Jun 28, 1994 - ation is extended to Joie Robinson and Patricia Peterson for typing this paper. REFERENCES. 1. K. F. Evans, J. Awros. Sci. 30,3111. (1993).
J. Quant,Spectrosc.Radiat.TransferVol. 53, No. 4, pp. 425W144, 1995 Cotwri~ht ~ 1995 Elsevier Science Ltd –..”= Printed in Great Britain, All rights reserved 0022-4073(95)00003-8
Pergamon
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THE PICARD ITERATIVE APPROXIMATION TO THE SOLUTION OF THE INTEGRAL EQUATION OF RADIATIVE TRANSFER—PART I. THE PLANE-PARALLEL CASE KWO-SEN
KUO,~
RONALD
C. WEGER,
and RONALD
M. WELCH
Institute of Atmospheric Scienees, South Dakota School of Mines and Technology, 501 E. St, Joseph Street, Rapid City, SD 57701-3995, U.S.A.
(Received 28 June 1994) Abstract—The present Picard Iterative (PI) approach is designed to be a compromise between the generality of Monto Carlo techniques and the numerical efficiency of the existing analytical approaches. The PI method, like the Successive Orders of Scattering approach, begins with the integral equation form of the radiative transfer equation (RTE). Starting with an approximate radiance field, the PI performs a fixed-point iteration. The solution converges in approx. 10 iterations for C 1 water cloud phase functions and in approx. 20 iterations for highly peaked cirrus-type phase functions. The PI iteration (1) is unaffected by vertical inhomogeneities; (2) converges even for highly peaked phase functions; and (3) can be extended to 3-D geometries. However, it is less efficient for (1) optically thick, but homogeneous, media; and (2) media with a weakly peaked scattering phase function.
1. INTRODUCTION
The solution to the radiative transfer equation (RTE) has a rich and varied history with extensive applications in such diverse areas as astrophysics, nuclear reactor design, and atmospheric science. In fact, the radiative energy balance lies at the heart of what is designated as planetary climate. Indeed, radiation is the driving force for atmospheric motions, and a detailed accounting of regional energy balances is critical for predictions of global climate change. At the present time, global prediction models have insufficient accuracy, due in part to the fact that they do not fully incorporate variations in those atmospheric parameters which affect the radiative balance. The extent and variability of cloud cover, cloud 3-D effects, cloud optical thickness and particle size, and aerosol properties are a few examples of the structural details required for accurate modeling. Satellite remote sensing provides a means for the determination of many of these important atmospheric features. These remote sensing applications require irradiances for the radiative energy balance studies, while accurate radiances are necessary for the retrieval of many of the physical parameters of the atmosphere. Good solution techniques of the RTE should incorporate the following characteristics: (1) determine accurate radiances; (2) handle 3-D geometries; (3) perform well over a wide range of phase functions, from smooth to highly peaked; (4) allow the optical characteristics of the medium to vary spatially; and (5) be computationally efficient. In order to place the present approach in proper perspective, it is first desirable to survey briefly the existing solution technique and to evaluate them in light of the above criteria. Until recently,”2 almost all solution techniques were limited to plane-parallel problems. Therefore, we begin with a survey of 1-D approaches. The general problem of radiative transfer in plane–parallel media is largely solved. However, highly peaked phase functions, whose series expansions involve a correspondingly large number of terms, pose computational difficulties. The standard computational approaches can be divided into one of five basic strategies: (1) differential equation approaches; (2) integral equation approaches; (3) adding/doubling approaches; TTo whom all correspondence
should be addressed. 425
426
Kwo-Sen Kuo et al
(4) invariant embedding approaches; briefly summarize these approaches. 1.1. The D@erential
and (5) stochastic
approaches.
The following
subsections
Equation Approaches
Examples of the Differential Equation Approach are the Discrete Ordinates (DO) Method’ and the Spherical Harmonics (SH) Method .’5-7 The DO method is among the most popular of the solution techniques. It discretizes the zenith angle into n finite values, with n equal to the number of terms used in phase function expansion, The multiple scattering integral is replaced by a quadrature over discrete angles. The resultant system of coupled differential equations is solved using an eigensystem approach. There exists two varieties of the SH method. One uses the finite difference approach”5 for which the inverse of a matrix is found. For the other,6’7 an eigensystem similar to that of the DO method is solved. The latter approach is more accurate, hence more preferable, than the former. However, in a medium of higher (2 or 3) dimensions, it would seem that only the finite difference approach is applicable.’ The eigensystem approach developed by Takeuchi7 is especially worth mentioning, since it reduces the size of the system by a factor of two. Moreover, it satisfactorily resolves the numerical instability that arises in the boundary value problem without resorting to artificial scaling transformations.4’6 These methods are widely used, are computationally efficient for many problems, and are accurate over a wide range of phase functions and optical thicknesses. However, they are not without limitations. When the phase function is highly peaked (i.e., a large number of expansion terms is required), the matrix becomes very large; in some cases, the number of rows and columns may exceed 1000. The inverse or the eigenvalue problem for such cases is computationally prohibitive on present-day workstations. Furthermore, both of these techniques, when applied to situations in which the optical characteristics vary with height, require that the atmosphere be divided into a series of homogeneous layers. A corresponding eigenvalue problem must be solved for each layer, and the associated boundary value equations must be solved at each interface of the adjacent layers. Another unfortunate disadvantage encountered in layering is that one must use the same number of terms in the spherical harmonics expansion of intensities in order to match boundary conditions at each interface. This means that the layer with the most anisotropic phase function determines the angular resolution necessary for the problem as a whole. For instance, a cirrus layer over a stratocumulus cloud might require hundreds of expansion coefficients for each layer. Thus the resulting boundary value equations may become unmanageable. And the computional expense significantly increases with the number of layers used. These methods also have other deficiencies. In the highly peaked case the number of discrete angles used in DO becomes large. This often results in quadrature angle(s) close to the solar angle, causing numerical singularities. By contrast, the SH approach,’ being an analytical technique, provides an approximate solution defined for all values of the zenith angle. One difficulty associated with this method has been a spurious oscillatory component resulting from the truncation of an infinite series. However, this difficulty can be largely overcome by an additional smoothing operation.8 1.2. The Integral Equation Approaches One example of the Integral Equation Approach is the Successive Orders of Scattering (SOS) method9 which uses an iteration approximation. The solar radiance is used as the initial term in a sequence of approximate solutions. Each term in the sequence is, in turn, substituted into the right-hand side of the integral equation, yielding the next term in the sequence. The integral equation approach can accommodate highly peaked phase functions and variable optical characteristics with little additional computational overhead. Therefore, it has a number of desirable properties. The principal drawback of this technique is that, even for smooth phase functions, the rate of convergence can be very S1OW,making the method computationally inefficient, except for cases in which the optical thickness is small.
427
PI approximation—I
1.3. Adding/Doubling
Approaches
The previous methods are based upon a perspective which models the interaction of radiation within infinitesimal volume elements. In contrast, the Adding/Doubling and Invariant Embedding approaches are derived from a macroscopic viewpoint. A homogeneous layer in the atmosphere is viewed as a linear two-port network (i.e., the Principal of Interaction). Scattering and absorption are described by a lumped parameter scattering matrix analogous to that employed in transmission line theory. In the Adding/Doubling method, each homogeneous layer within the atmosphere is divided into N thin layers. The scattering martrix for a single thin layer may be written directly in terms of the phase function. The scattering matrix relates the incoming intensities at the top and bottom of the layer to the outgoing intensities at those positions. That is, (1) The scattering matrix for two adjacent layers (not necessarily identical) is computed from those associated with the two layers: S = Sl * S1, where S1 and S1 and S2 are the scattering matrices of the individual layers and the (*) operation is an associative binary operator which turns the scattering matrices of the contributing layers into a semi-group. Thus, for an atmosphere consisting of N layers,
s=s~*sN_,
*”’”
*s,.
(2)
For computational efficiency, each homogeneous layer often is divided into N = 2k layers. The scattering matrix for the composite layer thus can be computed in k = Iogz N squaring operations (i.e., S = S, *S, ). For homogeneous layers, the Adding/Doubling method is computationally efficient. It also performs well for a wide variety of phase functions; however, for highly peaked phase functions, the radiances must be evaluated at a large number of discrete angles. If the optical properties of the atmosphere vary significantly with optical thickness ~, the computational burden may become large. Finally, it is not readily apparent how the method may be generalized to arbitrary 3-D geometries. 1.4. The Invariant Embedding
Approach
The Invariant Embedding Approach generates a differential equation scattering matrix S(?) for a layer of optical thickness &. Since S(7 + AT)= S(AT) * S(T), and using the Taylor expansion
from the macroscopic
(3)
of S(AT ) given by S(A~) = E + GAT,
(4)
where E is the identity matrix, and G = d’(T ) dz
T+o ‘
(5)
one obtains ~S(T)=G
*S(T),
(6)
The result is a set of differential equations for the components of the scattering matrix which may be solved numerically. The matrix G is determined by the optical properties of the medium and is independent of the optical thickness. 1.5. The Stochastic
Approach
An example of the Stochastic Approach is the use of Monte Carlo (MC) simulations for the computation of radiances. Typically, the trajectory of an ensemble of photons is traced through the atmosphere. The locations and outcomes of interactions with scattering events are determined QSRT53,+F
428
Kwo-Sen Kuo et al
randomly. Advantages of these stochastic approaches include: (1) arbitrary 3-D geometries can be simulated; (2) the introduction of variable optical characteristics causes no significant increase in computational expense; and (3) the run-time does not depend upon the nature of the phase function. The principal disadvantage of this technique is that it requires long run-times to achieve reasonable accuracies. In comparison to the other methods, it is so CPU intensive as to preclude its use in all but the most complex problems. In the present paper, a technique is presented which can accommodate both highly peaked phase functions and variable optical parameters in the atmosphere. In addition, it possesses a natural extension to 3-D geometries. This approach is based upon the integral form of the RTE. An initial estimate of the intensities is obtained by employing a low order expansion of the SH solution. Then, in a manner analogous with the SOS approach, improved solutions are obtained by substituting the previous solution into the right-hand side of the integral form of the RTE. Section 2 discusses the construction of the initial estimate, while Sec. 3 outlines the numerical details involved in solving the RTE. Section 4 presents selected results and comparisons obtained with the DO, SH, and MC methods. Section 5 discusses the extension of the present method to three dimensions, and Sec. 6 concludes. 2,
CONSTRUCTION
OF
THE
INITIAL
ESTIMATES
Although any of the above methods can be used to obtain the initial estimate of the solution, the SH is chosen here because it has a number of desirable characteristics. These are: (1) the solution in analytic and thus exists for all angular values; (2) as only low order approximations are required, the computational expense of the SH method is small in all cases; and (3) the SH approach is well established and is known to provide accurate solutions comparable with those obtained from the best of the other methods. The readers are referred to Takeuchi7 for the details of obtaining the SH solution. Only an outline of the method is given here for the continuity and clarity of the presentation. We start with the RTE of a plane parallel medium, (7) where r is the optical thickness, 0 and ~ are zenith angle and azimuth angle in a spherical coordinate, respectively, p is equal to cos (3, and I(T; p, @) is the radiance in the direction of (0, ~ ) at ~. The source function is given by J(T; p,@)=~
2X do’ (Ju
I dp’@(~, ~;y’,
@’)I(~; p’,@’)
–1
+~nFoexp(–t/uo)@
(p,@; –pO, #JO),
(8)
where nFo is the incicient solar flux, ~ ~ is the cosine of solar zenith angle, and ~ ~ is the solar azimuth angle. We choose to absorb the single scattering albedo fiO into the scattering phase function @(COS@) = @(~,@; p‘, @J ‘), where (p’ = cos 6‘, @‘) and (~ = cos 6, ~) denote the incident and scattered directions, respective y, and the angle between them is @. Thus, 1
J
(9)
d COS@@(COS@) = 2fio.
–1
@(p, @; ~‘, @‘) can be expressed using its cosine components,
@J”@,p ‘), as
@(p, (j; p’, (b’) = ~ @m(/f,p’)cosl’n(()
– ~’).
(lo)
m
We further expand Z(T; ~, @) using the cosine series. Due to the orthogonality functions, the cosine component RTEs are: dI”’(7 ; p) P
d~
=Im(T; p)–
Jm(T; #),
nZ=o,
1,2, . . . .
relation of the cosine
(11)
429
PI approximation—I
The cosine component r(r;p)=-
source function
1+67 4
‘
is
J
dp’@m(p, p ‘)lm(~; v’) + & nFo exp(–~/p
~)@”’(p, –vo).
(12)
–1
It is normal to focus on solving these differential equations instead of the original one. Then the final solution Z(T; ~, @) is constructed by taking the summation of the cosine components: I(7; p,q))=~zm(T, m
p)cosm(@
(13)
-40).
The SH method further expands the ,u and p‘ dependencies of Eqs. (1) and (2) in terms of renormalized Associated Legendre Polynomials (ALPs) and forms a system of coupled first-order ordinary differential equations (ODES). The coupling is found to be strictly between the coefficients of the odd ALPs and those of the even ALPs. A system of second-order ODES with half the size of the original first-order system therefore can be derived, thereby reducing the numerical cumbrance involved in obtaining the solution. Using an integration factor, Eqs. (11) and (12) become p = O, (14a)
rJm(~; O),
-1 I
Z~(T~; P)exP[– (~,. – T)/#] +
Im(7; ~)=
Im(O; p)exp(~/p)
–
‘Nd~ ‘ , ~Jm(T’;
J
:+J~(TL ,
p)exp[–(T’
,u)exp[-(~
– T)/P],
– T’)/~],
p >0,
(14b)
p 0,
(15a)
,
~m(T;/f)= Im(T – A; p)exp(A/~) {
–
T dT’ ~ Jm(~’; ~)exp[(~ – 7 ‘)/p], J T—A
p O), Eq. (16a) is applied, resulting in the computation of Zm(T~;~j) defined for all p,> O. Finally, a new value of J’”(T,l;p,) is computed from Eq. (17) and completes one Picard iteration (PI). This procedure is iterated until the maximum change in the radiances is smaller than some preassigned error value at all T. and p,. The downward radiances at the top of the atmosphere (T = O) and the upward radiances at the bottom of the atmosphere (~ = T,,,) cannot be computed from the above procedure. lm(O;pj) is zero for all p,< O(i.e., no downward diffuse fluxes at the top of the atmosphere). At the lower boundary, from the set of Z“’(z,v;p~), where p~ 0 is determined appropriate boundary condition. The surface boundary conditions typically are assumed to be either vacuum, Lambertian, or bidirectional reflecting. In the widely used DO and SH approaches, vertical inhomogeneities are treated as a composite of vertically homogeneous sublayers. As noted above, the impact of a large number of sublayers upon the boundary layer equations places practical limitations upon the number of sublayers that can be accommodated. Specifically, the storage requirements for the boundary layer equations depend quadratically upon the number of layers. In contrast, the iterative procedure outlined above accommodates vertical inhomogeneities in a natural way. In this approach, one represents the phase function at the level, Pm(TH;flj, ~,), as an A x A matrix, where A is the total number of discretized angles. In the vertically homogeneous case, the storage requirement is A ~. When there are M distinctly different regions within the atmosphere, each with its own phase matrix, the storage requirement is MA 2. Thus, the incorporation of layers in the iterative scheme is linear rather than quadratic in its storage requirements as a function of the number of layers and imposes no extraordinary computational burdens. 4. RESULTS
The Discrete Ordinate Method is a well-established, powerful, and efficient technique for the solution of the RTE in plane–parallel atmospheres. In order to evaluate the performance of the present algorithm, comparisons are made with the DO method, employing a variety of phase functions ranging from isotropic to highly anisotropic. In addition, cases are presented in which the solar zenith angle is varied from O to 60’ and for surface albedos of &30°/0. The DO approach is generally faster than the PI method for the cases tested, For 8 ~ = 60° and with a moderately peaked phase function typical of water clouds, the PI method is faster for ~ = 1 but slower for T = 10. For highly peaked phase functions typical of cirrus clouds, the present
431
PI approximation—1
method runs successfully while the DO method fails. In order to evaluate the performance of the Picard Iterative Method with highly peaked phase functions, comparisons are made with results obtained from Monte Carlo simulations and with the delta-m (b-m ) approximation incorporated with the DO approach. In all of the above cases, the two methods produce radiances which are in agreement within 0.5%. A detailed comparison of the two methods is presented in Sec. 4.1. Section 4.2 discusses the sensitivity of the PI method to the initial estimate, and Sees. 4.3 and 4.4. investigate the effect of angular resolution and step size upon accuracy, respectively. Finally, Sec. 4.5 compares the Monte Carlo, d-m DO and PI results for highly peaked phase functions. 4.1. Detailed
comparisons
with the Discrete
Ordinate
Method
The present section compares the performance of the DO and PI methods for the case of a single layer, plane–parallel atmosphere. Table 1(a) shows results obtained from a Rayleigh phase function for solar zenith angles of 8 ~ = O, 30, and 60’, and optical thicknesses of ~ = 1 and 10. For each of these cases, results are shown for albedos of a = O, 6, and sOO/o. For each choice of optical thickness, solar zenith angle and albedo, Table 1(a) displays the respective run-times obtained on a Silicon Graphics 4D/35 workstation and the percent flux conservation assuming a single scattering albedo do = 1 (i.e., no absorption). Table 1(b) provides corresponding information for the Diermendjian13 Cl water drop phase function with asymmetry parameter of g = 0.83.
Table 1, Comparison of run-time (in hh: mm: ss format) and flux deviation (in percentage) between the DO and P1 methods for the Rdyleigh (a) and Cl (b) scattering phase functions. . a, Rayleigh Phase Function Albedo a
Solar ZeniOl Angle
Optical Thickness T
o“
0.1
1
0.00%
o% 10
10
0.01%
0.00%
35.3
0.1
4.6
0.00%
——
0.1
-0.08%
-0.15%
-0.0170
-0.16%
0.0170
1
0.03% -15.1
10
0.13% 14.7
1 6%
0.07% 15.3
10
0.07% 14.9
1 30%
0.05% 15.6
10
OJXYXO 0.0290 -.— ‘43.2 1,4
-0.18%
-0.07%
-0.27%
—
30”
0“
DO 14.5
o%
-0.24% 10,8
0.4
Solar Zenith Angle
Optical TIlkkness
T
0.01% 33.9
-0.06%
49.4
b. Cl Size DiXrihticm
~—————
-0.01% 1.5
11.3
0.4
-0.24% 11.1
0.4
36.6
-o.17% - -0.09%
-0.09%
29.6
1.5 —— -0.16% -0.06%
0.4
O.oo% ~
0.01%
0.00%
0.00% -0.08%
0.01%
0.01%
10.8
0.4
48.6
P1 11.1
0,4
-30.8
““ 0.4
0,01%
0.1 –— -0.09%
DO
11.0
1,5
4.5
0.00%
10
Albedo a
0.4
0.1
1 3090
PI
4.5
-0.15?’
-0.06Y0
1 6V0
DO
28.0
0.1
w
30” — Pl
DO
0.00%
et
DO
9.2
20:45.6
0.00%
58:46,4
0,13% 10,3 0.00% 2:27.8 0,13% 11.6 0.0+3% 3:37.6 0,15%
9:44.6 0.00%
0.00%
1:08.2
4249,4
.0.089.
0.16%
20:39.7
9:39.1
-0.04% 24:24,2 0.08% 20:21.2 0.00’% ‘23:41.7 -0.10%
60° PI
0.04% M02,6 –
0.17% 9:41.5 0.05%
45:30.2 0.19%
DO 4%32.9 0.03% 5%51.0 -0.13% 4&56,5 -0,08% 31:39.9 -o.12% 47:34,6 -0.01% 3032.2 -0.16%
PI 15:09.4 0.08% 40:27.9 0.23% 15:05.5 0.08% 41:18.2 0.26% 15:00.3 0.09% 4226.0 0.29%
432
Kwo-Sen
Kuo et al
The run-time for DO is proportional to the cube of the number of angles, while for PI it is proportional to the number of 7 -steps multiplied by the square of the number of angles. Thus, one expects that when the number of angles is less than the number of ~-steps (i.e., for nearly isotropic case), then the DO method should be substantially faster. An inspection of Table 1(a) confirms this observation. Indeed, DO is at least an order of magnitude faster than PI for nearly isotropic phase functions (i.e., small numbers of angles). For the Cl phase function with r = 1, the number of angles is larger than the number of ~ -steps. Therefore DO becomes slower than PI, as confirmed in Table 1(b). However, for z = 10, the number of r-steps increases sufficiently that the DC) method regains the advantage. For both DO and PI, the flux conservation is essentially perfect for nearly isotropic phase functions, u = O% and ~ = 1. With increasing albedo and increasing solar zenith angle, there is a small decrease in flux conservation. Note that flux conservation in both methods is negatively impacted by increasing optical thickness. Both DO and PI produce nearly identical radiances using either the Rayleigh or C 1 phase functions for all values of optical thickness, solar zenith angle and albedo studied. Among all cases, the greatest discrepancy between DO and PI naturally arises for r = 10, 6 ~ = 60° and u = 30°A for the Cl phase function. Figure 1(a) shows reflected radiances at the top of the atmosphere for this worst-case scenario at @ – @~ = 45, 90135 and 180”. The corresponding transmitted radiances at the bottom of the atmosphere are shown in Fig. 1(b). The DO uses 232 angles, while PI uses 117 angles. For each case, the worst agreement between the methods occurs at the back-scattering direction. This discrepancy is saliently demonstrated in the lower-right plot of Fig. 1(a) and is believed to be due to the inaccuracy of the Do method. TO verify, the top boundary condition is plotted in Fig. 2 for two cases (see titles of the plots). The downward diffuse radiance should be zero at the top boundary, Indeed we find that while the PI method gives perfect boundary conditions everywhere, DO’s deviates from the truth the most near the solar zenith. The non-zero radiance for PI at p = O is due to the fact that PI is able to calculate the radiance using (14a) while DO has to avoid a quadrature angle at # = O. It has found that one-half as many angles as the terms in the phase function are always enough to guarantee good results using PI. This can be explained by the nature of the quadrature scheme used for the multiple-scattering integral, since the Lobato quadrature using N angles is perfectly accurate for a polynomial up to 2N terms. 4.2. Sensitivity
to initial estimate
The usefulness of a fixed-point iteration is strongly affected by the sensitivity of the convergence to the initial guess. If one must obtain initializations very close to the final solution, then the fixed-point iteration is of little value. For the PI method this is critical, since the algorithm is intended to be applied to 3-D problems, in which case it is difficult to obtain accurate initial estimates. In order to understand the sensitivity of PI to the accuracy of the initial estimate, a number of progressively less accurate initial estimates are employed. These are obtained by reducing the number of terms used in the SH solution. Figure 3 shows reflected radiances at the top of the atmosphere and transmitted radiances at the bottom of the atmosphere for ~ = 1, 0 ~ = 60°, the Cl phase function and # – @~ = 0° and 180°. The number of terms in the initial estimate derived from SH is varied from 58 terms (i.e., 1/4 of the full 232 terms) to 28 (i.e., 1/8) and finally to 14 (i.e., 1/16). From examination of Fig. 3, all of the Picard iterations converge to the same fixed-point (i.e., the DO solution). The only penalty to be paid by degrading the accuracy of the initial radiances is a slight increase in the number of iterations required to attain convergence, namely about 20°/0 for the 14-term expansion. 4.3. E~ect of angular resolution In general, the number of angles required to obtain accurate solutions from the Picard Iteration method depends upon the degree of anisotropy of the phase function. It is never necessary to use more angles than half the number of terms in the phase function expansion. As one progressively degrades the angular resolution, the flux conservation, and the accuracy of radiances grow steadily worse.
Reflected Radiances Cl (f30 = 60°, $ – (#JO = 45”, T = 10, cx = 30%) 1
r
1
Cl (6.
1 00,232 . . . . . . . . . . . PI. 117
= 60”,
Reflected Radiances #J – @e = 90°, T = 10, ~ = 30%)
0.60 ~
0.40
0.30
0.20
J-___—l 0.0
0.2
0.4 0.6 Observation Cosine, p
0.10 0.00 0.6
1.0
Reflected Radiances Cl (00 = 60°, @ – @a = 135”, 7 = 10, a = 30%) 1 1 I 0.50 k J I Do,232 0.40
E
........... PI,
117
1
0.0
0.2
0.4 0.6 Observation Cosiner p
0.8
1.0
Reflected Radiances Cl (00 = 60°, @ – @e = 180°, 7 = 10, a = 30%) 1 I , 1 I t DO,232 117
1
........... PI,
I
0.30 +
0.20
o.oo~ 0.10
0.0
0.2
0.4 0.6 Observation Cosine, p
0.0
1.0
o.o~ 0.0
Fig. 1(a). (Caption overleaf)
0.2
0.4 0.6 Observation Cosine, p
0,8
1.0
Transmitted Radiances Cl (00 = 60°, @ – @o = 45”, 7 = 10, a = 30%) , 1 1 I
0.s0
Transmitted Radiances Cl (6J0 = 60°} $ – ~a = 90°, T = 10, a = 30%) 1 1 0.50 , 1 00.Z22 . . . .. . . . . . . PI.117
Do,292
........... PI,
117
1
i
0.40 c 0.30 F
0.30
0.20
0.20
0.10
0.10
LJ.oo1-
-1.0
1
,
-0.8
,
3
I
-0,6
-0.4 Observation Cosine, p
-0,2
-1.0
0.0
0.50
I
1
1 Do,232 . . . . . . . . . . . PI, 117
0.40
0.30
0.30
0,20
0.20
0.10
0.10
0.00 -0.8
-0.6 -0.4 Observation Cosine, p
-0.6
-0.2
-0.4 Cosine, p
-0,2
I
0.00 -1.0
0.0
-0.8
-0.8 Observation
-0.4 Codne, #
-0.2
Fig. 1(b) Fig. 1, Retlecttxl rdianccs
at the top (a) and transmitted
0.0
Transmitted Radiances Cl (60 = 60°, @ – (#sO= 180°, ~ = 10, a = 30%) I 1 0.50 I 1 00,232 . . . . . .. . . . . PI,117 1
0.40
-1.0
-0.8
Observation
Transmitted Radiances Cl (Oa = 60°, # – #o = 135°, -r = 10, a = 30%) r
1
radiances at the bottom (b) for a cloud layer Cl droplet size distribution
and tio = 1.
0.0
435
PI approximation—I
Cl
(00 I
=
Boundary 60°, r$ – I
Condition $. = 180°,
at Top -r = 1, a
= 0%) I
I
. . . . . . . . . . . . . ..-
w
;-
0.005 –
0.000
& –0.005 –
1
-1.0
Observation
Cl
Boundary = 60°, @ –
(60
Cosine,
Condition (b. = 180°,
p
at Top T = 10, I
I
I
0.0
-0.2
–0.4
–0.6
–0.8
I
I
I
a
= 30Z) I
. . . . . . . .. . . . . . . . 9:
al $ .e z ~ .-N 1 E : z
;-
0.005 – !
(
0.000
I
/
-0.005 –
I
1 -1.0
–0.8
Cosine,
0.0
-0.2
–0.4
Observation Fig. 2. Comparisons
I
I
–0.6
p
of the boundary conditions of the downdwelling diffusive radiation thicknesses of 1 and 10 between DO and PI methods.
for optical
Figure 4 shows upwelling intensities at the top of the atmosphere and downwelling intensities at the bottom of the atmosphere for the C 1 phase function with ~ = 1, 9 ~ = 60°, and @ – @~ = 06 and 180’. In each case, the number of angles used in the Picard Iteration is varied from 117 to 103, 89, and finally to 75. As seen in Fig. 4, there is no appreciable loss of accuracy using 89 angles except in the back-scattering direction. However, when the number of angles is lowered to 75, the solution becomes unacceptable. Although the choice of the number of angles equal to one-half of the number of terms in the C 1 phase function expansion is always sufficient to yield accurate the minimum number of angles radiances, there appears to be no a priori means for predicting necessary
to
avoid
unacceptable
errors.
This
issue,
however,
is important
since
the
run-time
Reflected
Cl((?o= I
,
1.4 F
600, #-#, I
Radiances =0”,7= ,
l,a=
Reflected Radiances Cl (8C = 60°, @ - & = 1800, T = 1, a = ()%)
O%) 00, Pulr
PL 1/4 — PL 1/8
..... ......
.-.
—.___ -.
!m
PI, 1/16
-J
~~
0.0
0.2
0.4 0.6 Observation Cosine, p
0.8
1.0
o.o;~ 0.2
Transmitted Radiances Cl (0. = 60°, (j - #.=0 °,7=l, I 1 1
a=OZ) 1 DO,FUII . . . . . . . . . . . PI,1/4 —-—_ PL 1/9
~04 ,&
0.4 0.6 Observation Cosine, p
Transmitted Radiances Cl (8. = 60°, @ – @. = 180”, T =
0.0
1.0
1, a = 0%)
!
Id
‘~
*&
0.10 ,@
0.05
r F
@~ -1.0
-0.8
-0!6 -0.4 Observation Cosine, p
-0.2
Fig. 3. Sensitivity
0.0
cy~
of the radiances to initial estimates
. -0,6 -0.4 Observation Cosine, p
–0.2
0.0
L
Cl(00=600, 1
a= ,
Reflected
O%)
1
0.25 -
1
T
=
1, a = O%)
1
1 PI, 117
. . . . . . . . . . . PI, IOS ——— — P1.80 PI,76 —------
- L
Radiances
Cl (80 = 60°, #J – @o = 180°,
I
PI. 117
1.4 ●
Reflected Radiances #J-#0=0”,7 =1, 1 I
P1. 10s — PI, 89
...........
——.
–.–.
0.20 –
---
P1.76
0.15
0.10 -
0.05 0.2 r
o.oo~
. . I
0.0
0.2
Cl(f3e= r
10’
0.4 0.6 Observation Cosine, ~ Transmitted Radiances 600, @I#o=OO, T=l, I 1
0.8
a= I
1.0
O%)
I&
:.:
.—_:.
Cl 0.25
PI, 117 . . . . . . . . . . . PLI03 ~0s _
0.0
0.2
0.4 Observation
0,6 Cosine, ~
Transmitted Radiances (00 = 60°, @ – @e = 180°, ~ = I 1 I
0.6
1, a
. ... . .. .. . .
,. . .
U.GU
:.:
r
=
O%)
r
y
PI, CM PI, 75
1.0
.-– :.
PI. 117 PI. 103 PI. 89 PI. 76
-i
r
0.15 -
0.10 -
0.05
~ 0-1 r
-1.0
-0.6
-0.6 -0.4 Observation Cosine, p
-1
o.oo~
lo-’~ -0.2
Fig. 4, Sensitivity
0.0
-1.O
-0.8
of the radiances to angular resolution.
-0.6 -0.4 Observation Coeine, p
-0.2
0.0
438
Kwo-Sen
Cl 0.25
(60 = i
60°,
Kuo et al
Reflected Radiances # – @e = 180°, T = I I
1, ~
= O%] I
DO, 232 . .. ... . .. .. . .. .. PI, 103 0.20 1.
0.05
I o.oo~
I
0.0
0.2
0.6
0.4 Observation
Cosine,
1
1
0.6
1.0
w
Fig. 5. Comparison between DO results with full angular resolution and PI results with partial angular
resohstion.
requirement is proportional to the square of the number of angles. In all cases investigated, the loss of accuracy with decreasing number of angles appears abruptly at some critical value. The nature of this transition is, at present, not understood. To further illustrate the inaccuracy of the DO method near the back-scattering direction, the results of the PI method using 103 angles are plotted along with the results of DO using 232 angles in Fig. 5. It is seen that the Do results agree better with results from PI using a coarser angular resolution (i.e., 103 angles instead of 117 angles).
4.4. Eflect of -r-resolution Since the run-time as possible
is proportional
to the number
(i.e., as large a & as possible),
compatible
of T -steps,
it is desirable
with obtaining
reasonably
to use as few T-steps accurate
solutions.
6 shows reflected and transmitted radiances from the C 1 phase function with ~ = 1, 6 ~ = 0° and albedos of O, 6, and 30°/0. Solutions are shown for & = 0.03125, 0.04, 0.05, 0.0625, and 0.1. It is difficult to detect any significant differences in radiances in Fig. 6 for C5T
