Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–68

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Polarized radiative transfer of a cirrus cloud consisting of randomlyoriented hexagonal ice crystals: The 3�3 approximation fornon-spherical particles

S. Stamnes a,n, S.C. Ou b, Z. Lin c, Y. Takano b, S.C. Tsay d, K.N. Liou b, K. Stamnes c

a NASA Langley Research Center, MS 475, Hampton, VA 23681, United Statesb University of California at Los Angeles, Joint Institute for Regional Earth System Science and Engineering, 607 Charles E Young Drive East, Young Hall, Room4242, Los Angeles 90095-7228, United Statesc Stevens Institute of Technology, 1 Castle Point on Hudson, Physics and Engineering Physics, Hoboken, NJ 07030, United Statesd NASA Goddard Space Flight Center, Code 613, Greenbelt, MD 20771, United States

a r t i c l e i n f o

Article history:Received 9 May 2016Received in revised form1 July 2016Accepted 1 July 2016Available online 9 July 2016

Keywords:Vector radiative transfer modelDiscrete ordinate methodCirrus cloudsPolarization

x.doi.org/10.1016/j.jqsrt.2016.07.00173/& 2016 The Authors. Published by Elsevie

esponding author.ail addresses: snorre.a.stamnes@nasa.gov (S. Smos.ucla.edu (S.C. Ou).

a b s t r a c t

The reflection and transmission of polarized light for a cirrus cloud consisting of randomly oriented hex-agonal columns were calculated by two very different vector radiative transfer models. The forward peak ofthe phase function for the ensemble-averaged ice crystals has a value of order ×6 103 so a truncationprocedure was used to help produce numerically efficient yet accurate results. One of these models, theVectorized Line-by-Line Equivalent model (VLBLE), is based on the doubling–adding principle, while theother is based on a vector discrete ordinates method (VDISORT). A comparison shows that the two modelsprovide very close although not entirely identical results, which can be explained by differences in treatmentof single scattering and the representation of the scattering phase matrix. The relative differences in thereflected I and Q Stokes parameters are within 0.5% for I and within 1.5% for Q for all viewing angles. In 1971Hansen [1] showed that for scattering by spherical particles the 3�3 approximation is sufficient to produceaccurate results for the reflected radiance I and the degree of polarization (DOP), and he conjectured thatthese results would hold also for non-spherical particles. Simulations were conducted to test Hansen'sconjecture for the cirrus cloud particles considered in this study. It was found that the 3�3 approximationalso gives accurate results for the transmitted light, and for Q and U in addition to I and DOP. For these non-spherical ice particles the 3�3 approximation leads to an absolute error < × −2 10 6 for the reflected andtransmitted I, Q and U Stokes parameters. Hence, it appears to be an excellent approximation, which sig-nificantly reduces the computational complexity and burden required for multiple scattering calculations.

& 2016 The Authors. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Several ice cloud retrieval approaches have been developed andcarried out using radiometric data from airborne and space-bornesensors [2–7]. While success has been claimed by these approaches,the accuracy of their application to remote sensing data is subject touncertainties in surface reflectivity [8] and ice crystal shape [9]. Solarradiation becomes polarized when scattered by molecules, aerosols,water droplets or ice crystals. Many published analytical studies onthe behavior of polarized light scattering by ice crystals [10] havefound that remote sensing of ice clouds using polarimetric mea-surements of reflected sunlight [11,12] complements radiometricapproaches in the sense that polarization parameters contain

r Ltd. All rights reserved.

tamnes),

additional information about ice crystal shape and size. Furthermore,polarimetric data may result in more accurate cirrus cloud retrievalsbecause the degree of polarization has a high measurement accuracyon the order of 0.2% [13], and the signal produced by reflection fromplanetary surfaces in terms of the Stokes parameters Q and U is ty-pically wavelength-independent [14], which can help to furtherdistinguish cirrus clouds from highly reflective surfaces such as snowand ice. However, remote sensing of ice clouds using polarimetricmeasurements of reflected sunlight requires accurate polarizationsingle-scattering properties for ice crystals, and an accurate yet effi-cient polarized radiative transfer model with an adequate treatmentof the scattering phase matrix for non-spherical ice crystals.

The present work explores the capability of using computed icecrystal single-scattering properties in current polarized radiativetransfer models. Hansen found that the 3�3 approximation,which reduces the computational complexity of the radiativetransfer calculation, yielded accurate results for the polarized

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–6858

radiance reflected by liquid water clouds, and also conjecturedthat it would apply to large, non-spherical particles [1]. By using ahexagonal ice crystal model, we were able to verify Hansen'sconjecture, and we also found that the 3�3 approximation workswell for the transmitted light, which is important for remotesensing of ice clouds from ground-based instruments [15].

Because accurate modeling of multiple scattering by large, non-spherical particles is challenging, two different vector radiativetransfer codes were used in order to validate the results. In Section2, we explain the overall methodology, including the formulationof vector radiative transfer (Section 2.1), the single-scatteringproperties of ice crystals (Section 2.2), and the scaling of the phasematrix for strongly forward-peaked scattering (Section 2.3). InSection 2.4 we provide a discussion of the 4�4 calculations vs the3�3 approximation for the discrete ordinate method. In Section 3we discuss the results obtained by 4�4 calculation and the 3�3approximation including the impact of the Stokes scattering ma-trix element b2, and the scalar approximation, which further re-duces the complexity of the calculation, and can be useful for in-terpretation of radiance-only measurements. In Section 4 weprovide a summary of the results and recommendations for en-hancements of the vector discrete ordinate method.

2. Methodology

The polarized radiative transfer model VDISORT [16,17] is usedto calculate the reflectance and transmittance (i.e. apparent opticalproperties, AOPs) of a homogenous slab containing a collection ofice crystals (a cirrus cloud). The reflectance and transmittancewere first calculated by the VLBLE radiative transfer model[4,5,18], and a LUT (look-up-table) was originally produced foroptical depths of 4, 8, 16, and 32 at multiple solar zenith angles(θ0) for a wavelength of μ2.22 m.

The updated Cirrus AOP LUT contains the reflected and trans-mitted I, Q, and U Stokes parameters for optical depths of 0.25, 0.5,1, 2, 4, 8, 16, and 32 at solar zenith angles between °0 and °70 with1 degree resolution, and with azimuth angles over the full rangebetween °0 and °180 with 5 degree resolution. The surface ismodeled as a Lambertian reflector with an albedo of 0, 0.2, 0.4, 0.6,0.8, or 1.0. The scattering phase matrix for an ensemble of ran-domly oriented hexagonal ice crystals at λ = μ2.22 m is used, sothat absorption and scattering by molecules in the atmosphere atthis wavelength will be minor (there is weak absorption by watervapor, followed by methane and carbon dioxide), especially for icecloud optical depths ≥1.0.

The single-scattering properties of ice crystals were computedusing the geometric optics ray-tracing method, which is compre-hensively described in a series of papers by Takano and Liou [19–21]. For treatment of ice particles, there are several different typesof single-scattering models [22] such as the finite-difference timedomain technique (FDTD) [23,24], the improved geometric-optics-integral-equation hybrid method (GOM-2 or IGOM) [25], theAmsterdam discrete dipole approximation (ADDA) [26,27], and theT-matrix method [28]. Extensive datasets of ice cloud single-scattering properties have been created for many different types ofaspherical particle shapes and size distributions over spectralranges from 0.2 to μ100 m for both smooth and roughened particlesurfaces [29,30].

The elements of the scattering phase matrix including thescattering phase function do not have much wavelength depen-dence for a fixed type of ice particle if applied to calculation ofbroad-band quantities [4]. However, for ice cloud remote sensing

purposes, it is necessary to calculate the scattering phase functionand scattering phase matrix on a channel-by-channel basis be-cause subtle differences in the single-scattering properties canimpact retrieval accuracy. The single-scattering properties (Stokesscattering matrix, scattering phase matrix, asymmetry factor, sin-gle-scattering albedo, and extinction coefficient) also vary sig-nificantly with ice crystal shape and size. Note that a single size icecrystal was used in this study. Calculations for a single size shouldmatch closely to calculations that use a size distribution withuniform shape for the same effective diameter [20,3,4,6], sincethere is less sensitivity to the effective variance, but will not ne-cessarily match non-uniform size distributions with differentshapes or a size-dependent shape [11].

2.1. Formulation of polarized radiative transfer

In a plane-parallel (slab) geometry, the integro-differentialequation for polarized radiation, given in terms of the Stokes

vector→

= [ ]∥ ⊥I I I U V, , , T is [31–34]

∫ ∫

τ ϕτ

τ ϕ τ ϕ

ϖ τπ

ϕ τ ϕ ϕ

τ ϕ

→( ) =

→( ) +

→( )

− ( ) ′ ′ ( ′ ′)

→( ′ ′) ( )

π

−

ud I u

dI u S u

d du u u

I u

P

, ,, , , ,

4, , ; ,

, , 1

0

2

1

1

where ∥I and ⊥I are the radiances of the electromagnetic field thatare parallel and perpendicular to the scattering plane, U is thelinear polarization in the ° °45 /135 plane, and the degree of circularpolarization is V I/ . In Eq. (1), τ α σ= − ( + )d dz is the differentialoptical depth, α and s are the absorption and scattering coeffi-cients, dz is the differential vertical path length, andϖ τ σ α σ( ) = ( + )/ is the single-scattering albedo. The angles θ′ andϕ′ are the polar and azimuthal angles, respectively, of the directionof an incident parallel beam, while θ and ϕ are the polar andazimuthal angles of the observation direction, respectively. (Wedenote θ′ = ′u cos and θ=u cos .) The scattering phase matrix

τ ϕ ϕ( ′ ′)u uP , , ; , , describing the scattering properties of the med-ium [33,35–38], is related to the Stokes scattering matrix [see Eq.

(2)], and→S is the source term, proportional to the input solar flux

[16,17,34,39].

The Stokes vector→

= [ ]∥ ⊥I I I U V, , , T is related to→

= [ ]I I Q U V, , ,ST

by = +∥ ⊥I I I and = −∥ ⊥Q I I . Then Q is the linear polarization in the° °0 /90 plane, −Q I/ is the degree of linear polarization, and the

degree of polarization + +Q U V IDOP /2 2 2 . Note that the angles° °45 /135 and ° °0 /90 are measured with respect to the principal

meridional plane. Due to symmetry with respect to the meridionalplane, it does not matter which angle, e.g. °45 or °135 is chosen asthe positive direction in terms of I and Q. By convention the degreeof linear polarization is defined as −P P/12 11, so that −Q I/ corres-ponds to linear polarization of singly-scattered light.

2.2. Ice crystal single-scattering properties

The single-scattering properties (single-scattering albedo, op-tical depth, the Stokes scattering matrix, and the correspondingscattering phase matrix) are inherent optical properties (IOPs) sincethey depend only on the physical properties of the particles, andare thus properties of the medium itself and independent of theillumination. The single-scattering properties of the cirrus cloudwere calculated at multiple wavelengths λ for an ensemble ofrandomly oriented hexagonal columns whose diameter and length

0 30 60 90 120 150 18010−1

100

101

102cirrus a1

Θ

phas

e fu

nctio

n

a1 origa1 δ−fit 240a1 δ−fit 120

0 30 60 90 120 150 180−0.5

0

0.5cirrus −b1/a1

Θ

−b1/

a1

−b1/a1 orig−b1/a1 δ−fit 240−b1/a1 δ−fit 120

0 30 60 90 120 150 1800

0.5

1

1.5cirrus a2/a1

Θ

a2/a

1

a2/a1 origa2/a1 δ−fit 240a2/a1 δ−fit 120

0 30 60 90 120 150 180−1

0

1

2cirrus a3/a1

Θ

a3/a

1

a3/a1 origa3/a1 δ−fit 240a3/a1 δ−fit 120

0 30 60 90 120 150 180−0.2

0

0.2cirrus b2/a1

Θ

b2/a

1

b2/a1 origb2/a1 δ−fit 240b2/a1 δ−fit 120

0 30 60 90 120 150 180−0.5

0

0.5

1cirrus a4/a1

Θ

a4/a

1

a4/a1 origa4/a1 δ−fit 240a4/a1 δ−fit 120

Fig. 1. The 6 unique elements of the scattering matrix representing a cirrus cloudare shown, modeled by randomly oriented hexagonal columns with a diameter of

μ40 m and a length of μ100 m. The δ-fit method was used to optimally fit thescattering matrix elements for 240 and 120 scattering matrix moments, and theresulting δ-fit moments were used in the VDISORT calculations for efficiency. (Forinterpretation of the references to color in this figure caption, the reader is referredto the web version of this paper.)

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–68 59

are 40 and μ100 m, respectively, by using a geometrical optics ray-tracing program [19–21]. For this hexagonal ice crystal, the effec-tive diameter is de¼ μ44.29 m, and the single-scattering albedowas calculated to be 0.9586 at μ2.22 m. The angular scatteringdependence of the ice crystal is given by the Stokes scattering

matrix. In the representation→

= [ ]I I Q U V, , ,ST , the Stokes scatter-

ing matrix has the form

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Θ

Θ ΘΘ Θ

Θ ΘΘ Θ

( ) =

( ) ( )( ) ( )

( ) ( )− ( ) ( ) ( )

a b

b a

a b

b a

F

0 0

0 0

0 0

0 0 2

S

1 1

1 2

3 2

2 4

where Θ is the scattering angle (the angle between the incidentand scattered direction vectors).

Two rotations are required to connect the Stokes vector of thescattered radiation to that of the incident radiation. The first ro-tation through angle −i1 is from the meridian plane associated

with the Stokes vector→I S

inc, into the scattering plane. The second

rotation through angle π − i2 is from the scattering plane into the

meridian plane associated with the Stokes vector→I S

sca. Hence, the

Stokes vector for the scattered radiation is given by [31,33,34]

π Θ Θ→

= ( − ) ( ) ( − )→

≡ ( )→

( )I i i I IR F R P . 3S

sca

S S S S

inc

S S

inc

2 1

The matrix RS , called the Stokes rotation matrix, represents a ro-tation through an angle ω in the clockwise direction with respectto an observer looking into the direction of propagation. Thus RScan be written as ( ω π≤ ≤0 2 )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

ωω ωω ω

( ) =( ) − ( )( ) ( )

( )

R

1 0 0 00 cos 2 sin 2 00 sin 2 cos 2 00 0 0 1

.

4

S

According to Eq. (3), the scattering phase matrix, which connectsthe Stokes vector of the scattered radiation to that of the incidentradiation, is obtained from the Stokes scattering matrix Θ( )FS in Eq.(2) by

θ ϕ θ ϕ Θ( ′ ′ ) = ( − ) ( ) ( − ) ( )i iP R F R, ; , 5S S S S2 1

where π( − ) = ( − )i iR RS S2 2 since the rotation matrix is periodicwith a period π. Carrying out the matrix multiplications in Eq. (5)one finds [33,34]

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Θ( ) =

−˜ −

˜ −

− − ( )

a b C b S

b C a p b S

b S p a b C

b S b C a

P

0

0 6

S

1 1 1 1 1

1 2 2 23 2 2

1 2 32 3 2 2

2 1 2 1 4

where Θ= ( ) = …a a i, 1, , 4i i , Θ= ( ) =b b i, 1, 2i i ,

˜ = − = − − = + ˜

= − + = = =

=

a C a C S a S p C a S S a C p S a C C a S a

S a S C a C C i C i S i

S i

cos2 , cos2 sin2 ,

sin2 .

2 2 2 1 2 3 1 23 2 2 1 2 3 1 32 2 2 1 2 3 1 3

2 2 1 2 3 1 1 1 2 2 1 1

2 2

A comparison of Eqs. (2) and (6) shows that only the corner ele-ments of Θ( )FS remain unchanged by the rotations of the referenceplanes. The (1,1) element of the scattering phase matrix Θ( )PS , andalso of the Stokes scattering matrix Θ( )FS , is the scattering phasefunction. Since also the (4,4) element of the scattering phasematrix remains unchanged by the rotations, the state of circularpolarization of the incident light does not affect the radiance of thescatted radiation after one scattering event.

Also, if the b2 element in Eq. (6) is zero, the circular polarization

component decouples from the other 3. Then, the Stokes para-meter V is scattered independently of the others according to thephase function Θ( )a4 , and the remaining part of the scatteringphase matrix referring to I, Q, and U becomes a 3�3 matrix:

0 30 60 90 120 150 180

−200

0

200

Θ

rela

tive

diffe

renc

e %

100*(b1−b2)/max|b2| orignormalized to a1

0 30 60 90 120 150 180

−0.2

0

0.2

b1, b

2

Θ

b1 origb1/a1 origb2 origb2/a1 orig

Fig. 2. The scattering matrix elements b1 and b2 are plotted on the same scale. Toppanel: relative difference between the two quantities b1 and b2, and between thenormalized quantities b a/1 1 and b a/2 2. Bottom panel: b1, and b2 are plotted as thinsolid lines, while the ratios b a/1 1 and b a/2 1 are plotted as circles (thicker lines). Thehalos are visible in b1, most prominently at °20 , but generally both b1 and b2 have asimilar magnitude, so that the main difference between the two is their location inthe scattering matrix.

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–6860

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥Θ( ) =

−˜

˜ ( )

a b C b S

b C a p

b S p a

P .

7

S

1 1 1 1 1

1 2 2 23

1 2 32 3

For consistency, we will refer to the approximate solution thatoccurs from setting =b 02 as the “3�3 approximation” and referto the exact solution (where b2 is not taken to be 0) as the “4�4solution”.

Returning to Eq. (2), the calculated Stokes scattering matrix forthe cirrus cloud at μ2.22 m [19] is plotted in Fig. 1 by the red curve.The behavior of the scattering matrix as a function of scatteringangle, and how the single scattering properties influence themultiple scattering Stokes parameters are briefly discussed. Fromtop, first panel, the truncated strongly forward-peaked phasefunction (102) is plotted on a semi-log scale. In the visible [40], thescattering phase function, a1, displays a strong °22 halo peakcaused by two refractions through a °60 prism angle, a halo at °46due to two refractions through a °90 prism angle, and a third peakbetween °150 and °160 resulting from the superposition of rayssubject to one and two internal reflections. At μ2.22 m the °22 halois shifted to °20 and the °46 halo is situated near °38 . A lessprominent secondary peak also occurs near °170 , formed by thesuperposition of rays subject to higher-order reflections. In thesecond panel, −b a/1 1 has two local maxima near °20 and °38 ,corresponding to the halo features in a1. Between °110 and °160there is a broad peak with magnitude ranging from 0.1 to 0.2 forthe λ = μ0.865 m wavelength, and from 0.1 to 0.3 at λ = μ2.22 m(depicted here) due to the combined effects of external and in-ternal reflections. In the third panel, a a/2 1 is nearly 1 in the forwarddirection, so a2 is also very strongly peaked but decreases withincreasing scattering angle due to the non-sphericity of the icecrystal and the relative strength of external and internal reflec-tions: ≡a a/ 12 1 for a spherical particle or when only consideringthe externally reflected rays from a randomly-oriented, convexparticle [41]. In the fourth panel, a a/3 1 also decreases with in-creasing scattering angle, and crosses zero at an intermediateangle (the cross-over angle) because the sign of the Fresnel re-flection coefficient changes. This change, which also changes thesign of a a/3 1 for externally reflected light, is called the phase-shifteffect. If only external reflections are considered, the cross-overangle would be around °60 ; however, with the superposition ofinternally reflected rays the cross-over angle is shifted to between

°100 and °120 . In the fifth panel, the element b a/2 1 ranges between−0.2 and 0.2, and the angular variation is nearly the same as atvisible wavelengths [19,21]. The magnitude of b a/2 1 is smaller thanthat of the diagonal elements a a/i 1, so that its effect on the mul-tiply scattered I and Q Stokes parameters is weaker. In the sixthpanel, we note that the element ≥a a4 3. Similar to the behavior ofa a/2 1, ≡a a4 3 for a spherical particle or for the case when onlyexternally reflected rays are considered, so that differences be-tween a4 and a3 result from the non-sphericity of the scatteringparticle. A comparison of the b1 and b2 elements of the scatteringmatrix is provided in Fig. 2. The maximum value of b a/2 1 ap-proaches −0.2 between scattering angles of °60 to °90 and near

°180 , but was otherwise close to 0. Although b1 is similarly small invalue, it has significant impact on singly-scattered light, while b2only affects light that has been scattered at least twice. Thus if b2 issmall, setting it to zero should not have a significant impact on theresulting Stokes parameters since sunlight is initially unpolarized.Ou et al. [4] demonstrated that each scattering phase matrix ele-ment is sensitive to ice crystal shape, size, and surface roughnessdue to a variety of optical effects. The diagonal elements ai displaylarger sensitivity to ice crystal shape and size in the backscatteringdirection than in the forward scattering direction. This feature isuseful for remote sensing of ice crystal shape and size parameters,because polarimetric sensors onboard aircraft and satellites mostly

measure radiation in the backscattering direction during middayflight or overpass. In the backscattering direction, the off-diagonalelements b1 and b2 are more sensitive to particle surface rough-ness than to shape and size.

2.3. Scattering phase matrix moments and scaling

We can calculate τ ϕ ϕ( ′ ′)u uP , , ; , in Eq. (1), by decomposing theelements in Eq. (2) into generalized spherical function (GSF)coefficients, and use them to expand Θ( )P defined by Eq. (6) also inGSFs. We also transform from the scattering angle Θ to sphericalcoordinates ( ϕu, ) using an addition theorem for the GSFs [34–37].

First the scattering phase matrix P is expanded in a Fourierseries in azimuth angles using N2 terms:

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎫⎬⎭

∑

ϕ ϕ

ϕ ϕ ϕ ϕ

( ′ ′)

= ( ′) ( ′ − )+ ( ′) ( ′ − )( )=

−

u u

u u m u u m

P

P P

, ; ,

, cos , sin ,8m

N

cm

sm

0

2 1

where Pcm and Ps

m are matrices, the coefficients of the cosine andsine terms, respectively.

An addition theorem for the GSFs can be used to express theseFourier expansion coefficients directly in terms of the GSF ex-pansion coefficients of the Stokes scattering matrix [35,36]:

Δ Δ( ′) = ( ′) + ( ′) ( )u u u u u uP A A, , , 9cm m m

34 34

Δ Δ( ′) = ( ′) − ( ′) ( )u u u u u uP A A, , , 10sm m m

34 34

whereΔ = { − − }diag 1, 1, 1, 134 andtheskew-centrosymmetric ma-trix ( ′)u uA ,m is given by

∑ Λ( ′) = ( ) ( ′)( )ℓ=

−

ℓ ℓ ℓu u u uA P P,11

m

m

Nm m

2 1

where

Table 1The input parameters for the RT calculation of the cirrus cloud are optical thicknessτc, single-scattering albedo ϖ, solar zenith angle θ0, Lambertian bidirectional re-flectance distribution ρL, wavelength λ, and desired relative azimuth viewing angles

ϕΔ . Note that we do not consider gaseous absorption and molecular scattering(τ = 0gas ). The output is given in terms of the Stokes parameters (SPs). The reference

case is displayed in the second column, and since ρ = 0L the surface is completely

absorbing or black. The input parameters for the Cirrus AOP LUT are displayed inthe third column.

Input Reference case Cirrus AOP LUT

τc 4 0.25, 0.5, 1, 2, 4, 8, 16, 32ϖ 0.9586 0.9586θ0 °29.17 [ ° ° … °]0 , 1 , , 70ρL 0 0, 0.2, 0.4, 0.6, 0.8, 1.0λ μ2.22 m μ2.22 mτgas 0 0

ϕ ϕ ϕΔ = − 0 ° °0 , 180 [ ° ° … °]0 , 5 , , 180Output Reference case Cirrus AOP LUT

SPs I, Q, U, V I, Q, U

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–68 61

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟( ) =

( )

ℓ

ℓ

ℓ+

ℓ−

ℓ−

ℓ+

ℓ

u

P

P P

P P

P

P

0 0 0

0 0

0 0

0 0 0 12

m

m

m m

m m

m

,0

, ,

, ,

,0

= ( ± ) ( )ℓ±

ℓ−

ℓP P P 13m m m, 1

2, 2 ,2

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

α β

β α

α β

β α

Λ =

− ( )

ℓ

ℓ ℓ

ℓ ℓ

ℓ ℓ

ℓ ℓ

0 0

0 0

0 0

0 0 14

1, 1,

1, 2,

3, 2,

2, 4,

and

∑ α( ) = ( )( )ℓ=

−

ℓ ℓa x P x15

N

10

2 1

1,0,0

∑ α α( ) + ( ) = ( + ) ( )( )ℓ=

−

ℓ ℓ ℓa x a x P x16

N

2 32

2 1

2, 3,2,2

∑ α α( ) − ( ) = ( − ) ( )( )ℓ=

−

ℓ ℓ ℓ−a x a x P x

17

N

2 32

2 1

2, 3,2, 2

∑ α( ) = ( )( )ℓ=

−

ℓ ℓa x P x18

N

40

2 1

4,0,0

∑ β( ) = ( )( )ℓ=

−

ℓ ℓb x P x19

N

12

2 1

1,0,2

∑ β( ) = ( )( )ℓ=

−

ℓ ℓb x P x .20

N

22

2 1

2,0,2

Here Θ=x cos , ℓP0,0, ℓP2,2, ℓ−P2, 2, and ℓP0,2 are GSFs, ℓP0,0 are also

known as the Legendre polynomials, α ℓi, and β ℓj, are the GSF ex-pansion coefficients ( = =i j1, 2, 3, 4; 1, 2), and ai and bj are theelements of the Stokes scattering matrix in Eq. (2).

The elements of the Stokes scattering matrix have been pre-scaled by a Fraunhofer diffraction factor via [19]

∑Θ δ( ) = ( − ) +( )

( )F f F f F121

Skl

Dn

kln

kl D D

where = …k l, 1, , 4, ( )Fkln is the contribution from geometric optics

and FD is the contribution from Fraunhofer diffraction, and FSkl is

the (k l, ) element of the Stokes scattering matrix. This particularpre-scaling technique can be applied to any type of large particle,but the values of the Fraunhofer diffraction factors will depend onthe amount of light that can be represented as scattered by a δ-function. For particles that are large compared to the wavelength,e.g. λ≥a 20 , where a is the particle dimension, the extinction ef-ficiency = →

πQ 2C

aext

2 . In this limit, the total cross-section is an

equal combination of geometric optics and Fraunhofer diffractionso that the cross-section for the Fraunhofer diffracted light is halfthe total extinction cross-section, =C C

sca,diff 2ext . Then the ratio of

diffracted light to scattered light is the Fraunhofer diffractionfactor

ϖ= →

( − )=

( − )≥

( )δ δf

C

C

C

C f f11

2 112 22D

sca,diff

sca

sca,diff

sca

where ( − )δC f1sca is the total light scattered after considering thatlight scattered at Θ = 0 is not scattered, fδ is the fraction of lightthat can be represented as directly forward scattered by a δ-function, where =δf 0 for spherical particles and roughened ice

crystal particles [42], and ϖ = CC

sca

extis the single-scattering albedo.

For this particular hexagonal ice particle of length μ100 m anddiameter μ40 m, ≃δf 0.0384 leading to ≃f 0.542D for ϖ = 0.9586(see Table 1). Then the combined δ-function and Fraunhofer dif-fraction pre-scaling is given by [19]

′ = + − ( )δ δf f f f f . 23D D

The ′f pre-scaling is combined with the δ-M [43] truncation factor,α=δf N1,2M

, to arrive at the total scaling factor, ″f :

″ = ′ + − ′ ( )δ δf f f f f . 24M M

We used the δ-fit procedure [44,45] to optimally fit the ′f pre-scaled moments given by Eq. (24). The GSF moments(ℓ = … −N0, , 2 1) are then scaled by δ-M so that the final scalingis represented by ″f :

αα

′ =− ( ℓ + )

− ( )

δ

δℓ

ℓ f

f

2 1

1,

25ai

i,

, M

M

ββ

′ =− ( )δ

ℓℓ

f1.

25bj

j,

,

M

But for consistency, we also need to apply ″f scaling to thesingle-scattering albedo and optical depth:

ϖ ϖϖ

τ τ ϖ′ = ( − ″)− ″

′ = ( − ″ )( )

ff

f11

, 1 .26

The final (pre-scaled then δ-fit fitted) cirrus Stokes scatteringmatrix used in VDISORT is also depicted by the green and bluecurves in Fig. 1. To verify that we had a good fit, we comparedusing 240 moments and 120 moments with δ-fit scaling to fit theoriginal pre-scaled cirrus phase function. In Fig. 3 we plot the δ-fiterror for the phase function and the δ-fit relative difference fromfitting the remaining scattering phase matrix elements. The

0 30 60 90 120 150 180

10−3

10−2

10−1

100

101

Θ

|(a1 fit

/a1−

1)|*1

00

|(a1fit

/a1−1)|*100, 240

120

0 30 60 90 120 150 180

10−3

10−2

10−1

100

101

Θ

rela

tive

erro

r %

a2 fitting error, 240120

0 30 60 90 120 150 180

10−3

10−2

10−1

100

101

Θ

rela

tive

erro

r %

a3 fitting error, 240120

0 30 60 90 120 150 180

10−3

10−2

10−1

100

101

Θ

rela

tive

erro

r %

a4 fitting error, 240120

0 30 60 90 120 150 180

10−3

10−2

10−1

100

101

Θ

rela

tive

erro

r %

b1 fitting error, 240120

0 30 60 90 120 150 180

10−3

10−2

10−1

100

101

Θ

rela

tive

erro

r %

b2 fitting error, 240120

Fig. 3. The error introduced by performing δ-fit scaling to the ice crystal scattering matrix for 120 and 240 moments, which are used in the VDISORT calculation. At 240moments, the error from δ-fit scaling is generally less than 1.0% except in the forward direction where it approaches 8.75% for a1, a2, a3, and a4. At 120 moments, the fittingerror is significantly larger, but still provides a reasonable fit (generally within 1%) except for elements a1, b1 and b2 and in the forward direction.

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–6862

relative difference, which has the same shape as the absolutedifference | − |y x , is used to avoid division by small numbers, and isdefined by:

[ ] = | − |(| |)

×( )

y xx

relative difference %max

10027

where (| |)xmax is the maximum value of x after taking the absolutevalue. In Fig. 3, y is the δ-fit scattering matrix element (ai or bj)linearly interpolated to the quadrature of the original element x.For the phase function a1, the error is less than 0.7% except in theforward direction where it is ~9%, and in the backward directionwhere it approaches 1.5%. For b1 the relative difference is 0.8% inthe forward direction, and is typically less than 0.1% otherwise. Fora2, a3, and a4, there is a relative difference of about ~9% in theforward direction, and less than 0.1% otherwise. b2 has a relativedifference less than 1.0% except in the forward direction where itapproaches 2.5%. Thus the error is generally less than 1.0% exceptin the forward direction where it approaches ~9% for a1 andtherefore also a2, a3, and a4 since the error incurred from fittingthe phase function will propagate to the diagonal elements. At 120moments, the fitting error is significantly larger, but still provides areasonable fit (generally within 1%) except for elements a1, b1 andb2 and the forward direction. In the radiative transfer calculations

below we found that the difference between using 120 vs 240moments mainly affected the transmitted I and Q components,and we conjecture that we could achieve sufficient accuracy usingfewer moments by invoking an exact single-scattering correction.

2.4. The 4�4 solution and the 3�3 approximation

In the discrete ordinate method of radiative transfer, we needto determine homogenous and particular solutions to arrive at thegeneral solution. The particular solution is formulated as a set oflinear equations =Ax b that can quickly be solved using standardtechniques of linear algebra, for example Gaussian elimination.The most time-consuming step is solving the homogenous pro-blem, which is formulated as a standard algebraic eigenvalueproblem, λ( − ) =A x 0. In the case of the 4�4 solution, this ei-genvalue problem involves matrices of size ×N N4 4 , where N isthe number of quadrature points or “streams” in the upper andlower hemispheres, since a reduction of dimension step is used toreduce the size from ×N N8 8 . This step is completely analogous tothe reduction of dimension used in DISORT [46] to reduce ×N N2 2matrices to ×N N . Since the presence of the sign on b2 in Eq. (2)leads to a matrix A in the algebraic eigenvalue problem thatcannot be made symmetric, the eigenvalues and corresponding

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–68 63

eigenvectors in the 4�4 representation occur in complex con-jugate pairs. In the 3�3 approximation, the 4th row and columnin Eq. (2) are simply omitted, and the resulting impact on thescattering phase matrix is obtained by setting =b 02 in Eq. (6)which results in Eq. (7). Then the matrix A can be made symmetricimplying that the resulting eigenvalues and eigenvectors are real.Note that setting =b 02 in the 4�4 representation leads to I, Q andU parameters identical to those obtained in the 3�3approximation.

But since setting =b 02 decouples the V component from I, Q, andU, the eigenvalue problem required to solve the homogenous systemin N discrete ordinates is reduced from solving a ×N N4 4 system tosolving a ×N N3 3 system. Since the computational burden of solvingan eigenvalue problem scales like (n3) where n is the size of thematrix, this 3�3 approximation reduces the computational burdenby a theoretical factor of ≈4 /3 2.373 3 . Since the resulting eigenvec-tors and eigenvalues for the ×N N3 3 system are real, significantfurther computational savings are obtained by using an eigensolver,such as ASYMTX available in the DISORT package [46], that avoidscomplex arithmetic. In the 4�4 VDISORT calculation the eigenvec-tors and the eigenvalues are complex and complex solutions arerequired to obtain accurate solutions for I and Q.

3. Results

3.1. Apparent optical properties (AOPs)

The Stokes parameters (SPs) for reflected and transmitted ra-diation are AOPs in the sense that they depend on the ambientlight field. They are defined as [32]

0 20 40 60 800.1

0.2

0.3

0.4

0.5I, REFLECTION

viewing zenith

RI

VDISORT 4x43x3VLBLE

R

0 20 40 60 800

2

4

6

8I, TRANSMISSION

viewing nadir

T I

Fig. 4. Reference case: VDISORT vs VLBLE for an azimuth angle of ϕΔ = 0 (principal pla→

= [ ]I 1, 0, 0, 0S . The actual reflection and transmission for I and Q are scaled by a factor

quadrature points and Fourier moments are used in the VDISORT calculation. Top panels:Q components at nadir angles θT.

πμ

=( )

− − −RF

SP ,28a

SP0 0

out top of slab

πμ

=( )

− − −TF

SP ,28b

SP0 0

out bottom of slab

where = ISP , Q , U or V, μ0 is the cosine of the solar zenith angleand F0 is the incident solar irradiance (normal to the beam). RSPrepresents the Stokes parameter of reflected radiation escapingfrom the top of the slab, and TSP represents the Stokes parameter oftransmitted radiation escaping from the bottom of the slab.

3.2. The 4�4 calculation

The reference case from the look-up-table has input parametersdefined in Table 1. The SPs are calculated by solving Eq. (1), which inVDISORT is accomplished by the discrete ordinate method and inVLBLE by the doubling–adding method. A comparison of VDISORTand VLBLE results is plotted in Fig. 4, where the reflected compo-nents are plotted against the zenith angle θR, where θ = °0R is up-ward-looking, and θ = °90R is the horizon, and the transmittedcomponents are plotted against the nadir angle θT, where θ = °0T isdownward-looking, and θ = °90T is the horizon. The resulting solu-tions for I and Q in the 3�3 approximation are essentially identicalto the more computationally demanding 4�4 case.

The reflected I component increases with viewing zenith angle,but decreases near the horizontal direction. The reflected Q com-ponent generally decreases with viewing zenith angle, but exhibitslow-magnitude oscillations. The transmitted I component has astrong peak at °29 corresponding to the forward scattering peak of

0 20 40 60 80−0.02

−0.015

−0.01

−0.005

0Q, REFLECTION

viewing zenith

Q

0 20 40 60 80−0.01

0

0.01

0.02

0.03Q, TRANSMISSION

viewing nadir

T Q

ne). The input is described in Table 1, Reference case. The incident Stokes vector isπ

μ F0 0where =F 10 and μ0 is the cosine of the solar zenith angle θ0. A total of 120

reflected I and Q components at zenith angles θR; bottom panels: transmitted I and

0 20 40 60 800

0.1

0.2

0.3

0.4I, REFLECTION

viewing zenith

rela

tive

diffe

renc

e %

VDISORT 4x4VDISORT 3x3

0 20 40 60 800

1

2

3Q, REFLECTION

viewing zenith

rela

tive

diffe

renc

e %

0 20 40 60 80 800

2

4

6

8I, TRANSMISSION

viewing nadir

rela

tive

diffe

renc

e %

0 20 40 600

5

10

15Q, TRANSMISSION

viewing nadir

rela

tive

diffe

renc

e %

0 20 40 60 800

0.1

0.2

0.3

0.4I, REFLECTION

viewing zenith

rela

tive

diffe

renc

e % VDISORT 240

VDISORT 120

0 20 40 60 800

1

2

3Q, REFLECTION

viewing zenith

rela

tive

diffe

renc

e %

0 20 40 60 800

2

4

6

8I, TRANSMISSION

viewing nadir

rela

tive

diffe

renc

e %

0 20 40 60 800

5

10

15Q, TRANSMISSION

viewing nadir

rela

tive

diffe

renc

e %

Fig. 5. Top 4 panels: the relative difference between VDISORT and VLBLE for the reflected and transmitted I and Q Stokes parameters is plotted for (a) the 4�4 VDISORTcalculation using complex solutions with b2 from Fig. 1 and for (b) a VDISORT calculation performed after setting the scattering matrix element =b 02 in Eq. (2) (the 3�3approximation). Bottom 4 panels: the relative difference between VDISORT and V-LBLE using (a) 240 and (b) 120 moments and quadrature points in VDISORT.

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–6864

the phase function. The transmitted Q component has a strongpeak at °9 and °49 corresponding to the maxima of −b a/1 1 at thehalo angle of °20 . The peak at °67 corresponds to the maximum of−b a/1 1 at °38 . Moreover, the transmitted Q SP approaches 0 at °29as the light ray in the forward scattering angle should beunpolarized.

3.3. The 3�3 approximation

To test the 3�3 approximation, we set the b2 scattering matrixelement to zero in Eq. (2), and repeated the VDISORT calculation,which is also depicted in Fig. 4. In the top 4 panels of Fig. 5 we plotthe relative difference between VLBLE and VDISORT with the 4�4

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–68 65

calculation, as well as the VDISORT 3�3 calculation. Again, inorder to avoid division by small numbers, we use the relativedifference, Eq. (27), where y is the SP from VDISORT and x is the SPfrom VLBLE, and we have linearly interpolated the VDISORT SPs tothe VLBLE quadrature. In the principal plane, we find that theStokes parameters for reflected radiance, RI, agree within 0.5% forall viewing angles. The relative difference of the reflected Q-component, RQ is generally within 1% but can approach 1.5%. Forthe transmitted components, the relative difference of TI is gen-erally less than 1% but reaches 6% in the forward direction at ascattering angle of Θ = 0, presumably because VDISORT does notperform an exact single-scattering calculation and makes use of afitting for the Stokes scattering matrix. The VDISORT 4�4 solutionand 3�3 approximation for I and Q are indistinguishable.

In the bottom 4 panels of Fig. 5 we compare the relative dif-ference at 120 streams and 240 streams. The relative differencedecreases as the number of moments and streams is increasedfrom 120 to 240, particularly for the transmitted I and Q Stokesparameters. For the VDISORT 3�3 approximation with 120 mo-ments/streams, the relative difference of the transmitted TQ iswithin 5% (except at the halo angles °9 and °49 ), and increasing thenumber of moments/streams to 240 decreases the relative differ-ence, particularly at the halo angles. Though relative differencesfor the transmitted Q SP in the forward direction are somewhatlarger than in the sideward direction, the absolute magnitude ofthese differences is less than 0.004 for 120 moments/streams, andless than 0.002 for 240 moments/streams. In VDISORT, the single-scattering correction would be expected to improve accuracy atlower numbers of streams, particularly in the forward direction ordirections where the scattering phase matrix is strongly peaked,but would otherwise not significantly change the results. In thisexample, using only 120 streams is not enough to completely re-solve the transmitted components in the forward direction and the

0 20 40 60 800.1

0.2

0.3

0.4

0.5I, REFLECTION: VDISORT

viewing zenith

RI

0°

90°

180°

0 20 40 60 800

2

4

6

8I, TRANSMISSION

viewing nadir

T I

Fig. 6. I and Q SPs from the VDISORT 3�3 and VLBLE multiple scattering calculation at 3and blue curves; the corresponding VLBLE results are black, cyan, and magenta. (For intethe web version of this paper.)

halo directions, so the number of streams may need to be in-creased or the single-scattering correction would need to be im-plemented depending on the desired accuracy at those angles.

In Fig. 6 we plot the I and Q SPs obtained from VDISORT andV-LBLE at 3 different azimuth angles of ϕΔ = °0 (principal plane),

°90 , and °180 . The red, green, and blue curves are VDISORT resultsfor the 3�3 approximation and the black, cyan, and magentacurves are VLBLE results obtained from the 3�3 approximation.At ϕΔ = °90 the SP curves are generally smooth for both the re-flected and transmitted components.

3.4. Impact of b2 on U and V

To examine the impact that b2 has on U and V, we ran thefollowing test using the exact 4�4 solution compared to using the4�4 solution but setting =b 02 . In the latter case the results for I,Q, and U will exactly match those from the 3�3 approximation,but we can solve for V as well. But how good is this “approxima-tion” for V and what happens to U when b2 is not zero? In Fig. 7 weplot the multiple scattering calculation for U and V from VDISORTand VLBLE at an azimuth angle of ϕΔ = °90 , where the green curverepresents VDISORT 4�4, green crosses VDISORT 4�4 with

=b 02 , the cyan curve represents VLBLE 4�4, and the cyan circlesrepresent VLBLE 4�4 with =b 02 . Since =b 02 leads to a decou-pling of I, Q, and U from the V Stokes parameter, and the drivingradiation is unpolarized, then V must vanish when =b 02 . Thesomewhat less intuitive result is that for this reference case U islittle influenced by b2, so it does not matter whether or not it is setto 0. Note that U and V must vanish at ϕΔ = °0 and °180 .

3.5. Accuracy of the 3�3 approximation

The 3�3 approximation was investigated for spherical

0 20 40 60 80−0.02

−0.01

0

0.01Q, REFLECTION

viewing zenith

RQ

0 20 40 60 80−0.01

0

0.01

0.02

0.03Q, TRANSMISSION

viewing nadir

T Q

different azimuth angles of ϕΔ = °0 , °90 , and °180 corresponding to the red, green,rpretation of the references to color in this figure caption, the reader is referred to

0 20 40 60 80−0.015

−0.01

−0.005

0U, REFLECTION

viewing zenith

RU

VDISORT 4x43x3VLBLE 4x43x3

0 20 40 60 80−6

−4

−2

0

2

4 x 10−5 V, REFLECTION

viewing zenith

RV

0 20 40 60 80−10

−5

0

5 x 10−3 U, TRANSMISSION

viewing nadir

T U

0 20 40 60 80−4

−3

−2

−1

0

1 x 10−5 V, TRANSMISSION

viewing nadir

T V

Fig. 7. Comparison of U and V SPs from VDISORT and VLBLE multiple scattering calculation at an azimuth angle of °90 . Compare the green curve (VDISORT 4�4), greencrosses (VDISORT 4�4 with =b 02 ), to the cyan curve (VLBLE 4�4) and the cyan circles (VLBLE 4�4 with =b 02 ). Note that setting =b 02 (matching the 3�3 approx-imation) in VDISORT or VLBLE does not affect the reflected or transmitted U component. Also note that V is zero when =b 02 , and the U and V SPs vanish at ϕΔ = °0 and °180which is what we would expect. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–6866

particles in Hansen's 1971 paper [1] who found that “the3�3approximation introduces errors < −10 6 for the radiance, and errors≤ × −2 10 4 in the degree of polarization…”. Hansen concluded that“in computing the polarization properties for multiple scattering byspherical particles it is usually adequate to work with 3�3 matrices.”But Hansen only investigated errors in the reflected radiance andthe DOP, and not the error for the individual Stokes components Qand U, and he did not consider transmittances.

From Fig. 5 we can see that the 4�4 calculation and the 3�3approximation are very similar. For all cases simulated in thisstudy, we compared the 4�4 and 3�3 solution obtained fromVDISORT and found that I, Q, and U agree to within < × −2 10 6 inabsolute magnitude.

For the reference case we found that the degree of linear po-

larization ( )= +Q U IDOP /2 2 obtained from the 3�3 approx-

imation is almost identical (within 0.001%) to the degree of po-

larization ( )= + +Q U V IDOP /2 2 2 obtained from the 4�4 cal-

culation. Therefore the b2 element, which has a maximum of −0.2(after being normalized to a1) between scattering angles of °60and °90 , appears to have negligible effect on the reflected ortransmitted I, Q and U SPs or the DOP.

Hansen originally concluded: “Thus, for comparison to observa-tions of the polarization of sunlight reflected by water clouds, it isadequate to make theoretical computations with three-by-threematrices.” The present results suggest that this statement alsoholds for non-spherical ice crystals, and that it applies not just tothe reflected radiation, but also to the transmitted radiation, andthe Stokes parameters Q and U.

3.6. The scalar approximation

For the scalar assumption, Hansen makes the point that for

near-infrared wavelengths at 1.2, 2.25, 3.1, and μ3.4 m, the error inthe radiance obtained by the scalar assumption vs the 3�3 cal-culation is about 1–2% for water cloud droplets. “Thus, in most casesthe error in the scalar approximation should be ≤1%for reflection froma cloud of particles which are at least as large as the wavelength.”Additionally, Hansen conjectured that errors in the reflected ra-diance for non-spherical particles should also be small when usingthe scalar approximation. For the reference case we found therelative difference between the scalar approximation and the 4�4calculation to be less than 0.5% for the reflected radiance. Thus thescalar approximation works very well for calculating the reflectedand transmitted radiance, I, in this particular case of a homo-genous cirrus cloud. But we neglected the atmosphere so this re-sult may not hold at shorter wavelengths for which Rayleighscattering becomes significantly enhanced.

3.7. The exact single-scattering calculation/correction

In order to accurately calculate the polarized radiances Q, U,and V, Hansen emphasizes the importance of the single-scatteringcalculation: “However, spheres are less useful as an approximationfor nonspheres when the polarization, rather than the radiance, isconsidered. This is a result of the facts that multiple scattering doesnot wash out features in single-scattering polarization and that thesingle-scattering polarization is sensitive to the particle shape.”

Thus, the halos present in Q may be best resolved using thesingle-scattering correction, which makes use of the exact single-scattering solution. We also expect VLBLE to be correct for thetransmitted I component since (contrary to VDISORT) it includesan exact single-scattering calculation, which is needed to accu-rately calculate the total radiance in the forward direction. Sincepolarization effects are minimal in the direction of solar trans-mission, we achieve the expected result that Q¼0 in this direction

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–68 67

(at nadir angle θ = °29.17T ). However, compared to the full mul-tiple-scattering calculation it should be emphasized that use of thesingle-scattering correction mainly improves the efficiency of thecomputation, and aside from points located at difficult geometries(e.g. the forward direction) VDISORT results obtained with andwithout the single-scattering correction will be the same provideda sufficient number of streams is used.

4. Conclusion

Vector radiative transfer programs can produce apparent op-tical properties (Stokes parameters) from inherent optical prop-erties (IOPs) and appropriate boundary conditions. Two differentradiative transfer programs, VLBLE and VDISORT, were used toproduce results for the reflected I and Q Stokes parameters for acirrus cloud with a phase function peaking strongly in the forwarddirection. The SPs computed by the two codes were found to agreein terms of relative difference to within 0.5% for I and within 1.5%for Q (all viewing angles). Additionally, it was found that settingthe scattering phase matrix element =b 02 , the 3�3 approxima-tion provides accurate values of the I, Q, and U Stokes parametersof the reflected and transmitted fields. The input and output filesused in this study are available for download online [47]. Insummary:

� In VDISORT complex solutions are generally required in the4�4 representation ( ≠ )b 02 .

� However, if only I, Q, U and/or DOP are desired, then the 3�3approximation ( = )b 02 should be used because it is computa-tionally faster, requires only real solutions, and appears to workextremely well.

� The fact that this 3�3 approximation works so well also fornon-spherical particles has practical implications because itsuggests that for unpolarized incident light we can achieve acomputational savings of a factor 2.4 (Section 2.3) without anysignificant loss of accuracy. By using an eigensolver that avoidscomplex arithmetic (such as ASYMTX from DISORT) a significantincrease in efficiency can be achieved.

� In the scalar approximation, the scattering phase matrix in boththe particular solution and the homogenous solution is replacedwith the 1�1 approximation so that the only non-zero elementis a1, the scattering phase function. For the homogeneous cirruscloud considered in this study, the scalar approximation ap-pears to work very well for calculating the reflected andtransmitted radiance.

� The method used in this paper to deal with scattering fromhighly forward-peaked scattering phase functions, where thenormalized scattering phase function peak is on the order of

×6 103, namely a Fraunhofer diffraction peak pre-scaling fol-lowed by δ-fit scaling, appears to allow for an accurate scat-tering phase matrix representation for radiative transfer calcu-lations of the reflected Stokes parameters using only 120 phasefunction moments.

Future work would be to examine the efficiency gained frominvoking the exact single-scattering correction in VDISORT, whichwould not change the multiple scattering results but would allowone to investigate the minimum number of streams required toobtain accurate results.

Acknowledgments

Partial funding for this research came from the NASA Aerosol-Cloud-Ecosystems (ACE) project.

References

[1] Hansen JE. Multiple scattering of polarized light in planetary atmospheres. PartII: sunlight reflected by terrestrial water clouds. J Atmos Sci 1971;28(8):1400–1426.

[2] Ou SC, Liou KN, Gooch W, Takano Y. Remote sensing of cirrus cloud parametersusing advanced very-high-resolution radiometer 3.7-and 9-μm channels. ApplOpt 1993;32(12):2171–2180.

[3] Ou SC, Takano Y, Liou KN, Higgins GJ, George A, Slonaker R. Remote sensing ofcirrus cloud optical thickness and effective particle size for the national polar-orbiting operational environmental satellite system visible/infrared imagerradiometer suite: sensitivity to instrument noise and uncertainties in en-vironmental parameters. Appl Opt 2003;42(36):7202–7214.

[4] Ou SC, Liou KN, Takano Y, Slonaker R. Remote sensing of cirrus cloud particlesize and optical depth using polarimetric sensor measurements. J Atmos Sci2005;62(12):4371–4383.

[5] Liou KN, Ou SC, Takano Y, Liu Q. A polarized delta-four-stream approximationfor infrared and microwave radiative transfer: Part I. J Atmos Sci 2005;62(7):2542–2554.

[6] King MD, Platnick S, Yang P, Arnold GT, Gray MA, Riedi JC, et al. Remote sensingof liquid water and ice cloud optical thickness and effective radius in the arctic:application of airborne multispectral mas data. J Atmos Ocean Technol 2004;21(6):857–875.

[7] Ou S, Liou K, King M, Tsay S. Remote sensing of cirrus cloud parameters basedon a 0.63–3.7 μm radiance correlation technique applied to avhrr data. GeophysRes Lett 1999;26(16):2437–2440.

[8] Rolland P, Liou K. Surface variability effects on the remote sensing of thin cirrusoptical and microphysical properties. J Geophys Res: Atmos (1984–2012)2001;106(D19):22965–22977.

[9] Mishchenko MI, Rossow WB, Macke A, Lacis AA. Sensitivity of cirrus cloud al-bedo, bidirectional reflectance and optical thickness retrieval accuracy to iceparticle shape. J Geophys Res: Atmos (1984–2012) 1996;10(D12):16973–16985.

[10] Mishchenko MI. Vector radiative transfer equation for arbitrarily shaped andarbitrarily oriented particles: a microphysical derivation from statisticalelectromagnetics. Appl Opt 2002;41(33):7114–7134.

[11] Diedenhoven Bv, Cairns B, Geogdzhayev I, Fridlind A, Ackerman A, Yang P,et al. Remote sensing of ice crystal asymmetry parameter using multi-direc-tional polarization measurements—Part 1: methodology and evaluation withsimulated measurements. Atmos Meas Tech 2012;5(10):2361–2374.

[12] Diedenhoven Bv, Cairns B, Fridlind A, Ackerman A, Garrett T. Remote sensingof ice crystal asymmetry parameter using multi-directional polarizationmeasurements—Part 2: application to the research scanning polarimeter.Atmos Chem Phys 2013;13(6):3185–3203.

[13] Cairns B, Russell EE, Travis LD. Research scanning polarimeter: calibration andground-based measurements. In: SPIE's international symposium on opticalscience, engineering, and instrumentation. International Society for Optics andPhotonics; 1999. p. 186–96.

[14] Peltoniemi J, Hakala T, Suomalainen J, Puttonen E. Polarised bidirectional re-flectance factor measurements from soil, stones, and snow. J Quant SpectroscRadiat Transf 2009;110(17):1940–1953.

[15] Knobelspiesse K, van Diedenhoven B, Marshak A, Dunagan S, Holben B,Slutsker I. Cloud thermodynamic phase detection with polarimetrically sen-sitive passive sky radiometers. Atmos Meas Tech 2015;8(3):1537–1554.

[16] Sommersten ER, Lotsberg JK, Stamnes K, Stamnes JJ. Discrete ordinate andmonte carlo simulations for polarized radiative transfer in a coupled systemconsisting of two media with different refractive indices. J Quant SpectroscRadiat Transf 2010;111(4):616–633.

[17] Cohen D, Stamnes S, Tanikawa T, Sommersten ER, Stamnes JJ, Lotsberg JK, et al.Comparison of discrete ordinate and monte carlo simulations of polarizedradiative transfer in two coupled slabs with different refractive indices. OptExpress 2013;21(8):9592–9614.

[18] Ou S, Liou KN, Takano Y, Wong E, Hutchison K, Samec K. Comparison of theUniversity of California at Los Angeles Line-by-Line Equivalent RadiativeTransfer Model and the Moderate-Resolution Transmission Model for accuracyassessment of the National Polar-Orbiting Operational Environmental SatelliteSystem’s Visible–Infrared Imager–Radiometer Suite cloud algorithms, Appl.Opt. 2015;44(29):6274–6284, http://dx.doi.org/10.1364/AO.44.006274.

[19] Takano Y, Liou KN. Solar radiative transfer in cirrus clouds. Part I: single-scattering and optical properties of hexagonal ice crystals. J Atmos Sci 1989;46(1):3–19.

[20] Takano Y, Liou KN. Solar radiative transfer in cirrus clouds. Part II: theory andcomputation of multiple scattering in an anisotropic medium. J Atmos Sci1989;46(1):20–36.

[21] Takano Y, Liou KN. Radiative transfer in cirrus clouds. Part III: light scatteringby irregular ice crystals. J Atmos Sci 1995;52(7):818–837.

[22] Kahnert FM. Numerical methods in electromagnetic scattering theory. J QuantSpectrosc Radiat Transf 2003;79:775–824.

[23] Yee KS. Numerical solution of initial boundary value problems involvingMaxwells equations in isotropic media. IEEE Trans Antennas Propag 1966;14(3):302–307.

[24] Yang P, Liou KN. Finite-difference time domain method for light scattering bysmall ice crystals in three-dimensional space. J Opt Soc Am A 1996;13(10):2072–2085.

[25] Yang P, Liou KN. Geometric-optics integral-equation method for light scat-tering by nonspherical ice crystals. Appl Opt 1996;35(33):6568–6584.

S. Stamnes et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 193 (2017) 57–6868

[26] Purcell EM, Pennypacker CR. Scattering and absorption of light by non-spherical dielectric grains. Astrophys J 1973;186:705–714.

[27] Yurkin MA, Hoekstra AG. The discrete-dipole-approximation code adda: cap-abilities and known limitations. J Quant Spectrosc Radiat Transf 2011;112(13):2234–2247.

[28] Mishchenko MI, Travis LD. T-matrix computations of light scattering by largespheroidal particles. Opt Commun 1994;109(1):16–21.

[29] Baum BA, Yang P, Heymsfield AJ, Schmitt CG, Xie Y, Bansemer A, et al. Im-provements in shortwave bulk scattering and absorption models for the re-mote sensing of ice clouds. J Appl Meteorol Climatol 2011;50(5):1037–1056.

[30] Yang P, Bi L, Baum BA, Liou K-N, Kattawar GW, Mishchenko MI, et al. Spec-trally consistent scattering, absorption, and polarization properties of atmo-spheric ice crystals at wavelengths from 0.2 to μ100 m. J Atmos Sci 2013;70(1):330–347.

[31] Chandrasekhar S. Radiative transfer.New York: Dover Publications; 1960.[32] Liou KN. An introduction to atmospheric radiation, vol. 84. San Diego, Cali-

fornia: Academic Press; 2002.[33] Hovenier JW, van der Mee CV, Domke H. Transfer of polarized light in pla-

netary atmospheres: basic concepts and practical methods, vol. 318. Dor-drecht, Netherlands: Springer Science & Business Media; 2014.

[34] Stamnes K, Stamnes JJ. Radiative transfer in coupled environmental systems.Weinheim, Germany: Wiley-VCH; 2015.

[35] Siewert C. On the equation of the transfer relevant to the scattering of po-larized light. Astrophys J 1981;245:1080–1086.

[36] Siewert C. On the phase matrix basic to the scattering of polarized light. As-tron Astrophys 1982;109:195.

[37] De Haan J, Bosma P, Hovenier J. The adding method for multiple scattering

calculations of polarized light. Astron Astrophys 1987;183:371–391.[38] Mishchenko MI, Travis LD, Lacis AA. Multiple scattering of light by particles:

radiative transfer and coherent backscattering.Cambridge, UK: CambridgeUniversity Press; 2006.

[39] Jin Z, Stamnes K. Radiative transfer in nonuniformly refracting layered media:atmosphere–ocean system. Appl Opt 1994;33(3):431–442.

[40] Yang P, Fu Q. Dependence of ice crystal optical properties on particle aspectratio. J Quant Spectrosc Radiat Transf 2009;110(14):1604–1614.

[41] Van de Hulst H. Scattering of light by small particles.Moscow: Izd-vo inostr.lit.; 1961.

[42] Fu Q. A new parameterization of an asymmetry factor of cirrus clouds forclimate models. J Atmos Sci 2007;64(11):4140–4150.

[43] Wiscombe W. The delta-m method: rapid yet accurate radiative flux calcu-lations for strongly asymmetric phase functions. J Atmos Sci 1977;34(9):1408–1422.

[44] Hu Y, Wielicki B, Lin B, Gibson G, Tsay S-C, Stamnes K, et al. δ-fit: a fast andaccurate treatment of particle scattering phase functions with weighted sin-gular-value decomposition least-squares fitting. J Quant Spectrosc RadiatTransf 2000;65(4):681–690.

[45] Zhai P-W, Hu Y, Trepte C, Lucker P. A vector radiative transfer model forcoupled atmosphere and ocean systems based on successive order of scat-tering method. Opt Express 2009;17(4):2057–2079.

[46] Stamnes K, Tsay S, Jayaweera K, Wiscombe W. Numerically stable algorithmfor discrete-ordinate-method radiative transfer in multiple scattering andemitting layered media. Appl Opt 1988;27(12):2502–2509.

[47] Stamnes S, Ou S. Radiative transfer calculations for cirrus clouds (cirrus aop-lut), ⟨http://lllab.phy.stevens.edu/cirrus/⟩ [accessed 30.06.2016].