vSmartMOM: A vector matrix operator method-based radiative transfer model linearized with respect to aerosol properties Suniti Sanghavi n , Anthony B. Davis, Annmarie Eldering Jet Propulsion Laboratory, Caltech, 4800 Oak Grove Dr., Pasadena, CA 91109, USA article info Article history: Received 19 March 2013 Received in revised form 2 September 2013 Accepted 4 September 2013 Available online 14 September 2013 Keywords: Vector radiative transfer Benchmarking Matrix operator method Jacobian matrix Information content Aerosol Cloud abstract In this paper, we build up on the scalar model smartMOM to arrive at a formalism for linearized vector radiative transfer based on the matrix operator method (vSmartMOM). Improvements have been made with respect to smartMOM in that a novel method of computing intensities for the exact viewing geometry (direct raytracing) without inter- polation between quadrature points has been implemented. Also, the truncation method employed for dealing with highly peaked phase functions has been changed to a vector adaptation of Wiscombe's delta-m method. These changes enable speedier and more accurate radiative transfer computations by eliminating the need for a large number of quadrature points and coefficients for generalized spherical functions. We verify our forward model against the benchmarking results of Kokhanovsky et al. (2010) [22]. All non-zero Stokes vector elements are found to show agreement up to mostly the seventh significant digit for the Rayleigh atmosphere. Intensity computations for aerosol and cloud show an agreement of well below 0.03% and 0.05% at all viewing angles except around the solar zenith angle (601), where most radiative models demonstrate larger variances due to the strongly forward-peaked phase function. We have for the first time linearized vector radiative transfer based on the matrix operator method with respect to aerosol optical and microphysical parameters. We demonstrate this linearization by computing Jacobian matrices for all Stokes vector elements for a multi-angular and multispectral measurement setup. We use these Jacobians to compare the aerosol information content of measurements using only the total intensity component against those using the idealized measurements of full Stokes vector ½I; Q ; U; V as well as the more practical use of only ½I; Q ; U. As expected, we find for the considered example that the accuracy of the retrieved parameters improves when the full Stokes vector is used. The information content for the full Stokes vector remains practically constant for all azimuthal planes, while that associated with intensity-only measurements falls as we approach the plane perpendicular to the principal plane. The ½I; Q ; Uvector is equivalent to the full Stokes vector in the principal plane, but its information content drops towards the perpendicular plane, albeit less sharply than I-only measurements. Published by Elsevier Ltd. 1. Introduction Sanghavi et al. [38] demonstrated the linearization of a scalar radiative transfer model based on the matrix Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jqsrt Journal of Quantitative Spectroscopy & Radiative Transfer 0022-4073/$ - see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.jqsrt.2013.09.004 n Corresponding author. Tel.: þ1 818 641 9624. E-mail addresses: [email protected], [email protected] (S. Sanghavi). Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433
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Journal of Quantitative Spectroscopy &Radiative Transfer
Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433
0022-40http://d
n CorrE-m
suniti.sa
journal homepage: www.elsevier.com/locate/jqsrt
vSmartMOM: A vector matrix operator method-basedradiative transfer model linearized with respect toaerosol properties
Suniti Sanghavi n, Anthony B. Davis, Annmarie ElderingJet Propulsion Laboratory, Caltech, 4800 Oak Grove Dr., Pasadena, CA 91109, USA
a r t i c l e i n f o
Article history:Received 19 March 2013Received in revised form2 September 2013Accepted 4 September 2013Available online 14 September 2013
In this paper, we build up on the scalar model smartMOM to arrive at a formalism forlinearized vector radiative transfer based on the matrix operator method (vSmartMOM).Improvements have been made with respect to smartMOM in that a novel method ofcomputing intensities for the exact viewing geometry (direct raytracing) without inter-polation between quadrature points has been implemented. Also, the truncation methodemployed for dealing with highly peaked phase functions has been changed to a vectoradaptation of Wiscombe's delta-m method. These changes enable speedier and moreaccurate radiative transfer computations by eliminating the need for a large number ofquadrature points and coefficients for generalized spherical functions.
We verify our forward model against the benchmarking results of Kokhanovsky et al.(2010) [22]. All non-zero Stokes vector elements are found to show agreement up tomostly the seventh significant digit for the Rayleigh atmosphere. Intensity computationsfor aerosol and cloud show an agreement of well below 0.03% and 0.05% at all viewingangles except around the solar zenith angle (601), where most radiative modelsdemonstrate larger variances due to the strongly forward-peaked phase function.
We have for the first time linearized vector radiative transfer based on the matrixoperator method with respect to aerosol optical and microphysical parameters. Wedemonstrate this linearization by computing Jacobian matrices for all Stokes vectorelements for a multi-angular and multispectral measurement setup. We use theseJacobians to compare the aerosol information content of measurements using only thetotal intensity component against those using the idealized measurements of full Stokesvector ½I;Q ;U;V � as well as the more practical use of only ½I;Q ;U�. As expected, we find forthe considered example that the accuracy of the retrieved parameters improves when thefull Stokes vector is used. The information content for the full Stokes vector remainspractically constant for all azimuthal planes, while that associated with intensity-onlymeasurements falls as we approach the plane perpendicular to the principal plane. The½I;Q ;U� vector is equivalent to the full Stokes vector in the principal plane, but itsinformation content drops towards the perpendicular plane, albeit less sharply than I-onlymeasurements.
Published by Elsevier Ltd.
r Ltd.
,
1. Introduction
Sanghavi et al. [38] demonstrated the linearizationof a scalar radiative transfer model based on the matrix
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 413
operator method. In the present work, we briefly outlinethe formalism for the vectorization of the matrix operatormethod as also shown previously by Liu and Ruprecht [24]and Hollstein and Fischer [19]. Additionally, we provide aformalism for the linearization of the vector model withrespect to aerosol optical and microphysical parameters.We have named the resulting linearized vector radiativetransfer model “vSmartMOM”(vectorized Simulated mea-surement of the atmosphere using radiative transfer basedon the Matrix Operator Method). Other radiative transfermethods that have been used for this purpose include themethod of discrete ordinates [41], the Gauss–Seidelmethod [18] and the Markov chain formalism [45].
While intensity-only measurements have driven earlyaerosol remote-sensing missions [43,7,31], it has beenfound that the use of only the intensity component I ofthe Stokes vector I¼ ½I;Q ;U;V � is insufficient for constrain-ing the aerosol inverse problem. The information contentof a measurement is considerably enhanced by the use ofmultispectral and multi-angular observation strategies.However, it has been shown that the inclusion of theStokes vector elements Q and U (V is generally too small tomeasure reliably) can better constrain aerosol retrievalsfrom remote-sensing measurements [27,4]. This hasspurred new missions incorporating polarimetry in addi-tion to multispectral and multi-angular observations[28,42] as well as new retrieval algorithms developed totake full advantage of the information content of themeasurements [8,17].
Sanghavi et al. [38] were motivated by a need to find aspeedy and an accurate method for computing the Jaco-bian matrix necessary for optimized retrieval of aerosoland surface parameters from a multi-angular, multispec-tral satellite instrument like the Multi-angle ImagingSpectroRadiometer (MISR) [7]. The present work developsa similar framework for the Stokes vector I rather thanonly the component I.
We present the vector radiative transfer formulation ofthe matrix operator method in Section 2. We use bench-mark results from [22] to verify our computations, aspresented in Section 3. Our vector formulation differs fromour previous work, in that we adapt the delta-m trunca-tion method [44] for vector radiative transfer instead ofusing the more arbitrary delta truncation method [1] thatwe followed in Sanghavi et al. [38]. The vector form of thedelta-m method has been implemented into radiativetransfer codes previously by Rozanov and Kokhanovsky[37,41,32,26], however, we attempt to provide a morecomplete explanation in the form of a rigorous deriva-tion in Appendix A. Also, in order to eliminate error dueto interpolation between quadrature points, we haveenhanced the conventional method of direct raytracing(which uses the solar and/or viewing zenith angles asquadrature points carrying zero weight) by making use oftwo sets of Radau quadrature points (see Appendix B). TheJacobian matrix for the full Stokes vector is defined inSection 4. The linearization of the vector formulation ofthe radiative transfer equation is based on employing thechain rule of differentiation to obtain the derivative of theforward model. This has been summarized in Appendix C.We set up an aerosol-laden atmospheric scenario in
Section 5.1 to demonstrate both the forward model andthe results of linearization with respect to aerosol opticaland microphysical parameters in Sections 5.2 and 5.3,respectively. The gain in information afforded by use ofthe full Stokes vector compared to using only intensitymeasurements is quantified in Section 5.4.
2. The matrix operator method
The matrix operator method or discrete space theorywas first developed by Grant and Hunt [14,15]. It allows foran exact and a speedy computation of the radiativetransfer of turbid media, especially because of its encap-sulation of the infinite series of reflections into a singlematrix inversion [23]
ðE�XÞ�1�E¼XþX2þX3þ⋯; ð1Þwhere X is a matrix representing a pair of consecutivereflections between two layers and E is the identity matrix.This eliminates the issue of slow convergence for weaklyabsorbing atmospheres, i.e., at high values of single scat-tering albedo, ω0-1, that are faced by several othermethods. Also, there is minimal loss of computationalspeed for increasing optical thicknesses, making it suitablefor simulating aerosols and clouds alike. MOM can gen-erate the entire radiative field for a given scenario, bothinternal and at the boundaries of the atmosphere, forisotropic as well as anisotropic scatterers, and hence canbe used for the simultaneous computation of intensitiesmeasured using different viewing geometries. This makesit ideal for the simulation of backscattered light measuredby a multi-angle instrument like MISR aimed at quantify-ing the aerosol content of the atmosphere.
2.1. Vector formalism
For macroscopically isotropic media, the followingequation describes monochromatic, one-dimensional vec-tor radiative transfer for an infinitesimal layer in a plane–parallel atmosphere [3]:
μdLðτ ; μ;ϕ; μ0;ϕ0Þ
dτ¼�Lðτ ; μ;ϕ; μ0;ϕ0Þþð1�ω0ÞBðTÞ
þω0
4πZðμ;ϕ; μ0;ϕ0ÞS0 expð�τ=μ0Þ
þω0
4π
Z 2π
0
Z 1
�1Zðμ;ϕ; μ′;ϕ′ÞL
�ðτ ; μ′;ϕ′; μ0;ϕ0Þ dμ′ dϕ′ ð2Þwhere Lðτ ; μ;ϕ; μ0;ϕ0Þ represents the Stokes vector ofdiffuse light propagating along the direction ðμ;ϕÞ at anoptical depth τ in an atmosphere that can have twosources of light: the incident solar flux represented bythe Stokes vector S0 incident along the direction ðμ0;ϕ0Þand the thermal radiation BðTÞ ¼ BðTÞ½1;0;0;0�T, where B(T) is Planck's function at temperature T. Zðμ;ϕ; μ′;ϕ′Þ is thephase matrix which governs the scattering of the Stokesvector incident along ðμ′;ϕ′Þ in the direction ðμ;ϕÞ withrespect to the local meridian plane [20]. We computeZðμ;ϕ; μ′;ϕ′Þ using the formalism presented by Siewert[39]. (The over-score in the above equation is used todenote quantities averaged over different contributing
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433414
species, such as aerosols, molecular absorbers and scat-terers. This distinction has been made because of itsrelevance to linearization, considered later in this paper.)
The direction cosine μ is defined such that μ40 fordownward and μo0 for upward propagation of light. Allterms on the right hand side of Eq. (2) having a positivesign denote sources, while the negative sign denotes sinksof radiant energy.
The above radiative transfer equation constrains onlythe diffuse part of the total radiance
Restored to its original form, the radiative transfer equa-tion is
μdIðτ ; μ;ϕ; μ0;ϕ0Þ
dτ¼�Iðτ ; μ;ϕ; μ0;ϕ0Þþð1�ω0ÞBðTÞ
þω0
4π
Z 2π
0
Z 1
�1Zðμ;ϕ; μ′;ϕ′ÞIðτ ; μ′;ϕ′; μ0;ϕ0Þ dμ′ dϕ′; ð4Þ
where boundary conditions
Ið0; μ;ϕ; μ0;ϕ0Þ ¼ S0δðμ�μ0Þδðϕ�ϕ0Þ ð5Þfor μZ0 are used to represent the Stokes vector of theincoming solar radiation at the top of atmosphere,S0 ¼ ½1;0;0;0�T where δð�Þ is the Dirac delta function.
The atmosphere is bounded below by a surface ofemissivity ε and bidirectional polarization distributionfunction (BPDF) Rpðμ;ϕ; μ′;ϕ′Þ, so that denoting the totalatmospheric optical thickness as τs yields
Iðτs; μ;ϕ; μ0;ϕ0Þ
¼ εBðTÞþZ 2π
0
Z 1
0Rpðμ;ϕ; μ′;ϕ′ÞIðτs; μ′;ϕ′; μ0;ϕ0Þμ′ dμ′ dϕ′:
ð6Þwith the natural constraint
εþZ 2π
0
Z 1
0Rð1;1Þp ðμ;ϕ; μ′;ϕ′Þμ′ dμ′ dϕ′¼ 1; ð7Þ
as required by Kirchhoff's law.Following Siewert [39] and de Rooij and van der Stap
[6], we express the phase matrix as a sum of its Fouriercomponents:
Zðμ;ϕ; μ0;ϕ0Þ ¼12C0ðμ; μ′Þþ ∑
M
m ¼ 1½Cmðμ; μ0Þ cos mðϕ�ϕ0Þ
þSmðμ; μ0Þ sin mðϕ�ϕ0Þ�; ð8Þwhere M-1. Noticing that Cmðμ; μ0Þ is always a diagonalblock matrix while Smðμ; μ0Þ is always anti-block-diagonal,we can write
Zðμ;ϕ; μ0;ϕ0Þ ¼12C0ðμ; μ′ÞH0þ ∑
M
m ¼ 1½Cmðμ; μ0ÞþSmðμ; μ0Þ�Hm;
ð9Þwhere Hm is a 4�4 diagonal matrix such that
Hm ¼
cos mðϕ�ϕ0Þ 0 0 00 cos mðϕ�ϕ0Þ 0 00 0 sin mðϕ�ϕ0Þ 00 0 0 sin mðϕ�ϕ0Þ
266664
377775:
ð10Þ
Writing
Zmðμ; μ0Þ ¼1
ð1þδ0mÞ½Cmðμ; μ0ÞþSmðμ; μ0Þ�; ð11Þ
where δij ¼ 1 for i¼ j, and δij ¼ 0 for ia j, enables thesolution of each Fourier moment of the radiative transferequation:
μdImðτ ; μ; μ0Þ
dτ¼�Imðτ ; μ; μ0Þþ
ω0
2
Z 1
�1Zmðμ; μ′ÞImðτ ; μ′; μ0Þ dμ′
þð1�ω0ÞBðTÞδ0m; ð12Þyielding
Iðμ;ϕ; μ0;ϕ0Þ ¼ ∑M
m ¼ 0
1ð1þδ0mÞ
Hm
Imðμ; μ0ÞQmðμ; μ0ÞUmðμ; μ0ÞVmðμ; μ0Þ
266664
377775: ð13Þ
Using an Nquad�point quadrature to discretize thezenith angles, Eq. (12) can be rewritten in a supermatrixform as
ddτ
IþmI�m
" #¼ ð1�ω0ÞBðTÞδ0m
M�1
�M�1
" #
þω0
2M�1
Zðþ þÞm C�E Z
ðþ�Þm C
�Zð�þÞm C �Z
ð��Þm CþE
24
35 Iþm
I�m
" #:
ð14ÞIn the above equation, Iþm and I�m are supervectors respec-tively containing the mth Fourier components of theStokes vectors of the downwelling and upwelling inten-sities crossing the infinitesimal layer of optical thicknessdτ along each of the Nquad direction cosines, μi, defined bythe chosen quadrature rule, such that
where ci is the quadrature weight of the ith cosine μi.Dropping, for clarity, the subscripts m denoting the
Fourier moments, the solution [13] of Eq. (14) for anarbitrary layer of optical thickness Δ can be written fromfirst principles as
IþΔI�0
" #¼
JþΔ
J�0
" #þ
TΔ0 RΔ0
R0Δ T0Δ
" #Iþ0I�Δ
" #; ð18Þ
where TΔ0 denotes transmission of the Stokes vector ofradiation traversing the layer from top to bottom, T0Δ
denotes transmission of radiation traversing the layer frombottom to top. Similarly, R0Δ and RΔ0 denote reflectance atthe top and bottom of the layer, respectively. J�0 and Jþ
Δ
denote the Stokes vectors of up- and downwelling radia-tion sourced within the layer and emanating at the top andbottom of the layer, respectively. Iþ0 and I�0 denote thedown- and upwelling radiation at the top of the layer, IþΔand I�Δ denote the down- and upwelling radiation at thebottom of the layer, respectively.
In the limit of infinitesimal optical thickness, δ (not tobe confused with the Dirac or Kronecker delta functionsalso used in this work), the transmission, reflectance andsource functions can be expressed for a given Fouriermode as follows [34,14]:
Tδ0 ¼ E�M�1 E�ω0ðτÞ2
Zðþ þÞ
C
� �� �� δ
T0δ ¼ E�M�1 E�ω0ðτÞ2
Zð��Þ
C
� �� �� δ ð19Þ
Rδ0 ¼M�1 ω0ðτÞ2
Zð�þÞ
C
� �� δ
R0δ ¼M�1 ω0ðτÞ2
Zðþ�Þ
C
� �� δ ð20Þ
For m¼0,
Jδ0 ¼ δ0mðE�Tδ0�R0δÞ � BðTÞEJ0δ ¼ δ0mðE�T0δ�Rδ0Þ � BðTÞE ð21Þwhere E ¼ ½1;0;0;0;1;0;0;0;…;1;0;0;0� is a supervectorof length 4Nquad. We have used the convention Xba todenote a quantity associated with radiation incident in thedirection a-b. Thus, Tδ0 represents transmission 0-δ oflight from the upper boundary of the layer to the lowerboundary, while Rδ0 represents the reflection of this lightat the lower boundary. Jδ0 represents the emission ofradiation in the layer towards the lower boundary. Theopposite holds for T0δ, R0δ and J0δ.
The above equations have been shown to be applicablenot only to infinitesimal layers, but also to elemental layersof finite optical thickness, δ, such that
0oδominifμig1�ω0=2
; ð22Þ
which ensures that all matrix operators are non-negative[15].
Once the elemental layer has been defined as above,the following equations can be successively used to buildup an arbitrary layer of any desired thickness as in Eq. (18):
T20 ¼T21ðE�R01R21Þ�1T10 ð23Þ
R20 ¼R10þT01ðE�R21R01Þ�1R21T10 ð24Þ
T02 ¼T01ðE�R21R01Þ�1T12 ð25Þ
R02 ¼R12þT21ðE�R01R21Þ�1R01T12 ð26Þ
J20 ¼ J21þT21ðE�R01R21Þ�1ðJ10þR01J12Þ ð27Þ
J02 ¼ J01þT01ðE�R21R01Þ�1ðJ12þR21J10Þ ð28Þwhere the indices 0,1,2 indicate the boundaries of twoadjacent layers.
Computational time required for doubling [16] isreduced by the use of the following symmetry relation-ships [20,21] between the pairs Tab and Tba of transmis-sion matrices and Rab and Rba of reflection matricesbetween boundaries a and b of a homogenous layer:
Tab ¼DTbaD ð29Þand
Rab ¼DRbaD; ð30Þwhere D¼ diagfD;D;… Nquad timesg, and D is a 4�4diagonal matrix such that D¼ diagf1;1;�1;�1g. This sym-metry allows us to reduce the doubling computationsfollowing Eqs. (23)–(26) by half, in that we can use thesame equations (as in the scalar case) for light incident atthe top, as in T02 and R02, as for light incident from thebottom of a layer, as in T20 and R20 [5]. This is done by firsttransforming R10 into Rn
10 as
Rn
10 ¼DR10; ð31Þand then using Rn
10 in Eqs. (23) and (24) instead of R10
and R01, and replacing T01 by T10 to obtain T20 and Rn
20,
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433416
respectively. This can be repeated until the entire homo-genous layer has been reconstructed, yielding Tba and Rn
ba.All transmission and reflection matrices for the layer arethen obtained as
Tba ¼Tba;
Rba ¼DRn
ba;
Tab ¼DTbaD;
Rab ¼Rn
baD: ð32ÞThe medium is bounded by a surface having an emis-
sivity vector eg and a reflection matrix Rp at the bottomand by vacuum at the top of the atmosphere (TOA),denoted by the index 0.
The radiance at the boundary between the atmosphereand the surface, indexed s, is given by
Iþs ¼R0sI�s þTs0I
þ0 þJþ
0s ð33Þand
I�s ¼ egBðTsÞEþRpMIþs ; ð34Þwhere the emissivity eg is non-zero only for the 0thFourier moment.
Combining Eqs. (26) and (27), we get the boundarycondition at the surface:
The boundary condition at the top of atmosphere is
Iþ ¼ S0E2
C�1 ð36Þ
for the 0th Fourier mode and
Iþ ¼ S0EC�1 ð37Þfor other Fourier modes. Here, the Sun is treated as asource of completely unpolarized light.
Eqs. (18)–(37) provide a closed vector solution to theradiative transfer problem of the Earth–atmosphere sys-tem using the matrix operator method.
It should be noted that for an inhomogenous atmosphere,adding equations may equivalently be started at either thetop or the bottom of the atmosphere. However, when onlyupward values at TOA are required, as is the case for satelliteretrievals, the adding equations may be reduced whenstarting from the surface and adding upward. This is becauseall that is required for this computation is the reflectancematrix for incidence from above and the upward source.Thus, these are the only values required from the lower stackwith each additional adding.
2.2. Non-inclusion of the single scattering component of thesolar flux in the source term J
It should be noted that our treatment of the matrixoperator method in Section 2.1 as well as in Sanghavi et al.[38] differs inherently from various flavors of the discreteordinate method [29,41,40,32,26] and also from the matrixoperator formulation of Plass et al. [33] and subsequentlyof Fell and Fischer [10] (scalar) and Hollstein and Fischer[19] (vector), in that we do not include single scattering ofthe Sun's incident flux in our source terms J. This is
evident in Eq. (3), where we combine both the directand diffuse components into a single intensity term. As aresult, only thermal radiation (Planck's function) is con-sidered as a source within a given atmospheric layer. Thishas two main advantages:
1.
For UV–vis–NIR light, there is no source term J, as a resultof which our elemental, doubling as well as addingcomputations are reduced by a third (only Eqs. (23)–(27) have to be computed instead of Eqs. (23)–(33)).
2.
For thermal radiation, Eqs. (28) and (33) are consider-ably simplified due to the isotropic nature of the sourcematrices J. This makes it possible to reuse severalterms computed for the transmission and reflectionmatrices, T and R, respectively, reducing computingtime considerably. More importantly, the isotropicity ofthe source terms J requires them to be included nofarther than the 0th Fourier moment.
However, the exclusion of the source term also hithertoresulted in difficulties of accurately representing geome-tries involving off-quadrature solar and viewing zenithangles. By choosing not to separate our treatment of thesolar flux from that of flux due to diffuse radiation, we loseour ability to incorporate the solar flux integration (SFI)technique [2] in our radiative transfer computations.
The challenge encountered in the use of the dummy nodeinclusion (DNI) technique, as explained by Chalhoub andGarcia [2], is that, for a plane–parallel atmosphere illuminatedby a parallel source of incident light like the Sun, thecomputation of reflected or transmitted intensities involvesdivision of the corresponding reflection and transmissionmatrices by the quadrature weight associated with the direc-tion of incident flux (see Eqs. (36) and (37)). In the DNItechnique, a weight cðμ0Þ ¼ 0 is assigned to the additional(dummy) quadrature root μ0, if μ0 is an off-quadrature angle,which in our case would imply division by zero. In the specialcase where μ0 is an off-quadrature angle, but the viewingangle, μ, is a quadrature root, the principle of reciprocity [3]can be employed to get accurate normalized intensities. It isobvious, however, that the DNI technique is of limited use incase of completely arbitrary (off-quadrature) viewing as wellas incident zenith angles. The only hitherto alternative to thiswas interpolation between the quadrature roots [25,9,10] toobtain estimates at the solar and viewing zenith angles. Weintroduce a modification of the DNI technique using a block-Radau quadrature scheme (see Appendix B) to allow directraytracing of scattered light between the source and theobserver using our MOM approach, eliminating the need forerror-inducing interpolation.
3. Verification of the forward model
To verify our model, we use the benchmarking resultspublished by Kokhanovsky et al. [22]. We have used theabridged version of the benchmarking exercise, consisting ofthe six viewing zenith angles 01, 201, 401, 601, 801 and 891,instead of the 91 angles in each hemisphere used in the full-scale study. This choice has been made in view of the fact thatthe most challenging geometries are already covered in the
Fig. 1. The Stokes vector elements I, Q, U and V of reflected light as computed using SCIATRAN in Kokhanovsky et al. [22] for a Rayleigh atmosphere ofoptical thickness 0.3263 (dark blue), an aerosol-laden atmosphere of optical thickness 0.3262 (green) and a cloud of optical thickness 5.0 (cyan) for theviewing angles 01, 201, 401, 601, 801 and 891 in the principal plane in the forward scattering direction (ϕ�ϕ0 ¼ 0, solid line), perpendicular to the principalplane (ϕ�ϕ0 ¼ 901, dashed line), and in the aftward scattering direction of the principal plane (ϕ�ϕ0 ¼ 1801, dotted line). The sun is at 601 from the zenith.Red markers have been used to show vSmartMOM computations of these values. (For interpretation of the references to color in this figure caption, thereader is referred to the web version of this article.)
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 417
abridged version. The viewing geometry includes the azi-muthal planes represented by the principal plane in both theforward scattering (ϕ�ϕ0 ¼ 0) as well as backscattering(ϕ�ϕ0 ¼ 1801) directions, and the plane perpendicular tothe principal plane (ϕ�ϕ0 ¼ 901). The Sun is assumed to beat θ0 ¼ 601 from the zenith. The simulated intensities arenormalized as πI=μ0S0. These normalized intensities aresimulated for light that is both transmitted and reflected tothe bottom and the top of the atmosphere, respectively. The
surface bounding the atmosphere at the bottom is assumed tobe black. Three different atmospheric compositions are con-sidered:
1.
A purely Rayleigh scattering atmosphere of opticalthickness τ¼ 0:3262,
2.
A purely aerosol-laden atmosphere of optical thicknessτ¼ 0:3262, and
3.
A pure cloud of optical thickness τ¼ 5:0.
Fig. 2. Percentage differences between the Stokes vector elements I, Q, U and V of reflected light computed using vSmartMOM relative to SCIATRANcomputations of Kokhanovsky et al. [22] for a Rayleigh atmosphere of optical thickness 0.3263 (dark blue), an aerosol-laden atmosphere of opticalthickness 0.3262 (green) and a cloud of optical thickness 5.0 (cyan) for the viewing angles 01, 201, 401, 601, 801 and 891 in the principal plane in the forwardscattering direction (ϕ�ϕ0 ¼ 0, solid line), perpendicular to the principal plane (ϕ�ϕ0 ¼ 901, dashed line), and in the aftward scattering direction of theprincipal plane (ϕ�ϕ0 ¼ 1801, dotted line). The sun is at 601 from the zenith. Due to our use of a logarithmic scale, percentage differences of zero have beenomitted. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433418
ω0 ¼ 1.
All species are assumed to have a single scattering albedo
For this study, we set our elemental layer opticalthickness at δ¼ 10�6. However, this may not have beennecessary as the difference between results obtained atδ¼ 10�6 and δ¼ 10�3 was found to be marginal. Also, aftertruncation, all Fourier moments have been used for thebenchmarking computations.
TMS-correction [30] is implemented in both SCIATRANand vSmartMOM.
Inter-comparison results show agreement betweenvSmartMOM and SCIATRAN [36,37] mostly up to the 7thsignificant digit for the Rayleigh atmosphere at all viewingzenith and azimuthal angles, both for transmitted andreflected light. This does not hold true for some cases, forwhich the agreement cannot be ascertained beyond the
Fig. 3. Same as Fig. 1, for transmitted light.
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 419
6th significant digit because our results are currentlyoutput at a precision that occasionally falls below that ofthe SCIATRAN output. As a result, the maximum percen-tage difference between the first Stokes vector element Icomputed using vSmartMOM and SCIATRAN for bothtransmitted and reflected light is less than 6� 10�4%.The corresponding values for non-zero values of Q and Uare vanishing at most angles except at the SZA, 601 (for Q)and at the most oblique angles (for U) where the max-imum differences are found to be 3:5� 10�4% and2:1� 10�4%, respectively. As in the benchmarking study[22], we set Nquad ¼ 60 for the Rayleigh case.
For the aerosol, the largest difference between I com-puted using the two models is found to be 7.19% at theforward scattering peak at 601 (transmitted light, ϕ�ϕ0 ¼01). Transmitted light for ϕ�ϕ0 ¼ 901 and ϕ�ϕ0 ¼ 1801 andreflected light around ϕ�ϕ0 ¼ 1801 also shows increaseddifferences at 601 of 0.54%, 1.55% and 0.48%.
At all other angles, the difference is found to be below0.03%. The spherical harmonic series is truncated at L¼480(Nquad ¼ 240) for the plane ϕ�ϕ0 ¼ 01 and L¼360 (Nquad ¼180) for the other two azimuthal planes. The relativedifferences in Q, U and V between the two models arefound to increase as the absolute value of respectively Q,
Fig. 4. Same as Fig. 2, for percentage differences in transmitted light.
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433420
U or V decreases, as a result of which the percentagedifferences are higher for reflected light than for trans-mitted light and are least at oblique angles.
In case of the cloud, the largest difference between Icomputed using the two models is found to be 0.46% at theforward scattering peak at 601 (transmitted light, ϕ�ϕ0 ¼01). For light transmitted in the planes ϕ�ϕ0 ¼ 901 andϕ�ϕ0 ¼ 1801 and reflected in the plane ϕ�ϕ0 ¼ 1801,increased differences are seen at 601 of 0.09%, 0.14%and 1.82%.
At all other angles, the difference is again found to bemostly well below 0.05%. As with the aerosol, the relativedifferences in Q, U and V between the two models are
found to increase as the absolute value of respectively Q, Uor V decreases, as a result of which the percentagedifferences are higher for reflected light than for trans-mitted light and are least at oblique angles.
These results are summarized for reflected light inFig. 1 which shows the normalized intensities computedby SCIATRAN for Rayleigh scattering in dark blue, foraerosol in green and for cloud in cyan. vSmartMOM resultsare shown using red markers. Solid, dashed and dottedlines have been used to indicate the azimuthal planesrepresented by ϕ�ϕ0 ¼ 01, ϕ�ϕ0 ¼ 901 and ϕ�ϕ0 ¼ 1801,respectively. The percentage difference of vSmartMOMnormalized intensities relative to SCIATRAN computations
Table 1Aerosol microphysical properties.
Mode Fine Coarse
nr 1.42 1.38ni 10�4 10�3
rm ðμmÞ 0.05 0.10s 1.35 2.70
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 421
is depicted using the same color and line-style scheme todistinguish between the respective scenarios and azi-muthal angles in Fig. 2. Figs. 3 and 4 show transmittednormalized intensities and the corresponding percentagedifferences, respectively, between the two models.
It should be pointed out that the time required for thecloud computations (for the same number of sphericalharmonic moments and quadrature points) was less than10% longer than that needed for aerosol. The comparableperformance of vSmartMOM for both the aerosol andcloud scenarios reflects the inherent ability of the matrixoperator method to handle large optical thicknesses with-out requiring substantially more computational time,underlining the conclusion made by Lenoble [23] aboutthe special suitability of the method for the simulation ofscattering media, especially in remote sensing applicationswhere efficiency is of the essence in retrieval algorithms.
4. Linearization of the matrix operator method
The Jacobian matrix, K, is an m�n matrix containingthe derivatives of m independent measurements of anobserved atmospheric scene simulated by the radiativetransfer model, y¼ fy1; y2;…; ymg, with respect to the nparameters of the state vector, x¼ fx1; x2;…xng, to beretrieved using an optimized method [35], so that
K¼
∂y1∂x1
∂y1∂x2
… ∂y1∂xn
∂y2∂x1
∂y2∂x2
… ∂y2∂xn
⋮ ⋮ ⋱ ⋮∂ym∂x1
∂ym∂x2
… ∂ym∂xn
2666664
3777775: ð38Þ
Polarimetric measurements involving the full Stokes vec-tor I (or only the I, Q and U components) bring about afourfold (or threefold) increase in the length of themeasurement vector compared to intensity measurementsalone, and hence also a commensurate increase in the sizeof the Jacobian matrix, which can be expected to lead to anincreased information content.
By implementing an analytical formulation of thederivative of the radiative transfer equation, our modelcan simultaneously compute both the measured intensi-ties and their derivatives. The equations leading up tothe matrix operator formulation of vector radiative trans-fer as described in Section 2.1 are used to arrive at thecorresponding formulation of their derivative withrespect to aerosol optical and microphysical parametersin Appendix C.
In the following section, we simulate the scattering oflight due to an atmosphere consisting of a fine and acoarse mode aerosol in addition to molecular (Rayleigh)scatterers and examine the corresponding Jacobian matrixwith respect to aerosol parameters.
5. Linearization results
5.1. Setup
We have computed the intensities Iλ normalized withrespect to the incident solar flux simulated at TOA andtheir derivatives for viewing zenith angles ranging from
�751 through 01 to 751 at an angular resolution of 31 inthe principal plane (ϕ�ϕ0 ¼ 01) as well as perpendicular tothe principal plane (ϕ�ϕ0 ¼ 901). We use a viewing angleof �θ in a given azimuthal plane ϕ as shorthand for aviewing direction of (θ, 1801þϕ), not to be confused withthe formal sign convention for μ used in the radiativetransfer equations, where positive and negative valuesindicate down- and upwelling radiation, respectively. Thesolar zenith angle is 601. These intensities are computed at446 nm, 558 nm, 672 nm and 866 nm, for a black under-lying surface. (Linearization with respect to surface reflec-tion properties will be subject of a future study.)
In addition to Rayleigh scattering (depolarization factorassumed 0), we have assumed a bimodal aerosol distribu-tion, consisting of a coarse and a fine mode, both of opticalthickness 0.5 at 558 nm. The microphysical parameters,consisting of the complex refractive index (nr�niı) and thesize distribution parameters of both aerosol modes, havebeen summarized in Table 1. Both aerosol modes follow alognormal size distribution parametrized by the medianradius, rm, and antilogarithm of the lognormal distributionstandard deviation, s, such that the probability of anaerosol particle sized in the interval ½r; rþdr� is given by
pðrÞ dr¼ 1ffiffiffiffiffiffi2π
pr log s
exp �ðlog r�log rmÞ22 log 2 s
" #dr: ð39Þ
All scatterers are uniformly distributed in the atmosphereand have the optical thicknesses and single scatteringalbedos given in Table 2 for each wavelength consideredin this study. The phase matrices corresponding to eachmode for each of the wavelengths considered here areillustrated in Fig. 5 using solid lines for the coarse modeand dashed lines for the fine mode.
The number of quadrature points used are Nquad ¼ 24.Also, in contrast to our benchmarking computations, theFourier series calculations are stopped when the averagerelative difference in computed intensities for at least threesuccessive Fourier components m, mþ1 and mþ2 becomessmaller than 10�4%. Three successive components are used toavoid premature breaking-off of the Fourier series at potentialzero-crossings of the Fourier moments.
5.2. Forward model
The term “forward model” in this paper is used todescribe the simulation of the quantity being measured byan instrument [35]. In our case, this is the set of normal-ized intensities (defined as in [38] as πI=S0) measured atthe wavelengths and viewing angles characteristic of aninstrument, given the solar zenith angle prevalent at thetime of measurement.
Table 2Optical depths and single scattering albedos.
λ (nm) 446 558 672 866
Rayleigh OD 0.2359 0.0903 0.0436 0.0155Rayleigh SSA 1.0000 1.0000 1.0000 1.0000Aerosol fine mode OD 1.1052 0.5000 0.2616 0.0995Aerosol fine mode SSA 0.9981 0.9970 0.9954 0.9912Aerosol coarse mode OD 0.5045 0.5000 0.4897 0.4646Aerosol coarse mode SSA 0.9786 0.9825 0.9849 0.9876
Fig. 5. Elements f 11; f 33 ; f 12 and f34 of the phase matrix corresponding tothe coarse (solid line) and fine (dotted line) aerosol modes at 446 nm(blue), 558 nm (green), 672 nm (red) and 866 nm (purple). (For inter-pretation of the references to color in this figure caption, the reader isreferred to the web version of this article.)
Fig. 6. The Stokes vector IðλÞ at λ¼ 446 nm (blue), 558 nm (green),672 nm (red) and 866 nm (purple) for a black underlying surface. (Forinterpretation of the references to color in this figure caption, the readeris referred to the web version of this article.)
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433422
The normalized intensities for the atmospheredescribed in Section 5.1 are illustrated in Fig. 6. The solidlines show the intensities measured in the principal plane,while the dashed lines depict measurements in the planeperpendicular to the principal plane. The lines have beencolored to reflect the wavelength they represent: blue for446 nm, green for 558 nm, red for 672 nm and magenta for886 nm (NIR).
As expected, the values of I, U and V coincide at nadir forboth ϕ�ϕ0 ¼ 01 and ϕ�ϕ0 ¼ 901, while Q changes its sign. Inthe principal plane, both I and Q become more pronouncedaround the backscatter peak (at negative viewing angles)and have U ¼ V ¼ 0 for all viewing angles. I has elevated
values at larger viewing angles due to increased multiplescattering. In the absence of the backscatter peak, thepredominant multiple scattering produces a cloud-like effect(at large positive viewing angles), causing intensities of allwavelengths to converge. In the plane perpendicular to theprincipal plane, I and Q are symmetrical about nadir while Uand V become antisymmetric.
5.3. Jacobians
Fig. 7 shows the linearized computation of derivativesof Iλ with respect to the aerosol optical thicknesses of thecoarse (left panels) and fine (right panels) aerosol modes.Derivatives with respect to the microphysical parametersnr, ni, rm and s are shown in Figs. 8, 9, 10 and 11,respectively, each depicting the coarse mode on the leftpanel and the fine mode on the right. Again, the lines arecolored according to wavelength, with solid lines forintensities in the principal plane and dashed lines for theplane perpendicular to it.
As expected, the Jacobians show the same symmetryrelationships as the intensities measured in the principalplane and the one perpendicular to it. Each of thesederivatives is found to agree within 10�5% of valuesobtained using finite difference estimation. That linearizationprovides better accuracy and computational efficiency than
Fig. 7. Jacobians with respect to AOT. (Left) Coarse mode: ∂I=∂τc for all λ. (Right) Fine mode: ∂I=∂τf for all λ
Fig. 8. Jacobians with respect to the real part of the refractive index nr. (Left) Coarse mode: ∂I=∂nr;c for all λ. (Right). Fine mode: ∂I=∂nr;f for all λ.
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 423
the finite difference method has been proven for the scalarcase in Sanghavi et al. [38]. The same conclusions apply forvector computations. Currently a linearized Jacobian compu-tation with respect to 10 parameters takes about 5 times the
computing time of a forward run. This figure, though betterthan at least 11 computations required for the finite differ-ence method, could be improved with better optimization ofour code in the future.
Fig. 9. Jacobians with respect to the imaginary part of the refractive index ni. (Left) Coarse mode: ∂I=∂ni;c for all λ. (Right) Fine mode: ∂I=∂ni;f for all λ.
Fig. 10. Jacobians with respect to the median radius r. (Left) Coarse mode: ∂I=∂rc for all λ. (Right) Fine mode: ∂I=∂rf for all λ.
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433424
5.4. I vs. I: information gain
As discussed in Section 1, the main motivation for usingthe full Stokes vector instead of only the total intensitycomponent I is the additional information contained in the
components Q, U and V, which can be used to betterconstrain the aerosol inverse problem. We have made useof the above illustrative example to compute the a poster-iori covariances Sx, information content H, and the degreesof freedom, d [35] associated with the aerosol parameters
Fig. 11. Jacobians with respect to the median antilogarithm of the lognormal distribution standard deviation s. (Left) Coarse mode: ∂I=∂sc for all λ. (Right)Fine mode: ∂I=∂sf for all λ.
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 425
of the scenarios considered in Section 5.1 for measure-ments y consisting, respectively, of
1.
only the total intensity component I, 2. only components I, Q and U of the Stokes vector, which
is conventional due to the difficulty of measuring thevery small value of V, and
3.
the full Stokes vector I¼ ½I;Q ;U;V � in the ideal casewhere V can be measured with the same precision asthe other three components,
under the same assumptions of measurement error, Sε,and a priori covariance, Sa.
For the geometry considered here, the measurementvector for the scalar case sy consists of sm¼4 (number ofbands) �51 (number of viewing angles)¼204 elements,while in the vector cases, the measurement vector exclud-ing the V component, v3y, is 3 times as large so thatv3m¼612, whereas using the full Stokes vector would yieldv4y with 4 times that number, viz., v4m¼816. Assuming,for simplicity, a measurement error whose variance isgiven by the square of 3% of the absolute value of themeasurement with a noise floor of 10�6, we get an sm� smdiagonal measurement covariance matrix sSε, for whichthe ith diagonal element is the square of 0:03� syiþ10�6.Translating this assumption to the vector cases yields thev3m� v3m and v4m� v4m diagonal measurement covar-iance matrices v3Sε and
v4Sε, whose ith diagonal elementsare the squares of 0:03� v3yiþ10�6 and 0:03� v4yiþ10�6,respectively.
The aerosol parameters that constitute the state vectorare the same for both the scalar and the vector case, so thatthe state vector x consists of the same 10 parameters forthe bimodal aerosol distribution of Section 5.1. Assuming avariation of both the fine and coarse mode aerosol opticalthicknesses between 0�6, of nr for both modes between1.2 and 1.7, ni for both modes between 0 and 0.2, of rm forboth modes between 0.02 and 1.0, and s values between1.005 and 3.0, we roughly estimate the a priori covariancematrix Sa to be a 10�10 diagonal matrix whose square-root is given by
S1=2a ¼ 1ffiffi2
p diag 6;6;0:5;0:2;0:1;2:0;0:5;0:2;0:1;2:0f g; ð40Þ
where the first two values represent the uncertainty in theoptical thicknesses of the two modes, the 3rd and 7thvalues denote the uncertainties associated with nr for eachmode, the 4th and 8th values denote the uncertaintiesassociated with ni for each mode, the 5th and 9th valuesdenote the uncertainties associated with rm for each mode,and the 6th and 10th values denote the uncertaintiesassociated with s for each mode. In other words, we groupthe aerosol optical thicknesses of the two modes first,followed by the microphysical parameters associated withthe coarse and fine modes.
Given the measurement and a priori covariancematrices, we can use the Jacobian matrix as computed inSection 5.3 to compute the a posteriori covariance matrixSx as follows:
Sx ¼ ðS�1a þKTS�1
ε KÞ�1: ð41Þ
Table 3
Comparison of the square roots of the a posteriori covariance matrices S1=2x for scalar and vector measurements both within ðϕ�ϕ0 ¼ 01Þ and perpendicularto ðϕ�ϕ0 ¼ 901Þ the principal plane.
Parameter Scalar: I only Vector: ½I;Q ;U� � ½I;Q ;U;V � Vector: I¼ ½I;Q ;U� Vector: I¼ ½I;Q ;U;V �
sf (1.35)79.77�10�02 (1.35)71.76�10�01 (1.35)71.97�10�02 (1.35)74.78�10�02 (1.35)75.33�10�03
Table 4Information content H and degrees of freedom d for scalar and vector measurements both within (ϕ�ϕ0 ¼ 01) and perpendicular to (ϕ�ϕ0 ¼ 901) theprincipal plane.
Parameter Scalar: I only Vector: ½I;Q ;U� � ½I;Q ;U;V � Vector: I¼ ½I;Q ;U� Vector: I¼ ½I;Q ;U;V �
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433426
The square roots of the diagonal elements of this matrixprovide a measure of the accuracy with which the corre-sponding parameter of the state vector can be retrieved.These values have been tabulated as error bounds aboutthe true values of the state vector in Table 3, for both scalarand vector measurements, consisting of only the I compo-nent and the full Stokes vector I, respectively. Bothmeasurements within the principal plane and perpendi-cular to it have been considered.
As expected, the accuracy afforded by use of the fullStokes vector I is consistently higher than that due toI-only measurements, with the accuracy gains beingremarkably higher for measurements made perpendicularto the principal plane, where only 50% of the I-onlymeasurement vector provides independent informationdue to its symmetry about nadir. For full Stokes-vectormeasurements, this is made up for not only by Q but alsoby the non-zero U and V components (see Figs. 6–11), sothat all azimuthal planes remain roughly comparable inthe quantity of independent information they can afford.½I;Q ;U� measurements provide the same accuracy as thefull Stokes vector in the principal plane (since U ¼ V ¼ 0),but have reduced sensitivity in the perpendicular plane.
Given the measurement error considered in this illus-trative example, I-only measurements in the principalplane result in retrievals of ni;c, ni;f and rm;c that areswamped by uncertainties comparable to or larger thanthe absolute values of these parameters. For I-only mea-surements perpendicular to the principal plane, the uncer-tainties become roughly 2–7 times larger, so that theretrieval of sc also becomes unreliable. The tabulatedvalues reveal that I measurements are also not perfect,
where the retrieval of ni; f continues to be swamped by theuncertainty associated with it, albeit to a lesser extent (10times lesser for the principal plane and 40 times for theplane perpendicular to it) than I-only measurements.Under the present instrumental assumptions which aresimplistic and for illustrative purposes only, the uncer-tainty associated with I�based retrievals are seen to be 1–2 orders of magnitude smaller than those based on I only.½I;Q ;U� measurements incur a loss of accuracy comparedto full I measurements in the perpendicular plane, so thatthe uncertainties associated with ni;c, ni;f and rm;c continueto be larger than their true values. However, it can be seenfrom Table 3 that the accuracy gains for ½I;Q ;U� measure-ments in the perpendicular plane are already better thanor comparable to I-only measurements in the principalplane.
This can be seen to be a direct result of the informationcontent, H, of the respective measurement (shown inTable 4) which encapsulates the ratio of the uncertaintyassociated with the state vector before to that after themeasurement
H¼ 12 log 2 S�1
x Sa :j��� ð42Þ
This quantity is found to be 1.5–2 times larger for fullStokes vector measurements. Also, while the informationcontent for the perpendicular azimuthal plane is 20%smaller than in the principal plane for I-only measure-ments, measurement of the full Stokes vector I in theperpendicular plane slightly surpasses the correspondingprincipal plane value. This occurs because, even thoughthe effective length of measurement vector I contributing
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 427
independent information remains the same as in theprincipal plane (due to symmetry about the nadir ofmeasurements at 901 to the principal plane concurrentwith non-zero values for U and V), the total length of themeasurement vector gets doubled. This has a co-addingeffect, which effectively scales down the measurementnoise by a factor
ffiffiffi2
p. Confirming the results of Table 3, we
observe that ½I;Q ;U� measurements afford a lower infor-mation content in the principal plane than ½I;Q ;U;V �measurements, but the information content continues tobe larger than for principal plane I-only measurements.
The inability of the I-only measurements to retrieve allparameters in the perpendicular plane is underlined bythe degrees of freedom associated with the measurement,d (also shown in Table 4). d is a rough measure, obtained asthe number of diagonal elements of the singular valuedecomposition of S�1=2
ε KS1=2a that are greater than unity.
5.4.1. CaveatThe above conceptual analysis is simplified to assume a
constant percentage measurement uncertainty throughoutthe measurement vector. If, instead, we assume the detec-tor size for both scalar and vector measurements to beconstant, and make the additional simplifying assumptionthat each element of the Stokes' vector (whether inclusiveof V or not) occupies the same amount of detector space,the uncertainty associated with the scalar measurementwould be scaled down by a factor of
ffiffiffi3
pcompared to an
½I;Q ;U� measurement, or become half that of an ½I;Q ;U;V �measurement. This diminishes the gain in the informationcontent of vector measurements relative to scalar mea-surements. It should also be noted that Q and U are smallerthan and V is negligible compared to the I component ofStoke's vector, and hence carry potentially much largermeasurement uncertainties than those considered in oursimplified conceptual example.
In addition, real retrievals involve modeling error dueto the differences between reality and model simulations.They also follow a non-linear iterative path from the initialguess to the solution, which in the case of aerosol isfrequently ridden with local minima. This makes the actualuncertainties in the retrieved parameters much larger thanthose shown in Table 3, which have been computeddirectly at the solution in the absence of model error.Hence, the tabulated values should be seen as a lowerbound and not as values guaranteed by the measuringinstrument.
6. Summary and outlook
In this paper, we have outlined the matrix operatorformulation of the vector radiative transfer equation for ascattering atmosphere and derived the correspondinggeneral formulation for its derivatives with respect toaerosol optical and microphysical parameters, by buildingup on the work presented in Sanghavi et al. [38]. We havereplaced the delta truncation method used in smartMOMwith an adaptation of the delta-m method to vectorradiative transfer. This greatly reduces both the computa-tion time and the memory requirements of the model,while allowing minimal compromises on accuracy. We
have also replaced the method of obtaining the intensitiesfor the measurement geometry of interest by interpolationbetween quadrature points in smartMOM. The new block-Radau technique provides exact raytracing between anyuser-defined solar and viewing zenith angles, therebyeliminating the need for a large number of quadraturepoints in order to avoid interpolation error.
Comparison with the results of the benchmarkingexercise carried out by Kokhanovsky [22] reveal an accu-racy of up to the 6th–7th significant digit for Rayleighscattering. The aerosol and cloud scenes showed anagreement of better than 0.03% and 0.05% for most angles,except in the directions corresponding to the forwardscattering and backscattering peaks, which remain aregion of general disagreement across RTMs.
Using the example of a multi-angle satellite instrumentsimilar to MISR measuring normalized intensities in theVIS–NIR, we have demonstrated the use of our model tocompute both the forward model consisting of all ele-ments I, Q, U and V of the Stokes vector and their Jacobianswith respect to aerosol optical and microphysical para-meters for a bimodal aerosol distribution consistingof a coarse and a fine mode. We use these Jacobians tocompare the aerosol information content of measurementsusing only the total intensity component against use ofeither the ½I;Q ;U� components of the Stokes vector (prac-tical case) or the full Stokes vector (ideal case). We findthat the accuracy of the retrieved parameters improves by1–2 orders of magnitude when the full Stokes vector isused. The information content for the full Stokes vectorremains nearly constant for all azimuthal planes, whilethat associated with intensity-only measurements falls by20% from the principal plane to the plane perpendicular to it.The ½I;Q ;U� measurements are equivalent to ½I;Q ;U;V �measurements in the principal plane, but lose information(and thus also accuracy) increasingly towards the planeperpendicular to it. Nevertheless, for the example we con-sidered, both the accuracy and information content for½I;Q ;U� measurements in the perpendicular plane werefound to equal or exceed those of principal plane I-onlymeasurements.
The combined gains in speed and accuracy afforded bydelta-m truncation, direct raytracing and linearizationmake our model well-suited for use in optimized retrie-vals, especially for scattering media like aerosols andclouds, which rely on fast and accurate computations ofthe forward model and their Jacobians.
Acknowledgements
The authors thank two anonymous reviewers whowere instrumental in greatly improving the quality ofour manuscript. The authors are grateful to Feng Xu andVijay Natraj for helpful discussions and inter-comparisondata during the formative stages of this model. AndréHollstein is thanked for constructive and congenial dis-cussions. Alexander Kokhanovsky made verification of ourmodel much easier by having made his vector benchmark-ing results publicly available online. We thank RajeevJoshi for his timely support with programming issues.
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433428
David Diner and John Martonchik are thanked. Manythanks are due to Nathaniel Livesey and Michelle Santeefor their time and support.
This research was carried out at the Jet PropulsionLaboratory, California Institute of Technology, under con-tract with NASA. This work has been partially supported bythe NASA Aerosol-Cloud-Ecosystem (ACE) mission project.
Copyright 2013 California Institute of Technology. Gov-ernment sponsorship acknowledged.
Appendix A. Delta-m truncation
Truncation involves the removal of a fraction βT of theoriginal phase matrix from its forward scattering peak sothat it can be treated as unscattered light. This is equiva-lent to removing βTδðμ�1Þ from the phase function [44] orβTδðμ�1ÞE from the phase matrix, where δ denotes Dirac'sdelta function. As a result, the RTE Eq. (4) takes the form
Iðτ ; μ′;ϕ′; μ0;ϕ0Þ dμ′ dϕ′: ðA:1ÞIn the above, we have added and subtracted βTδðμ�1ÞEfrom Z in the first step and normalized the truncatedphase matrix by dividing it by ð1�βTÞ in the second step.Dividing both sides by ð1�ω0βTÞ yields for a homogenousatmospheric layer
μdIðτ ; μ;ϕ; μ0;ϕ0Þdð1�ω0βTÞτ
¼�Iðτ ; μ;ϕ; μ0;ϕ0Þþ 1�ω0ð1�βTÞ1�ω0βT
� �BðTÞ
þ 14π
ω0ð1�βTÞ1�ω0βT
;
Z 2π
0
Z 1
�1
Zðμ;ϕ; μ′;ϕ′Þ�βTδðμ�1ÞE1�βT
" #
Iðτ ; μ′;ϕ′; μ0;ϕ0Þ dμ′ dϕ′; ðA:2Þwhere ð1�ω0βTÞ dτ ¼ dð1�ω0βTÞτ since ω0 and βT areconstant within the layer.
Writing
τn ¼ ð1�ω0βTÞτ ;
ωn
0 ¼1�βT
1�ω0βTω;
Znðμ;ϕ; μ′;ϕ′Þ ¼ Zðμ;ϕ; μ′;ϕ′Þ�βTδðμ�1ÞE
1�βTðA:3Þ
gives Eq. (A.2) the regular form of the radiative transferequation, with the single scattering parameters of thelayer modified according to Eqs. (A.3) as a result oftruncation.
We use the Fourier decomposition method recom-mended by Siewert [39] to obtain Zm from Z. This entailsthe computation of the Greek coefficients β l, δ l, γ l, ε l, ζ land α l that make up the matrix B l (not to be confused withthe vector form of Planck's function BðTÞ). To obtain Z
n
m, weneed to obtain the matrix Bδ
l corresponding to the Dirac
delta function, consisting of the Greek coefficients βδl , δδl , γ
δl ,
εδl , ζδl and αδl . Wiscombe's delta-m truncation makes use of
the fact that δðμ�1Þ can be expressed in terms of Legendrepolynomials PlðμÞ as
δðμ�1Þ ¼ ∑1
l ¼ 0
2lþ12
PlðμÞ; ðA:4Þ
so that
βδl ¼2lþ12
: ðA:5Þ
Using the same logic for δðμ�1ÞE, we get
δδl ¼2lþ12
: ðA:6Þ
Also, since the off-diagonal terms of δðμ�1ÞE are zero,
εδl ¼ 0;γδl ¼ 0: ðA:7ÞTo find ζδl and αδl , we notice that
ðl�2Þðlþ2Þ
� �1=2R2l ð1Þ ¼
ðl�2Þðlþ2Þ
� �1=2T2l ð1Þ ¼
12:
This, along with relationships presented in Siewert [39]yields
ζδl ¼2lþ12
ðl�2Þ!ðlþ2Þ!
� �1=2 Z 1
�1δðμ�1ÞðR2
l ðμÞþT2l ðμÞÞ dμ
¼ 2lþ12
ðl�2Þ!ðlþ2Þ!
� �1=2ðR2
l ð1ÞþT2l ð1ÞÞ
¼ 2lþ12
: ðA:8Þ
The same process leads to
αδl ¼2lþ12
; ðA:9Þ
so that
Bδl ¼
2lþ12
E: ðA:10Þ
Now, we can use Wiscombe's approach to truncate theinfinite spherical function series at l¼L, by setting
βT ¼2
2Lþ1βL; ðA:11Þ
as a result of which the matrix B l takes the modified form
Bn
l ¼Bl�βTB
δl
1�βT¼Bl�
2lþ12Lþ1
βLE
1� 22Lþ1
βL
; 8 lrL: ðA:12Þ
Appendix B. Block-Radau quadrature for direct ray-tracing between the sun and the observer
The standard implementation of the RTM as in San-ghavi et al. [38] computes the intensity field only along thedirections given by the quadrature roots, μq, representingthe streams of the model. Since the viewing geometryconsisting of the solar zenith angle (SZA ¼ cos �1 μ0) andthe view zenith angles (VZAs ¼ cos �1 μv) (assuming amulti-angle viewing instrument like MISR) does not
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 429
generally coincide with the quadrature angles, interpola-tion techniques have to be used to obtain the values ofinterest.
Direct raytracing allows us to bypass error-prone inter-polation by including the direction cosines of interest inthe quadrature scheme with a zero quadrature weight(also called “dummy node inclusion” (DNI) technique by[2] and used within a MOM framework by [10]). Thisallows the computation of intensities scattered into eachof the additional directions (μ0 and μv) from the standardquadrature directions μq. The principle of reciprocity [3]would allow us to work out intensities propagating fromthe additional cosines μ0 and μv into the standard quad-rature cosines μq. However, due to the zero quadratureweight associated with both μ0 and μv, it is not possible todirectly estimate the intensities between two anglesunless at least one of them is a standard quadrature point.
Thus in order to include both the source, μ0, and theviewing directions, μv, in a direct raytrace, we devise aquadrature scheme that always includes μ0 as a standardquadrature point, i.e., carrying a non-zero quadratureweight, while allowing the μv's to be treated as dummynodes with zero weights. For this, we construct our mainquadrature scheme using two Radau quadrature blockswith the limits ð0; μ0� and ðμ0;1� in addition to the dummynodes consisting of Nview view angle cosines μv;1; μv;2;…;
μv;Nview. These dummy nodes are appended to the end of
the set of main quadrature points and hence can beconveniently excluded from computations where theywould carry zero weight. This minimizes the computa-tional cost of our method even when Nview is large.
In the following discussion, we disregard the dummynodes and focus only on the main quadrature scheme. Ifwe use N1-point quadrature for the interval ð0; μ0� and N2-point quadrature for the interval ðμ0;1�, the roots, μn
i , andweights, cni , of the composite ðN1þN2Þ�point quadratureare given by
μn
i ¼ μ0 � μi;cni ¼ μ0 � ci; i¼ 1;2;…N1 ðB:1Þ
and
μn
ðN1 þ iÞ ¼ μ0þð1�μ0Þ � μi;
cnðN1 þ iÞ ¼ ð1�μ0Þ � ci; i¼ 1;2;…N2; ðB:2Þ
where μi and ci denote the roots and weights of regular N1-and N2-point Radau quadratures (each having limits ð0;1�)in Eqs. (B.(1) and B.2), respectively.
Given the same viewing geometry, the above schemecan be extended to include more solar zenith cosines (forsimultaneous computations) by using quadrature blocksð0; μ0;1�, ðμ0;1; μ0;2� and so on up to ðμ0;n;1� for n differentsolar zenith cosines. The advantage of this method is thatthe solar zenith cosines actually contribute to the actualquadrature, allowing the total number of main quadraturepoints to be kept small. If the number of solar positions tobe computed exceeds the number of viewing directions,view zenith cosines can be included in the main quad-rature scheme instead of solar zenith cosines (which arethen treated as dummy nodes) in combination with the
principle of reciprocity to enable more economicalcomputations.
Appendix C. Linearization of the vector form of thematrix operator method
C.1. General formalism
To analytically compute the derivative of the forwardmodel used to simulate the measurement vector y, weneed to solve the derivative of the radiative transferequation Eq. (4)
μd_Iðτ ; μ;ϕ; μ0;ϕ0Þ
dτ�dIðτ ; μ;ϕ; μ0;ϕ0Þ
dτ∂_τ
∂τÞ
¼�_Iðτ ; μ;ϕ; μ0;ϕ0Þþð1�ω0Þ _BðTÞ� _ω0BðTÞ
þ_ω0
4π
Z 2π
0
Z 1
�1Zðμ;ϕ; μ′;ϕ′ÞIðτ ; μ′;ϕ′; μ0;ϕ0Þ dμ′ dϕ′
þω0
4π
Z 2π
0
Z 1
�1
_Z ðμ;ϕ; μ′;ϕ′ÞIðτ ; μ′;ϕ′; μ0;ϕ0Þ dμ′ dϕ′
þω0
4π
Z 2π
0
Z 1
�1Zðμ;ϕ; μ′;ϕ′Þ_Iðτ ; μ′;ϕ′; μ0;ϕ0Þ dμ′ dϕ′;
ðC:1Þwhere the dotted variables denote the derivative of thatvariable with respect to any element of the state vector x.The second term on the left hand side contains thederivative ∂ _τ=∂τ , which vanishes for all aerosol singlescattering parameters (optical thickness τ, single scatteringalbedo ω, phase function P and the truncation factor β),and only plays a role in the computation of derivativeswith respect to the aerosol microphysical parameters, forwhich _τ is indeed a function of τ , as will be demonstratedin the following.
Simple differentiation of Eqs. (5), (6) and (8) yields theboundary conditions to the above equation:
_Ið0; μ;ϕ; μ0;ϕ0Þ ¼ 0 ðC:2Þfor μZ0 at the TOA, since the incoming solar flux at theTOA is constant.
At the surface,
_Iðτs; μ;ϕ; μ0;ϕ0Þ ¼ _εðμÞBðTÞþεðμÞ _BðTÞ
þZ 2π
0
Z 1
0ð _Rpðμ;ϕ; μ′;ϕ′ÞIþRpðμ;ϕ; μ′;ϕ′Þ_IÞμ′ dμ′ dϕ′;
ðC:3Þwith the constraint
_εþZ 2π
0
Z 1
0
_Rð1;1Þp ðμ;ϕ; μ′;ϕ′Þμ′ dμ′ dϕ′¼ 0: ðC:4Þ
In order to find a matrix representation for the aboveequations, we use the following relationships for differ-entiation with respect to an arbitrary scalar parameter xfor any given matrices A and B:
∂ðAþBÞ∂x
¼ ∂A∂x
þ∂B∂x
; ðC:5Þ
∂ðA � BÞ∂x
¼A � ∂B∂x
þ∂A∂x
� B; ðC:6Þ
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433430
and
∂A�1
∂x¼�A�1∂A
∂xA�1: ðC:7Þ
The above equations help us obtain the derivatives ofEqs. (19)–(21) for an elemental layer of optical thickness δ:
where the indices 0,1,2 indicate the boundaries of twoadjacent layers.
The symmetry relationships of Eqs. (29) and (30) alsohold for their derivatives, so that
_Tab ¼D _TbaD ðC:17Þand
_Rab ¼D _RbaD: ðC:18ÞHence, for linearized doubling, computational time can
again be halved using the transformation
_Rn
10 ¼D _R10; ðC:19Þalong with Eq. (31). Using _R
n
10 in place of _R10 and _R01, _T10 inplace of _T01, Rn
10 in place of R10 and R10, and T10 in place ofT01, Eq. (C.(11) and C.12) can be used to construct thederivatives of the transmission matrix _Tba and the trans-formed reflection matrix _R
n
ba. As in Eqs. (32), the derivativesof all the transmission and reflection matrices associatedwith the homogenous layer can be reconstructed as
_Tba ¼ _Tba;
_Rba ¼D _Rn
ba;
_Tab ¼D _TbaD;
_Rab ¼ _Rn
baD: ðC:20ÞUsing the above equations in a doubling-adding frame-
work allows the computation of the intensity derivativesin a form similar to Eq. (18) for an arbitrary layer of opticalthickness Δ (the subscripts m denoting the Fouriermoments have been dropped for clarity):
_IþΔ
_I�0
" #¼
_JþΔ
_J�0
" #þ
_TΔ0 _RΔ0
_R0Δ _T0Δ
" #Iþ0I�Δ
" #þ
TΔ0 RΔ0
R0Δ T0Δ
" #_Iþ0_I�Δ
" #:
ðC:21Þ
C.2. Derivatives with respect to individual aerosolparameters
We wish to obtain derivatives of the measured inten-sities with respect to five aerosol parameters per aerosolmode for spherical particles. We consider a bimodalaerosol, consisting of a fine and a coarse mode. The aerosolparameters for each mode are
1.
The aerosol optical thickness at a chosen wavelength, λ0. 2. The complex refractive index, nr�niı. 3. The lognormal size distribution parameters, rm and s,
as in Eq. (39).
We represent each aerosol mode out of a total of Naer
modes by the index i. The radiative transfer equation (4)and its derivative, Eq. (C.1), only involve the aerosol singlescattering parameters, viz. τi, ω0;i and Zi, which areembedded within the layer-averaged values as follows:
τ ¼ τrþτgþ ∑Naer
i ¼ 1τi; ðC:22Þ
ω0 ¼τrþ∑Naer
i ¼ 1ω0;iτiτ
; ðC:23Þ
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433 431
Z ¼ τrZrþ∑Naeri ¼ 1τiω0;iZi
ω0τ; ðC:24Þ
where the subscripts ‘r’ and ‘g’ represent Rayleigh scatter-ing and trace gas absorption, respectively. If the aerosolphase matrix is truncated (see Appendix A), the aerosolsingle scattering properties τi, ω0;i and Zi in Eqs. (C.22)–(C.24) get replaced by their modified versions τni , ω
n
i andZn
i , respectively.The terms τi;ω0;i;Zi, and additionally βi when trunca-
tion is employed, depend on the microphysical parametersof the aerosol, viz. nr;i;ni;i; rm;i and si and thus encapsulatetheir effect on the radiative transfer of light. The elementalderivatives in Eqs. (C.8)–(C.10) have an explicit form withrespect to τi;ω0;i;Zi, and βi through Eqs. (C.22)–(C.24) andadditionally Eqs. (A.3) for the truncated case, whichenables the calculation of further derivatives culminatingin the intensity derivatives expressed in Eq. (C.21).
If we express the quantities derived in Eqs. (C.11)–(C.16)by the general variable X, we can write by virtueof the dependence of the aerosol single scattering para-meters, represented here collectively using a vectorxSS ¼ fτi;ω0;i;Zi; βig, on the microphysical parameters, like-wise represented collectively using a vector xμ ¼fnr;i;ni;i; rm;i; sig:∂X∂xμ
¼ ∂X∂xSS
� ∂xSS
∂xμ; ðC:25Þ
or, in an explicit form,
∂X∂xðkÞμ
¼ ∑4
j ¼ 1
∂X∂xðjÞSS
� ∂xðjÞSS
∂xðkÞμ
; ðC:26Þ
where xðkÞμ represents the kth element of xμ and xðjÞSSrepresents the jth element of xSS. If the aerosol can beassumed to be spherical, Mie theory may be used to obtainthe relationship between xSS and xμ as well as thederivative of xSS with respect to xμ [12,18]. Once theelemental derivatives are defined for τi and the micro-physical parameters xμ, the same doubling-adding treat-ment leads up to the intensity derivatives of Eq. (C.21).
In the following, we provide expressions for the deri-vatives _δ, _ω0 ,
_Z occurring in Eqs. (C.8)–(C.10) with respectto τi, ω0;i, Zn
i and βi. Let δ be chosen such that ndbl
doublings are needed to produce a homogenous layer ofoptical thickness τ (after truncation), i.e.
δ¼ τ
2ndbl: ðC:27Þ
Thus, from Eqs. (A.(3) and C.22), we get
∂δ∂τi
¼ 1�βiω0;i
2ndbl: ðC:28Þ
Similarly,
∂δ∂ω0;i
¼�βiτi2ndbl
; ðC:29Þ
∂δ∂Zn
i¼ 0; ðC:30Þ
and
∂δ∂βi
¼�ω0;iτi2ndbl
: ðC:31Þ
Eqs. (A.(3) and C.23) yield
∂ω0
∂τi¼ ðω0;i�ω0Þ�ω0;iβið1�ω0Þ
τ; ðC:32Þ
∂ω0
∂ω0;i¼ ð1�βið1�ω0ÞÞ �
τiτ; ðC:33Þ
∂ω0
∂Zn¼ 0; ðC:34Þ
and
∂ω0
∂βi¼�ω0;iτið1�ω0Þ
τ; ðC:35Þ
Eqs. (A.(3) and C.24) yield
∂Z∂τi
¼ ð1�βiÞω0;i
τi� ðZn
i �ZÞ; ðC:36Þ
∂Z∂ω0;i
¼ ð1�βiÞτiτ
� ðZn
i �ZÞ; ðC:37Þ
∂Z∂Zn
i¼ ð1�βiÞω0;iτi
τðC:38Þ
and
∂Z∂βi
¼�ω0;iτiτ
� ðZn
i �ZÞ: ðC:39Þ
It may be noted that for the non-truncated case, βi ¼ 0 andτni ¼ τi, ωn
i ¼ω0;i and Zn
i ¼Zi in the above equations.Linearization of Mie theory [18] can provide the deri-
vatives of the extinction cross-section, the single scatteringalbedo and the series of Greek coefficients, viz. _κ , _ω0 and_B l, of the aerosol with respect to its microphysical para-meters, xμ, which can be coupled to the single scatteringparameters as shown in Eqs. (C.(25) and C.26). FollowingSiewert [39], we can construct _Z from
_Zmðμi; μjÞ ¼2
ð1þδ0mÞ∑L
l ¼ mΠm
l ðμiÞ _BlΠml ðμjÞ; ðC:40Þ
where δom is the Kronecker delta function, μi and μj are anytwo direction cosines represented by the chosen quad-rature and Πm
l represent normalized matrices containingthe generalized spherical harmonics [11].
However, for the truncated case, we need to find a formfor _B
nwith respect to xμ since _B is no longer applicable in
its original form (see Appendix A). From Eq. (A.11) it isclear that the truncation factor β associated with theaerosol is differentiable with respect to the elements ofxμ, such that
_β ¼ 22Lþ1
_βL; ðC:41Þ
for truncation at l¼L. This can be substituted in thederivative of Eq. (A.12), so that
_Bn
l ¼ð2Lþ1Þ _Blþ _βL½2Bn
l �ð2lþ1ÞE�ð2Lþ1Þ�2βL
; ðC:42Þ
thus allowing the computation of _Z with respect to xμ.
S. Sanghavi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 412–433432
In the case where the optical thickness of the aerosol isknown at a given wavelength λ0, the optical thickness, τλ,at all other wavelengths λ varies as
τλ ¼ κλκλ0
� τλ0 ; ðC:43Þ
where τλ0 is known. Thus, τλ for all λaλ0 is a function ofthe microphysical parameters of the aerosol and its deri-vative can be written as
_τλ ¼τλ0κλ0
� _κλ� _κλ0κλκλ0
� �: ðC:44Þ
Note that the number density, Naer, of the aerosoldistribution representing the average number of aerosolparticles per unit volume within an atmospheric layer ofthickness, δz, is given by
Naerδz¼ τλ0κλ0
¼ τλκλ: ðC:45Þ
This quantity is invariant with respect to wavelength andis implicitly present in Eqs. (C.(43) and C.44).
Now we have explicit forms for all the equationsinvolved in the computation of the derivative of theradiative transfer with respect to aerosol parameters.
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