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Backward Monte Carlo Calculations of the PolarizationCharacteristics of the Radiation Emerging fromSpherical-Shell Atmospheres

Dave G. Collins, Wolfram G. Blttner, Michael B. Wells, and Henry G. Horak

A Monte Carlo procedure, designated as FLASH, was developed for use in computing the intensity andpolarization of the radiation emerging from spherical-shell atmospheres and is especially useful for investi-gating the sunlit sky at twilight time. The procedure utilizes the backward Monte Carlo method and iscapable of computing the Stokes parameters for discrete directions at the receiver position. Both molec-ular and aerosol scattering are taken into account as well as ozone, aerosol, water vapor, and carbon dioxideabsorption within the atmosphere. The curvature of the light path due to the changing index of re-fraction with altitude is taken into account. Some comparisons are made between FLASH calculations fora pure Rayleigh atmosphere, a combined Rayleigh and aerosol atmosphere, and calculations reported byother authors for plane-parallel atmospheres. The comparisons show that the FLASH calculations forspherical-shell atmospheres are in good agreement with those for plane-parallel atmospheres within therange of zenith angles for which no differences could be attributed to the difference in the geometries of thetwo atmospheric models.

IntroductionDuring the past decade several Monte Carlo codes

have been developed for use in studying the effects ofmultiple scattering of light in plane-parallel atmospheres.Skumanich and Bhattachargiel used this method tocalculate time-dependent light scattering by a cloudlayer for a pulsed point source. Wells and Collins2

developed the LITE codes to study problems in atmo-spheric optics; these codes were later modified to in-clude the effects of water vapor and carbon dioxideabsorption3 and to allow for molecular and aerosolpolarization. Plass and Kattawar5 have also reportedMonte Carlo calculations that were obtained from acomputer code that is similar to that reported in Ref.2. Subsequently Kattawar and Plass6 extended theircode to allow for molecular and aerosol polarization.Kastner7 has reported Monte Carlo calculations ofresonance-line polarization of sodium D and hydrogenLyman-a in the upper atmosphere.

The Monte Carlo method is especially well adapted tosituations involving atmospheres whose propertieschange with altitude and for phase functions that arehighly anisotropic. The restriction to plane-parallel

The first three authors named are with Radiation ResearchAssociates, Inc., Fort Worth, Texas 76107; H. G. Horak is withthe University of California, Los Alamos Scientific Laboratory,Los Alamos, New Mexico 87544.

Received 7 February 1972.

geometry is not a limitation on the method but is donefor mathematical simplicity and to enable the com-puter runs to be of moderate duration. However, it isnecessary to resort to spherical geometry in order tosolve certain problems, e.g., the illumination of thesky at twilight or during a total solar eclipse or the ex-tension of the cusps of Venus at inferior conjunction.

One of the earliest reported applications of the MonteCarlo method to problems involving the scattering ofsunlight in a spherical-shell nonhomogeneous atmosphereis that reported by Marchuk and Mikhailov.5 Theseauthors used the Monte Carlo method to estimate theintensity of atmospheric scattered sunlight at a satellitepositioned above the atmosphere. Later, Collins andWells9 developed the FLASH Monte Carlo procedure thattreated the scattering of sunlight in a spherical-shellnonhomogeneous atmosphere. Their procedure usesthe backward Monte Carlo method and treats theeffects of polarization on the scattered intensity andthe absorption of light by water vapor, carbon dioxide,ozone, and aerosol particles in the atmosphere. FLASH

calculations of light scattering under twilight condi-tions are reported in Ref. 10. Subsequently, Kattawaret al."1 have modified their Monte Carlo procedure totreat the reflection of light by a symmetric planetaryatmosphere.

The application of the Monte Carlo method to spheri-cal-shell atmospheric models is not as simple as theapplication to plane-parallel models because of the in-terdependence of the position of entrance into the atmo-sphere and the angle of incidence at that position. The

2684 APPLIED OPTICS / Vol. 11, No. 11 / November 1972

net flux incident at the top of a spherical-shell atmo-sphere is not uniformly distributed. Therefore, in aforward Monte Carlo approach, it is necessary to sam-ple either a position of entrance for each photon historyor an angle of incidence for the photon history. Thisintroduces a problem that is not easily solved, viz., de-termining the bounds of the surface area within whichthe incident radiation significantly contributes to thescattered intensity at a given receiver location. Arelated question is whether some importance functionsshould be developed to concentrate the sampling of theentrance positions of the photon histories in those areasin which the incident radiation is most likely to scatterinto the receiver. Utilization of a backward MonteCarlo approach eliminates the necessity of having tosolve the above problems.

This paper describes a backward Monte Carlo pro-cedure, designated as FLASH, which computes theStokes parameters I, Q U, and V for a spherical-shellatmosphere. Comparisons are shown between FLASHcalculations for a Rayleigh atmosphere and a combinedRayleigh and aerosol atmosphere and data computed byother authors for plane-parallel atmospheres.

Backward Monte Carlo

We shall adopt the notation of Chandrasekhar"2 andcharacterize a beam of polarized light by the Stokesvector (I,I,,U,V), where 1 and r refer to two mutuallyorthogonal directions in the plane transverse to theoncoming light. It is convenient to define the propaga-tion plane as the plane containing the earth's center andthe direction of a light ray. The line of intersection ofthe propagation plane with the transverse plane at agiven point determines the direction 1, and the intensityobserved for an analyzer oriented in this direction is I;Ir is then the intensity for the analyzer oriented atright angles to the direction.

The sunlight incident on the top of the atmosphere isassumed to be unpolarized. The Stokes vector Ioused in FLASH to describe the characteristics of thesource radiation is defined by

Io = (II = 0.5, I = 0.5, U = 0.0, V = 0.0).

II(R) = fA If i ... fi JR - RnJ-2

X OT'P(Q. - Q)Tg(R. - R) T(R. - R)P(Qn. -Q

T(Rn-- R.)T(R., -l R.)... P(Q, - Q2 )T,(Ri - R2)T(R1 - R2)P(Q 0 - Q-)T-(Ro - R1) T(Ro - R1)

IOdrndQ._,drn-. ... d 2dr2d~2drjdA. (2)

A is the area of the source plane, Ro is a position on thesource plane, and Q0 is the direction of emission of thesource. The position of the nth collision is designatedby R., and the distance between the (n- 1)th collisionand the nth collision, R,- - R4, is r. The receiveris positioned at R, and the direction of incidence of thescattered radiation at the receiver is a. The parame-ter 0

T is the pseudoextinction coefficient of the atmo-sphere. It represents all the interactions undergone byphotons in the atmosphere except gaseous absorption.The scalar function T(R-, - R,,) describes the ab-sorption of photons by gaseous absorption along thedistance between the (n - 1)th and the nth collision, andthe scalar function T(Rn,-* R,,) describes the attenu-ation of the photons along the distance between the(n - 1)th and nth collision that results from scatteringinteractions and absorption interactions other thangaseous absorption. P(Qn_1 - an) is the phase matrixfor polarized scattering interactions. For atmospheresin which aerosol particle and Rayleigh scattering occur,the scattering matrix is given by

S(Qn - ) =AT

X [ Sp(Qn- On) + SR(Qn-1 .) (3)-o's O's

where o is the sum of the aerosol scattering coefficient,ap, and the Rayleigh scattering coefficient, R andSP(Qn-l -> -Qn), and f Q,)are the aerosolparticleand Rayleigh scattering matrices, respectively. Thedifference between O-T and s represents the absorptioncoefficient for all absorption processes other than gas-eous absorption.

The integrated Stokes's vector at the receiver is givenby

(1)

We shall define the integral over solid angle at thereceiver position of the Stokes vector I(R,Q), whichdescribes the characteristics of the polarized scatteredlight at position R in the direction Q by

II(R) = LI I(RQ)dQ.

Let us write in symbolic integral form the transportequation giving the integrated Stokes vector, I(R),for photons that have undergone exactly n collisionswithin the atmosphere, for the case where the source isa plane monodirectional, monochromatic source emit-ting one photon per unit area in the direction o.

(4)II(R) = E nn(R).n =1

Equation (2) can be evaluated by the use of MonteCarlo techniques by sampling a source position R, fromthe density function for the source area, A, and n dis-tances between collision, r, from the density function:

O.TT(R-l - R,) = 0JT exp(-crTRn - Rn|).

The type of scattering interaction at each of the n colli-sions can be determined at random from a knowledge ofthe Rayleigh and aerosol particle scattering coefficients.The directions after each of the first (n - 1) collisionscan then be selected from either the Rayleigh or aerosolparticle phase matrix, depending on the type of scatter-ing event selected for that collision. The directionfrom the nth collision to the receiver position is fixed

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2685

once the nth collision position is selected. An estimateof the integrated Stokes vector, In(R), is then givenby

EST = 7rpmax2R - R|J-2 ( S) exp(-rTjR - RJ)

X T(R. - R) [I T(Ri-i - Ri)0 P(Q. - Q)Io, (5)

where the phase matrix P' is taken as either Pp or PRdepending on the type of scattering event selected forthe nth collision, and Pmax is the largest radius on thesource plane for which any significant contribution toInI(R) can be expected.

The Stokes vector, I(R,Q), characterizing the polar-ization for scattered radiation at the receiver as a func-tion of direction can be obtained by accumulating theestimates for each of the polarization parameters, IIr,U,V, given by Eq. (5) into a set of solid angles definedabout the receiver position and then dividing the accu-mulated values by the size of the solid angles and thenumber of histories.

Equation (2) can be transformed into another integralequation in which the variables of integration definingthe source area, A, and the distance to the first collision,r1, are replaced with the variables Q and r defining thedirection of incidence at the receiver and the distancefrom the nth collision position to the receiver position.Applying such a transformation to Eq. (2) results in thefollowing equation:

(R) = f~ fl ... fn fi f f1 0TnP(n `-

X T(Rn - R)T,(Rn - R) P(Qn-l - n)T(Rn 1 - Rn)

X T(Rn- 1 - Rn) - P ...- Q2) T(R 1 - R2)

X T(R 1 - R2)P(Qo -. l)T(Ro - R1)Tg(R, - R1)1o

* dndrdrnd~ndr,_1dhn-* *. dr2d2. (6)

Equation (6) can be evaluated by Monte Carlo tech-niques by sampling (1) a direction of incidence at thereceiver, 5, from an isotropic distribution, (2) a distancer from the nth collision to the receiver from the density

-TT (Rn -- R), (3) n - 1 distances between collisions, I.,,from the density 0rTT(Rl, -- R), (4) the type of scat-tering event for each collision, and (5) n - 1 directionsafter collision from either PR or Pp depending on thetype of scattering event selected for each collision andthen evaluating the estimating function

EST2 = 4r (s Tg (Rn 1

X P'(Qo - Q,) exp(-JR - R,1)In (7)

for the sampled value. The value of Q, is determinedby a knowledge of Qo, RI, and Q2.

The phase matrix P(Qn 1 - Q,,) is a bivariate distri-bution in the polar and azimuthal scattering angles.Since it is not possible in the backward Monte Carlomethod to sample ,, directly from this distribution, amethod similar to that later used by Kattawar andPlass6 was developed for use in the FLASH program in

which the polar and azimuthal scattering angles areselected from a biased distribution. The Stokes vec-tor is then divided by this distribution function. Adiscussion of the sampling technique is given in the nextsection.

The Stokes vector for the scattered radiation thatreaches the receiver in the solid angle Q' after exactlyn collisions can be obtained from Eq. (6) by changingthe limits of integration over Q to correspond to thelimits of the solid angle Q'. When the limits for 0 and4 that define Q are changed in Eq. (6) to correspond tothe limits of a given solid angle interval (1,02 and 1,42),

the normalized density functions for 0 and are

sino/(cosl - COS02) (8)

and

(9)

respectively. The estimating function for Eq. (6) be-comes

EST3 = (EST2/47r)(cosOi - COS02)(02 - 1) (10)

An estimate of the Stokes vector for the scatteredradiation in the solid angle defined by 01,02,01,01 isobtained by dividing the estimate defined in Eq. (10)by the size of the solid angle. If the receiver solidangle is defined by a discrete direction, 0,4, the esti-mating function

EST4 = (EST2/47r) (11)

gives an estimate of the Stokes vector for the scatteredradiation at the receiver in the discrete direction 0,4 forphotons that have undergone exactly n collisions. Themethod described above for evaluating Eq. (6) is de-noted as the backward Monte Carlo method since thesampling of the photon trajectory for each history isstarted at the receiver position, and the histories aretraced backward along the random path.

From an examination of Eq. (2) it is seen that priorknowledge of the maximum size of the radius of thesource area is required in order to be able to select posi-tions of emission on the source plane. Also, the estimat-ing function of Eq. (2) as given by Eq. (5) has infinitevariance since JR - R0 -2 increases without limit as JR -RnJ approaches zero. It is not feasible to place limits onthe integration variables in Eq. (2) so as to computethe Stokes vector for the scattered radiation for afinite solid angle or a discrete direction at the receiver.

The use of the backward Monte Carlo method toevaluate the scattered intensity removes the necessityof determining the maximum radius of the source areathat needs to be considered and results in the use of anestimating function that has finite variance. In addi-tion, the backward Monte Carlo method allows limitsto be placed on the directions at which the photonsenter the receiver. The capability of computing thescattered intensity for a discrete direction at the receiverposition eliminates the argument advanced by Dave 3

and Hansen' 4 against the use of finite solid angle inter-vals in Mfonte Carlo calculations for recording the an-gular distribution of the polarization parameters at the


1/(02 - 1),

SOURCE PLANE

DETECTOR

Fig. 1. Geometry for spherical-shell atmosphere.

receiver. Although the random distribution of thecollision centers generated by the forward Monte Carloprograms developed by the authors, Plass and Katta-war, 5 and Kastner7 for plane-parallel atmospheres, donot allow one to control the direction of incidence at thereceiver position, it is possible to control the direction ofincidence at the receiver position in forward MonteCarlo programs for plane-parallel problems by usingstatistical estimation methods. Use of the backwardMonte Carlo approach in a spherical shell geometryallows all the photon histories to be concentrated in agiven field of view of particular interest at the receiverposition. Thus, no computer time is wasted in comput-ing the polarization parameters outside that field of view.

Flash Program

A computer program employing the backward MonteCarlo approach has been developed to aid in light trans-port studies involving spherical-shell geometries. Theprogram evaluates the four Stokes parameters, I,Q,U,V, characterizing the scattered radiation incident at areceiver position within each of a set of finite solid angleintervals (view angles); the size and direction of theview angles are arbitrary and may be limited to a set ofdiscrete directions if desired. The program assumesthat a plane-parallel unpolarized source of monochro-matic radiation is incident at the top of a spherical-shellatmosphere (Fig. 1). The receiver polar position angle,OD, between the radial through the receiver position andthe radial that is normal to the source plane, is equivalentto the zenith angle of incidence for a plane-parallelatmosphere.

The FLASH program treats the scattering of mono-chromatic radiation by molecules and aerosol particlesand the absorption of monochromatic radiation byaerosols, ozone, water vapor, and carbon dioxide in theatmosphere. The spherical-shell atmosphere is dividedinto concentric spherical-shell layers, each havingdifferent scattering and absorption coefficients. Thismethod for describing an atmosphere allows for arbi-trary and individual variation of each of the scatteringand absorption coefficients with altitude, which is essen-tial for twilight studies. In addition to the variationof the aerosol scattering coefficient with altitude, the

aerosol phase matrix may also be varied with altitude todefine the different angular scattering distributionsexpected in haze layers, clouds, and clear regions of theatmosphere.

Another feature of the program that enhances itsapplication to twilight conditions is the considerationof the bending of the photon paths by the change in theatmospheric index of refraction with altitude. Theindex of refraction is specified for each of the concentricspherical-shell layers, and the direction of the photon'spath is modified using Snell's formula at each crossingof a boundary between the spherical-shell layers.

The polarization parameters at the receivers arecomputed as a function of the ground albedo where thealbedo may vary from 0.0 to 1.0. The ground surfacemay be defined in FLASH as a Lambert type reflectingsurface or alternatively by an arbitrary reflection dis-tribution.

Water Vapor and CO2 AbsorptionProcesses such as aerosol scattering and absorption,

Rayleigh scattering, and ozone absorption are treatedseparately in FLASH from processes such as water vaporand carbon dioxide absorption which are dependent onthe amount of the gas present and the atmospherictemperature and pressure along the photon's path. Analtitude dependent pseudoextinction coefficient, o-T,which is taken to be the sum of the aerosol and Rayleighscattering coefficients and the ozone and aerosol ab-sorption coefficients, is used to compute optical dis-tances between collisions. The attenuation betweencollisions, in each of the concentric spherical layersthrough which the photon path passes that results fromwater vapor and carbon dioxide absorption is used toaccount for these processes. The transmission throughthe water vapor contained in each of the atmosphericlayers is based on the use of a transmission equationdeveloped by Howard et al. 15 The transmissionthrough the CO2 in each of the atmospheric layers iscomputed with a transmission equation developed byGreen and Griggs. 6

Selection of Scattering EventAt each collision in the atmosphere a scattering event

is forced to occur. The photon weight before collision ismultiplied by the ratio of the scattering coefficient tothe pseudoextinction coefficient at the collision altitudeto correct for the bias introduced by forcing all colli-sions to be scattering collisions. The type of scat-tering collision, Rayleigh or aerosol particle, is deter-mined at random from a knowledge of the ratio of theRayleigh scattering coefficient to the sum of the Ray-leigh and aerosol scattering coefficients at the collisionaltitude.

Optical Distance Between CollisionsIn the FLASH program optical distances are defined as

the integral of the pseudoextinction coefficient, T, overthe physical distance, taking into account the fact thatthe pseudoextinction coefficient varies with altitude.


Two methods are used to sample the distances be-tween the receiver and the nth collision point and thesuccessive collisions that occur along the photon's back-ward path. If the photon's extended path intersectsthe ground surface, the optical distance is sampled fromthe distribution,

r = -ln( - RN), (12)

where r is the optical distance between collisions andRN is a random number sampled from a uniform dis-tribution between 0.0 and 1.0. If the optical distancesampled is greater than the optical distance along thepath to the ground surface, a reflection is forced at thepoint where the photon path intersects the ground sur-face. The weight parameter is then multiplied by thealbedo for the ground surface. If the sampled path-length is less than the optical distance to the groundsurface, a collision is taken to occur at the point on thepath where the optical distance equals the sampledoptical distance. The photon is forced to scatter atthis point, and the photon weight is multiplied by theratio of the scattering-to-pseudoextinction coefficientevaluated for the collision altitude.

If the photon's extended path intersects the upperbound of the spherical shell atmosphere before inter-secting the ground surface, a collision is forced beforethe photon escapes the atmosphere by selecting theoptical distance to the next collision from the truncatedexponential distribution,

r = -n - N[1 - exp(-Tmax)J }, (13)

where max is the optical distance along the path to theupper bound of the atmosphere. When the collision isforced to occur before the photon escapes the atmo-sphere, the photon weight before collision, W, isadjusted after the collision to remove the bias introducedby forcing the collision with the expression

S(V) = (/o0T)[(oR/o'S)S(1P) + (/oas)SP(V1)] (16)

for atmospheric scattering and by

S(P) = SG(Vp) (17)

for ground reflection. The matrices SR(4l), S,(4,), andSG(il) are the scattering matrices for Rayleigh scatter-ing, aerosol particle scattering, and ground reflection,respectively. For Rayleigh scattering, SR(4) is givenby

cos2 0

SR(P) = 3 08,r 0

0

0 01 00 cost0 0

0

00

cost/

(18)

For aerosol particle scattering, Sp(tl) is given by

Sp( ) =

P2(P) 00 P1(#)0 00 0

00

P3(,P)P4(V,)

00

-P4() P3(Q)

(19)

where P1(4/), P2(t), P3(4,), and P4(J/) are the fourelements of the normalized aerosol phase matrix com-puted with Mie theory. When light reflects from aLambert surface, SG(,) is defined by

0.5SG () = cospt 0.5

00

0.5 0 00.5 0 00 0 00 0 0

(20)

The selection of a polar and azimuthal scattering anglefrom the phase matrix P(Qni_ - n) is accomplishedwith the use of a biased sampling scheme. Randomvalues of X,, and a,,, the polar and azimuthal scatteringangles at the nth collision, are sampled from the densityF(,/,a), where F(4l, a) is the phase function for unpolar-ized light. The Stokes vector after scattering is thengiven by

Wn = Wn-1[1 - exp (-rm.. )]-

A program option is available for systematically sam-pling the positions of the first order collisions and forusing the combined Rayleigh and aerosol phase func-tion to determine the Stokes parameters at the receiverresulting from first order scattering. Use of this optionminimizes the statistical fluctuations in the FLASH cal-culations since the Stokes' parameters for single orderscattering are calculated in a straightforward numericalmethod.

Scattering of Polarized Light

The bivariate phase matrix P(Qni_ - n) for polar-ized light is defined by the relation' 2

P(Qn1- )- 62n) = R(i,)(nR¢nl^ (15)

where R(O)nl) is the matrix for rotation of the Stokesvector from the plane of propagation before collisioninto the scattering plane, S(ql',,) is the scattering matrix,and R(4)n,2) is the matrix for rotation of the Stokesvector from the scattering plane to the plane of propa-gation after scattering. The scattering matrix SM isgiven by

where

C,.2 - a.)

Cn = 1F(V,..ce) (22)

and Il is the Stokes vector in the reference planebefore the nth collision. The rotation angle On,, iscalculated from a knowledge of the direction of propaga-tion in the reference plane after scattering and thescattering angles l/n and an.

The density function F(l, a) is defined in FLASH forRayleigh scattering by the Rayleigh phase function

F(P,cz) = (3/16r)(1 + cos'4k) (23)

for unpolarized light. For aerosol scattering F(V,,a) isdefined by the normalized aerosol particle phase func-tion for unpolarized light as given by

F(+,~) = [Pl(4,) + P2(4)]/2, (24)

and for reflection by a Lambert surface F(Vl,a) is definedby

F(,p,a) = cos/ir. (25)


(14) In = CnR(0n2)S(n)R(0n,1)In-1, (21)

The correction factor C, for removing the bias intro-duced in the sampling scheme outlined above is

Cn = 1/F(knan) (26)

where An, and a,, are the polar and azimuthal scatteringangles selected at random from the density functionF(4l,a). The estimating function used in the FLASHprocedure to determine the Stokes vector for the jthhistory in the discrete direction at the receiver forphotons that have undergone exactly n collisions isgiven by

Ijn f Wi] T(Rn R) [f[ Tg(Rn-, . Rn)]

X exp(-0-TR - R) [ Ci P(Q-n1- D)i =2

X P(an-2 -* Qn-1)- .. P(Q1 I Q2)P(Q0 - Q)l0. (27)

The parameter W is a weight correction factor toaccount for the use of biased sampling in selecting theoptical distance to collision and to account for forcingthe interaction at the collision to be a scattering event.When the photon's direction of motion after the(n- 1)th collision is upward, the parameter Wn is givenby

Wn = (s/IOT)[1 - exp(-rma)],

where max is the optical distance along the path to theupper bound of the atmosphere. When the photondirection of motion after the (n - 1)th collision is down-ward and the nth collision position is within the at-mosphere, W,, is given by W = S/rT. If the photonhits the ground surface, W = A, where A is the magni-tude of the ground albedo. The phase matrixP( Qn-i 3- C) used in Eq. (27) for each of the n ordersof scattering is the phase matrix for either Rayleighscattering, aerosol scattering, or ground reflection,depending on which type of scattering event was se-lected for the nth collision. It should be noted thatthe parameters giving the collision positions as computedfor the nth collision for n > 1 become the parametersfor the (n + 1)th collision when the above estimatoris used to compute I,,n+,. This results from the factthat after each estimate is made a new position for thefirst collision and a new direction of motion before thesecond collision are selected at random. The positionof the first collision for the previous estimate becomesthe position of the second collision. The position ofthe second collision for the previous estimate becomesthe collision position for the third collision etc.

All the quantities in Eq. (27) are scalars except thematrix product

I = P(a-l Q)P(Qn-2 Qn-1)-..P(Q2 2)

X P - Q)Io, (28)

which is a vector. The product vector is obtained inFLASH by multiplying the P matrices in Eq. (28) fromthe left to the right. The matrix resulting from theproduct of the P matrices is then multiplied by thevector Io.

* The estimator functions for both the forward andbackward Monte Carlo methods when the densityfunction F(4ta) is used to select the polar and azi-muthal scattering angles at each collision will both con-tain the matrix product given by Eq. (28). From anexamination of Eq. (28) it is seen that the order inwhich the polar and azimuthal scattering angles wereselected is not important as long as Eq. (28) is evaluatedin the manner described above. This results from thefact that the matrix product is associative.

When the Stokes vector is computed by FLASH forsingle scattering and the option for systematicallysampling of the first order collision positions is used,the estimator for the jth history is

la, = Ls exp[-7(!R - R1 + R - R)

X T(Ro RI)T,(RI - R)P(Qo Q )o (29)

where L is the distance through the atmosphere alongthe direction from the receiver position and P(Qo -LI) is defined by Eq. (15). Distances r between thefirst collision and the receiver are selected systematicallyfrom the density L-l.

The Stokes vector for the scattered radiation in thedirection at the receiver position is obtained from theestimates [Eq. (27) ] by use of the relation

I(Il'i,7 UV) = e E E Ijn(IIIrUV)j

The Stokes's parameters I and Q are then given by

I = II + I,

Q = I - II.

(30)

(31)

(32)

The degree of polarization (%) is computed with theexpression

a = 100(Q 2+ U

2+ V2)1/I, (33)

and the orientation of the plane of polarization is com-puted with the expression

x = 0.5 tan-1 (U/-Q). (34)

It should be. noted that the above expression for differsfrom the expression

a = 100Q/I, (35)

as used for certain purposes by Plass and Kattawar.17

As pointed out by Dave,13 serious errors can result ifone confuses the degree of polarization given by the useof Eq. (33) with that given by the use of Eq. (35) whenthe direction of propagation is not parallel or perpen-dicular to the plane containing the direction of the scat-tered radiation. At receiver azimuthal angles of =300 and 1500, for example, the values of Q and U areof the same magnitude (see results reported in a latersection of this paper). Use of Eq. (33) would thereforegive values of the degree of polarization that are ap-proximately 40% higher than those obtained using Eq.(35).


0.7 t r =ucso0 23.1'

0 6 A FLASH, A = 0.80

.\ * FLASH, A= 0.25 10.5 * FLASH, A= 0.00

0 \4 COULSON, et al

0.2

0

800 60' 40 20' 00 200 40 600 800,6=150° =30°

POLAR ANGLE ()

Fig. 2. Scattered intensity transmitted through a Rayleighatmosphere with an optical thickness of 0.25 as a function of the,

receiver zenith angle: OD = 23.1°, q = 300 and 1500.

Program EfficiencyThe FLASH program is written in FORTRAN IV and

has been compiled and executed on the CDC-6600computer. The execution time is highly dependentupon the number of spherical-shell atmospheric layersused to define the atmosphere and on the total numberof photon histories run. For the homogeneous Ray-leigh atmosphere with an optical thickness of 0.25,calculations of the polarization parameters for sixty-four different receiver directions of view at each of tworeceiver positions were performed in approximately 20min of CDC-6600 computer time. Each of the prob-lems run to calculate the transmitted or reflected polar-ization parameters were for 12,800 histories. Thisrepresented 200 histories per receiver view angle.(One history provides an estimate of the polarizationparameters incident at a given direction for all receiverpositions being considered.) The polarization param-eters for up to ten ground albedos may be computedsimultaneously with effectively no increase in computertime.

Calculations for a Rayleigh AtmosphereSeveral calculations of the polarization parameters

for the scattered radiation transmitted and reflectedfrom a spherical-shell Rayleigh atmosphere have beenmade with the FLASH program. Comparison of theresults of these calculations with those obtained withplane-parallel atmospheric models' 8"19 reveals the abilityof the program to calculate accurately the transmittedand reflected polarization parameters and indicates thedifferences one would expect to find between calcula-tions for these two atmospheric models.

Receivers were placed at the bottom and top of anhomogeneous molecular atmosphere having an opticalthickness of = 0.25. The ground surface was as-sumed to be a Lambert reflector. Calculations wereperformed for receiver positions of O = 23.1° and84.30 (receiver position in the spherical-shell atmo-

spheric model corresponds to zenith angle of incidencein the plane-parallel models). The scattered polariza-tion parameters were computed as a function of thereceiver zenith angle for receiver azimuthal directionsof = 00, 300, 150°, and 180°, and the ground albedoA. The azimuthal angle toward the incident radiationwas taken to be 00. Two hundred photon historieswere run for each receiver view angle.

The polarization parameters I, Q, and U, as plottedin the following figures, are defined having a dimensionof intensity or radiance (photons cm-2 sr-') with theincident flux density taken as one (photon cm-2 ) normalto the direction of incidence. The calculated quantitiesare given as a function of the receiver zenith angle, 0,and the receiver azimuthal angle, . The 00 azimuthdirection is contained in the plane of the incidentradiation. The zenith angle for the transmitted radi-ation varies between 00 and 900. For the reflectedradiation, the receiver zenith angle varies between 900and 1800.

Intensity CalculationsCalculations of the receiver zenith angle distribution

of the transmitted scattered intensities in the 300 and1500 azimuthal planes for several values of the groundalbedo are compared in Figs. 2 and 3 with datafor a plane-parallel Rayleigh atmosphere by Coulsonet al. 8 for incident angles of OD = 23.10 and SD = 84.30,respectively. The transmitted scattered intensitiesin the 00 and 1800 azimuthal plane for ground albedosof 0.0 and 0.8 are compared in Fig. 4 with the datacomputed by Coulson et al. for an incident zenith angleof 84.30. For comparison purposes, smooth curveshave been drawn through Coulson's et al. data for aplane-parallel atmosphere, while the calculations forthe spherical-shell geometry are shown as point data.The agreement with the data for a plane-parallel atmo-sphere is excellent when OD = 23.10, and indicates that,

015

010$t4

0.05

0.00800 600 40 200 0 200 400 600

=150° 0=300

POLAR ANGLE ()

800

Fig. 3. Scattered intensity transmitted through a Rayleighatmosphere with an optical thickness of 0.25 as a function of the

receiver zenith angle: OD = 84.3°, 0 = 300 and 1500.


0.18 , -7-;8D =84.3°

0.16 - A FLASH, A= 0.8 A

* FLASH, A= 0.00.14 -COULSON, et al A

0.12

''0.0 A

0.08~~~~0.

0.06

0.02

0.0080° 60° 40° 20° 00 20° 400 600 80°

0=180° 0=0.

POLAR ANGLE (8)

Fig. 4. Scattered intensity transmitted through a Rayleighatmosphere with an optical thickness of 0.25 as a function of the

receiver zenith angle: D = 84.30, = 0 and 1800.

0.7

0.6

0.5

-I 0.4~

0.3

0.2

0.1

, zO.25 AA'0.80 82 *23.1 A A :

A A FLASH, A'0.800 FLASH, A 0.0

COULSON, et i

-0

0 n 0I AAO.0 I

9_0 120° 50 180° 150° 1200

#0180O POLAR ANGLE () #000

9_0

Fig. 5. Intensity reflected from a Rayleigh atmosphere with anoptical thickness of 0.25 as a function of the receiver zenith angle:

GD = 23.10, - = O° and 1800.

except for receiver zenith angles close to 900, there isvery little difference between the scattered intensitiestransmitted through the plane-parallel and spherical-shell atmospheres for that angle of incidence.

Good agreement between the transmitted scatteredintensities for the plane-parallel and the spherical-shellatmospheric models have been obtained for other azi-muthal planes and other values of D less than 700.

A difference in the transmitted scattered intensitiescomputed for the plane-parallel and spherical-shellatmospheric models is noted in Figs. 3 and 4, for radia-tion incident at D = 84.3°. The scattered intensitiestransmitted through the spherical shell atmosphere arehigher at all receiver zenith angles, except for thosegreater than 700 in the 1500 and 1800 azimuthal planes.As seen in Fig. 4, the most significant differences be-tween the results for the two atmospheric models occur

at receiver zenith angles between 800 and 900 in the 00azimuthal plane. The scattered intensities computedfor these angles in the spherical-shell atmosphere exceedthose computed by Coulson et al. for the plane-parallelatmosphere by more than 40%. Bearing in mind thedifferences in the plane-parallel and spherical-shellgeometries, one would expect the differences in thetransmitted scattered intensities to be larger in the 00azimuthal plane when the receiver zenith angle ap-proaches 90°.

The receiver zenith angle distribution of the intensityreflected in the 00, 300, 1500, and 1800 azimuthal planesfrom 0.25 optically thick homogeneous plane-paralleland spherical-shell Rayleigh atmospheres for groundalbedos of 0.0 and 0.8 are compared in Figs. 5-7 for

0.40

0.36A

0.32 -0.25

0.28 A FLASH, A0.80 A* FLASH, A 0.0

- COULSON, et al . A024

0.20

0 0.80 10.16 A

0.12 A

AA

008 * A AA AAA

0.04 0

01900 1200 150' 1800 1500 120' 900

0 8.POLAR ANGLE .(9) 00

Fig. 6. Intensity reflected from a Rayleigh atmosphere with anoptical thickness of 0.25 as a function of the rec eiver zenith angle:

OD = 84.30, k=Oo and 1800.

Hto

90° 120° 150° 1800 15000 = 150°

POLAR ANGLE ( )

120° 90°

0 = 30°

Fig. 7. Intensity reflected from a Rayleigh atmosphere with anoptical thickness of 0.25 as a function of the receiver zenith angle:

OD = 84.3°, ( = 30° and 1500.


(IB ,

80° 60° 40° 20° O' 20° 40° 60° 80°0-180° o 0

POLAR ANGLE (8)

Rayleigh atmosphere have also been compared withcorresponding parameters computed for the plane-parallel atmosphere by Coulson et al. For a zenithangle of incidence of D = 23.10, good agreement wasobtained between the two different calculations of theparameters Q and U for the transmitted and reflectedradiation, except for receiver zenith and receiver nadirangles, respectively, near 900.

Figures 8 and 9 show comparisons of the parameter Qfor the transmitted scattered radiation when the sourceradiation is incident at 84.30 to the top of the two atmo-spheric models and the ground albedo was either 0.0or 0.8. The parameter Q for the transmitted scatteredradiation in the 00 to 1800 azimuthal plane, as shownin Fig. 8 for the spherical-shell atmosphere, appears

Fig. 8. Parameter Q for the scattered intensity transmittedthrough a Rayleigh atmosphere with an optical thickness of 0.25as a function of the receiver zenith angle: D = 84.30,4) = 00 and

1800.

-0.01 .AZ44A £~~

-0.02 L L80° 60° 40° 20° O' 20° 40° 60° 80°

#=1500 #=300

POLAR ANGLE ()

Fig. 10. Parameter U for the scattered intensity transmitted

through a Rayleigh atmosphere with an optical thickness of 0.25

as a function of the receiver zenith angle: D = 84.30, 4 = 30°

and 150°.

100Fig. 9. Parameter Q for the scattered intensity transmittedthrough a Rayleigh atmosphere with an optical thickness of 0.25as a function of the receiver zenith angle: OD = 84.30, 0 = 300

and 1500. 80

60

40

incident zenith angles of 23.1° and 84.30. As notedpreviously for the transmitted scattered intensities, thereflected intensities for the two atmospheric modelsagree reasonably well for an incident zenith angle ofOD = 23.10 (see Fig. 5), except for receiver zenith anglesnear 900 where the intensities reflected from the spheri-cal-shell atmosphere underpredict those reflected fromthe plane-parallel atmosphere. The reflected intensityfor the radiation incident upon the spherical-shell atmo-sphere at a zenith angle of 84.30 tends to overpredictthat reflected from the plane-parallel model (Fig. 6,for the 00, 1800 azimuthal plane and Fig. 7 for the 300and 1500 azimuthal planes), particularly for receiverdirections oriented toward the sun.

Parameters 0 and U

Values of the Stokes parameters Q and U computedfor the 0.25 optically thick spherical-shell homogeneous

20

080° 60° 40° 20° 01 20° 400 600

0= 150° #=300

POLAR ANGLE (0)

800

Fig. 11. Percent of polarization of the scattered intensity trans-mitted through a Rayleigh atmosphere with an optical thicknessof 0.25 as a function of the receiver zenith angle: OD = 23.1°,

4 = 30 ° and 1500.


002

0.01

0.00a

-001

-0.02

0.02

0.01

002

0 01

000

ato -0.01

-0.02

-0.03

to 0.00

80° 60° 40° 20° O 20 40° 60° 80°= 150° #=300

POLAR ANGLE (8)

Ze

" 60 L

50

40 -A =O.8

30

20 A

10 I l l l 80° 600 400 200 O° 200 400 600 800

#0150. #=300

POLAR ANGLE (8)

Fig. 12. Percent of polarization of the scattered intensity trans-mitted through a Rayleigh atmosphere with an optical thicknessof 0.25 as a function of the receiver zenith angle: D = 84.3°, 4 =

30° and 1500.

to be slightly larger in absolute value than the param-eter Q for the plane-parallel atmosphere. However,the transmitted scattered parameter Q in the 300 and1500 azimuthal planes is seen in Fig. 9 tobe in betteragreement with the parameter Q for the plane-parallelatmosphere at all receiver zenith angles, except forthose zenith angles close to 900. It should be notedthat the parameter Q as given in Figs. 8 and 9, is in-dependent of the magnitude of the ground albedo.

The transmitted scattered parameter U in the 0-180°azimuthal plane for radiation incident to the spherical-shell atmosphere at zenith angles of D = 23.10 and84.30 was computed to be approximately zero for allreceiver zenith angles. This agrees favorably with thevalue of zero for the parameter U in the 0-180° azi-muthal plane as given by the plane-parallel atmospheredata reported by Coulson et al. The transmitted scat-tered parameter U in the 300 and 1500 azimuthalplanes that is shown in Fig. 10 when the source radia-tion is incident at a zenith angle of 84.30 to the spheri-cal-shell atmosphere appears to be slightly larger inabsolute value than the corresponding parameter Ufor the plane-parallel atmosphere. It is seen in Fig. 10that the magnitude of the parameter U is independentof the value of the ground albedo.

Degree and Plane of Polarization

Comparisons of calculations of the percent of pola-rization, , of the scattered transmitted radiation in the300 and 1500 azimuthal planes when 6 = 23.10 and84.3° for the spherical-shell and plane-parallel atmo-spheres are shown in Figs. 11 and 12 for several valuesof the ground albedo. Except for the statistical fluc-tuation in the percent of polarization data shown in

Fig. 11 for the spherical-shell atmosphere when theground albedo was 0.8, the agreement with theplane-parallel atmospheric data for an incident zenithangle of D' = 23.10 is excellent. It appears that thepercent of polarization of the transmitted radiation isnot as highly affected by the differences in the spherical-shell and plane-parallel atmosphere as were the intensi-ties and the parameters Q and U when the angle ofincidence is 84.30 (see Fig. 12). Differences in thepercent of polarization near its maximum value areattributed to the statistical fluctuation of the MonteCarlo results rather than to the difference in the atmo-spheric geometries.

Comparisons of data giving the angle x between theplane of polarization and the vertical plane in the ap-propriate azimuth for the transmitted scattered radia-tion in the 300 and 1500 azimuthal planes for the twoatmospheric geometries when the ground albedo is 0.0and 0.8 are shown in Fig. 13 for radiation incident atG = 84.30. The agreement between the data for thetwo atmospheric models is excellent, and thus, it ap-pears that the atmospheric geometry has little influenceon the values of the angle X for all incident zenithangles.

The FLASH calculations of the plane of polarizationin the 0 and 1800 azimuthal plane were erratic, alter-nating between approximately 0 and 900. This is dueto the fact that the values computed for the parameterU, while insignificant in absolute value, fluctuatedabout 0, and therefore, the ATAN2(U, - Q) functionused in FLASH for computing the plane of polarizationgave values of approximately 0 where U/ - Q wasslightly greater than 0 and values of approximately rwhen U/ - Q was slightly less than 0.

It should be pointed out that the FLASH calculationsfor the spherical-shell atmospheric model give the op-posite sign of the angle X than that reported by Coulsonet al.'6 Using the equation given in Ref. 18 for calcu-

I I I I -

_, = 0.25

8D= 84.3°A FLASH, Ao.8* FLASH, Ao.0

- COULSON, et al30

0

-30 _

600 80800 60 400 200 O' 200 40°

0 = 150's #=30°

POLAR ANGLE ()

90

60

-60

Fig. 13. Plane of polarization for the scattered intensity trans-mitted through a Rayleigh atmosphere with an optical thicknessof 0.25 as a function of the receiver zenith angle: D = 84.30,

4 = 30° and 150°.


I I I I

I I �

I I - I I I

0.7,

0.6

0.51

0.4

.31

021

0.

90 6OLA 30 A 00 300 60

#0-1500 POLAR ANGLE (9 #0300

_0

Fig. 14. Scattered intensity transmitted through a model at-mosphere with a turbidity of 2 as a function of the receiver

zenith angle: D = 36.90, 0 = 30° and 1500.

0.6h

0.5

0.4

'0

0.3

0.2

0.I

~~~~~~Ij/ A 0.80 \"

A FLASH *j

i' S ESCHELBACH

A Ai

* t ~ a . . . 4 .AA0 .0 i.

10 120° 50 180 150° 120° 90°

#.150° - .. - - #0300

Fig. 15. Intensity reflected from a model atmosphere with aturbidity of 2 as a function of the receiver zenith angle: D =

36.9°, 4) = 300 and 1500.

lating the angle X and the calculated values for theparameters U and Q, the FORTRAN statement

CHI = 0.5*ATAN2(U,-Q)*180.0/PI,

where PI = 'r, will also give a sign of the angle X (CHI)opposite to that reported in Ref. 18.

Calculations for an Atmosphere ContainingBoth Molecules and Aerosol Particles

The FLASH program was also used to calculate polar-ization data for a spherical-shell atmosphere containingboth molecules and aerosol particles. The atmospherewas assumed to be homogeneous with a turbidity factorof T = 2, i.e., the total optical thickness of the atmo-sphere was taken to be twice that of a Rayleigh atmo-sphere for 0.55-A wavelength light.

The size distribution of the aerosol particles was as-sumed to be described by dN(r )/dr = 7-4 with limitingradii of 0.04 jA and 10.0 A. The index of refraction forthe aerosol particles was taken to be 1.50-0.OOi, thewavelength of the source was chosen to be 0.55 A.Since the optical thickness for a Rayleigh atmosphereat that wavelength is 0.09874, the total optical thick-

ness of the atmosphere was taken to be 0.19748. Cal-culations were performed for a zenith angle of the inci-dent light of 36.90. Receivers were placed at the bot-tom and the top of the atmosphere for recording thetransmitted scattered and reflected radiation in the30° and 1500 azimuthal planes for several receiverzenith directions; 350 histories were run for each re-ceiver view angle.

The results calculated for the spherical-shell at-mosphere with a turbidity of T = 2 have been com-pared with results reported by Eschelbach' 9 for aplane-parallel atmosphere using the same aerosolscattering data. Eschelbach employed a modifiedversion of the method reported by Herman,' 0 wherethe mathematical approach is derived from the integralform of the equation of radiative transfer as given byChandrasekhar,2 which, without any circuitous seriesdevelopment, is directly transformed into a system oflinear equations for the upward and downward radia-tion.

Figures 14 and 15 show the transmitted and reflectedscattered intensity in the 30° and 1500 azimuthal planesas a function of the receiver zenith angle for groundalbedo values of 0.0 and 0.8. The circled points are

a

U.Ub *I

A\. A FLASH IA0.05 A 0 ESCHELBACH I

0.04 -\ .

0.03 -h

0.02 - 1 _a \ A - .0

Q01 a\ i QOV '0~~~~.\A'A .

0- _ Asa a

-0.01 I I ._ n ,

90° 60°50 0 PO L 309

0=5. POLAR ANGLE (9)

60'

0 s30.

90'

Fig. 16. Parameter Q for the scattered intensity transmittedthrough a model atmosphere with a turbidity of 2 as a function of

the receiver zenith angle: OD = 36.90, 4) = 30° and 1500.

I.

0.06 I I IA FLASH I

0.05A * ESCHELBACH I!

0.04 \ 2

0.03 / X

10.02- \

0.01- -,,,A=0.0 4

n~~~ ~~ At

0 . A_ ,

-00'~ ~ ~ ~ A-00 60 30 00 30 090° 60° 30'

+ 150 _

0° 30 # 603

POLAI- AN(.+ L30W

90'

Fig. 17. Parameter U for. the scattered intensity transmittedthrough a model atmosphere with a turbidity of 2 as a function of

the receiver zenith angle: GD = 36.9°, 4) = 300 and 1500.


a

A FLASH A

* ESGHELBACH l

A A

A e ~ -\ ./:I_ -*/

a, *o<s0.80 ,AA/A-O.0

- * ^ *.A- AS

I

n, Io __

rM 71 l

/^11 ! I _

I

u1.r

POLA AILL tU0

POLAR ANGLELMe

6C

4C

2'

20

0

JO I I I

A FLASHO _ as 11 ESCHELBACH

2 1 a~~~~~~~~~~~~

°I \Ar -0.0 .0,

A=.8 10,~ \o S S a

-"-

a.~~~~~~~~~~~~~~-00

90' 60' 30' 0' 30' 60°

#0150' POLAR ANGLE (9) 0300

90

Fig. 18. Percentage of polarization of the scattered intensitytransmitted through a model atmosphere with a turbidity of 2as a function of the receiver zenith angle: D = 36.9°, ) = 300

and 1500.

80

60 A FLASH I 2a ESCHELBACH /

~40/20 '\A.0 A

*r*.* m o no A~ rs 11 Ao.800

VUL IU- ILU- Mu IDLE lie

#~ 150' POLAR ANGLE () #0300

90'

Fig. 19. Percentage of polarization of the intensity reflectedfrom a model atmosphere with a turbidity of 2 as a functionof the receiver zenith angle: D = 36.9°, = 300 and 1500.

the results computed by Eschelbach for the plane-parallel atmospheric model, and the triangles give thedata computed for the spherical-shell atmosphericmodel. The agreement between the two calculations isexcellent, except for receiver zenith angles close to 900where the spherical-shell model seems to underpredictslightly the data for the plane-parallel model. (Betterstatistics were obtained for the combined Rayleigh andaerosol atmosphere than for the pure Rayleigh atmo-sphere due to the larger sample size of 350 histories perreceiver view angle.)

Figures 16 and 17 show a comparison of the param-eters Q and U for the scattered radiation transmittedthrough the plane-parallel and spherical-shell atmo-spheres for a ground albedo of 0.0. Again an excellentagreement was observed. Similar to the calculatedintensity data, the parameters Q and U as computedfor the spherical-shell model seem to underpredictslightly the data obtained for the plane-parallel modelfor receiver zenith angles near 900. Similar comparisonwas found for an albedo of A = 0.8. The parametersQ and U as calculated for both atmosphere geometrieswere found to be nearly independent of the groundalbedo. For clarity, the points for A = 0.8 were there-fore not plotted in Figs. 16 and 17.

No favorable agreement was found for the parameterV as computed for the two atmospheric models. WhereEschelbach 9 claims that his values for the parameter

V are at least of the correct magnitude, the FLASHdata show large statistical variations about Eschel-bach's data. In the 300 and 1500 azimuthal planes theparameter V, however, is very small when comparedwith the parameters Q and U, and therefore no statis-tically better data for v can be expected using MonteCarlo methods.

Excellent agreement was obtained between theFLASH data and Eschelbach's calculations of the per-centage of polarization (Fig. 18 for the transmitted andFig. 19 for the reflected radiation). There seems to beno apparent difference in the percentage of polarizationwhen calculated for the spherical-shell and plane-parallel models for a zenith angle of incidence of 36.90.

Remarks

Additional comparisons between the results reportedby Coulson et al.'8 and Eschelbach 9 and FLASH cal-culations have been made for incident angles otherthan those reported in this paper and for a molecularatmosphere with an optical thickness of 1.0 and atmo-spheres with turbidities of 4 and 6. These comparisonsalso showed good agreement for angles of incidenceless than 800.

The FLASH calculations show that orders of scatteringgreater than 1 contribute significantly to the sky radia-tion. The relative importance of multiple scattering is,however, dependent upon several parameters, i.e.,the receiver polar position angle, D, the optical thick-ness of the atmosphere, the ground albedo, and thecoordinates of the receiver view angle. A detaileddescription of the influence of multiple scattering on thesky radiation in a molecular atmosphere is given by deBary and Bullrich2l and by Dave.22 For the atmosphericmodels considered in this paper, the ratio of the total tothe single scattered intensities varies between 1.1 and1.9 for a ground albedo of 0.0 and between 1.3 and 5.3for a ground albedo of 0.8.

Conclusions

The good agreement found between the polarizationcalculations obtained with the FLASH program and thedata for plane-parallel atmospheres reported in Refs.18 and 19 for zenith angles of incidence D = 23.10 andOD = 36.90, indicates that the FLASH program correctlyaccounts for the polarization of atmospheric scatteredlight. Where one may be doubtful about the accuracyof polarization calculations averaged over rather largeangle increments, i.e., 300 over azimuth (as used inRefs. 4 and 17), the utilization of the backward MonteCarlo method allows one to compute polarization datafor discrete directions at a receiver position.

It appears, from the comparisons shown in thispaper, that the parameters I,Q, and U at low sunelevations are more dependent upon the geometricmodel being used than are the degree and plane ofpolarization. The most significant differences betweenthe polarization parameters computed for the twoatmospheric odels occur for receiver view anglesoriented near the sun's direction.


FlE

It should be pointed out that the differences in thediffuse transmitted and reflected polarization param-eters for plane-parallel and spherical-shell atmospheresshown in this paper would have been smaller if realisticatmospheric models had been considered. That is,if the scattering properties of the atmospheres had beendefined to decrease with altitude, the differences be-tween the results for the two atmospheric models wouldbe expected to be smaller for the low sun elevations.Such being the case, however, the FLASH program stillprovides a means for identifying the limitations ofplane-parallel atmospheric models.

The FLASH program, therefore, provides a computa-tion tool for performing studies of the transmittanceand reflectance of twilight radiation from realisticspherical-shell atmospheric models and will extendcomputations of the sky radiation to the twilight sky.The program is currently being used in studies ofthe effect of dust layers upon the color ratios computedfor various sun depression angles.' 0

The flexibility of the program also allows the place-ment of receivers above the spherical shell atmosphereand therefore will prove a valuable method for anoptical analysis of planetary atmospheres and for theinterpretation of data taken by satellites and otherspacecraft.

The backward Monte Carlo approach is not limitedto spherical-shell geometries or to plane-parallel sources.Programs2 3 similar to the FLASH program have beendeveloped to treat both point and volume sources.

The work described in this paper was partially sup-ported by the U.S. Atomic Energy Commission and theAir Force Systems Command.

References1. A. Skumanich and K. Bhattachargie, J. Opt. Soc. Am. 51,

484 (1961).

2. D. G. Collins and M. B. Wells, Monte Carlo Codes for Study ofLight Transport in the Atmosphere, Report RRA-T54 (Radia-tion Research Associates, Inc., Fort Worth, Texas, 1965),Vols. 1 and 2.

3. D. G. Collins, K. Cunningham, and M. B. Wells, Monte CarloStudies of Light Transport, Report RRA-T74 (Radiation Re-search Associates, Inc., Fort Worth, Texas, 1967).

4. D. G. Collins, J. Opt. Soc. Am. 57, 1423 (1967); also se6 D. G.Collins, Report RRA-T86 (Radiation Research Associates,Inc., Fort Worth, Texas, 1968).

5. G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 361 (1968).

6. G. W. Kattawar and G. N. Plass, Appl. Opt. 7, 1519 (1968).

7. S. 0. Kastner, J. Quant. Spectrosc. Rad. Transfer 6, 317(1966).

8. G. I. Marchuk and G. A. Mikhailov, Bull. Acad. Sci. USSR,Atmos. Oceanic Phys. Ser. 3, No. 2 (1967).

9. D. C. Collins and M. B. Wells, FLASH, a Monte Carlo Procedurefor Use in Calculating Light Scattering in a Spherical-ShellAtmosphere, Report RRA-T704 (Radiation Research Asso-ciates, Inc., Fort Worth, Texas, 1970).

10. W. G. Blittner, D. G. Collins, and M. B. Wells, Monte CarloCalculations in Spherical-Shell Atmospheres, Report RRA-T7104 (Radiation Research Associates, Inc., Fort Worth,Texas, 1971).

11. G. W. Kattawar, G. N. Plass, and C. N. Adams, Astrophys.J. 170, 371 (1971).

12. S. Chandrasekhar, Radiative Transfer (Clarendon Press,Oxford, 1950).

13. J. V. Dave, Appl. Opt. 9, 2673 (1970).

14. J. E. Hansen, J. Atmos. Sci. 28,123 (1971).

15. J. N. Howard, D. E. Burch, and D. Williams, Sci. Rept. 1,GRD, AFCRL (Ohio State University Research Foundation,1954).

16. A. E. S. Green and M. Griggs, Appl. Opt. 6, 561 (1963).

17. G. N. Plass and G. W. Kattawar, Appl. Opt. 9, 1122 (1970).

18. K. L. Coulson, J. V. Dave, and Z. Sekera, Tables Related toRadiation Emerging from a Planetary Atmosphere with Ray-leigh Scattering (University of California Press, Berkeley,1960).

19. G. Eschelbach, private communication (University of Mainz,1971). The method is described in: K. Bullrich, R. Eiden,G. Eschelbach, K. Fischer, G. Hainel, K. Heger, H. Scholl-mayer, and G. Steinhorst, Research on Atmospheric RadiationTransmission, University of Mainz, Final Report, AFCRL-69-0266 (January, 1969).

20. B. M. Herman, J. Geophys. Res. 70, 1212 (1965).

21. E. de Bary and K. Bullrich, J. Opt. Soc. Am. 54, 1413 (1964).

22. Jw V. Dave, J. Opt. Soc. Am. 54, 307 (1964).

23. D. G. Collins and M. B. Wells, Computer Procedures for Cal-culating Time Dependent Light Scattering in Spherical-ShellAtmospheres, Report RRA-T7017 (Radiation Research As-sociates, Inc., Fort Worth, Texas, 1970).


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