, which belong to the radiative transfer problem includinginelastic Raman scattering, satisfy the relation

G = G0 − GULG0: (26)

This may be shown in a straightforward manner by substituting UL = L − L0 and using relation(20) for the radiative transfer problems de=ned by the operators L and L0, respectively. Eq. (26)can be solved formally with respect to the Green’s operator G, viz.

G= G0 − G0ULG0 + G0ULG0ULG0 − · · ·

=∞∑k=0

(−1)kG0[ULG0]k ; (27)

where we used the de=nition of the inverse operator [1 + ULG0]−1 through the geometric serialexpression

[1+ULG0]−1 = 1− ULG0 + [ULG0]2 − [ULG0]3 + · · · : (28)


= + ...Go

Go

Go Go

Go

Go∆L

∆L

∆LG

Rayleigh Rayleigh + 1. order Raman correction

Rayleigh + 2. order Raman correction

- -

Fig. 1. Illustration of the perturbation series in Eq. (27). The Green’s operator G may be approximated by the unperturbedGreen’s operator G0 plus higher order corrections. Here, each correction term consists of a chain of elements built upfrom the unperturbed Green’s operator G0, which describes the propogation of light to the point of perturbation, and theperturbation of transport itself, represented by the operator UL.

Eq. (27) may be interpreted in a physical manner. The propagation of light between two pointsof the phase space including inelastic Raman scattering is given by a series expression, where eachterm of the series consists of a k-fold application of the operator

ULG0 =∫

dz′d′d�′′d�′UJ(z;�; | z;�′′; ′)G0(z;�′′; ′ | z′;�′; ′) ◦ (29)

times the monochromatic Green’s operator G0 on the left. The Green’s function G0 in Eq. (29)describes the propagation of light from a phase point (z′;�′; ′) to a point (z;�′′; ′). At this pointin the phase space the radiation =eld is perturbed and a Rayleigh scattering process is replacedby a corresponding Cabannes and Raman scattering event. Both processes cause a change in thepropagation direction of light, �′ → �′′, and Raman scattering provides a wavelength shift ′ → as well. Finally, the most left Green’s operator G0 in Eq. (27) denotes the propagation of theradiation =eld to the target point of consideration without change in wavelength. Thus the kth termof the perturbation series represents the propagation of light including k Raman scattering processes.The perturbation series is illustrated in Fig. 1 for the =rst three orders.

Substitution of Eq. (27) in Eq. (22) gives a corresponding series expansion of the radiation e:ect,viz.

E = E0 +∞∑k=1

(−1)k〈R; G0[ULG0]kS〉: (30)

Here, the Green’s operator G0 has to be known in order to calculate any correction to the unperturbede:ect E0. G0 is determined by the solution of Eq. (25) and even for a discretized problem in heightand propagation directions of light, its numerical calculation is very time consuming, because of themany di:erent sources for which the radiative transfer problem has to be solved. This restricts theapplication of the perturbation series for numerical use. However, the so-called adjoint formulationof radiative transfer facilitates the calculation to =rst order in the perturbation series.

3.2. The adjoint transport problem

The transport operator adjoint to L, which is called L†, is de=ned by requiring [17,18,22]

〈I2; LI1〉 = 〈L†I2; I1〉 (31)


for arbitrary vector functions I1 and I2. Here, the adjoint vector =eld I† is the solution of the adjointtransport equation

L†I† = S† (32)

with any suitable adjoint source S†. For homogeneous boundary conditions of the adjoint problem

I†(ztop;�; ) = [0; 0; 0; 0]T for ¿ 0;

I†(0;�; ) = [0; 0; 0; 0]T for ¡ 0; (33)

the operator L† is given by [29,30]

L† =∫ ∞

0d

∫4�

d�

{�(− )

{[− @

@z+ �e(z; )

]�(�− �)E

− A()�

�(z)�()||�(−)||}

− J†(z; �; | z;�; )}

◦ (34)

with

J†(z; �; | z;�; ) = �(− )�cabs (z; )

4�Zcab;T(;�; �) − �rrs (z; ; )

4�Zrr;T(;�; �): (35)

Compared to the forward operator L the adjoint operator L† di:ers in the sign of the streamingterm. Furthermore, the scattering phase matrices Zcab(; �;�) and Zrr(�;�) are replaced by theirtransposed matrices Zcab;T(;�; �) and Zrr;T(�; �), respectively. Analogous expressions hold forthe monochromatic radiative transfer equation (16).The adjoint formalism can be applied also to the Green’s function method. Similar to Eq. (25)

the Green’s function G† is de=ned by

L†g†i (z;�; | z′;�′; ′) = �(z − z′)�(�−�′)�(− ′)ei ; (36)

where g†i represents the ith column vector of G†. Next, we de=ne the operator G† analogous to

Eq. (23). So the adjoint intensity =eld, which belongs to an adjoint source S†, may be calculated by

I† = G†S†: (37)

The forward and adjoint formulations of radiative transfer do not describe two independent radia-tive transfer problems. The corresponding intensity =elds I and I† are linked by the relation

〈S†; I〉 = 〈I†;S〉; (38)

which can be derived using Eqs. (2), (31) and (32). For the simulation of a radiation e:ect E wechoose the corresponding response function R as the adjoint source S†. In this particular case, theleft-hand side of Eq. (38) represents the de=nition of the radiation e:ect E in Eq. (12) and theright-hand side provides a second recipe to calculate E using the adjoint =eld. Here, one weighsthe source S by the adjoint intensity =eld I† and integrates the product over the phase space ofthe problem. In other words, the adjoint =eld I†(z;�; ) can be interpreted as the importance ofthe source S(z;�; ) with respect to the radiation e:ect E [31,32]. In particular, due to the speci=cresponse function Ri in Eq. (14) the solution of the adjoint problem

L†0I

†i;0 = Ri (39)


has the form

I†i;0(z;�; ) = I†i;0(z;�; v)�(− v): (40)

The use of the response function as an adjoint source allows one to rewrite the perturbation seriesin Eq. (30) to =rst order as

Ei = Ei;0 + 〈I†i;0;ULI0〉 + O(�2); (41)

where I†0 and I0 are the adjoint and forward solutions of the unperturbed radiative transfer problem. 1

Here, O(�2) indicates higher orders of perturbation. Thus, the =rst order perturbation e:ect is solelydetermined by the adjoint and forward intensity =eld. This represents a signi=cant simpli=cation forany computational e:ort, because the numerically expensive calculation of the full Green’s functionis avoided.

For the numerical calculation of I†i;0 another advantage of the adjoint formulation is that both theforward and the adjoint formulation can be solved with one radiative transfer code. The de=nitionof the vector =eld

'i(z;�; ) = I†i;0(z;−�; ) (42)

allows one to transform the adjoint equation (32) into a corresponding pseudo forward equation[17,22,30], viz.

Lps0 'i = Sps

i (43)

with the source

Spsi (z;�; ) = Ri(z;−�; ): (44)

Here Lps0 is the same as L0, except that the phase matrix Zray(;�; �) is replaced by the matrix

QZray(; �;�)Q with Q = diag[1; 1; 1;−1]. Thus, only one algorithm is needed to determine theforward as well as the adjoint intensity =eld.

3.3. The perturbation integrals of 9rst order

Given the forward and adjoint intensity =eld I0 and I†0 as well as the perturbation operator ULin Eq. (18), the perturbation integrals of =rst order can be evaluated further, viz.

UEi = 〈I†i;0;ULI0〉

=∫ ∞

0d[K rr

i (; v) + Kcabi (; v) − K ray

i (; v)]: (45)

The integral kernels K rri , K

cabi , and K ray

i are of the same type and are given by

Ki(; v) =∫ ztop

0dz�s(; v)

4�

∫4�

∫4�d�d�'T

i (z;−�; v)Z(; �;�)I0(z; �; ): (46)

1 To derive Eq. (41) one makes use of the fact that the operator G† is the adjoint operator of G. To show this, wewrite Eq. (38) in the form 〈S†; GS〉 = 〈G†S†;S〉 using Eqs. (21) and (37). This is true for any sources S† and S andthus G† is the adjoint operator of G, according to the de=nition in Eq. (31).


Here the adjoint intensity =eld is already replaced by its pseudo-forward correspondent 'i, givenin Eq. (42). Depending on the speci=c kernel, Z represents the phase matrix for Raman, Cabannesand Rayleigh scattering, respectively. Furthermore, the scattering coeOcient �s is given by

�s(; v) =

�rrs (; v) for K rr

i (; v)

�(− v)�rays (v) for K rayi (; v)

�(− v)�cabs (v) for K cabi (; v):

(47)

Subsequently, the kernels Ki in Eq. (46) are expanded in a cosine and sines Fourier series using theFourier expansions of the scattering phase matrix and of the forward and pseudo-forward intensity=eld (see [16,33] and Appendix B). For the radiation e:ects Ei with i=1; 2, thus for radiation e:ectsrepresented by the Stokes components I(ztop;�v; v) and Q(ztop;�v; v), the kernels are given by thecosine series

Ki(; v) =�4

∞∑m=0

(2 − �m0) cosm(’v − ’0)Kmi (; v) (48)

with coeOcients

Kmi (; v) =∫ ztop

0dz�s(z; ; v)

∫ 1

−1

∫ 1

−1d d'+mT

i (z; ; v))Zm(; ; )I+m0 (z; ; ): (49)

Here, ) = diag[1; 1;−1;−1] and Zm, '±mi and I+m are the Fourier components of the scattering

phase matrix, the pseudo-forward and forward intensity =eld, respectively (see Eq. (B.1) and (B.6)in Appendix B). Analogously, we obtain a corresponding sines expansion for the radiation e:ectsEi, with i = 3; 4,

Ki(; v) =�4

∞∑m=0

(2 − �m0) sinm(’v − ’0)Kmi (z; ; v) (50)

with coeOcients

Kmi (; v) =∫ ztop

0dz�s(z; ; v)

∫ 1

−1

∫ 1

−1d d'−mT

i (z; ; v))Zm(; ; )I+m0 (z; ; ): (51)

To determine the coeOcients in Eqs. (49) and (51), any vector radiative transfer model may be usedthat calculates the Fourier components of the internal intensity =elds I and 'i. A further evaluationof the perturbation integrals can be performed in a straight-forward manner, but it relies on thespeci=c solution technique used for the radiative transfer problem. Appendix C gives an analysis ofthe integrals utilizing the Gauss–Seidel iteration method to solve the radiative transfer problem. Thissolution method is used for all numerical simulations presented in this paper.

4. Simulation of re+ectance spectra

4.1. Rotational Raman scattering and polarization spectra

To study the e:ect of Raman scattering on the intensity vector of backscattered light, we considersimulated spectra of I , Q and U of the re;ected light in a nadir viewing direction at the top of


Fig. 2. Re;ectivity spectra I=F0, Q=F0, and U=F0 at the top of a Rayleigh scattering atmosphere. Simulations are performedfor solar zenith angles #0 = 10◦ (left panel) and #0 = 70◦ (right panel), for a viewing zenith angle #v = 10◦ and for arelative azimuthal angle of 120◦. The vertical distribution of ozone is adopted from the US standard atmosphere [51],where the model atmosphere is divided in 60 homogeneous layers of the geometric thickness of 1 km. The Lambertiansurface albedo is 5%. The solar spectrum F0 is given by Chance and Spurr [4].

cloud- and aerosol-free model atmospheres. The simulations are performed on a 1 cm−1 spectralgrid from 290 to 405 nm. Due to the speci=c form of the phase matrix for Raman, Rayleigh andCabannes scattering (see Eq. (A.10) of Appendix A), the Stokes parameter V always vanishes forthese model atmospheres. The spectra are convolved with a Gaussian slit function with a FWHMof 18 cm−1, which corresponds to a FWHM of 0.15–0:29 nm depending on wavelength. Fig. 2gives two examples of simulated spectra I=F0, Q=F0, and U=F0 for solar zenith angles #0 = 10◦and 70◦. Ozone absorption is included in the model calculations, which causes a low re;ectivity atwavelengths below 320 nm. The decrease in re;ectance toward longer wavelength is due to the −4

wavelength dependence of Rayleigh scattering. Overall the polarization components Q and U areone order of magnitude smaller for #0 = 10◦ than for #0 = 70◦, which is primarily caused by thedi:erence in the single scattering geometry.

The corresponding Ring spectra in I , Q and U are depicted in Fig. 3. Here, a Ring spectrum isde=ned by

RI =Irr − IrayIray

(52)

for the radiance component and corresponding expressions hold for the other Stokes parameters. Irayrepresents the re;ected intensity, simulated for a Rayleigh scattering atmosphere, and Irr denotes thesame intensity but taking Raman scattering into account in the simulation. At the center of the strongCa II K and H Fraunhofer lines at 393.5 and 396:8 nm rotational Raman scattering causes a =lling-inof the Fraunhofer lines in I of 10% for #0 = 10◦ but of 16% for #0 = 70◦. This dependence onsolar zenith angle has been noted before [2,5] and can be explained by a di:erence in the scatteringgeometry.


Fig. 3. Relative Ring spectra RI , RQ, and RU simulated with the =rst order perturbation theory approach in its vectorformulation. Same model atmosphere and same solar and viewing geometries as in Fig. 2. Left panels #0 = 10◦, rightpanels #0 = 70◦.

Ring structures in RQ are relatively small for #0 = 70◦ (less than 1.5%) but are more pronouncedfor the solar zenith angle #0 =10◦ (up to 5%). Structures in RU are weak for both solar geometries.To understand the di:erent amplitudes of polarization Ring spectra, we have to consider two

causes for Ring lines in the polarization components Q and U of the backscattered light. First, theseRing structures can be induced by polarization properties of Raman scattering itself. The importanceof this e:ect can be estimated by a look at the scattering phase matrices Z rr

ij and Zcabij for rotational

Raman and Cabannes scattering. An analysis of the scattering matrices shows that

Z rr11(�)

Zcab11 (�)

¿Z rrij (�)

Zcabij (�)

(53)

holds for all i = 1 or j = 1. In Fig. 4 this relation is demonstrated for i = 1 and j = 2. Due to thenormalization of the scattering phase matrix the ratio Z rr

11=Zcab11 oscillates around 1. The dependence

on scattering angle originates primarily from the angle distribution of Cabannes scattering, whereasRaman scattering is nearly isotropic. The ratio Z rr

12=Zcab12 =3=40 does not depend on scattering angle �


Fig. 4. Ratio Z rr1j=Z

cab1j of elements of the scattering phase matrix for rotational Raman and Cabannes scattering as a function

of scattering angle �. For j = 1; 2 the ratios are given by corresponding ratios of the scattering phase matrix P de=nedwith respect to the plane of scattering (see Eqs. (A.10) and (A.12) of Appendix A).

at all and is signi=cantly smaller than Z rr11=Z

cab11 . Relation (53) implies that Raman scattered light is

less polarized than Cabannes scattered light. Thus it suggests that the outgoing light at a scatteringpoint shows weaker Ring structures in Q and U compared to structures in the radiance component.Second, Ring structures in Q and U can be caused as well by elastic scattering processes following

a Raman scattering event. This way Ring structures in I propagate to Q and U because of thecoupling between the Stokes parameters in the Cabannes scattering phase matrix. This e:ect can bedemonstrated most easily by a simulation with a modi=ed Raman scattering phase matrix, whichneglects any polarization properties, so Z rr

ij = 0 for i = 1 or j = 1. Although this assumption biasesthe simulation of the re;ected light, which causes the slight o:set in the Ring spectra RQ andRU in Fig. 5, it allows us to estimate the propagation e:ect. Fig. 5 shows structures in RQ of upto 3.7% for #0 = 10◦, whereas for the other polarization spectra those are much weaker. Hence,from a comparison of Figs. 3 and 5 we conclude that for #0 = 10◦ the Ring structures in RQ arepredominately caused by the polarization of light due to Cabannes scattering following a Ramanscattering processes.

Despite the relatively large Ring structures in RQ for #0 = 10◦ one has to keep in mind that theyare related to small values of the Stokes parameter Q. Only due to the small degree of polarization ofthe re;ected light becomes the propagation e:ect visible, while for stronger polarization the e:ect isoverwhelmed by the polarization contribution of Cabannes scattering. Based on this fact we concludethat the simulation of Ring structures in the polarization components is of minor importance for mostapplications.

Rotational Raman scattering does not a:ect only the structures of Fraunhofer lines in the re;ectedsunlight, also the =lling-in of absorption lines from atmospheric constituents may take place [2].At the longwave ultraviolet between 300–340 nm lie the weak Huggins absorption bands of ozone.


Fig. 5. Relative Ring spectra RQ and RU for a arti=cial Raman scattering phase matrix with Z11 = Z rr11 and Zij = 0 for the

other elements. Same model atmosphere and same solar and viewing geometries as in Fig. 2.

The =lling-in of these absorption lines can be demonstrated for a constant solar source F0() = 1in a model simulation, which suppresses the e:ect of Fraunhofer lines. The corresponding Ringspectra are depicted in Fig. 6. For the radiance components the =lling-in of the Huggins absorptionbands reaches 0.5% for #0 = 10◦ and 1.5% for #0 = 70◦ and are thus signi=cantly smaller than thecorresponding Ring structures of Fraunhofer lines. Furthermore, the Ring e:ect for the polarizationparameters Q and U is again weaker than for the radiance Ring structures. Here, most pronouncedRing structures are present in RQ for #0 = 10◦ due to the reasons given above.

4.2. Approximation methods

The calculation of Ring spectra with the proposed vector radiative transfer model are numericallyexpensive and thus approximation methods are desirable for applications to large data sets. One ofthe most simple approximations of radiative transfer is the single scattering approach. Numerically itcan be implemented in a straightforward manner and simulations require only minor computationale:ort. However, the contribution of multiple scattered light �msc to the Stokes parameters I , Q, andU can be signi=cant with a clear dependence on wavelength. This is demonstrated in Fig. 7. Below300 nm the Stokes parameters are mainly determined by singly scattered light due to the strongozone absorption in this spectral range. But at about 320 nm the multiple scattering contributionreaches its maximum of about 55% for the radiance component. For the polarization components Q


Fig. 6. Contributions of the Huggins ozone absorption bands to the Ring spectra of the Stokes parameters I; Q, and U .Same model atmosphere, and same solar and viewing geometries as in Fig. 2.

Fig. 7. Relative contribution of multiple scattering �msc to the Stokes parameter I; Q and U . Same model atmosphere andsame solar and viewing geometries as in Fig. 2.


Fig. 8. Absolute di:erences DI=RI; ssc−RI;vec and analogous expressions for DQ and DU . Here, RI; ssc is the single scatteringRing spectrum and RI;vec is the corresponding Ring spectrum calculated with the vector radiative transfer model. Samemodel atmosphere and same solar and viewing geometries as in Fig. 2.

and U the contribution is somewhat less. So one expects signi=cant errors in Ring spectra that arecalculated in the single scattering approximation.

To compare Ring spectra that are calculated with di:erent radiative transfer models, it is essen-tial that they are normalized with respect to the same reference spectrum. Otherwise artifacts maybe introduced in the analysis originating from the di:erent normalization. Thus, in this study wenormalize all Ring spectra to the corresponding vector radiative transfer simulation, e.g.,

RI;app =Irr;app − Iray;app

Iray;vec; (54)

where Irr;app and Iray;app are approximated radiance spectra and Iray;vec is the corresponding radiancespectrum simulated with the vector model. Fig. 8 shows the errors in the single scattering Ringspectra RI; ssc, RQ; ssc, and RU; ssc. As expected, below 300 nm Ring structures are reproduced well bythis approximation due to the small fraction of multiple scattering events at these wavelengths. Withthe increase of multiple scattering toward longer wavelengths also the error in the single scattering


Ring spectra increases. For the radiance component the absolute error in the Ring spectra is generallyless than 6% but can reach 10% at the Ca II lines for #0 = 70◦. Keeping in mind that the =lling-inof this Fraunhofer lines is about 16%, this represents a clear bias of more than a factor of 2.5 in thesimulation of Ring spectra. For the polarization components relatively large errors occur in RQ; sscfor a solar zenith angle #0 = 10◦. This fact con=rms the interpretation that for this particular caseRing structures are mainly produced by elastic scattering processes following a Raman scatteringevent. Obviously, this e:ect cannot be simulated in the single scattering approximation. For the otherpolarization Ring spectra, where this propagation e:ect is of minor importance, the di:erences aremuch smaller and generally below 0.4%.

Another widely used approximation method, if only radiances need to be simulated, is the scalarradiative transfer approach. The advantage of this approach is that it takes into account multiplescattering of light but at the same time greatly simpli=es the calculations, which reduces the com-putational cost. However, neglecting polarization can cause errors in the modeled radiance of up to10% [9,15,34,35]. The upper panel of Fig. 9 shows the error in the radiance component I . In the

Fig. 9. (upper panel) Relative error �Iray in simulated spectra Iray due to the scalar approximation of radiative transfer,�Iray = (Iray; sca − Iray;vec)=Iray;vec. (lower panel) Di:erence DI; sca in the radiance Ring spectra RI; sca and RI;vec for scalarand vector radiative transfer simulations, respectively, DI; sca = RI; sca − RI;vec. Here, both Ring spectra are normalized tothe radiance spectrum of a Rayleigh scattering atmosphere simulated with the vector radiative transfer model. The modelatmosphere, surface albedo, and solar and viewing geometry is the same as in Fig. 2.


single scattering domain below approximately 300 nm the error is very small. For singly scatteredlight the scalar approximation yields the same radiance as the vector approach, because the incomingsunlight can be assumed unpolarized [36]. However, if a second scattering process takes place, whichis very likely for wavelengths longer than 300 nm (see Fig. 7), the incoming light for this process isstrongly polarized (see e.g. [34]). The radiance of this second order scattered light does not dependonly on the radiance component of the incoming light but also on its Stokes parameters Q andU . Hence, a neglect of polarization leads to an incorrect value of the modeled radiance. The sameis true for higher order scattering but here the e:ect is smaller. In Fig. 9 magnitude and sign ofthe error in the scalar radiative transfer depend additionally on the scattering geometry and on theorientation of successive scattering processes [34].

For the simulation of Ring spectra RI; sca the scalar approach is much more exact than for sim-ulations of the continuum. The errors, shown in Fig. 9 are mostly below 0.15% and reach theirmaximum of 0.2% for the Ca II lines for #0=70◦. Again, compared with the =lling-in of the Fraun-hofer line of 16% this represents a bias of only a factor of about 1.01. The high accuracy can beexplained by the fact that Raman scattered light is less polarized and so the coupling of the Stokesparameters due to scattering processes is of minor importance for the simulation of Ring spectra in I .

5. Simulation of polarization sensitive measurements of GOME including Ring structures

The GOME spectrometer on board of ESA’s ERS-2 satellite measures the backscattered sunlightin the spectral range 240–790 nm. Although the aim of GOME is to measure the radiance componentI only, the measurements are also sensitive to the state of polarization of the backscattered sunlight.Furthermore, due to the spectral resolution of GOME of about 0:2 nm, Raman scattering clearlya:ects the spectral structures of the measurement. Thus for an extensive simulation of polariza-tion sensitive GOME measurements a vector radiative transfer model is needed that simulates Ringstructures for all relevant Stokes parameters in addition to the continuum spectrum for a Rayleighscattering atmosphere. In this section it is investigated to which extent approximate Ring spectra,simulated with the single scattering and scalar radiative transfer approach, are acceptable within theerror margin of the GOME instrument.

To describe the radiance spectrum measured by the detectors of a GOME-type instrument, weneed both the intensity vector of light illuminating the instrument and the transmission properties ofthe spectrometer. The radiance detected by a certain detector pixel i is given by

Idet; i =M11Ii +M12Qi +M13Ui; (55)

where M11 to M13 are the elements of the =rst row of the Mueller matrix of the instrument char-acterizing its transmission properties. For sake of simplicity we omit the dependence of the Muellermatrix on the pixel index i. The Stokes parameters Ii, Qi, and Ui are components of the intensityvector

Ii =∫ ∞

0d'i()I(): (56)

Here I denotes the intensity vector of the re;ected sunlight at the entrance slit of the spectro-meter. The integration over weighted with the instrument spectral response function 'i describes


both the e:ect of the instrument slit and the sampling by the detector pixel i [37]. For all sim-ulations we assume a constant sensitivity within the spectral range of the detector pixel. In Eq.(55) the contribution of Stokes parameter Vi is neglected due to its small value for atmosphericapplications.

A radiometric calibration of Idet; i yields the polarization sensitive measurement

Ipol; i =1M11

Idet; i = Ii + m12Qi + m13Ui: (57)

Here Ipol; i denotes the radiance measurement not corrected for the instrument polarization sensitivityand m12 and m13 are the relative elements of the Mueller matrix M12=M11 and M13=M11, respectively.Any simulation of Ipol; i requires these instrument characteristics, which have to be determined beforelaunch of the instrument. For GOME in particular it is assumed that the instrument is not sensitivewith respect to Stokes parameter U , thus m13 = 0. For a nadir viewing geometry of GOME thesensitivity m12 ≈ 0:5 below 315 nm, which belongs to channel 1 of the instrument. In the secondinstrument channel m12 changes approximately linearly from 0.30 to 0.46 in the spectral range315–405 nm.

An additional calibration step of the standard GOME data processing is the so-called polarizationcorrection, which aims to eliminate the polarization dependence of Ipol; i in Eq. (57). In other words,Ipol; i has to be converted to a corresponding radiance spectrum, viz.

Iconv; i = CconvIpol; i (58)

with a conversion factor

Cconv =1

1 + m12qi: (59)

Here, qi is the relative Stokes parameter Qi=Ii. In case of an unbiased conversion the spectrumIconv; i is identical to Ii in Eq. (57). This conversion requires polarization measurements to determineqi at the spectral resolution of the measurement Iconv; i. However, GOME only performs broadbandpolarization measurements and so polarization spectra have to be reconstructed from measurementsat lower spectral resolution. Here, errors of the order of 10% may be introduced in the convertedradiances, in particular in parts of the spectrum with strong variation of qi due to atmosphericabsorption [6–8]. Additionally, qi is a:ected by Raman scattering. Ring lines in qi are mainly causedby corresponding Ring structures in the radiance component Ii and only to a minor extent by Ringstructures in Qi. Fig. 10 shows that the neglect of the Ring structures in qi can cause errors of upto 1.4% in the converted radiance spectrum Iconv; i, depending on the degree of polarization.

To investigate the relevance of this error, we consider the instrument noise of GOME re;ectivityspectra r= Iconv=F0 between 290–405 nm, at 12 geo-locations (±60◦ and ±20◦ latitude, 0◦, 120◦ and240◦ longitude) for the period April, 1996–March, 1998. Fig. 11 shows the frequency distributionof the mean error + for these measurements, where + is de=ned by

+=

√√√√ 1N

N∑i=1

[eiri

]2

: (60)

Here ei denotes the absolute 1 , error of a re;ectivity spectrum ri for a detector pixel i, and N is thetotal number of detector pixels of the considered spectral range. For about 97% of all measurements


Fig. 10. Relative error �Iconv in the polarization corrected radiances Iconv due to the neglect of Ring structures in therelative Stokes parameter q in Eq. (58). Calculations are performed for the same model atmosphere as in Fig. 2 with#0 = 70◦, #v = 10◦, and for relative azimuthal angle U’= 120◦.

Fig. 11. (left panel) Frequency distribution of the mean error of GOME re;ectance measurements for the period April,1996–March, 1998. Measurements are considered at 12 geo-locations, viz. ±60◦ and ±20◦ latitude and 0◦; 120◦ and 240◦

longitude. Here, the distribution is binned in intervals of width 0.01%. (right panel) Two GOME error spectra representingan upper and a lower error margin for GOME measurements. The solid line shows a 1 , error spectrum with a meanerror of + = 0:43%, the dotted line represents a corresponding spectrum with + = 0:19%.

the mean error lies between 0.19–0.43%. From this distribution we select two spectra emin with+=0:19% and emax with +=0:43%, representing a lower and an upper error margin of GOME. Theerror spectra are shown in the right panel of Fig. 11. Using these error scenarios, we consider therelative bias bi=ei, where bi = Iconv; i − Ii is the absolute di:erence between the polarization corrected


radiances and the unbiased radiance spectrum. The mean relative bias

bmean =1N

N∑i=1

∣∣∣∣biei∣∣∣∣ (61)

can be interpreted as a measure of the conversion bias. Obviously, bmean depends on the selectedspectral range of the measurement. In the following we consider two examples: (a) the spectral range290–313 nm, which can be used for ozone pro=le retrieval (see e.g. [20]) and (b) two detector binsrelated to the center of the Ca II K and H Fraunhofer lines at 393.5 and 396:8 nm, which containinformation on the cloud distribution of the observed ground scene [38]. In the case of ozone pro=leretrieval the mean relative bias bmean due to the neglect of Raman scattering in the polarizationcorrection is 0.3 and 0.8 for the error scenarios emax and emin, respectively. This indicates that themean bias is below the instrument noise for this application. However, at the center of the Ca IIFraunhofer lines bmean reaches values of 11.4 and 11.9 for both error scenarios. So in this case thebias is 11 to 12 times as large as the instrument noise, which can cause serious problems for theinterpretation of the =lling-in of these Fraunhofer lines.

For retrieval purposes the polarization correction can be avoided, if one attempts to reproduce thepolarization sensitive measurement Ipol; i instead of a polarization corrected radiance measurement.For GOME this requires the simulation of Stokes parameters Ii and Qi of the backscattered sunlight.Hasekamp et al. [9] have demonstrated that this approach provides a clear improvement for ozonepro=le retrieval from GOME re;ectance measurements, compared to a retrieval approach using po-larization corrected radiances. The retrieval approach utilizes measurement simulations, which areperformed for a Rayleigh scattering atmosphere. To include Ring structures in the simulations onemay introduce Ring spectra in the di:erent Stokes parameters of the polarization sensitive GOMEmeasurement in Eq. (57), viz.

Ipol = (1 + RI;vec)Iray;vec + m12(1 + RQ;vec)Qray;vec: (62)

To simplify matters we omit the pixel index i on the spectra.For an eOcient simulation of polarization sensitive GOME measurements one can try to approx-

imate the Ring structures of Ipol by a simpli=ed calculation scheme. The results of the previoussection suggest to neglect the Ring spectrum RQ;vec in Eq. (62) because of its small value. Thus we=rst consider the approximation

Iapp1 = (1 + RI;vec)Iray;vec + m12Qray;vec: (63)

Subsequently, we replace the Ring spectrum RI;vec by the respective scalar and single scatteringspectra, which yields the approximations

Iapp2 = (1 + RI; sca)Iray;vec + m12Qray;vec; (64)

Iapp3 = (1 + RI; ssc)Iray;vec + m12Qray;vec: (65)

In the context of GOME measurement simulations the goodness of these approximations can beassessed using the mean relative bias bmean in Eq. (61) with bi = Iapp; i − Ipol; i. Again we considerthe mean bias for the spectral range 290–313 nm as well as at the center of the Ca II Fraunhoferlines. Fig. 12 shows bmean for the spectral window of ozone pro=le retrieval as functions of thesolar zenith angle, of the viewing zenith angle and of the relative azimuthal angle for a =xed model


Fig. 12. Mean relative bias bmean for simulated GOME measurements Ipol and model simulations Iapp1; Iapp2, and Iapp3 inEqs. (63), (64) and (65) in the spectral range 290–313 nm. Here, the dotted, dashed and solid lines show the averagedvalue for the error scenarios emin and emax. The di:erence between the mean biases for the two error scenarios is less than0.04 for model simulations Iapp1, and less than 0.1 and 1.9 for Iapp2 and Iapp3, respectively. (left panel) bmean as a functionof solar zenith angle #0 for viewing zenith angle of #v = 10◦, and for a relative azimuthal angle U’= ’v − ’0 = 120◦,(middle panel) bmean as a function of #v for #0 =40◦ and U’=120◦, (right panel) bmean as a function of U’ for #0 =40◦

and #v = 10◦. The model atmosphere is adopted from Fig. 2.

atmosphere. The dependence on atmospheric parameters is demonstrated in Fig. 13, which showsthe dependence on the vertically integrated ozone column, on the Lambertian ground albedo, andon the truncation height for the lower model boundary. The latter combined with a high Lambertianalbedo of A= 0:8 simulates the e:ect of di:erent cloud top heights.The di:erent scenarios show that in average the neglect of polarization Ring spectra between

290 and 313 nm insigni=cantly biases the simulation of GOME measurements with a mean biasof about two orders of magnitude below the instrument noise level. Also the approximation inEq. (64), which employs scalar radiance Ring spectra, shows a small bias with bmean¡ 0:1. There-fore, the bias is overwhelmed mostly by instrument noise. Contrary to that, the single scatteringapproximation produces a clear bias in the simulation with 1¡bmean¡ 4. Here, the dependence onthe di:erent parameters is related to the amount of multiple scattered light contributing to the detectedsignal. This becomes obvious for the increase of bmean with increasing ground albedo. A change ofthe lower boundary height of the model atmosphere shows a similar e:ect, where the smallest bmean

values are related to the highest truncation height, which corresponds to a high cloud top. In thiscase the optically dense troposphere is cut o: and the radiative transfer becomes dominated by singlescattering.

The error assessment di:ers, if one considers the bias for the Ca II Fraunhofer lines only. Fig. 14shows the corresponding relative mean bias bmean as functions of solar zenith angle. For the singlescattering approximation Iapp3 we obtain a large bias with 40¡bmean¡ 70. Also the approximationsIapp1 and Iapp2 are clearly biased in this case. For Iapp1 the relative mean bias bmean increases toward


Fig. 13. Same as Fig. 12 but for di:erent atmospheric parameters. The calculations are performed for a solar and viewinggeometry of #0 =40◦, #v=10◦, U’=120◦. (left panel) Dependence on ozone column C=200−400 DU for an albedo ofA= 0:1. (middle panel) Dependence on Lambertian ground albedo A= 0:0− 1:0 for a vertically integrated ozone columnof C = 300 DU. (right panel) Truncated atmosphere at a truncation height ztrun = 0 − 10 km for a Lambertian groundalbedo =0.8. The di:erence in bmean for the error scenarios emin and emax is less than 0.02 for Iapp1, and less than 0.03and 3.00 for Iapp2 and Iapp3, respectively.

Fig. 14. Mean relative bias bmean as function of solar zenith angle #0 for simulated GOME measurements Ipol and modelsimulations Iapp1; Iapp2, and Iapp3 in Eqs. (63), (64) and (65) at the center of the Ca II K ans H Fraunhofer lines at 393.5and 396.8 nm. The model atmosphere and the viewing geometry are the same as for the left panel of Fig. 12. Thedi:erence in bmean for the error scenarios emin and emax is less than 0.04 for Iapp1, and less than 0.04 and 2.43 for Iapp2and Iapp3, respectively.


larger solar zenith angles and reaches the level of the instrument noise at #0 =70◦. This dependenceis caused by the increase of the degree of polarization of the re;ected sunlight for large solar zenithangles (see e.g. Fig. 2). The approximation Iapp2 has its largest bias for #0 = 0◦ with bmean ≈ 1.Here, the bias decreases toward larger solar zenith angle because of the cancellation of errors dueto both the scalar simulation of the radiance Ring spectrum and the neglect of the polarizationRing spectrum RQ. We achieve similar results for di:erent viewing zenith angles and for di:erentrelative azimuthal angles as well as for a change of atmospheric parameters (not shown). Thus thesimulation of the =lling-in of the Ca II Fraunhofer lines in GOME radiance measurements, whichuses the approximations Iapp1 and Iapp2, can cause biases in the order of the instrument noise.Overall, the suitability of the proposed approximation techniques depends clearly on the speci=c

application. Here, we have considered two examples: For ozone pro=le retrieval from GOME po-larization sensitive measurements an eOcient simulation can be achieved using the approach of Eq.(64). This includes (a) the calculation of Ring spectra with a scalar radiative transfer approachand (b) the use of a vector radiative transfer model to determine the continuum spectra of Stokesparameters I and Q for a Rayleigh scattering atmosphere. Contrary, for the retrieval of cloud prop-erties from the =lling-in of the Ca II H and K Fraunhofer lines, an accurate simulation of the Ringstructures within the bounds of the instrument noise can only be achieved using a vector radiativetransfer simulation of both the Ring spectra RI and RQ.

6. Summary

A vector radiative transfer model is presented that includes the simulation of inelastic rotationalRaman scattering in addition to elastic Cabannes scattering. The model describes the re;ectanceproperties of a vertically inhomogeneous, plane parallel model atmosphere bounded from below bya Lambertian surface. Rotational Raman scattering is treated as a perturbation to Rayleigh scattering.This perturbation is presented by a classical perturbation series using the Green’s function formalismof the unperturbed problem. The nth contribution of the perturbation series describes the e:ect ofn orders of Raman scattering. The model is worked out in more detail for =rst order perturbations.In this case the forward and the adjoint solutions of the radiative transfer problem replace theexpensive calculation of the corresponding Green’s function, which provides a signi=cant reductionin the numerical requirements. Based on this a numerical vector radiative transfer model is presentedincluding one order of Raman scattering. For the calculation of the perturbation e:ect any vectorradiative transfer model can be used that calculates the internal forward and adjoint intensity vector=eld of the atmosphere. In this paper the Gauss–Seidel iteration method is employed, which provideseOcient simulations for cloud-free atmospheres.

Simulations of Ring spectra are presented for the spectral range 290–405 nm and the e:ect ofRaman scattering on the polarization components Q and U is discussed. Here, Ring structures in theStokes parameters Q and U originate both from the polarization of light by Raman scattering itselfand from the polarization by Cabannes scattering following a Raman scattering process. In general,polarization Ring spectra of Q and U are much weaker than those of the radiance componentdue to the low polarization of Raman scattered light. Only for a low polarization of the re;ectedsunlight pronounced Ring structures occur in the polarization spectra. Those are mainly caused bythe propagation of Ring structures from the radiance component to the polarization components by


Cabannes scattering processes. However, because of the small degree of polarization of the re;ectedlight, this e:ect is of minor importance for most applications.

The accuracy of two common approximation techniques is investigated for an eOcient simulationof Ring spectra: the single scattering approximation and the scalar approximation. The single scat-tering approach shows a clear bias in the simulation of Ring structures. For example, the =lling-inof the Ca II Fraunhofer lines at about 395 nm is underestimated by more than a factor of 2.5. Thecorresponding =lling-in simulated with the scalar approach is only biased with a factor of about1.01. The relevance of these biases depends on the spectral range of the measurement and on thespeci=c application. For example, for ozone pro=le retrieval from GOME polarization sensitive radi-ance measurements between 290–313 nm the combination of both a vector radiative transfer model,which simulates the measurement for a Rayleigh scattering atmosphere, with a scalar radiative trans-fer approach to account for the e:ect of inelastic Raman scattering on the measurement provides aneOcient simulation of the measurement with only a minor bias. However, for the retrieval of cloudproperties using the =lling-in of the Ca II Fraunhofer lines of GOME radiance measurements werecommend comprehensive vector radiative transfer simulations of the Ring spectra in both Stokesparameters I and Q.In this paper we have investigated polarization aspects of one order of Raman scattering. The

relevance of multiple order of Raman scattering for the simulation of GOME-type measurementswill be subject of future work.

Appendix A. Optical properties of rotational Raman scattering, Cabannes scattering, and Rayleighscattering

In general, scattering of light by an ensemble of random oriented molecules is characterized bya scattering coeOcient �s and a scattering phase matrix Z. Here, the scattering coeOcient may bewritten as

�s = ,.; (A.1)

where , is the scattering cross section and . the volume density of scattering molecules. TheRayleigh scattering cross section for a constituent of the atmosphere is given by [39–41]

,ray() =24�2c4

N 20 4

(n2() − 1)2

(n2() + 2)2F(); (A.2)

where is wavelength, n is the refraction index, N0 is the molecular number density and F is theKing factor. Here, the refraction index n and the number density N0 have to be taken for the sameconditions, i.e., same temperature and same pressure. The King factor and the refraction index forN2 and O2 molecules are given by Bates [39]. For inert gases like Ar the King factor is 1. Therefraction index of Ar is adopted from Peck and Fisher [42].

Cabannes scattering cross sections may be calculated from the corresponding Rayleigh cross sec-tions in a straightforward manner, viz.

,cab() =18 + +()18 + 4+()

,ray(): (A.3)


The factor f = (18 + +)=(18 + 4+) describes the fraction of scattered light that is contained in theCabannes line [10]. Here + = (1=2)2, 1 is the anisotropy of the polarizability, and 2 is the averagepolarizability, which can be determined from the relation

2() =9

16�2N 20

(n2() − 1)2

(n2() + 2)2: (A.4)

For N2 and O2 a parameterization of 1 is given by Chance and Spurr [4]. However, + may becalculated as well directly from the King factor using the relation F = 1 + 2+=9.Rotational Raman scattering consists of Stokes and anti-Stokes scattering. Stokes lines are charac-

terized by the change in rotational angular momentum of a molecule J → J ′=J+2 for J=0; 1; 2; : : : ;anti-Stokes lines are indicated by the transition J → J ′ = J − 2 for J = 2; 3; 4; : : : . The scatteringcross sections for pure rotational Raman scattering are [4,43].

,rr(; ′) =256�2

27′ 12()fJbJ;J ′ ; (A.5)

where is wavelength of the incoming light and ′ represents the corresponding wavelength of thescattered light. fJ is the fractional population in the initial state and can be approximated by [2].

fJ =1Nf

gJ (2J + 1) exp[−EJkT

]; (A.6)

where gj is the nuclear spin statistical weight factor, k is the Boltzmann’s constant, T is the tem-perature, and EJ = J (J + 1)hcB is the rotational energy (h is Planck’s constant, c is the speed oflight, and B is the rotational constant). The coeOcient Nf can be determined from the normalizationcondition∑

J

fJ = 1: (A.7)

bJ;J ′ represents the Placzek–Teller coeOcient for the transition J → J ′. For simple linear moleculesthese coeOcients are given by

bJ;J+2 =3(J + 1)(J + 2)2(2J + 1)(2J + 3)

; (A.8)

for Stokes lines and

bJ;J−2 =3J (J − 1)

2(2J − 1)(2J + 3); (A.9)

for anti-Stokes lines. The change in wavelength U= ′ − due to rotational Raman scattering canbe calculated easily from the energy di:erence between the initial and =nal state, UE = EJ ′ − EJ .The scattering phase matrix P, de=ned with respect to the plane of scattering, has the same

structure for Rayleigh, Cabannes and rotational Raman scattering, viz.

P(�) =1N

A(1 + cos2�) + C −A(1 − cos2�) 0 0

−A(1 − cos2�) A(1 + cos2�) 0 0

0 0 2Acos� 0

0 0 0 Bcos�

(A.10)


Table 1Parameters A; B; C, and normalization constant N of scattering phase function (A.10) for Rayleigh, Cabannes and rotationalRaman scattering

A B C N

Rayleigh 3(45 + +) 30(9 − +) 36+ (180 + 40+)Cabannes 3(180 + +) 30(36 − +) 36+ 40(18 + +)rot. Raman 3 −10 36 40

with parameters A; B, and C, and a scattering angle � [11,13]. The factor N guarantees the normal-ization of the scattering phase matrix,∫

P11(�) d� = 4�: (A.11)

The parameters A; B; C and N are listed in Table 1 for Rayleigh, Cabannes, and rotational Ramanscattering.

For radiative transfer simulations it is necessary to choose one common plane of reference forthe Stokes parameters, which in this paper is the local meridian plane. The scattering phase matrixde=ned with respect to the local meridian plane may be obtained from P(�) by means of tworotations [15,36], viz.

Z(�;�) =

1 0 0 0

0 cos 2i2 −sin 2i2 0

0 sin 2i2 cos 2i2 0

0 0 0 1

P(�)

1 0 0 0

0 cos 2i1 −sin 2i1 0

0 sin 2i1 cos 2i1 0

0 0 0 1

(A.12)

with two appropriate rotation angles i1 and i2. Here, � and � represent the incoming and outgoingdirection of light.

Appendix B. The Fourier expansion

For an evaluation of a plane parallel radiative transfer problem a common approach is to describethe azimuthal dependence of the problem by a Fourier decomposition. The Fourier expansion of thescattering matrix may be given by [16,33]

Z(z; �;�) =12

∞∑m=0

(2 − �m0)[B+m(’− ’)Zm(z; ; )(E+ ))

+B−m(’− ’)Zm(z; ; )(E− ))]; (B.1)

with the Kronecker delta �m0,

)= diag[1; 1;−1;−1]; (B.2)


and

B+m(’) = diag[cosm’; cosm’; sinm’; sinm’]; (B.3)

B−m(’) = diag[ − sinm’;−sinm’; cosm’; cosm’]: (B.4)

The mth Fourier coeOcient of the phase matrix can be calculated by

Zm(z; ; ) = (−1)m∞∑l=m

Plm(−)Sl(z)Plm(−); (B.5)

where Sl is the expansion coeOcient matrix and Plm is the generalized spherical function matrix[16,33]. For Rayleigh, Cabannes and rotational Raman scattering these coeOcient matrices can bedetermined straightforwardly [13]. In the corresponding Fourier expansion of the intensity =eld

I(z;�) =∞∑m=0

(2 − �m0)[B+m(’0 − ’)I+m(z; ) + B−m(’0 − ’)I−m(z; )]; (B.6)

two types of Fourier coeOcient vectors are needed

I+m(z; ) =12�

∫ 2�

0d’B+m(’0 − ’)I (z;�);

I−m(z; ) =12�

∫ 2�

0d’B−m(’0 − ’)I (z;�): (B.7)

An expansion analogous to (B.6) holds for the radiation source S. The Fourier coeOcients of thesolar source in Eq. (9) are given by

S+m(z; ; ) =12�0�(z − ztop)�( + 0)F0();

S−m(z; ; ) = [0; 0; 0; 0]T: (B.8)

On the other hand, the Fourier coeOcients of the source Sps in Eq. (43), which is de=ned by theresponse function Ri in Eq. (14), are

S+m' (z; ; ) =

12��(z − ztop)�( + v)ei ;

S−m' (z; ; ) = [0; 0; 0; 0]T: (B.9)

for i = 1; 2, but

S+m' (z; ; ) = [0; 0; 0; 0]T;

S−m' (z; ; ) =

12��(z − ztop)�( + v)ei ; (B.10)

for i = 3; 4. Hence, due to the Fourier coeOcients of the solar source in Eq. (B.8) the FouriercoeOcients I−m of the forward intensity =eld are zero. For the adjoint =elds the coeOcients '−m

are zero for the response function Ri with i = 1; 2, while for the response function Ri with i = 3; 4the coeOcients '+m are zero. This simpli=cation is possible due to the special form of the Fourier


expansion in Eqs. (B.1) and (B.6). For the types of radiation sources considered in this paper, thisreduces signi=cantly the numerical e:orts required for solving the radiative transfer equation.

With the Fourier expansion of Z; I and S the monochromatic radiative transfer equation (16)and the pseudo forward equation (43) decompose into corresponding equations for every Fouriercomponent [16],

Lm0 I±m0 = S±m; (B.11)

where Lm is the transport operator

Lm=∫ ∞

0d

∫ 1

−1d

{�(− )

{[@@z

+ �e(z)]�( − )E

− �0mA()� �(z)�()||�(−)||}

− J±m(z; ; | z; ; )}

◦ (B.12)

with

J±m(z; ; | z; ; ) = �(− )�rays (z; )

4�Zray;±m(; ): (B.13)

A similar expression holds for the pseudo-forward problem in Eq. (43).Due to the special form of the forward and adjoint radiation sources in Eqs. (9) and (14), it is

common to separate the uni-directional beam from the di:use part of the intensity =elds, viz.

I+m(z; ) = I+md +12��( + 0)F0e−;=0 (B.14)

and

'+m(z; ) ='+md +

12��( + v)

1veie−;=0 ;

'−m(z; ) = 0 (B.15)

for i = 1; 2, and

'+m(z; ) = 0

'−m(z; ) ='−md +

12��( + v)

1veie−;=0 (B.16)

for i = 3; 4. Here Id and 'd denotes the di:use part of the forward and pseudo-forward =eld,respectively. This allows one to use standard techniques, e.g. the discrete ordinate approach [44–46]and the doubling adding technique [33,47,48], to solve the radiative transfer equation (B.11). Inthis paper we use the Gauss–Seidel iteration technique [30,49,50], which provides an accurate andeOcient solution technique for clear sky and aerosol loaded model atmospheres.

Appendix C. Evaluation of the perturbation integrals

In Section 3.3 we have shown that the perturbation in Eq. (45) can be expressed by the coeOcientsKm in Eqs. (49) and (51). Subject of this section is the further evaluation of these coeOcients, where


we make use of the speci=c representation of the internal intensity =elds, provided by the vectorradiative transfer model of Hasekamp and Landgraf [30].

Due to the splitting of the forward and pseudo-forward intensity =eld in their di:use and directcomponents in Eqs. (B.14) and (B.15), we can rewrite coeOcient Km in Eq. (49) as

Km(; v) =∫ ztop

0dz�s(z; ; v)

×

N∑i; j=−Nj �=0

aiaj'+mTd (z;−i; v))Zm(; j; i)I+md (z; j; )

+12�

e;(z;)=0N∑

i=−Ni �=0

ai'+mTd (z;−i; v))Zm(;−0; i)F0

+12�

1v

e−;(z;v)=vN∑

i=−Ni �=0

aieTi Zm(; i; v)I+md (z; i; )

+14�2

1v

e−;(z;)=0 e−;(z;v)=veTi Zm(;−0; v)F0

: (C.1)

Here, all integrations over are approximated by a double Gaussian quadrature of the order 2Nwith Gaussian weights ai and Gaussian streams i (i = −N; : : : ;−1; 1; : : : ; N ).To calculate the remaining integrals over height z, the model atmosphere is divided in homo-

geneous layers with height independent scattering coeOcients and scattering phase matrices. Theintensity =eld within the model atmosphere is taken from a monochromatic vector radiative transfermodel using the Gauss–Seidel iteration approach (for more details see e.g. [30,50]). Here each at-mospheric layer is split into optically thin sublayers, which are chosen to be thin enough so that anyheight dependence of the intensity =eld can be neglected within a sublayer. In this way the modelatmosphere is subdivided into M homogeneous, optically thin layers indicated by an index k. Theintensity =eld is calculated at the layer boundaries and its height dependence within these layers isdescribed by the layer average. So the coeOcient Km in Eq. (C.1) are given by

Km(; v)≈M∑k=1

�sk(; v)[=mkUzk +>

mk

∫ zk

zk−1

e;(z;)=0 dz + ?mk

∫ zk

zk−1

e−;(z;v)=v dz

+�mk

∫ zk

zk−1

e−;(z;)=0 e−;(z;v)v dz]

(C.2)


with coeOcients

=mk =N∑

i; j=−Nj �=0

aiaj〈'+mTd (−i; v)〉k)Zmk (j; i)〈I+md (j; )〉k

>mk =

12�

N∑i=−Ni �=0

ai〈'+mTd (−i; v)〉k)Zmk (−0; i)F0

?mk =12�

1v

N∑i=−Ni �=0

aieTi Zmk (i; v)〈I+md (i; )〉k

�mk =

14�2

1veTi Z

mk (−0; v)F0: (C.3)

Here zk−1 and zk indicate the lower and upper layer boundary, respectively, �sk represents the scatter-ing coeOcient, and Zmk is the Fourier component of the scattering phase matrix in layer k. Quantitiesof the form 〈f〉k denotes a layer average de=ned by

〈f〉k = 12[f(zk−1) + f(zk)]; (C.4)

where f is a function of altitude z. The remainder of the integrals in Eq. (C.2) can be calculatedin a straight forward manner, which gives

Km(; v)≈M∑k=1

�sk(; v)[=mk +>m

k0�ek()

[1 − eU;k ()=0 ]e−;k−1()=0

+?mk1

�ek(v)[1 − eU;k (v)=v]e−;k−1(v)=v

+�mk0e−;k−1(v)=ve−;k−1()=0

�ek(v)0 + �ek()v

[1 − eU;k (v)=veU;k ()=0 ]]: (C.5)

An analogous approach holds for coeOcients (51), which allows one to evaluate the perturbationintegral in Eq. (45).

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