Comparison of the Gauss Seidel spherical polarized radiative transfer code with other radiative transfer codes

Transcription

1 Comparison of the Gauss Seidel spherical polarized radiative transfer code with other radiative transfer codes B. M. Herman, T. R. Caudill, D. E. Flittner, K. J. Thome, and A. Ben-David Calculations that use the Gauss Seidel method are presented of the diffusely scattered light in a spherical atmosphere with polarization fully included. Comparisons are made between this method and the Monte Carlo calculations of other researchers for spherical geometry in a pure Rayleigh atmosphere. Comparisons with plane parallel atmospheres are also presented. Single-scatter intensity comparisons with spherical geometry show excellent agreement. When all orders of scattering are included, comparisons of polarization parameters I, Q and U as well as the plane of polarization show good agreement when allowances are made for the statistical variability inherent in the Monte Carlo method. Key words: Atmospheric radiative transfer, spherical shell radiative transfer, polarization, Rayleigh scattering. 1. Introduction In a recent paper by Herman et al., 1 a new way of using the Gauss Seidel iterative method for numerically solving the equation of radiative transfer in a spherical shell atmosphere was presented. The results of that scalar code, which neglected polarization, were compared with a plane parallel Gauss Seidel code to show the differences caused by geometrical considerations only. The reflected and transmitted intensities were also compared with other spherical models to test the accuracy of the new method. Those comparisons showed agreement to within the estimated few percent statistical fluctuation of the Monte Carlo solution. It was estimated that the new method was accurate to within,1% for cases neglecting polarization. 1 However, it has been shown that the total intensity calculated by the use of the scalar Rayleigh phase function may differ substantially from the Rayleigh phase matrix solution under B. M. Herman, T. R. Caudill, and D. E. Flittner are with the Institute of Atmospheric Physics and K. J. Thome is with the Optical Sciences Center, the University of Arizona, Tucson, Arizona A. Ben-David is with the Science and Technology Corporation, 2719 Pulaski Highway, Edgewood, Maryland T. R. Caudill is also with Phillips Laboratory, Hanscom Air Force Base, Massachusetts Received 12 September 1994; revised manuscript received 11 January @95@ $06.00@0. r 1995 Optical Society of America. certain conditions. 2 This paper presents results that use the conical boundary Gauss Seidel method of Herman et al. 1 for a spherical atmosphere with polarization fully included, and it compares these new results with previously developed methods in greater detail. Various numerical techniques for solving the equation of radiative transfer for spherical shell atmospheres have already been developed. These include invariant imbedding, 3 stream approximations, 4 the discrete ordinate method, 5 the source-function model, 6 the Monte Carlo 7 9 method, and the dodecaton approach to radiative transfer 1DART2 method. 10 Of these, only the Monte Carlo and DART methods have been extended to include polarization for spherical atmospheres Because only the Monte Carlo method has been presented extensively in the literature, the accuracy of the spherical Gauss Seidel model is tested by comparing it with three different Monte Carlo models. 9,12,13 The DART method does not appear to be well suited for angular comparisons, and little published data are available for comparison. The Monte Carlo results of Marchuk et al. 11 are not used because earlier comparisons showed problems with some of their results. 14,15 For a review of most of the above methods for spherical atmospheres, see Lenoble Theory Using the notation of Chandrasekhar, 2 the characteristics of the radiation field can be represented in 20 July Vol. 34, No. APPLIED OPTICS 4563

2 the equation for the pth parameter at point s in direction u, f can be written as I p 1s, u, f2 5 I p 1s 0, u, f2exp32t1s, s 0 24 s 1 es 0 J p 1s8, u, f2exp32t1s, s824krds8, 122 where p 5 1, 2, 3, 4. Here I p 1s 0, u, f2 is the polarization parameter at point s 0, k is the mass extinction coefficient, r is the mass density, and t1s, s82 is the optical depth along the path between points s8 and s defined as s t1s, s82 5 krds es8 Fig. 1. Coordinate system that shows the variables used to define point s and a line of sight at point s. terms of the polarization parameters as I 5 1I l, I r, U, V2 5 1I 1, I 2, I 3, I The solar radiation incident at the top of the atmosphere is assumed to be a completely unpolarized, parallel beam of unit irradiance, i.e., F , 1 2, 0,02. The geometrical coordinate system used here is shown in Fig. 1. The z axis is the radial line from the center of the planet to the top of the atmosphere 1also called the zenith2 along which the solution is desired. The principal plane is defined by the z axis and the radial line from the center of the planet in the direction of the Sun. Any arbitrary point s is defined by polar angle C, azimuthal angle h, and distance z above the planet surface at R 0. Angle h is measured between the plane containing the z axis and point s, and the principal plane. A line of sight from point s is defined by the polar view angle, u, and the azimuthal view angle, f. The polar view angle at point s is measured relative to the z axis, and the azimuthal view angle is measured relative to the plane, parallel to the principal plane, containing point s. Using these coordinates, an observer looking in the u50 direction on the z axis is viewing the zenith sky, while u5180 is the nadir direction. With these coordinates, the pth polarization parameter at any point can be represented by I p 1s, u, f2, where s 5 s1z, C, h2. The source function, J p 1s8, u, f2, for a conservative scattering atmosphere can be written J p 1s8, u, f2 5 P pq 1s8, u, f, u 0, f 0 2F q 1s8, u 0, f p p P pq 1s8, u, f, u8, f82i q 1s8, u8, f82 3 sin u8du8df8, 142 where F q 1s8, u 0, f 0 2 and I q 1s8, u8, f82 are the qth polarization parameters of the incident solar irradiance in direction u 0, f 0, attenuated to the point s8, and the diffuse radiation in direction u8, f8, at point s8, respectively. The repeated q subscript implies a summation over q. The scattering phase matrix, P pq 1s8, u, f, u8, f82, determines the redistribution of radiation from the u8, f8 incident direction into the u, f scattered direction. The solar zenith angle, u 0,is measured from the z axis, and the incident solar azimuthal angle is f 0 ; 0 by definition of the principal plane. In general, the phase matrix is a matrix as given by Sekera 17 and may be of any physically correct form for the polarized spherical Gauss Seidel code. For Rayleigh scatter, as noted by Chandrasekhar, 2 the phase matrix is reducible with respect to I 4 and, because the incident solar radiation is unpolarized, the phase matrix can be reduced to a matrix. This reduces the number of calculations significantly and contributes to a considerable time savings. The phase matrix for Rayleigh scatter is given by P pq 1s8, u, f, u8, f cos 2 c µ 2 sin 2 Df 2µ cos c sin Df µ8 2 sin 2 Df cos 2 Df µ8sin Df cos Df, 152 8p 2µ8 cos c sin Df 22µ sin Df cos Df 2µµ8 sin Df 1 cos c cos Df4 Because each polarization parameter must satisfy the scalar form of the equation of radiative transfer, where µ 5 cos u,µ85cos u8, Df5f2f8, and cos c5 sin u sin u8 1 cos u cos u8 cos Df APPLIED Vol. 34, No. 20 July 1995

3 Substituting Eq. 142 into Eq. 122, we find that each polarization parameter at point s can be written as the sum of an incident term at the beginning of the path and the single- and multiple-scatter contributions from points along the line of sight. It is assumed that there is no polarization parameter incident at the top of the atmosphere, and thus the first term of Eq. 122 is zero for paths starting at the top of the atmosphere. For radiation traveling in the upward direction, if the path begins at the surface of the planet, the incident term due to reflection is added. The surface for this study is assumed to be a Lambertian reflector, although any type of function may be used in the Gauss Seidel code. For the polarization parameters along a zenith of the spherical atmosphere to be computed, a conical boundary making an angle C 0 with the zenith is defined, within which the radiative transfer equation is solved. For the equation to be solved, the polarization parameters must be known within the conical volume and the incident parameters must be known over the surface of the boundary. The numerical solution is accomplished by the division of the atmosphere into homogeneous spherical shells, and grid points along the zenith are established at the intersection of these shells with the z axis. Grid points along the conical boundary are given by the intersection of the spherical shells with the lines defined by C5C 0 at fixed intervals of h, Dh. Figure 2 shows a sample grid-point layout for downward-traveling radiation. Because points along the zenith only depend on z, the grid-point notation on the zenith can be simplified such that s i 5 s1z i,0,02. The z dependence for grid points on the cone is handled in a similar manner, i.e., s i 1C 0, h2 5 s1z i, C 0, h2. For downward-traveling radiation from the top of the atmosphere 1s to point s i, the single-scatter component, which refers to the first scatter event out of the solar beam, is given by I p ss 1s i, u, f2 5 s i P pq 1s8, u, f, u 0, f 0 2F q 1s8, u 0, f exp32t1s i, s824krds The comparable downward multiple-scatter component refers to all subsequent scattering of the diffuse radiation and can be written as s i 2p p I ms p 1s i, u, f2 5 P pq 1s8, u, f, u8, f82i q 1s8, u8, f82 3 sin u8du8df8 exp32t1s i, s824krds Calling the second term on the right-hand side of Eq. 142 the multiple-scatter source function, J ms p 1s8, u, f2, we may break up the integral over ds8 in Eq. 172 into a sum of integrals over each of the layers. Thus for point s i, I p ms 1s i, u, f2 5 I p ms 1a, u, f2exp32t1s i, a24 1 eas i J p ms 1s8, u, f2exp32t1s i, s824krds8, 182 where the first term on the right-hand side is the sum of the integrals from the top of the atmosphere to point a in Fig. 2 along the line of sight, and the second term gives the contribution from point a to point s i. 3. Method As described in Herman et al., 1 the intensity field is broken into discreet u and f directions as required to meet the necessary accuracy requirements. This dictates that more angles be used for near-tangent directions than for other directions in which a coarser angular grid is adequate. The single-scattered polarization parameters at every grid point within and on the cone are exactly calculated by the use of Eq. 162 or the analogous equation for upward radiation to give an initial radiation field. This field is used as the first-guess solution in the Gauss Seidel procedure. Using an iterative technique, one computes the multiple scatter at each grid point to satisfy the radiative transfer equation, as described below. Fig. 2. Grid system that shows the zenith and conical boundary used to compute polarization parameters on the zenith. This two-dimensional cross section shows the boundary associated with the h and h1180 lines. Labeled points are discussed in the text. A. Zenith Calculation To calculate the multiple scatter at point s i along the zenith 1see Fig. 22, we assume that calculations have already been performed for the radiation field at the grid points along the z i21 shell. Because the incident multiple-scatter term, I ms p 1a, u, f2, lies between grid 20 July Vol. 34, No. APPLIED OPTICS 4565

4 points, it is not explicitly calculated. It is estimated by the multiplication of an interpolated multiple scatter-to-single scatter ratio at point a with the previously computed single scatter at point a as shown in I p ms 1a, u, f2 5 I p ss 1a, u, f2r1a, u, f Here I ss p 1a, u, f2 is the single scatter at point a and R1a, u, f2 is the ratio of the multiple to single scatter interpolated by the use of a quadratic fit to the known values of the ratio at the grid points s i21 1C 0, h2, s i21 and s i21 1C 0, h The ratio is used because it was found to be a smoother function between the calculated directly because the angular distribution of the polarization parameters is not known along the integration path from b to s i 1C 0, h2, because this path is outside our conical solution volume. Therefore, the multiple scatter for the pth polarization parameter for a grid point such as s i 1C 0, h2 on the cone boundary is approximated with the use of the following assumptions. First, for the q 5 1, 2 elements of the phase matrix, a weighted average value is defined for the zenith between points a and s i. This weighting is over both the transmission and the polarization parameter distribution through the layer. The averaging of the source function over transmission between a and s i is given by J pq ms 1u, f2 5 ea s i 2p p P pq 1s8, u, f, u8, f82i q 1s8, u8, f82sin u8du8df8 exp32t1s i, s824krds s i exp32t1s i, s824krds8 ea points at all angles than the multiple scatter itself. The incident term is then attenuated through the layer to the zenith at point s i as per the first term on the right-hand side of Eq The multiple scatter on the zenith caused by scattering within the layer from a to s i is given by the second term on the right-hand side of Eq The optical depth integration is done with the assumption that the phase matrix is constant through the layer, and that the polarization parameter, I q 1s8, u8, f82, varies linearly between a and s i. Because the single-scatter component is known only for the u and f directions at point a, the ratioing technique used above cannot be employed here. Instead the single plus multiple scatter is interpolated by the use of the values at points s i21 1C 0, h2, s i21 and s i21 1C 0, h11802, whereas the value at s i is from the previous iteration. B. Cone Calculation Once the zenith solution has been calculated for s i, these new values are used to modify the cone values at s i 1C 0, h2. The cone incident multiple-scatter term at point b is calculated in a fashion similar to that of the incident zenith term 3Eq However, at point b only the single scatter is known, but the ratio of the multiple to single scatter is not. Therefore, the ratio at point s i21 1C 0, h2, which is known, is used for the value of R in Eq. 192 to determine the incident multiple scatter at b. The multiple-to-single scatter ratio at point s i21 1C 0, h2 applied to point b was found to introduce acceptably small errors into the boundary approximations. Under most conditions, these boundary approximations produce less than 1% error in the zenith solution. 1 The multiple-scatter contribution from within the layer for the cone grid point at s i 1C 0, h2 cannot be It should be noted that unlike the more general multiple-scatter equation, Eq. 172, here the repeated indices are not summed over. The polarization parameter weighted angular averaging is then done, yielding an average phase matrix element given by P pq 1u, f2 5 2p p J pq ms 1u, f2 I q 1s, u8, f82sin u8du8df8, 1112 where s is the midpoint on the zenith between s i21 and s i. Second, this average phase matrix element on the zenith P pq is assumed to be valid on the cone as well. The assumption here is that by defining a phase matrix average, P pq, which expresses the multiple scatter between a and s i in terms of the polarization parameter average between points s i21 and s i,wemay use the same P pq for the multiple scatter on the cone between b and s i 1C 0, h2, using the intensity average along s i21 1C 0, h2 and s i 1C 0, h2. Using the values of I q 1u8, f82 at midpoint s1c 0 2 between points s i21 1C 0, h2 and s i 1C 0, h2, we use P pq to estimate the first two polarization parameters contribution to the multiple scatter on the cone caused by scattering between b and s i 1C 0, h2, as shown in the first two terms on the right-hand side of Eq below. Finally, because the I 3 polarization parameter can be represented as a sine function with respect to f, its integral over f is identically zero, which would make ms Eq undefined. For this case, the value of J p3 at the zenith is used directly from Eq without any angular averaging. Therefore, the multiplescatter contribution for the cone can be written in the 4566 APPLIED Vol. 34, No. 20 July 1995

5 form eb s i1c 0, h2 J p ms 1s8, u, f2exp52t3s i 1C 0, h2, s846krds8 5 P p1 1u, f2 3 eb s i1c 0,h2 2p 1 P p2 1u, f2 3 eb s i1c 0,h2 p I 1 3s1C 0 2, u8, f84sin u8du8df8 exp52t3s i 1C 0, h2, s846krds8 2p 1 J p3 ms 1u, f2 eb p I 2 3s1C 0 2, u8, f84sin u8du8df8 exp52t3s i 1C 0, h2, s846krds8 s i1c 0,h2 exp52t3s i 1C 0, h2, s846krds It should be noted that this method maintains the vector identity for each component in the source integral. It is clear that for the scalar case, when polarization is neglected, there is only one average source term given by Eq and therefore only one term in the source-function integral. Under these conditions, it is easily shown that Eq reduces to the form given in Ref. 1 3Eq Results As stated above, the polarized spherical Gauss Seidel 1PSGS2 model accuracy has been compared with three different Monte Carlo models. 9,12,13 Some of the Monte Carlo models use the plane parallel atmosphere calculations of Coulson et al. 18 for comparison, and those results are presented as well. For comparison with the other models available, the polarization parameters used in the PSGS code are converted to the Stokes parameters. The Stokes parameters can be computed from the above polarization parameters by the use of and the plane of polarization angle, x, is computed from tan 2x 5 U Q As mentioned above, the total intensity calculated by the use of a scalar Rayleigh code may be significantly different from the Rayleigh solution with polarization. An example of this is shown in Fig. 3 for a vertically inhomogeneous 81-km atmosphere with the Sun directly overhead 1u The atmosphere has a total vertical optical depth of 1.0, and the surface reflectivity is 0.0. The unpolarized spherical Gauss Seidel and PSGS codes differ by,10% when viewing the nadir 1180 view angle2 from the top of the atmosphere, whereas the polarized plane parallel and PSGS codes agree very well. As the line of sight moves toward the horizon, the unpolarized spherical Gauss Seidel code changes from underestimating the total intensity near the nadir to overestimating near the horizon. The differences between a plane parallel and a spherical atmosphere are apparent when viewing through the limb of the atmosphere 1i.e., view angles Note that the abrupt drop-off in the intensity at 110 would be smoother with the addition of extra viewing angles. Figure 4 shows the same codes for an 80 solar zenith angle and surface reflectivity of 0.5. It is clear that even though the differences between the two spherical codes are smaller because of the nonzero surface reflectivity, the unpolarized is still noticeably different from the polarized. It is also obvious from Fig. 4 that the plane parallel atmosphere is a significantly worse approximation than in the previous case 1Fig. 32, because of the low solar elevation angle. For a more complete discussion of these geometry effects, see Ref. 1. I 5 I 1 1 I 2, 1132 Q 5 I 1 2 I 2, 1142 U 5 I 3, 1152 V 5 I 4, 1162 as shown in Chandrasekhar. 2 Because V ; 0 for Rayleigh scatter of natural sunlight, the radiation is always linearly polarized. Under these conditions, the percent polarization is given by P 5 1Q2 1 U 2 2 1@2 I 3 100, 1172 Fig. 3. Comparison of reflected total intensity at the top of atmosphere calculated by three versions of the Gauss Seidel code for optical depth t51.0, solar zenith angle u , and surface reflectivity A July Vol. 34, No. APPLIED OPTICS 4567

6 Fig. 4. Same as Fig. 3 except for u and A Fig. 6. Same as Fig. 5 except for t51.0. The first model comparison is with the Monte Carlo code of Adams and Kattawar for the reflected singlescatter intensity. 9 Adams and Kattawar used a scalar code for a vertically homogeneous 100-km atmosphere with a planetary radius of 6371 km, a surface reflectivity of 0.0, and an input solar flux of p units. Because the incident solar radiation is completely unpolarized, from Eq. 152 it can be shown that the single-scattered intensity for the polarized and unpolarized cases must be equal. Figures 5 and 6 show that the spherical Monte Carlo and PSGS codes are nearly identical except for very small random fluctuations. The largest difference between the two codes is,0.5%, while most points agree to three significant figures. The differences between the spherical codes and the plane parallel code, as shown in Figs. 5 and 6, are due to differences of line-of-sight path length and solar zenith angle between the plane parallel and Fig. 5. Reflected single-scatter intensity at the top of a homogeneous 100-km conservative Rayleigh atmosphere with t50.25 and u ; the crosses are the spherical shell Monte Carlo calculations of Adams and Kattawar 9 and the dashed curve is their plane parallel solution. spherical atmospheres. In a spherical atmosphere the local solar zenith angle changes at each point along a line of sight but is constant along the line of sight in a plane parallel atmosphere. The local solar zenith angle is defined as the angle between the solar beam and the radial line through the desired point on the line of sight. This change makes the spherical solar attenuation either larger or smaller than that of the plane parallel one, depending on the altitude and the local solar zenith angle, but for solar zenith angles,60, the difference in attenuation between the spherical and the plane parallel atmospheres is very small. In contrast, increasing the line-of-sight path length gives a longer path over which to collect single-scattered photons. For u greater than,95, the reflected spherical pathlengths are longer than the plane parallel ones. Figure 5 shows that for an optical depth of 0.25, the spherical atmosphere single-scattered reflected intensity increases steadily from the nadir to,100, at which angle the line of sight becomes tangent to the Earth s surface. At that point the intensity levels off sharply as the line of sight looks through the limb of the atmosphere. As the line of sight looks higher in the atmosphere, the spherical path length begins to decrease rapidly to zero. With an optical depth of 1.0, as shown in Fig. 6, the spherical intensity is still larger than the plane parallel one for angles between approximately 95 and 135 because of the longer path length, but the differences are smaller because the larger optical depth produces greater extinction along the line of sight in the spherical case. The Adams and Kattawar intensity does not show the decrease in intensity near the limb because their data only extend to 92. For a solar zenith angle of 84.26, shown in Figs. 7 and 8, the agreement between the two spherical codes is excellent whereas the plane parallel code is significantly different, especially for the t50.25 case near the horizon 1i.e., viewing angles near For the t51.0 case, the plane parallel and spherical codes are closer but the spherical one gives larger intensi APPLIED Vol. 34, No. 20 July 1995

7 Fig. 7. Same as Fig. 5 except for u ties at all angles out to within <2 of the horizon. In these figures the effect of changing the local solar zenith angle along the line of sight is clearly shown by the spherical model intensity differences between the f50 and f5180 directions. This occurs because the local solar zenith angle at all points along the f5 0 line of sight will be less than that along the f5 180 line of sight. This makes the attenuation of the solar beam smaller for points along the f50 line of sight, which gives rise to larger single-scatter intensities. The second model comparisons are for the 50-km vertically homogeneous atmosphere of Collins et al., 12 based on results first reported by Blättner et al. 19 The values used for this comparison were estimated from the plotted values shown in Blättner et al. 19 that were generated by their Monte Carlo code. The results of Coulson et al. 18 for a plane parallel atmosphere are also shown for comparison. The planetary radius and surface reflectivity are the same as in the previous case. All cases were run for a solar zenith angle of and an optical depth of Fig. 9. Transmitted total intensity in the f plane at the bottom of a 50-km homogeneous conservative Rayleigh atmosphere with t50.25 and u ; the crosses are the spherical shell Monte Carlo calculations of Blattner et al. 19 and the dashed curve is the Coulson et al. 18 plane parallel solution. As previously shown by Collins et al., 12 there is very little difference between the spherical and plane parallel atmospheres for small solar zenith angles except near the horizon. 12 There is a fairly large statistical fluctuation in the Monte Carlo results because of the low number of photon histories per receiver angle 1, Figures 9 and 10 show the transmitted intensity for the f and f planes, respectively. Except for the statistical variability, there is good agreement between the Monte Carlo and PSGS codes. There are clearly large differences between the spherical codes and the plane parallel code, especially near the horizon. In the f 5 0 direction the difference is approximately 25% at 80 and nearly a factor of 2 at Figure 11 shows similar agreement between the spherical models for the reflected intensities. The reflected radiation does show better agreement between the Monte Carlo and Fig. 8. Same as Fig. 5 except for t51.0 and u Fig. 10. Same as Fig. 9 except for f plane. 20 July Vol. 34, No. APPLIED OPTICS 4569

8 Fig. 11. Reflected total intensity at top of the atmosphere for same conditions as in Fig. 9. Fig. 13. Transmitted Q parameter for the same conditions as in Fig. 9 except for f plane. PSGS codes on the magnitude and angle of the peak intensity at the limb. For comparisons of Q and U, because Blättner et al. 19 and Coulson et al. 18 used the opposite definition for Q 3see Eq , the sign of Q was changed to agree with their notation. Furthermore, the sign of U must be changed as well to preserve the correct plane of polarization angle 3Eq Figures 12 and 13 show the transmitted Q parameter. The PSGS results confirm the Blättner et al. 19 results that in the f plane the Q parameter is slightly larger in magnitude for spherical geometry than for the plane parallel one, except in the f5180 direction near the horizon. Figure 13 also shows that the plane parallel and spherical codes are much closer over a larger range of angles for the f plane than for the f plane. The discrepancies between the spherical codes near the horizon appear to be due to large statistical fluctuations in the Monte Carlo code. Figure 14 generally shows good agreement between the PSGS and Monte Carlo codes for the U parameter. The spherical codes show a slightly greater absolute value than the plane parallel code, as noted by Collins et al. 12 There appears to be an anomalous value for U at 0 shown in the Blättner et al. 19 results that is not shown in the paper published by Collins et al. The reason for this discrepancy is not known. There also seems to be some problem with the percent polarization results presented by Blättner et al. for u Those results do not agree with the published percent polarization curves of Collins et al. Comparisons of the PSGS and Coulson et al. results show that for both the transmitted and reflected cases, the results of the two codes are virtually identical. This finding agrees with the assertion of Collins et al. that the percent polarization is not highly affected by differences in the spherical and plane parallel atmospheres. 12 However, the small differences do confirm the geometrical effect on the Fig. 12. Fig. 9. Transmitted Q parameter for the same conditions as in Fig. 14. Fig. 13. Transmitted U parameter for the same conditions as in 4570 APPLIED Vol. 34, No. 20 July 1995

9 Fig. 15. Transmitted plane of polarization for the same conditions as in Fig. 13. ratio of the single scatter to total intensity. It has been well established that the percent polarization of the total intensity is smaller than that of the singlescattered radiation because multiple scattering generally depolarizes the single-scattered light. For transmitted light the spherical atmosphere has a slightly larger percent polarization than the plane parallel one, which agrees with the larger single-scatter ratio observed in the spherical atmosphere. For reflected light the situation is reversed. The comparison of the plane of polarization angle 1i.e., the angle between the plane of polarization and the vertical plane2 for radiation in the f plane is shown in Fig. 15. As pointed out by Collins et al., 12 the angle calculated with Eq gives the opposite sign of the angle reported in Ref. 18. As in the previous case, the geometry seems to have very little effect on the values of x. Fig. 17. Same as Fig. 16 except for u The final comparison is with Blättner and Wells 13 for a 100-km-thick vertically inhomogeneous atmosphere based on the 1962 U.S. Standard Atmosphere. 20 Figures 16 and 17 show the transmitted total intensities and the single-scatter intensities. The differences in the results between the codes might be due to refraction that is included in the Monte Carlo model but not in the PSGS model. With u , for a view angle of 5, the singlescatter intensity difference between the codes is,0.4%, whereas the PSGS is 4.2% smaller than the Monte Carlo for an 85 view angle. When the solar zenith angle is increased to for the same view angles as above, near the zenith the PSGS single scatter is larger by 2.5% but at the horizon is smaller by 2.5%. The total intensity differences between the codes are generally of the same order and pattern as the single-scatter results. However, a problem with the Monte Carlo code for viewing angles near the vertical was noted previously 21 and is clearly shown in Fig. 16. Transmitted total intensity and single- and multiplescatter intensity for 100-km vertically inhomogeneous Rayleigh atmosphere for t50.25 and u The curves represent the results of the PSGS whereas the symbols are for the Blättner and Wells 13 Monte Carlo calculations. Fig. 18. Comparison of the transmitted total intensity for 100-km vertically inhomogeneous and homogeneous Rayleigh atmospheres for t50.25 and u The dashed curve represents the results of Ref July Vol. 34, No. APPLIED OPTICS 4571

10 the figures here. As with the previous cases, in general the agreement between the PSGS and the Monte Carlo model of Blättner and Wells is quite good. A study by Blättner et al. 22 showed that a homogeneous spherical atmosphere is actually less realistic for modeling low Sun conditions than a plane parallel atmosphere. This is clearly shown in Fig. 18. The intensity for the vertically inhomogeneous atmosphere is much closer to the plane parallel than that for the homogeneous spherical atmosphere, especially for view angles Conclusions We presented results that use the Gauss Seidel method for a spherical atmosphere with polarization fully included and compared these new results with previously developed methods. These comparisons show excellent agreement between the new model and the Monte Carlo method for all polarization parameters. Comparisons of the single-scatter intensities show maximum differences of less than 0.5%, while most points are identical to three significant figures. There are also differences caused by statistical fluctuations in the Monte Carlo method, but these appear to be very small. There is also generally good agreement between the polarized spherical Gauss Seidel method and the Monte Carlo method for the total intensity, Q, U, and the plane of polarization when all orders of scattering are included. Problems with the data of Blättner et al. 19 prevented a direct comparison of the percent polarization. Clearly the statistical variability of the Monte Carlo method of Collins et al. 12 and Blattner and Wells 13 is significantly larger than for the single-scatter calculations of Adams and Kattawar, 9 but the overall agreement between the Monte Carlo and PSGS codes is clear when compared with a plane parallel result. One obvious advantage to the new method is that it avoids the statistical fluctuations that are inherent in the Monte Carlo method. Another advantage is that it gives all the polarization parameters at all grid points along the zenith for all grid-specified directions. Once the overall radiation distribution is known, it is easy to interpolate between the grid points for any desired position and line-of-sight direction. This method also has the advantage of incorporating any desired phase matrix, which permits aerosols to be included easily without any changes to the code. With the addition of a special extraangle subroutine, the solar aureole can be accurately calculated as described in Herman et al. 23 This research was partially supported by NASA contract NAG References 1. B. M. Herman, A. Ben-David, and K. J. Thome, Numerical technique for solving the radiative transfer equation for a spherical-shell atmosphere, Appl. Opt. 33, S. Chandrasekhar, Radiative Transfer 1Dover, New York, 19602, pp. 42, R. E. Bellman, H. H. Kagiwada, and R. E. Kalaba, Diffuse reflection of solar rays by a spherical shell atmosphere, Icarus 11, S. J. Wilson and K. K. Sen, Light scattering by an optically thin inhomogeneous spherically symmetric planetary atmosphere, Astrophys. Space Sci. 69, A. Dahlback and K. Stamnes, A new spherical model for computing the radiation field available for photolysis and heating at twilight, Planet. Space Sci. 39, D. E. Anderson, Jr., The troposphere-stratosphere radiation field at twilight: a spherical model, Planet. Space Sci. 31, G. I. Marchuk and G. A. Mikhailov, The solution of problems of atmospheric optics by Monte Carlo method, Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 3, D. G. Collins and M. B. Wells, FLASH, a Monte Carlo procedure for use in calculating light scattering in a sphericalshell atmosphere, Rep. AFCRL Radiation Research Associates, Inc., Fort Worth, Tex., C. N. Adams and G. W. 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