numerical technique for solving the radiative transfer ...vijay/papers/rt models... · efficiency,...

Numerical technique for solving the radiativetransfer equation for a spherical shell atmosphere

B. M. Herman, A. Ben-David, and K. J. Thome

A method for numerically solving the equation of radiative transfer in a spherical shell atmosphere ispresented. The method uses a conical boundary and a Gauss-Seidel iteration scheme to solve for allorders of scattering along a single radial line in the atmosphere. Tests of the model indicate an accuracybetter than 1% for most Earth-atmosphere situations. Results from this model are compared withflat-atmosphere model results for a scattering-only atmosphere. These comparisons indicate thatexcluding spherical effects for solar zenith angles greater than 850 leads to errors larger than 5% at opticaldepths as small as 0.10.

Key words: Spherical shell radiative transfer, atmospheric radiative transfer.

Introduction

Several methods exist for finding the distribution ofscattered radiation in a plane-parallel atmosphereilluminated by a plane-parallel solar beam."2 Inthese techniques the atmosphere is assumed to con-sist of homogeneous plane-parallel layers of infiniteextent. This flat-atmosphere approximation is nor-mally adequate for treating radiation traveling in theEarth's atmosphere, and because of computationalefficiency, plane-parallel models are used wheneverpossible. There exist situations for which the plane-parallel model is not applicable. For example, aplane-parallel model cannot address subhorizon solarbeams. Significant differences between results fromplane-parallel models and spherical models also arisefor solar zenith angles greater than 750 and forobservation angles within 100 of the horizon, par-ticularly for reflected radiation exiting the top of theatmosphere.3 Analyses of these situations require aradiative transfer model that considers multiple scat-tering in a spherical shell atmosphere with altitude-dependent scattering and absorption. A comprehen-sive model would also include the effects of refractionand polarization.

B. M. Herman is with the Institute of Atmospheric Physics andK. J. Thome is with the Optical Sciences Center, University ofArizona, Tucson, Arizona 85721; A. Ben-David is with the Scienceand Technology Corporation, 2719 Pulaski Highway, Edgewood,Maryland 21040.

Received 14 December 1992; revised manuscript received 21June 1993.

0003-6935/94/09176011$06.00/0.6 1994 Optical Society of America.

The differential form of the radiative transferequation in a spherical shell atmosphere, thoughsimple in form, has no analytical solution.4 However,numerical solution techniques have since been devel-oped. These include such methods as invariant im-bedding,5-9 stream approximations and moments ofintensity,' 0 -'4 and Monte Carlo techniques.3" 5-'7Whitney et al. 18 and Lenoble'9 review these and othermethods in further detail.

Approximations that simplify the problem are alsoused, such as approximate expressions for the meanintensity in a Rayleigh atmosphere;2 0 iterative meth-ods based on the Eddington approximation;21 22 solv-ing the single scatter exactly and approximatinghigher-order scattering in terms of this exact singlescatter;23 and a quasi-spherical iterative method uti-lizing the plane-parallel equations but with all anglesand distances corrected for spherical geometry.24

The current study addresses cases for incidentsolar zenith angles between 0 and 900. Futureresearch, in preparation, will address the case of asubhorizon Sun. The atmosphere is divided intohomogeneous spherical shells. The intensity at anypoint is then a function of five independent variables:three for location in the atmosphere and two for thedirection of the radiation. The model includes aero-sol scattering as well as gaseous and aerosol absorp-tion, but it neglects polarization and atmosphericrefraction. Additionally, an accurate model for theaerosol forward peak has yet to be included.25 Foreach shell, the model uses the integrated form of thegeneral radiative transfer equation and computes alldistances and angles by assuming spherical geometry.

1760 APPLIED OPTICS / Vol. 33, No. 9 / 20 March 1994

The technique utilizes a Gauss-Seidel iterationscheme to establish a steady-state solution that in-cludes all significant orders of scattering.2 6 In-tensities are found at a discrete set of directions atdiscrete height levels along a selected radial line inthe atmosphere. A conical boundary is introducedabout this radial line, and approximate' solutions arefound for each level along selected radii around theboundary. These solutions are used as boundaryvalues in an interpolation scheme to find the requiredintensities. This approach permits more versatilityand accuracy than previous spherical models thatstrived for computational efficiency. It is also morecomputationally efficient than other methods strivingfor the same accuracy as the current one.

Theory

Figure 1 illustrates the coordinate system used in thecurrent study. The origin of this system is taken tobe the center of the planet. Intensities are found atselected heights along the z axis; the z axis is theradius along which the solution is desired. This lineis also referred to as the zenith. In other words, thezenith is any radial line from the center of the planetto the top of the atmosphere along which intensitiesare found. Height above the planet's surface ismeasured along the z axis and is designated by thevariable z. Values for z range from zero at thesurface to Zo at the top of the atmosphere. Theradius of the planet is designated by Ro. An arbi-trary point s in the atmosphere is further defined bythe variables + and q. The variable 4) is the polarangle between the radial line through s and the z axis,whereas is the azimuth angle between s and thezero-azimuth plane, defined by the z axis and theincoming solar beam. It should be noted that q) and'r do not represent latitude and longitude but aremeasured with reference to the principal plane andthe radial line along which the intensities are desired.The polar angle 0, measured relative to the z axis, andthe azimuth angle 4), measured relative to the zero-azimuth plane, define the line of sight direction.Then the intensity in an arbitrary direction 0, ) and

z axisof Sight

Principal Plane

4, ri =0

Fig. 1. Coordinate system used in this study indicating thevariables used to describe the point s and define the direction of aline of sight.

through a point z, l, 'q is represented by I(z, qj, A, 0, 4)).Through the use of the previously defined notationfor s, this becomes I(s, 0, 4)). The solar zenith angle00 and solar azimuth angle 4)o, which is taken to bezero, define the direction of the incident solar beam.

Following the notation of Chandrasekhar,27 wemay write the integral form of the general radiativetransfer equation as

I(s, 0, 4) = I(s = 0, 0, ,)exp[-T(s, 0)]

+ J(s', 0, )exp[-T(s, S')]Kpds', (1)

where T(S, S') is the optical depth along the pathbetween the points s and s',

T(S, s') = Kpds', (2)

I(s, 0, 4)) and I(s = 0, 0, 4)) are the total intensities atthe points s and s = 0 in the direction 0 and X), s = 0 isthe point where the path begins, which can occur atthe top or bottom of the atmosphere or at some pointwithin the atmosphere, K is the mass extinctioncoefficient with the units of area per unit mass, and pis the density of the attenuating medium. HereJ(s', 0, 4)) is the source function and for a scatteringatmosphere is

J(s', 0, 4) = P(s', 0, ; o, 4o)Fj(s' Oo, Jo)

2fs+ J JP(S" H. -; O, v)i(s ', or, d,,)

X sin 0'dO'd4',

where

2r

(3)

P(s', 0, 4); 0', ')sin 0'dO'd4' = wiuo, (4)

P(s', 0, 4); 0', 4)') is the phase function describing theangular distribution of scattered light, wo is thesingle-scatter albedo that is unity for conservativescattering, and F,(s', Oo, G)o) is the reduced solar irra-diance at the point s'.

In this study the atmosphere is broken into homog-enous shells. The total intensity at point s is takento be the sum of single- and multiple-scatteringcontributions. The term single scattering refers tothe first scatter out of the incoming solar beam;multiple scattering refers to the subsequent scatter-ing of the diffuse radiation and does not include singlescattering. The sum of the single- and multiple-scattering terms constitutes the total intensity.Thus the total intensity at point s along the level i inthe direction 0 and 4) is written as

I(si, 0,)) = ss(s 0, ) + Im(Si, 0, 4)). (5)

The first term on the right-hand side, Is(si, 0, 4)), is

20 March 1994 / Vol. 33, No. 9 / APPLIED OPTICS 1761

the single-scatter intensity, given by

ISS(si, 0, 4) = j P(s', 0, 4); o, 4o)F(s', o, o)

x exp[-'r(si, s')]Kpds'.

Here I is also known as the primary scatter and issimply the integral over optical depth of the singlescatter at all points s' transmitted to the point s.In the case where the path strikes the surface of theplanet, a reflection term is added. Equation (6)assumes that the incident single-scatter intensity atthe top of the atmosphere is zero.

The multiple-scatter contribution, Is(Si, ()), issimilarly the integral over optical depth of the secondterm in Eq. (3) transmitted to s. In the currentresearch the multiple scatter is calculated within eachlayer, so that at the level i it is the intensity caused bythe scattering of diffuse radiation in the layer be-tween the levels i and (i - 1) and an attenuatedincident term from the previously calculated multiplescatter at the level (i - 1). The level (i - 1) refers toeither the level above or below the level i, dependingon whether the radiation is traveling downward orupward. With this in mind we write the multiple-scatter contribution at point s as

Ims(Si, 0, 4)) = Im(si 0, ))exp[-T(si, si-,)]

psi e27rA+J J J1'(si0, 0', 4)')I(s 0 4)

x exp[-T(Si,S')]Kp sin 0'do'dv'ds', (7)

where the first term on the right-hand side is theattenuated multiple-scatter intensity from the points:_1 in the direction 0 and 4), and the second term isthe intensity caused by the scattering of diffuseradiation within the layer into the direction 0 and 4.The intensity used in the integral of the second termis the total intensity, and the intensity used for theincident term is the multiple scatter only.

Method

The single-scatter term, though complicated by thespherical geometry, is calculated in a straightforwardmanner by approximating the path integral of Eq. (6)with a layer-by-layer summation. The difficulty incomputing the total intensity in a spherical atmo-sphere is in finding the multiple-scatter contribution,partly because of the lack of horizontal homogeneity.This lack of horizontal homogeneity is caused by thevariation of the solar zenith angle at the top of theatmosphere caused by the spherical nature of theproblem, as shown in Fig. 2 (i.e., 00 • 00' • 00).Sphericity also causes parallel paths beginning at thesame height to have differing optical paths, as shownin Fig. 2(a). Both of these factors cause the intensityfields at the same height but at varying locations todiffer. In a plane-parallel atmosphere, shown in Fig.2(b), these problems do not occur and the solution

Incoming Solar Beam

(6)

(a)

Incoming Solar Beam

(b)

Fig. 2. Differences between flat and spherical atmospheres's pathlengths and incident solar angles: (a) incident solar beam at thetop of a spherical atmosphere (note the change in incident angleand path length caused by sphericity); (b) same as in (a) except forthe flat-atmosphere case (note all angles and path lengths areidentical).

need only be found along one vertical axis to describethe intensity field in the entire atmosphere. In aspherical atmosphere, computational problems arisebecause solutions have to be found over the entiresunlit hemisphere to solve the problem completely.

The current research avoids solving throughoutthe entire hemisphere by using an approximatemethod that employs a conical boundary surroundingthe z axis. The size of this conical boundary isdefined by the angle 0, where the is the angleformed by an arbitrary radial line and the z axis [Fig.1(a)]. Approximate solutions are found along thisboundary, and these solutions are used in an interpo-lation scheme to solve for the intensity field along thezenith. This procedure reduces the number of com-putations significantly. The accuracy of the methodhighly depends on the accuracy of the boundarysolution and how errors on the boundary affect thez-axis solution.

The atmosphere is divided into shells, and themethod uses discrete angles. A grid system is set upwith the points of the grid defined by the intersec-tions of the zenith and boundary radial lines with the


discrete height levels. The boundary radial lines areat intervals of 300 about the zenith. Figure 3shows grid points for the zenith and two of the conicalboundary radial lines, corresponding to = 0 andvi = 1800. The process of solution begins by comput-ing the single scatter at all grid points on the zenithand boundary for all discrete directions defined by theset of 0's and 4's.

The multiple scatter on the zenith is calculated byiterating to a stable solution in a fashion similar tothat of the flat-atmosphere situation.26 Because ofthe lack of horizontal homogeneity, the intensities ata given level cannot be assumed to be constantaround a shell at a given height (points a, b, and c inFig. 3). Instead, interpolation techniques are usedto determine the needed intensities for finding thezenith solution. To compute Ims(si-i 0, 4)) in Eq. (7),we interpolate between intensities in a given 0 and 4)direction at surrounding grid points. Using Fig. 3 asan example, we obtain the intensity at the point si1by means of a quadratic interpolation of the intensi-ties at points a, b, and c. The source integral, thesecond term on the right-hand side of Eq. (7), iscomputed by assuming that the total intensity withinthe layer (i.e., single- plus multiple-scattered light)varies linearly along the path with the value at thepoint si-1 obtained by interpolation of grid-pointvalues from the previous iteration. The mass extinc-tion coefficient, density of the attenuating medium,and phase function for a given direction are taken tobe constant within the layer. Then a summationapproximates the angular integration and the opticaldepth integration is performed analytically withineach layer by using the assumptions stated above.Implicit in finding the multiple scatter on the z axis isthe knowledge of the intensities on the boundary usedin the interpolation technique.

Once an intensity value on the zenith for a specifieddirection is found, values for parallel directions at allgrid points around the cone at the same level arecomputed. To do this, the multiple scatter is againtreated as two parts, an incident term and the source

Zenith

Ro +zi.1

Ro +zi

Liner = 0 of

r00 Sight

11 d; l (wtp 0)

VFig. 3. Grid system and conical boundary used to compute theintensity field on the zenith. Two-dimensional view shows theradial lines on the conical boundary associated with X = 0° and q =1800. Labeled points are used for discussion in the text.

term integral, as in Eq. (7). Because the intensitiesoutside the cone are unknown, interpolation tech-niques cannot be used as they were in the case of thezenith. Extrapolation techniques were attemptedand found to introduce large errors. Instead, to findthe incident multiple scatter for points on the cone,the first term on the right-hand side of Eq. (7), weassume that the ratio of multiple to single scatteroutside of the cone is constant along any shell. Thisapproach limits the model to cases in which thesingle-scatter term is not zero. Future research willexamine methods for solving cases in which singlescatter terms are zero (e.g., the subhorizon Sun case).Then using the known single-scatter intensity at the(i - 1) point outside the cone, point sj(4 = q)o) inFig. 3, we find an incident multiple scatter by multi-plying this single-scatter intensity by the ratio ofmultiple to single scatter at the (i - 1) level on thecone from a previous iteration. Referring to Fig. 3we find that this would be

Ims[Si-l(4) = 40), 0, 4)]

= ISJsi-A(4 = 4 o 0, 4))] IMs(SC, 0, 4*) (8)

Iss(sc, 0, 4))

Assuming the ratio of multiple to single scatter tobe homogeneous along a shell outside the conicalboundary is better than assuming the intensity fieldto be homogenous, because the intensity field variesmuch more rapidly along a shell than the ratio of themultiple to single scatter. This method also permitsthe known single scatter to be used for added informa-tion.

To compute the source term integral for the pathwithin the layer, from the (i - 1) point outside thecone to the point on the cone boundary, the secondterm on the right-hand side of Eq. (7), we find itnecessary to know intensity as a function of anglealong the path of the integral [line si-L1(4 = 4)o) tosj(t) = *0) in Fig. 3]. However, these intensities arenot known outside the cone. We therefore approxi-mate the source integral in the following manner;On the zenith (point si in Fig. 3), the source integral isgiven by

f J(s', 0, 4))exp[-T(Si, S')Kpds'

Si-i

= J J J P(s', 0, ; 0, ')I(s 0', 4)')sin(0')Si-i

x exp[-T(si, S')]d0'd4)'KpdS'. (9)

This is simply a restatement of the second term onthe right-hand side of the Eq. (7) integral and issolved by using the method previously described.On the cone [point si(* = t)o) in Fig. 3], we assume thesource integral to be the source integral for the zenithmodified by the ratios of the path integrations and the


intensity fields between the cone and zenith. Thatis,

Pif J[s'(a) = 4)o), 0, ]exp{-T[SA( = 4o), s( = 0)]Si-l

X KpdS'(a = 4to) = f J(s', 0, 4)exp[-r(si, s')]Sij 1

f27rPI(4 = Bo, 0', 4')sin(0')d0'd)'

X Kpds' f 2 � rf�rI(0', 4)')sin(0')dW'd'

x

fSiexp-T[Si(N = )o) S'(4 = qJo)]}KpdS'(q) = t)o)

Si-i

SeXp[-T(Si, S)]KpdsSi-l

where (0', 4)') is the layer average of intensity atthe points s and b in the direction ' and 4)' andI(N = q0, ' ') is the layer average of intensity at thepoints si(i) = 4)o) and c in the direction ' and 4)'.The path integrations of optical depth in the ratios inEq. (10) are performed analytically. As stated previ-ously, Eq. (10) may be thought of as the product ofthree separate terms. The first term is the sourceintegral for a point on the zenith at the selected levelin the given direction. The second term is the ratioof the integrated intensities between the cone andzenith, and the third term is a ratio of the pathlengths between the cone and zenith. The validity ofthis technique depends on the size of the cone bound-ary angle 0.

The size of the cone is a compromise consideringseveral criteria. On one hand if the cone is small, theinterpolated values used in the zenith computationsare more accurate, parallel paths on the zenith andcone are more similar, and the approximations usedin computing the boundary intensities are more valid.On the other hand, if i)o is too small, then paths fromthe zenith exit the side of the conical boundary beforeentering the next layer. This creates an interpola-tion situation that is less accurate then if the pathenters the next layer before exiting the conical bound-ary. A small cone also encompasses less of theatmosphere, thus the iteration technique used to findthe solution is less accurate. Finally, if the bound-ary is too close to the zenith, errors in the approxi-mated boundary intensities will have a larger effecton the final zenith solution. A value of 1 degree wasfound to be an adequate compromise of the abovepoints and is used in all computations presented inthis paper, where a planet with a radius of R = 6380km and an atmosphere of 50 km is used.

Checking the Model

The model was first tested for errors in the code itself.This was done by ensuring energy conservation in ascattering-only atmosphere (o = 1.0) by computingthe flux divergence assuming each layer in the atmo-sphere to be a cylinder. Also, model runs performedby using a radius of 638,000 km (i.e., 100 x the radiusof the planet used in this study) were compared withflat-atmosphere results for the same atmosphere.Last, intensities were calculated for the case of asmall solar zenith angle (00 = 50) and low opticaldepth (0.05). Results were compared with flat-atmosphere model results, and differences were exam-ined for consistency with geometrical differences.The results from these three tests indicate the modelis free from errors in the code.

The effect of the size of 4o for a planet with a radiusof 6380 km was examined by comparing results forvalues of 4)o ranging from 0.250 to 2.0°. For 0.75° <q) < 1.25°, solutions differed from one another byless than 1%. For 0 < 0.750 or 4 > 1.25°, solutionsdiverged rapidly. Thus a value of 1.00 appears appro-priate and is used in all calculations presented in thispaper.

To test the effect of boundary errors, we positivelybiased the approximate boundary solutions for 4)o =10 by 10%. The final zenith solution using thesebiased boundary values differed by less than 2.5%from the unbiased case. For nontangent lines ofsight (paths that intersect the surface), these differ-ences were less than 1%. An additional check of theboundary approximations was performed by examin-ing the approximate accuracy of the boundary solu-tions.

Through the proper adjustment of the geometry, aboundary solution to one problem can become azenith solution for a different problem. These testsindicate that the boundary solution for transmittedlight is accurate to better than 3% at all angles, andfor 0 < 80 (where 0 = 0 indicates radiation travelingvertically downward, i.e., an observer looking up-ward) the boundary is accurate to better than 1%.Boundary solutions for reflected light for nontangentlines of sight were accurate to better than 3%.

For tangent lines of sight (an observer lookingdown from the top of the atmosphere), the differencesbetween the two solutions (i.e., the boundary and thezenith solutions) ranged 3-12%, with the largestdifferences occurring for 930 < 0 < 950 (where aline-of-sight angle of 1800 indicates radiation travel-ing vertically upward along the zenith). These anglesare the worst because the paths between the levels iand (i - 1) are the longest at these lines of sight.

Because the errors on the boundary in the com-puter code used by the model are not biased positivelyor negatively, and because these errors are much lessthan the 10% used in testing the boundary's effects,we conclude that the boundary approximations im-pose less than a 1% error of the final zenith solution.

To test the model's accuracy we compared it with


three previously developed models, a quasi-sphericalapproach 2 4 and two Monte Carlo approaches. 17 2 8

The transmitted intensity results were comparedwith those of Asous24 and Marchuk et al.

2 8 For thereflected intensities, the models of Adams andKattawar17 and of Asous2 4 were used. These compari-sons were well within the Monte Carlo techniqueuncertainties of 3%. Thus the current model istaken to be accurate to at least 3%. Because of theclose agreement found between the current researchand other models, and because the current approachdoes not suffer from the statistical fluctuations of theMonte Carlo techniques, the current approach isbelieved to be accurate to better than 1% for mostEarth-atmosphere situations.

1.04

i 1.03

4-C

M 1.02LL

> 1.01a)C

C 1

C)._@ 0.99

U)sL

_ A

0 10 20 30 40 50 60 70 80 9(

Line of Sight Angle (degrees)(a)

2

Sr 1.9

C2D 1.8C- 1.7t-U 1.6

> 1.5U)

r 1.44-C

- 1.3

'ao 1.20)C 1.1

(1

0 10 20 30 40 50 60 70 80 90

Line of Sight Angle (degrees)(b)

Results

Here we present results of comparisons betweenintensities calculated from the current spherical atmo-sphere model and those of a flat-atmosphere modelusing a Gauss-Seidel approach.2 6 Two cases areexamined: in one the solar zenith angle is varied forconstant optical depth, and in the other the opticaldepth is varied at constant solar zenith angle. Theresults for the transmitted light and the reflectedlight are treated separately.

All of the results presented here assume the top ofthe atmosphere Z0 to be 50 km, a planetary radius Roof 6380 km, and a surface reflectance of zero. Theoriginal 0 angles of 0, 5, 15,..., 165, and 1750 and 4)

1.04.6-i

U')C 1.03U-

" 1.02LL

'5; 1.01.)

CC 1

in

Q 0.99

a-Un(Ino

O

2

1-

r 1.6

1.4

4-

C

- 1.3

C

S 1.2en

r-14

0

0 10 20 30 40 50 60 70 80 90

Line of Sight Angle (degrees)(C)

Transmitted intensities at the surfaceOptical Depths:

Aeroo = 0.17Rayleigh = 0.33

- View Azimuth = 180 degrees-- Solar Zenith = 80 degrees

- - - Solar Zenith = 85 degrees-Solar Zenith 88 degrees

10 20 30 40 50 60 70

Line of Sight Angle (degrees)(d)

80 90

Fig. 4. Ratio of spherical to flat-atmosphere transmitted intensities at the surface for a scattering atmosphere with an 0.50 optical depth:(a) - = 0°, (b) + = 00, (c) 4 = 180', (d) k = 1800.


Transmitted Intensities at the SurfaceOtclDeyths:Apierosol 0.1 7Rayleigh = 0.33

View Azimuth = 0 degrees

-- SolarZenith- 5degrees-- Solar Zenith = 35 degrees

Solar Zenith = 65 degrees-Solar Zenith = 75 degrees

......... ....~~~~~~~~~-.__,I................................ I-

Transmitted Intensities at the SurfaceOptipal Depths:

Aerosol , 0. 17Rayleigh - 0.33

View Azimuth . 180 degrees

-- Solar Zenith - 5 degrees- - Solar Zenith =35 degrees

Solar Zenith = 65 degrees- Solar Zenith = 75 degrees

I-

Transmitted intensities at the surfaceOptical Depths:

Aeroo = 0.174ayleigh .0.33

- View Azimuth = 0 degrees- - Solar Zenith = 75 degrees

.Solar Zenith = 80 degreesSolar Zenith = 85 degrees

- -Solar Zenith = 88 degr

/-li

/._

I , . .

| g | s n ~~~~~~~I - ---------. <*............__----

0.98s u.U0

I

I

angles of 0, 30, 60,..., 150, and 1800 from Hermanet al.25 are used, plus an additional 14 0 anglesbetween 80 and 1000 that are varied to delineatemaximums and minimums in the intensity field.Twelve angles are used at 30° intervals, and asmentioned previously, 4)o = 1.

The vertical distribution of scatterers is assumed tofollow the 1976 U.S. Standard Atmosphere for amidlatitude winter and 23-km surface visibility.The aerosol relative size distribution is constantwith height and follows a Junge power law[dn/dlog(r) - cr-] with v = 3.0 and minimum andmaximum radii of 0.01 and 5.0 plm, respectively.Aerosols are assumed to scatter as Mie particles withrefractive index 1.54 + 0.0i.

Transmitted Light at the Surface

For an infinitely thick atmosphere or for an infinitesi-mally thin atmosphere, the downward-scattered inten-sities at the surface are zero. Thus, initially thescattered light at the surface increases with increas-ing optical depth. The scattered light will reach amaximum value as a result of this gain mechanismand then decrease as a result of two loss mechanisms:(a) energy from scattering within a layer is attenuatedby subsequent layers, and (b) the incident solarenergy, which is the source for all scattering events, isreduced as it penetrates into deeper layers and thusless energy is available for scattering. The increasein intensity as a result of increased optical depth isfrom the integration over optical depth of the sourceterm in the second terms of Eqs. (1) and (7). Thedecrease of the transmitted intensity with the in-crease of optical depth is from the reduced transmis-sion in the exp[-r(s, s')] term. It should be kept inmind that large optical depths can result from bothlarge zenith angles (long geometric paths) and anoptically thick atmosphere.

The ratios of the transmitted intensity of thecurrent spherical atmosphere model to a flat-atmo-sphere model are given in Figs. 4 and 5. Excludingthe case of the subhorizon Sun, optical paths arealways longer for transmitted light in a flat atmo-sphere. For small optical depths this leads to largerintensities in a flat atmosphere relative to a sphericalatmosphere for the reasons given above. At largeroptical depths the loss mechanisms decrease thetransmitted intensity in the flat atmosphere fasterthan in the spherical atmosphere. Thus intensitiesfor the flat atmosphere are smaller at larger pathlengths or optical depths caused by reduced transmis-sion. The shorter solar path in the spherical atmo-sphere also permits deeper penetration of solar en-ergy, and the transmitted intensities for large solarpaths will be greater for the spherical atmospherethan for the flat one.

The first case, that of varying Sun angle at constantoptical depth, is shown in Fig. 4. Here the verticaloptical depth is 0.50 (Rayleigh optical depth of 0.33and aerosol optical depth of 0.17). The solar zenithangle 00 takes values of 5, 35, 65, 75, 80, 85, and 88°,

1.5

r1.4

4)

4-

--U-M; 1.3I._

'*5 1.2

_ . .

.0.

Q nL- HA

0.90 10 20 30 40 50 60 70 80 90


1.5

(a<X 1.4

-_C

.L 1.3

U-

'3 1.2Ca)

-1

.1

-i

a-a) 1

en-i

0 10 20 30 40 50 60 70 80 cLine of Sight Angle (degrees)

(b)Fig. 5. Ratio of spherical to flat-atmosphere transmitted intensi-ties at the surface for a molecular scattering atmosphere with solarzenith angle of 85° and varying optical depths from 0.10 to0.80: (a)4= 0°, (b)4i= 180°.

where 00 = 0° indicates the Sun is directly overheadon the zenith. Figures 4(a) and 4(b) show results for50 < 0 < 88.90 and 4 = 00, whereas 4 = 180° is shownin Figs. 4(c) and 4(d). In these figures a line-of-sightangle of 00 indicates that the radiation is travelingvertically downward along the zenith. An azimuthangle of zero' indicates that the radiation is travelingin the same azimuthal direction as the solar beam.

For 0 < 750 Figs. 4(a) and 4(c), the differencebetween flat and spherical models is less than 4%.It is also noticed that for 00 < 350, the flat-atmosphere results exceed the spherical ones foralmost all lines of sight. This is due to the longer


Transmitted Intensities at the SurfaceSolar Zenith = 85 degreesView Azimuth = 0 degreesOptical Depths:

Aerosol = 0.00-o- Rayleigh-0.10- - Rayleigh - 0.20

Rnvlh - n An

Rayleigh 3 0.60 ., 1-Rayleigh = 0.80

_____..__.___......

…-- - - - - - - - - - - - - - - -

- _ _ _ 9I:

Transmitted Intensities at the SurfaceSolar Zenith . 85 degreesView Azimuth , 180 degreesOptical Depths:

Aerosol - 0.00Rayleigh =0.10

- Rayleigh = 0.20-- Rayleigh = 0.40

Rayleigh = 0.60-Rayleigh = 0.80

. .

…. .--. S

, , , , , , , | X-'

: :II'

I

I

90

paths of the flat atmosphere enhancing the gainmechanism relative to the spherical one, as discussedpreviously.

The minimum in the curves at 800 is due to thetrade-off between the loss and gain mechanisms.For 0 < 800, scattering into the line of sight is moreimportant. Thus because the line-of-sight opticaldepths are increasing as 0 increases, and because thisincrease occurs faster in the flat atmosphere, theratio has a decreasing trend. For 0 > 800, the lossmechanism becomes dominant and is larger in theflat atmosphere because of the longer paths. As aresult the ratio increases. These factors becomemore pronounced for 4) = 180° because the solar pathis larger in this case than it is for 4 = 0. Thisminimizes the differences caused by solar path be-

tween the flat and the spherical atmospheres (dis-cussed further below), and effects caused by line-of-sight path differences become more pronounced [Fig.4(c)].

In Figs. 4(b) and 4(d), the peaks in the curves can beexplained by using the above loss and gain mecha-nisms. For large enough line-of-sight paths, the lossmechanism dominates the gain mechanism. Thisoccurs first in a flat atmosphere and the ratio thenincreases with the line-of-sight angle. Eventually,the loss mechanisms dominate in the spherical case,and from this point on the ratio decreases with theline-of-sight angle. This effect is more pronouncedin the case of larger solar zenith angles, in which thesolar attenuation is larger and effects caused by lossesare more evident.

Reflected IntensitesOptica Depths

VerosiolrRayleigh =

View Azimuth Solar Zenith =Solar Zenith :Solar Zenith = iSolar Zenith =

Ic I I

at Top of Atmosphere

0.170.33* 0 degrees5 degrees A

35 degrees -65 degrees ---75 degrees

l

U)C

._

C

1,

C

.-

0_O

eo

1.5

1

0.5 -

n9090 100 110 120 130 140 150 160 170 180


100 110 120 130 140 150 160 170 180Line of Sight Angle (degrees)

(c)

3.5

._

U)C 3

_2.-.. 2.5

._

, 1.5._C

co0.5a-)

90 100 110 120 130 140 150 160Line of Sight Angle (degrees)

(b)

C

C

E

C._

CI-

a)-.

s)

3.5

3

2.5 1

2

1.5

1

0.5

.I ' OU170 180 90 100 110 120 130 140 150 160

Line of Sight Angle (degrees)(d)

Fig. 6. Ratio of spherical to flat-atmosphere reflected intensities at the top of the atmosphere for a scattering atmosphere with an 0.50optical depth: (a) + = 00, (b) + = 0°, (c) 4, = 180°, (d) 4, = 180°.


1.5

U)

C-.C

EC._,

a)C

C-=

a-U)Co

(1n

0.5

0

A,,

Reflected Intensites at Top of Atmosphere

P erosoF 0. 17Rayleigh - 0.33

View Azimuth = 180 degreesSolar Zenith 5 degrees A

Solar Zenith = 35 degrees -Solar Zenith = 65 degrees - - -

Solar Zenith = 75 degrees

Reflected Intensites at Top of Atmospherel Opti~cal Deyths:01

Rayleigh = 0.33View Azimuth = 0 degreesSolar Zenith = 75 degrees -----Solar Zenith = 80 degrees - - -

Solar Zenith = 85 degrees - -


Vi 0\s

Reflected Intensites at Top of AtmosphereOptia Depths:

Aerosol 0.17Rayleigh = 0.33

View Azimuth = 180 degreesSolar Zenith = 75 degrees - -Solar Zenith = 80 degrees - - - -Solar Zenith = 85 degrees - - -


4.- , . .-

170 180

- -- o call A_ - SF - - .!, iF - S

. . . . .

I

For long solar optical paths, the effects of sphericitybecome less noticeable as a result of the large amountof attenuation of the incoming energy. This ex-plains why the ratios in Figs. 4(c) and 4(d) are closerto 1.0 than for the corresponding curves in Figs. 4(a)and 4(b). However, there is always some differencebetween the flat and spherical atmosphere intensitiesbecause of the shorter solar path in the sphericalatmosphere, and thus the spherical to flat ratio willalways exceed unity for large enough vertical opticaldepths.

The case for varying optical depth with constantSun angle is shown in Fig. 5. A pure Rayleighatmosphere is used with optical depths of 0.1, 0.2, 0.4,0.6, and 0.8. The solar zenith angle 00 is 850.Figure 5(a) shows the ratio for 4 = 0, and the ratiofor 4 = 1800 is given in Fig.5(b). The shapes ofthesecurves can be explained by using the previous argu-ments and will not be repeated. The most importanteffect is for 0 less than 75°, where the ratio in-creases as the optical depth increases. This is aresult of larger loss mechanisms for the flat-atmo-sphere case. As optical depth increases to larger andlarger values, the differences between the flat andspherical atmospheres become minimized. Thus theincrease in the ratio is larger at smaller opticaldepths. For near-horizon lines of sight, the inter-play of the loss and gain mechanisms complicates thisdiscussion.

Reflected Light At the Top of the Atmosphere

In a conservative atmosphere (i.e., no absorption), thetotal reflected energy increases with increasing opti-cal depth. As the optical depth increases, the re-flected energy increases at the expense of the transmit-ted energy. This qualitative explanation for thereflected energy is useful when one examines thereflected intensities as a function of geometry (i.e.,solar angle and line-of-sight angle). The same twocases used above for the transmitted case are used toexamine the reflected light exiting the top of theatmosphere and are presented in Figs. 6 and 7. Herea line-of-sight angle 0 of 1800 indicates radiationtraveling vertically upward along the zenith. Theinadequacies of a flat-atmosphere model caused bythe lack of tangent lines of sight (paths that do notintersect the planet's surface) become readily appar-ent when one examines the results of these compari-sons. Because of tangency, the longest path in aspherical atmosphere occurs for that line of sight thatis tangent to the surface ( = 97.15° with Z = 50km). In the flat atmosphere, the longest path occursin the limit as 0 approaches 90°. For 0 less than-950 (for Z = 50 km), the flat-atmosphere path

lengths are longer than the spherical ones, whereasthe spherical path lengths are longer for all otherline-of-sight angles. Finally, as in the transmittedcase, solar paths are always shorter in the sphericalatmosphere.

The case of constant optical depth (aerosol opticaldepth of 0.17 and Rayleigh optical depth of 0.33) is

a)

CCU0

C

4._

-a)

._

C

a-U)

2.5

2

1.5

1

0.5

0C

1.2

. _

C 1a)C

W 0.8U-

5; 0.6

Ca)C 0.4

C)._"- 0.2

0)n

Reflected Intensites at Top of AtmosphereSolar Zenith = 85 degreesView Azimuth = 0 degreesOptical Depths:

,1,; ~~~~Aerosol' 0.00

{i g~~~~~ayleigh 018 0 ------; ~~~~~Rayleigh . 0.40 - -

Rayleigh = 0.60- , ark ~~~~Rayleigh 0.80 A

10 100 110 120 130 140 150 160 170 18Line of Sight Angle (degrees)

(a)

0

90 100 110 120 130 140 150 160 170 180Line of Sight Angle (degrees)

(b)Fig. 7. Ratio of spherical to flat-atmosphere reflected intensitiesat the top of the atmosphere for a molecular scattering atmospherewith solar zenith angle of 850 and varying optical depths from 0.10to 0.80: (a) 4 = 0, (b) , = 1800.

shown in Fig. 6. The zero aximuth case is shown inFigs. 6(a) and 6(b), whereas the case for 4 = 180 isshown in Figs. 6(c) and 6(d). The dominating fea-tures in Fig. 6 are the sharp peaks caused by tangenteffects. As described above, these effects result inpath lengths in the spherical atmosphere that aremuch smaller than those in the flat atmosphere forangles less than 950. Thus, using the same argu-ments presented for the transmitted light case, we seethat the intensities in the spherical atmosphere caseare smaller, and the ratio is less than one. Theseeffects become greater as the solar zenith angleincreases, except in Fig. 6(d). In this case for anazimuth angle of 1800 (i.e., looking away from theSun) and for the Sun near the horizon, some of the


Reflected Intensites at Top of AtmosphereSolar Zenith = 85 degrees

View Azimuth =180 degreesOptical Depths:

Aerosol = 0.00Rayei = 0.10 ..------

Rayleigh = 0.20 - -Rayleigh = 0.40 -*-Rayleigh - 0.60 ARayleig h = 0.80

lines of sight cross the planet's shadow, and there isno single-scatter contribution [Eq. (6)] from thisportion of the path in the spherical atmospheremodel. Because there is no shadow in a flat-atmosphere model, the single-scatter contributionwill always exceed the single-scatter component ofthe spherical atmosphere model, and the ratio will besmaller. The increase in the peaks as 00 increasesarises from less solar energy being available to bescattered in the flat atmosphere (as discussed above),and the longer paths in the spherical case permittingmore light to be scattered into the line of sight. Thecurves are smoother for 4 = 1800 because of theminimization of spherical effects in the solar path (aspreviously discussed) in addition to shadow effects.

Figure 7 shows the varying optical depth case.Results for 4) = 0 are given in Fig. 7(a), whereas Fig.7(b) shows the case for 4) = 1800. The most notablefeatures are (a) the peaks in the ratio decrease andmove to lower line-of-sight angles with increasingoptical depth; (b) for 0 greater than 1100, the ratioincreases with increasing optical depth; and (c) theratio increases rapidly for small changes in verticaloptical depth for tangent lines of sight. As theoptical depth increases the peak intensity occurs atsmaller line-of-sight angles because of the attenua-tion along the line of sight. Because the intensitypeak has shifted to lower angles, the ratio peak shiftsas well. The increase in the ratio with increasingoptical depth at larger line-of-sight angles is due to anincrease of scattering into the line of sight. Becausethe paths are longer in the spherical atmosphere,intensities should increase faster relative to those of aflat atmosphere. The cause for the smoothness ofthe curves in Fig. 7(b) has been previously explained.

Conclusions

A method for numerically solving the equation ofradiative transfer in a spherically symmetric atmo-sphere is presented. The method uses a conicalboundary with an interpolation scheme to solve forthe intensity field along a single radial line in theatmosphere. Tests of the model indicate that it isaccurate to better than 1% for most Earth-atmo-sphere situations.

Comparisons with flat-atmosphere results are pre-sented for two cases. The first case examines thedifferences in a scattering-only atmosphere with atotal optical depth of 0.50 and by using various solarzenith angles. The second case examines the effectof changing optical depth at a solar zenith angle of850. Differences greater than 10% are found in thetransmitted intensities at the surface for all lines ofsight when the solar zenith angle is greater than 850and spherical effects are neglected. For an opticaldepth of 0.50, a solar zenith angle greater than 800,and view angles within 100 of the horizon, sphericaleffects must be included to avoid errors of the order of5%. For a solar zenith of 850, errors of 5% are foundfor optical depths as small as 0.10.

These flat-atmosphere comparisons also indicate

the need for including spherical effects when oneexamines the intensity field at the top of the atmo-sphere. For lines of sight that do not strike theEarth's surface, there is no true comparison with theflat-atmosphere model. Thus for a 50-km-thick at-mosphere and 6380-km-radius planet, a sphericalmodel must be used for lines of sight greater than 820from nadir. There is a similar interplay betweensolar zenith and view angle as described above for thereflected intensities as well. Again, for solar zenithangles greater than 850, spherical effects must beincluded to obtain results accurate to better than 10%at all veiw angles.

These results are obtained by using a surfacereflectance of zero. Future research will includestudies to examine the ratio of spherical to flatatmosphere model results as a function of changingsurface reflectance. Also planned are more in-depthstudies of the type presented here. Future researchwill include examining methods for the subhorizonSun. This will eventually permit further researchinvestigating the light scatter at twilight.

The authors thank S. R. Browning for his contribu-tion to the basic development of ideas leading to thisproject.

References1. J. Lenoble, Radiative Transfer in Scattering and Absorbing

Atmospheres: Standard Computational Procedures (Deepak,Hampton, Va., 1985), Part 1, Chaps. 1-8, pp. 1-144.

2. H. C. van de Hulst, Multiple Light Scattering: Tables, Formu-las and Applications (Academic, New York, 1980), Vol. 1,Chaps. 4-6, pp. 34-123.

3. D. G. Collins, W. G. Blattner, M. B. Wells, and G. H. Horak,"Backward Monte Carlo calculations of polarization character-istics of radiation emerging from spherical-shell atmospheres,"Appl. Opt. 11, 2684-2696 (1972).

4. J. Lenoble and Z. Sekera, "Equation of transfer in a planetaryspherical atmosphere," Proc. Nat. Acad. Sci. USA 47, 372-378(1961).

5. P. B. Bailey, "A rigorous derivation of some invariant imbed-ding equations of transport theory," J. Math. Anal. Appl. 8,144-169 (1964).

6. R. E. Bellman, H. H. Kagiwada, and R. E. Kalaba, "Invariantimbedding and radiative transfer in spherical shells," J. Comp.Physiol. 1, 245-356 (1966).

7. R. E. Bellman, H. H. Kagiwada, and R. E. Kalaba, "Diffusetransmission of light from a central source through an inhomo-geneous spherical shell with isotropic scattering," J. Math.Phys. 9, 909-912 (1968).

8. R. E. Bellman, H. H. Kagiwada, and R. E. Kalaba, "Diffusereflection of solar rays by a spherical shell atmosphere," Icarus11, 417-423 (1969).

9. A. Uesugi and J. Tsujita, "Diffuse reflectance of a searchlightbeam by a slab, cylindrical and spherical media," Publ. Astron.Soc. Jpn. 21, 370-383 (1969).

10. V. V. Sobolev, Light Scattering in Planetary Atmospheres(Pergamon, New York, 1975), Chap. 8, pp. 161-164.

11. J. Schmidt-Burgk, "Radiative transfer through spherically-symmetric atmospheres," Astron. Astrophys. 40, 249-255(1975).

12. W. Unno and M. Kondo, "The Eddington approximationgeneralized for radiative transfer in spherically symmetricsystems: I. Basic method," Publ. Astron. Soc. Jpn. 26,347-354 (1976).


13. S. J. Wilson and K. K. Sen, "Light scattering by an opticallythin inhomogeneous spherically-symmetric planetary atmo-sphere," Astrophys. Space Sci. 69, 107-113 (1980).

14. S. J. Wilson and K. K. Sen, "Light scattering by an opticallythin inhomogeneous spherically-symmetric planetary atmo-sphere: brightness at zenith near terminator," Astrophys.Space Sci. 71, 405-410 (1980).

15. D. G. Collins and M. B. Wells, "FLASH, a Monte Carloprocedure for use in calculating light scattering in a sphericalshell atmosphere," Rep. 70-0206 (U.S. Air Force CambridgeResearch Laboratory, Cambridge, Mass., 1970), pp. 1-57.

16. W. G. Blattner, G. H. Horak, D. G. Collins, and M. B. Wells,"Monte Carlo studies of the sky radiation at twilight," Appl.Opt. 13, 534-547 (1974).

17. C. N. Adams and G. W. Kattawar, "Radiative transfer inspherical shell atmospheres: I. Rayleigh scattering," Icarus35, 139-151 (1978).

18. C. K. Whitney, R. E. Var, and C. R. Gray, "Research intoradiative transfer modeling and applications," Tech. Rep.73-0420 (U.S. Air Force Cambridge Research Laboratory,Cambridge, Mass., 1973), pp. 1-91.

19. Ref. 1, Part 2, Chap. 3, pp. 247-255.20. 0. I. Smoktiy, "Multiple light scattering in the spherical

planetary atmosphere," Pure Appl. Geophys. 72, 214-226(1969).

21. R. 0. Chapman, "Radiative transfer in extended stellaratmospheres," Astrophys. J. 143, 61-74 (1966).

22. D. G. Hummer and G. B. Rybicki, "Radiative transfer inspherical symmetric systems. The conservative gray case,"Mon. Not. R. Astron. Soc. 152, 1-19 (1971).

23. N. B. Divari and L. I. Plotnikova, "Computed brightness of thetwilight sky," Sov. Astron. 9, 840-852 (1966).

24. W. A. Asous, "Light scattering in spherical atmospheres,"Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1982),Chap. 3, pp. 16-60.

25. B. M. Herman, S. R. Browning, and R. J. Curran, "The effectof atmospheric aerosols on scattered sunlight," J. Atmos. Sci.28, 419-428 (1971).

26. B. M. Herman and S. R. Browning, "A numerical solution tothe equation of radiative transfer," J. Atmos. Sci. 22, 559-566(1965).

27. S. Chandrasekhar, Radiative Transfer (Dover, New York,1960), Chap. 1, p. 9.

28. G. K. Marchuk, G. A. Miknilov, M. A. Nazaraliev, R. A.Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte CarloMethods in Atmospheric Optics (Springer-Verlag, New York,1980), Chap. 4, pp. 109-146.


numerical technique for solving the radiative transfer ...vijay/papers/rt models... · efficiency,...

Documents