Finite-element algorithm for radiative transfer in vertically inhomogeneous media: numerical scheme and applications Viatcheslav B. Kisselev, Laura Roberti, and Giovanni Perona The recently developed finite-element method for solution of the radiative transfer equation has been extended to compute the full azimuthal dependence of the radiance in a vertically inhomogeneous plane-parallel medium. The physical processes that are included in the algorithm are multiple scattering and bottom boundary bidirectional reflectivity. The incident radiation is a parallel flux on the top boundary that is characteristic for illumination of the atmosphere by the Sun in the UV, visible, and near-infrared regions of the electromagnetic spectrum. The theoretical basis is presented together with a number of applications to realistic atmospheres. The method is shown to be accurate even with a low number of grid points for most of the considered situations. The FORTRAN code for this algorithm is developed and is available for applications. Key words: Atmospheric optics, radiative transfer, finite-element method. r 1995 Optical Society of America 1. Introduction In a recent paper 1 the finite-element method 1FEM2 2,3 was applied to solve the problem of the transfer of monochromatic radiation in a homogeneous slab with absorbing boundaries. The results for the zero-order Fourier component of the intensity were compared with those obtained with the discrete ordinate method 1DOM2. 4–7 It was demonstrated that the FEM algo- rithm, being practically equivalent to the DOM for low values of the asymmetry factor of the phase function, with the same number of grid points, pro- vided an accuracy of at least an order of magnitude higher in the case of highly asymmetric phase functions. This sharp increase in the FEM accuracy is due to the fact that the application of the FEM numerical scheme makes it possible to overcome the main problem of the DOM, namely, the necessity to trun- cate the Legendre series for the phase function, omitting the terms of the order higher than the number of grid points on the polar angle interval. 8 At first glance the truncation seems logical: discard- ing the high-order terms is in a sense equivalent to smoothing, making it possible to assume that it eliminates the unimportant subgrid frequencies of the angular dependence of the phase function. If this were so it would only improve the performance of the scheme without affecting the overall accuracy. A more detailed analysis, however, shows that this is not the case. As the truncation point in the DOM is determined not from a consideration of admissible phase function smoothing but from the necessity to provide flux conservation, the number of remaining terms in the series is too low. Because of this, truncation of the Legendre series filters out not only the subgrid but also the grid size frequencies, which leads to nonphysical oscillations of the solution and a decrease in the accuracy of the DOM scheme as a whole. The FEM scheme is free of this disadvantage: it provides flux conservation independent of grid size and of the number of terms in the Legendre expansion of the phase function. The Legendre series can be truncated in an arbitrary manner 1or even not trun- cated at all2, giving the possibility to perform calcula- tions with highly asymmetric phase functions, requir- ing a large number of Legendre polynomials for the approximation, on comparatively sparse grids. As high asymmetry is typical for the phase func- tions related to the atmospheric aerosol in the optical V. B. Kisselev is with the Saint Petersburg Institute for Informat- ics and Automation, Academy of Sciences of Russia, 14-th linea, 39 St. Petersburg, Russia. L. Roberti and G. Perona are with the Department of Electronics, Polytechnic of Torino, Corso duca degli Abruzzi 24, 10129 Torino, Italy. Received 20 January 1995; revised manuscript received 20 July 1995. 0003-6935@95@368460-12$06.00@0. r 1995 Optical Society of America. 8460 APPLIED OPTICS @ Vol. 34, No. 36 @ 20 December 1995
Transcript
Finite-element algorithm forradiative transfer in vertically inhomogeneousmedia: numerical scheme and applications
Viatcheslav B. Kisselev, Laura Roberti, and Giovanni Perona
The recently developed finite-element method for solution of the radiative transfer equation has beenextended to compute the full azimuthal dependence of the radiance in a vertically inhomogeneousplane-parallel medium. The physical processes that are included in the algorithm are multiplescattering and bottom boundary bidirectional reflectivity. The incident radiation is a parallel flux on thetop boundary that is characteristic for illumination of the atmosphere by the Sun in the UV, visible, andnear-infrared regions of the electromagnetic spectrum. The theoretical basis is presented together witha number of applications to realistic atmospheres. The method is shown to be accurate even with a lownumber of grid points for most of the considered situations. The FORTRAN code for this algorithm isdeveloped and is available for applications.Key words: Atmospheric optics, radiative transfer, finite-element method. r 1995 Optical Society of
America
1. Introduction
In a recent paper1 the finite-element method 1FEM22,3
was applied to solve the problem of the transfer ofmonochromatic radiation in a homogeneous slab withabsorbing boundaries. The results for the zero-orderFourier component of the intensity were comparedwith those obtained with the discrete ordinatemethod1DOM2.4–7 It was demonstrated that the FEM algo-rithm, being practically equivalent to the DOM forlow values of the asymmetry factor of the phasefunction, with the same number of grid points, pro-vided an accuracy of at least an order of magnitudehigher in the case of highly asymmetric phase functions.This sharp increase in the FEM accuracy is due to
the fact that the application of the FEM numericalscheme makes it possible to overcome the mainproblem of the DOM, namely, the necessity to trun-cate the Legendre series for the phase function,omitting the terms of the order higher than the
V. B. Kisselev is with the Saint Petersburg Institute for Informat-ics and Automation, Academy of Sciences of Russia, 14-th linea, 39St. Petersburg, Russia. L. Roberti and G. Perona are with theDepartment of Electronics, Polytechnic of Torino, Corso duca degliAbruzzi 24, 10129 Torino, Italy.Received 20 January 1995; revised manuscript received 20 July
number of grid points on the polar angle interval.8At first glance the truncation seems logical: discard-ing the high-order terms is in a sense equivalent tosmoothing, making it possible to assume that iteliminates the unimportant subgrid frequencies ofthe angular dependence of the phase function. Ifthis were so it would only improve the performance ofthe scheme without affecting the overall accuracy.A more detailed analysis, however, shows that this isnot the case. As the truncation point in the DOM isdetermined not from a consideration of admissiblephase function smoothing but from the necessity toprovide flux conservation, the number of remainingterms in the series is too low. Because of this,truncation of the Legendre series filters out not onlythe subgrid but also the grid size frequencies, whichleads to nonphysical oscillations of the solution and adecrease in the accuracy of the DOM scheme as awhole. The FEM scheme is free of this disadvantage:it provides flux conservation independent of grid sizeand of the number of terms in the Legendre expansionof the phase function. The Legendre series can betruncated in an arbitrary manner 1or even not trun-cated at all2, giving the possibility to perform calcula-tions with highly asymmetric phase functions, requir-ing a large number of Legendre polynomials for theapproximation, on comparatively sparse grids.As high asymmetry is typical for the phase func-
tions related to the atmospheric aerosol in the optical
region, the FEM seems to be promising for practicalcalculations. To verify this assumption the numeri-cal scheme has been extended to compute all theharmonics of the intensity in a vertically inhomoge-neous plane-parallel medium. Bidirectional bottomboundary reflection is also included. A fast recursionprocedure is introduced for evaluation of the elementsof the involved matrices. All these innovations areused in the development of the resulting FORTRANcode.Testing the code is performed on a set of realistic
atmospheric parameters taken from the LOWTRANprogram.9,10 Being reliable and well known this setprovides a wide variety of natural situations thatincludes a broad range of practical situations encoun-tered in atmospheric optics.
2. Theory
A. Finite-Element Method for Solution of the RadiativeTransfer Equation
Considering a plane-parallel atmospherewithL homo-geneous layers, the monochromatic diffuse intensityI1t, h, f2 at optical depth level t along the directiondefined by h 5 cos u and f, where u is the polar angleand f is the azimuthal angle, can be represented by aFourier cosine series 1for more details on the subjectthe reader is referred to Refs. 8 and 11–142:
I1t, h, f2 5 om50
M
Im1t, h2cos mf. 112
The optical depth is zero at the top of the atmosphereand increases downward, being t1L2 at the surfacelevel. The polar angle is measured from the positivedirection of the optical depth, i.e., h . 0 for down-welling radiation and vice versa. Because of thestepwise change of Im at h 5 0, it is useful to separatethe downwelling and upwelling radiation according tothe formulas
5Im11t, h2 5 Im1t, h2 if h . 0,
Im11t, h2 5 0 if h , 0,122
5Im21t, h2 5 Im1t, h2 if h , 0,
Im21t, h2 5 0 if h . 0.132
The resulting equation for the mth Fourier compo-nent in the lth layer 1l 5 1, . . . , L2 is
hdIm61t, h2@dt 5 2 Im61t, h2
1 L1l2/2 e21
0
plm61h, h82Im21t, h82dh8
1 L1l2/2 e0
1
plm61h, h82Im11t, h82dh8
1 L1l2S@4plm61h, h02exp12t@h02,
142
where plm11plm22 is equal to plm for h . 0 1,02 and zerootherwise,
plm1h, h82 5 1@12p2 e0
2p
pl1g2cos1mf2df, 152
where g is the scattering angle whose cosine is givenby
cos g 5 cos u cos u8 1 sin u sin u8 cos1f 2 f82, 162
L1l2 is the single scattering albedo, pl1g2 is the phasefunction, h0 is the cosine of the polar angle of the Sun,and S is such that ph0S is the incident solar flux.The choice of direction f is arbitrary and is defined tobe zero in the solar plane in the direction opposite theSun. In the remainder of this subsection layer indexl is omitted and the resulting formulas apply for eachl. In accordance with the FEM, we seek a solution inthe form
Im21t, h2 5 oi51
N
Iim1t2bim1h2,
Im11t, h2 5 oi5N11
2N
Iim1t2bim1h2, 172
where bim1h2, i 5 1, . . . , 2N, is a system of basisfunctions that are nonzero only in a finite interval.Furthermore it is desirable to have zero values fornonzero intensity harmonics for downward and up-ward directions, i.e., for h 5 61, as follows from thetheory. These conditions are fulfilled with the follow-ing set of basis functions:
bim1h2 5 Pmm1h251 2 0h 2 hi 0@h if 0h 2 hi 0 , h
0 otherwise,
182
where m is the harmonic number, Pmm1h2 are the
spherical harmonics that are proportional to theassociated Legendre polynomials but with the normal-izing condition
e21
1
Pkm1h2Pk
m1h2dh 5 2@12k 1 12, 192
hi 1i 5 1, . . . , 2N2 are the nodes of the homogeneousgrid in the interval 321, 14, and h 5 1@1N 2 12 is thegrid step. The descending order of nodes for h . 0and that ascending for h , 0 are adopted, i.e., h1 5
21, hN 5 0, hN11 5 1, h2N 5 0, in order to haveidentical blocks in the resulting matrices given in Eq.1112. For hi 5 61, 0 only half of the correspondingtriangular function in Eq. 182 is of importance be-cause of Eqs. 122 and 132. Substituting Eqs. 172 intoEq. 142 and projecting them onto the conjugate space
where matrices A m and Bm are block matrices of theform
A m 5 1Am112 0
0 2A m1122 , Bm 5 1Bm112 Bm122
Bm122 Bm1122 .1112
The formal expression of the elements of matricesA m and B m in terms of double integrals has beengiven in Ref. 1. The numerical estimation of theseintegrals is a time-consuming procedure, but if thephase function is expanded in Legendre polynomials,i.e.,
p1g2 5 om50
`
xmPm1g2, 1122
then the matrix elements can be estimated analyti-cally and recursion formulas can be used in thecalculations. The explicit expressions for the ele-ments of Bm are presented here as an example.Using the addition theorem the coefficients of the
Fourier expansion of the phase function can be writ-ten in the form 1Ref. 112
pm1h, h825 ok5m
`
xkmPkm1h2Pk
m1h82, 1132
and therefore the elements of matrix Bm 5 3Bi, jm4 can
be expressed as
Bi, jm 5 2Bi, j
m 1 L/2 ok5m
`
xkJN2j11k,mJN2i11
k,m
for i 5 1, . . . , N; j 5 1, . . . , N,
Bi, jm 5 1212mL/2 o
k5m
`
1212kxkJ2N2j11k,mJN2i11
k,m
for i 5 1, . . . , N; j 5 N 1 1, . . . , 2N, 1142
where
Jik,m5 e1i222h
ih
b2N2i11m1h2Pk
m1h2dh, i 5 1, . . . , N, 1152
Bi, jm5 e
nj21
nj11
bim1h2bjm1h2dh, i, j 5 1, . . . , N. 1162
For i 5 11N2 the lower1upper2 limit must be substi-tuted by 0112 in Eq. 1152 and by 21102 in Eq. 1162.The J integrals in Eq. 1152 can in turn be easilyexpressed in terms of theR, S, and T integrals defined
The recursion formulas for these integrals are of theform
Rik11,m1t2
5 51@31k 1 m 1 121k 2 m1 12461@2
3 512k 1 12Sik,m 2 31k 1 m21k 2 m241@2Ri
k21,m6,
1182
Sik11,m1t2
5 51@31k 1 m 1 121k 2 m1 12461@2
3 512k 1 12Tik,m 2 31k 1 m21k 2 m241@2Si
k21,m6,
1192
Tik,m
5 1@12m 1 325231m 1 12@12m 1 1241@2Qik,m 1 Ri
k,m6
Qik,m
5 1@1k 1 m1 325tPm12m1t2Pk
m1t2
1 31k1m21k2m241@2@1m2 k1 22Pm12m1t2Pk21
m1t2
2 12Œm 1 12@1m 2 k 1 22Pm11m1t2Pk
m1t261i212hih,
1202
and the analytical expressions for the initial values ofthe index, i.e., k 5 m, m 1 1, can be readily obtained.
B. Solution of the Matrix Equation
In this and in the following subsection the superscriptm 1m 5 0, . . . ,M2, corresponding to a harmonic num-ber, is omitted, and the resulting formulas apply foreach value ofm, whereas superscript l 1l 5 1, . . . , L2,corresponding to the layer number, is reintroduced.The general solution of Eq. 1102, obtained when onedeals with the radiative transfer in the slab,1 is
I 1l21t2 5 X 1l25L1l2S@4 exp12t@h02W1l2X 1l2tR1l2
1 D 1l21t2N11l26, 1212
where N11l2 is a vector with arbitrary elements, D 1l21t2
and W 1l2 are diagonal matrices
D 1l21t2 5 diag5exp32aj1l2t46, 1222
W 1l2 5 diag51@3aj1l2 2 1@h046 1232
for j 5 1, . . . , 2N, aj1l2 are the eigenvalues, X 1l2 are the
matrices that consist of column eigenvectors of ageneralized eigenvalue problem:
2a1l2AX1l2 5 B1l2X1l2 1242
for matrices A and B1l2, and superscript t stands fortranspose. It can be shown that the eigenvectormatrices X 1l2 are of the form
X 1l2 5 1X 11l2 X 2
1l2
X 21l2 X 1
1l22 . 125
The singularities for certain values of the solar angleand the exponential growth of the matrix elements ofD1l21t2 when the optical thickness tends to infinity inEq. 1212 create certain difficulties for a numericalestimation of the solution. To avoid the problems itis necessary to 1a2 subtract explicitly the singularityfrom the first term of Eq. 1212 and make it compensatethe singularity of the second term 1it is clear that N1
l
are singular on h0, as the intensity should be regular2and 1b2 change the origin of the t axis for each layer, sothat t equals 0 at the top of the layer for downwellingradiation and at the bottom of the layer for upwellingradiation. The transformation similar to 1b2 has beenproposed and has been used effectively.6,15 Thesetransformations can be introduced by substitutingthe arbitrary vector N1
1l2 by another arbitrary vectorN1l2, given by
N1l2 5 1D11l23t1l2124 0
0 D11l232t1l242
3 3N11l2 1 L1l2S/41 E 0
0 0 2W 1l2X 1l2R1l24 ,1262
where D11l2 is the upper left-hand part of D1l2 from Eq.
1222, E is a unit matrix, and t1l2 is the cumulativeoptical thickness of the l upper layers. The generalsolution of Eq. 1212 becomes
I1l21t2 5 X 1l251D11l23t 2 t1l2124 0
0 D11l23t1l2 2 t42N1l2
1 L1l2S/4 1K 1l21t2 0
0 E exp12t@h022
3 1E 0
0 W21l22X 1l2tR1l26 , 1272
where
K 1l21t2 5 3 E exp12t@h02 2 D11l21t24W1
1l2, 1282
and W11l2 and W2
1l2 are, respectively, the upper left-hand part and lower right-hand part of diagonalmatrix W 1l2. It can readily be seen that Eq. 1272 has
2
no singularities and no exponentially growing terms.
C. Boundary Conditions for the Multilayer Problem
The interlayer boundary conditions are those of conti-nuity and are readily obtained by equating the expres-sions in Eq. 1272 for the lth and 1l 1 12th layers at theinterface, i.e.,
I1l23t1l24 5 I1l1123t1l24. 1292
The boundary conditions at the upper boundary forthemultilayer problem are exactly the same as for thehomogeneous slab, i.e.,
I1121102 5 0. 1302
The lower boundary conditions are based on a generalexpression for the relationship between the incidentand reflected radiation for the reflecting surface, i.e.,
I3t1L2, h, f4 0h,0 5 1@p e0
2p
df8 e0
1
r1h, f; h8, f82
3 Iin3t1L2, h8, f84h8dh8, 1312
where t1L2 is the total optical thickness of L layers andIin is the total incident radiation, i.e., the sum of thedirect and diffuse radiation. The reflection coeffi-cient is normalized in such a way that, when r1h, f;h8, f82 5 r, i.e., it does not depend on angular vari-ables, r 5 LS, where LS is the surface albedo. Herewe consider only a subclass of reflecting surfaces,namely, those with the reflection coefficient being afunction of only the angle between the directions ofthe incident and reflected beams. The boundarycondition for themth harmonic of the intensity in thiscase can be written as
I 1L2m23t1L2, h4 5 2 e0
1
rm1h, h82I1L2m13t1L2, h84h8dh8
1 Sh0rm1h, h02exp32t1L2@h04, 1322
where
rm1h, h82 5 1/12p2 e0
2p
r1g2cos mfdf, 1332
and rm1h, h82 can be presented as
rm1h, h825 ok5m
`
rkPkm1h2Pk
m1h82, 1342
where rk are coefficients of the Legendre expansion ofr1g2. Substituting into Eq. 1322 the intensity Im1t, h2from Eq. 172 and projecting the resulting equation onthe conjugate space corresponding to the FEM, weobtain 1superscriptm is omitted in the following2
where B and S are N 3 N matrices, RS is an Nvector, and I11L2 and I21L2 are the upper and lower partsof I1L2. The expressions for elements of B are givenby Eq. 1162, and the elements of S and RS can beestimated in terms of the R, S, and T integrals in Eqs.1182–1202.Conservation of the energy flux in relation to the
boundary conditions in the form of Eq. 1352 needs somediscussion. The expression for flux of diffuse radia-tion, which follows immediately from the integrationof Eq. 172 and was presented earlier in Ref. 1, can betransformed to
F 5 F 1 2 F 2,
1
2pF 1 5 e
0
1
I01t, h2hdh 5h
2 51hN11 21
3h2IN11
0
1 2 oj5N12
2N21
hjIj0 1 1h2N 11
3h2I2N06 5 F ? I01
1
2pF 2 5 e
0
21
I01t, h2hdh 5 2h
2 51h1 11
3h2I10
1 2 oj52
N
hjIj0 1 1hN 21
3h2IN06 5 F ? I02,
1362
i.e., both downwelling flux F 1 and upwelling flux F 2
can be presented as an inner 1scalar2 product of thecorresponding intensity and a vector F 5 3 T i4 1i 5 1,. . . , N2, which depends only on grid parameters.If the reflection coefficient r does not depend onangular variables, it can be shown that the numericalrelationship in Eq. 1352, similar to the analyticalexpression in Eq. 1322, gives the ratio of reflected andincident fluxes, i.e., the albedo, which is exactly equalto r, and r , 1 provides the stability for the scheme.If r is angular dependent, albedo LS is a functional ofthe angular distribution of the downwelling intensity,and for stability of the scheme it is necessary that, forany distribution of the downwelling intensity, LS doesnot exeed 1. It can be shown that this condition is
where G1i2 is the ith column vector of matrix B21 S.The fulfillment of inequality 1372must be provided forall values of i by a proper choice of coefficients rk in Eq.1342.Substituting Eq. 1272 into Eq. 1352, one can readily
obtain the equation forN1L2:
3 BX11L2 2 SX2
1L24D11L23t1L2 2 t1L2124N1
1L2
1 3 BX21L22 SX1
1L24N21L2
5 2SLL@4 53 BX11L2 2 SX2
1L24K 1L23t1L24
3 3X11L2tR1
1L2 1 X21L2tR2
1L24 1 exp32t1L2@h04
3 3 BX21L2 2 SX1
1L24W21L23X2
1L2tR11L2 1 X1
1L2tR21L246
1 Sh0 exp32t1L2@h04RS, 1382
where R11L2, R2
1L2, N11L2, and N2
1L2 are, respectively, theupper and lower parts of vectors R1L2 and N1L2 and X1
1L2
and X21L2 are the blocks of eigenvector matrix X 1L2 from
Eq. 1252.In order to obtain the final solution of the radiative
transfer equation 1RTE2 in a multilayer media thevalues of vector constants N1l2 must be determinedfrom a set of linear algebraic equations,
UN 5 B, 1392
formed by 1L 2 12 expressions of the type in Eq. 1292,each consisting of 2N equations, N equations of thetype in Eq. 1302, andN equations of the type in Eq. 1382,so thatN is anN 3 L dimensional vector:
N 5 1N112
N122
···N1L22 , 1402
andU is a block matrix:
1X2
112 Y1112 0 0 · · · 0 0 0
Y1112 X2
112 2X1122 2Y2
122 · · · 0 0 0
Y2112 X1
112 2X2122 2Y1
122 · · · 0 0 0
0 0 Y1122 X2
122 · · · 0 0 0
0 0 Y2122 X1
122 · · · 0 0 0
· · · · · · · · · · · · · · · · · · · · · · · ·
0 0 0 0 · · · X21L212 2X1
1L2 2Y21L2
0 0 0 0 · · · X11L212 2X2
1L2 2Y11L2
0 0 0 0 · · · 0 G1 G2
2
2
where
Y11221l2 5 X1122
1l2D11l23t1l2 2 t1l2124,
G1 5 3 BX11L2 2 SX2
1L24D11L23t1L2 2 t1L2124,
G2 5 BX21L2 2 SX1
1L2. 1412
The expression for vector V can be obtained from Eqs.1272, 1292, 1302, and 1382 and is not presented herebecause of its large size.For the case of a homogeneous slab with absorbing
boundaries the resulting solution is equivalent to thatobtained earlier in Ref. 1. It is presented here in anexplicit form in order to correct the typing error in themanuscript of Ref. 1 3Eq. 1452 of that paper4. If L 5 1and r1h, f; h8, f82 5 0, as is the case considered in Ref.1, only Eqs. 1302 and 1382 are used, matrix S and vectorRS are null, t1L2 5 t0, t1L212 5 0, vector N 5 N112, andmatrixU takes the form
A 112 5 1E 0
0 B21X2 X1D11t02
X1D11t02 X22 . 14
The right-hand side vector V of the system can betransformed in this case to
V 5 2LS/4 1 E 0
0 B210 X1
X 1K 1t02 exp12t0@h02X22
3 1 E 0
0 W 22X tR, 1432
and it can readily be seen from Eqs. 1272, 1422, and 1432that the expression for the intensity is
I 5 X 31K 1t2 0
0 exp12t0@h022Z1
1 1D11t2 0
0 D11t0 2 t22Z24 , 1442
where
Z1 5 LS/41 E 0
0 W 22X tR,
Z2 5 2LS/41 X2 X1D11t02
X 1D11t02 X2221
3 1 0 X1
X1K 1t02 exp12t0@h02X22
3 1 E 0
0 W 22X tR. 1452
D. Interpolation Procedure
The solution we obtain by choosing the finite elementsin the form as in Eq. 182 is piecewise linear, so itsderivative is not continuous at angular grid points.In order to make the solution smoother and in such away closer in analytical properties to an exact solu-tion, it is reasonable to substitute Eq. 1272 into thescattering term of Eq. 142 and to solve the resultingequation analytically. Use of the so-obtained formu-
2
2
las in a numerical estimation does not lead to asignificant increase of computer time.For h $ 0 an interpolated solution of the RTE is
found as
Im1t, h2 5 exp12t@h2@h e0
t
exp1t8@h2Qm1h, t82dt8,
1462
where
Qm1h, t2 5 L/2 e21
1
pm1h, h82Im1t, h82dh8
1 SL@4pm1h, h02exp12t@h02, 1472
and Im1t, h2 is the solution given in Eq. 172. Takinginto account Eqs. 172 and 182 and after some manipula-tion 1the superscript m is omitted everywhere but inthe expressions connected with the spherical harmon-ics2 the expression forQm1h, t2 can be written as
Q 1l21h, t2 5 F1l2 ? I1l2 1 SL1l2@4
3 exp12t@h02 ok5m
`
xk1l2Pkm1h2Pk
m1h02,
1482
where l 1the layer number2 depends on t, I1l21t2 is thevector of the intensity values in the grid points aspresented in Eq. 1272:
Fi1l21h2 5 L1l2/2 o
k5m
`
1212k1mxk1l2Pkm1h2JN2i11
k,m,
i 5 1, . . . , N,
Fi1l21h2 5 L1l2/2 o
k5m
`
xk1l2Pkm1h2J2N2i11
k,m,
i 5 N 1 1, . . . , 2N. 1492
Substituting Eqs. 1492 into Eq. 1462 allows us to writethe interpolated solution in the following form:
for t1l212 , t , t1l2, l 5 1, . . . ,L.For h , 0 the interpolated solution of the RTE is
found in a similar way, differing only in the directionof integration:
I1t, h2 5 2exp12t@h2@h et
t1L2
exp1t8@h2Q 1l21h, t82dt8
1 I3t1L2, h4exp53t1L2 2 t4@h6, 1532
where I 3t1L2, h4 is given by Eq. 1322. Using thisequation and the expression for Q 1l21h, t82, which is ofthe same form as in Eq. 1482, one can obtain theexpression for the interpolated upgoing intensity.The matrices H i
1l21t2, i 5 1, . . . , 3, and the scalarH4
1l21t2 in this case are 3 j 5 1, . . . , N; t1l212 , t , t1l24
3. Data Extraction from the LOWTRAN Program
Our main aim in this paper is to test the developedalgorithm on realistic atmospheric models and toestimate the accuracy of the method and its computa-tional requirements in these cases. As the radiativetransfer in the atmosphere is of practical importancefor remote sensing in the visible region of the spec-trum and because realistic atmospheric parametersare readily available from many sources, for example,from the LOWTRAN data set, the atmosphere was
PLIED OPTICS @ Vol. 34, No. 36 @ 20 December 1995
chosen as a test medium. The use of these dataallows one to consider different situations that cover awide variety of natural situations with particularattention to the atmospheric boundary layer 10–2 km2
that greatly influences radiative transfer in the vis-ible region. The test calculations have been per-formed for wavelengths of 0.55 and 0.4 µm, but, inorder to make better comparisons between the re-sults, a unit value for solar constant S has beenadopted. The atmosphere from 0 to 100 km is di-vided into 20 layers and the heights of the layerboundaries are 0, 0.5, 1, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15,20, 25, 30, 40, 50, 70, and 100 km. For each layer l 5
1, . . . , L the optical thickness t1l2, the single scatter-ing albedo L1l2, and phase function pl1g2 are estimatedfrom the LOWTRAN program. The gaseous composi-tion in all the considered cases corresponds to the120th day of the 1976 U.S. Standard Atmosphere.The LOWTRAN code has different execution modes.
The two that are most convenient for extraction ofoptical parameters were used: the transmittancemode and the radiance mode with only primary solarscattered radiance included. By running theLOWTRAN code in the transmittance mode, the follow-ing parameters can be obtained for each layer l1l 5 1, . . . , L2:
where Ke is the extinction coefficient,Ka is the absorp-tion coefficient, Ks is the scattering coefficient, super-scripts mo and ae 2 hy refer, respectively, to mol-ecules and aerosol and hydrometeors, and h is theheight of the layer. In the transmittance mode scat-tering is treated only as a loss mechanism and thisinvolves an error in the computation of the aboveparameters. Nevertheless the main aim here is totest the FEM algorithmwith realistic, even if not real,parameters. Using Eqs. 1552–1582 estimation of theoptical thickness and of the single scattering albedo isstraightforward. Since LOWTRAN is not a single-frequency code, the obtained parameters are theweighted average values over a spectral interval of 5cm21 centered on the frequency of interest.One can determine the aerosol–hydrometeors phase
function p1g2ae2hy for each layer by running theLOWTRAN
code in the radiance mode, with only primary scatter-ing of the solar radiation included. The internalLOWTRAN database of 70 Mie phase functions may beused in such a way. The phase function for eachlayer is estimated as
p1g2 5 3p1g2ae2hy Ksae2hy 1 p1g2mo Ks
mo4@Ks, 1592
where p1g2mo is the LOWTRAN Rayleigh phase functionand Ks 5 Ks
ae2hy 1 Ksmo. After the phase function is
determined, the moments of expansion in Legendrepolynomials are computed. The number of momentsdepends on the type of phase function with a maxi-mum of 500 moments for phase functions with astrong forward peak.
4. Numerical Results
The possibilities of the FEM are demonstrated herefor six typical sets of atmospheric parameters, includ-ing the presence of all types of aerosol particle in theatmosphere, i.e., anthropogenic aerosol in the cities,natural dust, volcanic emissions, clouds, and fogs.The accuracy of the method in other situations can beestimated by interpolating or extrapolating the pre-sented results. The intensity of the scattered radia-tion is estimated in the solar plane, in which thevariations in intensity with respect to angle are thelargest and at which, as a consequence, the highestnumerical errors occur. We primarily used a solarangle of 50° 1h0 5 0.642, but the dependence of errors
on the solar polar angle is also presented for one of thecases. The Henyey–Greenstein approximation forthe phase function is used for surface scattering:gs and Ls denote, respectively, the surface asymmetryfactor and albedo. For a Lambertian surface gs 5 0.A quantitative evaluation of the accuracy is made
through computation of the mean and maximumabsolute deviations with respect to the referencesolution. With a high number of grid points 1N . 1002,the FEM and DOM solutions practically coincide.For all the cases examined the FEM solution withN564 may be considered as fully converged, so it hasbeen assumed as the correct reference solution againstwhich the accuracies are estimated. The interpo-lated solution is computed in 20 polar angles in theinterval 321, 14 at three different levels: the groundlevel 1z 5 0 km2, the top level 1z 5 100 km2, and anintermediate level that corresponds to the boundaryof a layer characterized by a strongly forward peakedphase function. The numbers in all the tables corre-spond to the real values of the deviations multipliedby 100.
A. Rural Aerosol
The following results refer to a rural type boundarylayer 10–2 km2 atmosphere with visibility equal to 23km, l 5 0.55 µm, Ls 5 0.5, and gs 5 0.6. The phasefunction that corresponds to this type of extinction atthis frequency is i 5 4 of the LOWTRAN Mie database1Fig. 12. Qualitative and quantitative comparisonsbetween the FEM and DOM results have already beenpresented for the zero-order Fourier component ofthe intensity,1 and the DOM has already been com-pared with other well-established methods.4,5 Herewe intend to demonstrate that only the higher accu-racy of the FEM in the case of intense forwardscattering is maintained for the intensity and notonly for the zero-order Fourier harmonic1 of theintensity. The downgoing and upgoing intensities atz 5 0 km, z 5 2 km, and z 5 100 km are computed.
Fig. 1. Some of the phase functions from the LOWTRAN database1i 5 1, 702 used in the computations: ......., i 5 4 rural aerosol;–––, i 5 18 urban aerosol; –·–, i 5 51 advective fog; ——, i 5 55volcanic particles.
Table 1 lists the mean absolute deviations for theFEM and DOM for different numbers of grid points.The higher accuracy obtained with the FEM, with alow number of grid points, is evident even in a casesuch as this in which the phase function is notextremely forward peaked. The lowest accuracy forboth methods is for the downgoing intensity 1h . 02 atz 5 0 km and is due to the peak in the direction of thesolar angle that results from the intensive forwardscatter in the boundary layer. However, the highestmean deviation for the FEM corresponds toN 5 2 andis of the order of 1023. A similar accuracy can beobtained with the DOM but withN 5 32.
B. Urban Aerosol
The following results refer to an urban type boundarylayer 10–2 km2 atmosphere with visibility equal to 5km, l 5 0.55 µm, Ls 5 0.5, and gs 5 0.6. The phasefunction that corresponds to this type of extinction atthis frequency is i 5 18 of the LOWTRAN database 1Fig.12. In order to demonstrate the angular dependenceof the error and of the character of convergence, theresults for this case are presented in graphic form fordifferent solar angles. In this case the convergenceof the solution has also been studied for different solarpolar angles. The downgoing intensities obtainedwith the FEM and DOMwith h0 5 0.2, 0.64, 0.9 at z 50 km are plotted versus the cosine of the polar anglefor different values of N 1Fig. 22. For the upgoingintensity the curves with different N are almostindistinguishable and for this reason they are notreported.The graphs show a smaller interval of the angular
variable centered upon h0, because this is the point atwhich the largest gradients and consequently thelargest errors occur. The interpolated solution iscomputed in 100 polar angles in the interval 3h0 2 0.1,h0 1 0.14.
Table 1. Mean Deviations of the FEM and DOM Intensity Values forDifferent Numbers of Grid Points for Rural Aerosol with 23-km Visibility, l
5 0.55 mm, LS 5 0.5, and gS 5 0.6a
LayerLevel Angle Scheme
Grid Points
N 5 2 N 5 4 N 5 8 N 5 16 N 5 32
z 5 0 h . 0 FEM 0.457 0.224 0.093 0.031 0.001km DOM 37.421 18.071 10.642 4.410 0.617
h , 0 FEM 0.057 0.018 0.000 0.000 0.000DOM 5.46 0.967 0.019 0.000 0.000
z 5 2 h . 0 FEM 0.090 0.055 0.010 0.000 0.000km DOM 17.789 4.395 0.733 0.138 0.020
h , 0 FEM 0.229 0.091 0.000 0.000 0.000DOM 3.847 0.713 0.077 0.034 0.022
z 5 100 h , 0 FEM 0.182 0.039 0.007 0.001 0.000km DOM 3.673 0.570 0.042 0.008 0.009
aThe numbers in all the tables correspond to the real valuesmultiplied by 100. In all the tables the deviations are computedwith respect to the FEM solution with N 5 64, which is consideredas converged. The columns of the tables in which all the devia-tions are lower than 1024 are removed.
For the downgoing intensity, the convergence of thesolution obtained with the FEM is relatively fast andnonphysical oscillations do not appear. The solutionobtained with the DOM with N 5 16 does not evenresemble the correct solution.In general the accuracy decreases with increasing
cosine of the solar angle. This is the result of anarrowing of the peak in the angular dependence ofscattered radiation because of the decrease of theoptical path for direct solar light. N . 16 grid pointsare necessary to obtain an average error 1mean abso-lute deviation2 lower than 1023 with h0 5 0.2, 0.64 at z5 0 and h . 0, whereas, for h0 5 0.9, the error is stillof the order of 1022 withN 5 32.
C. Advective Fog
The boundary layer extinction in this case is charac-terized by advective fog with 0.2-km visibility, l 5
Fig. 2. Downgoing intensity at z5 0 km corresponding to the casein Subsection 4.B 1urban aerosol with visibility of 5 km, l 5 0.55µm, Ls 5 0.5, and gs 5 0.62 with A, h0 5 0.2; B, h0 5 0.64; C, h0 5
0.9.
Table 2. Mean and Maximum Deviations of the FEM Intensity Values forDifferent Numbers of Grid Points for Advective Fog with 0.2-km Visibility, l
5 0.55 mm, LS 5 0.8, and gS 5 0
Layer Level Angle Deviation
Grid Points
N 5 2 N 5 4 N 5 8 N 5 16
z 5 0 km h . 0 Mean 0.059 0.013 0.002 0.000Max 0.112 0.295 0.003 0.000
h , 0 Mean andmax 0.014 0.007 0.001 0.000
z 5 2 km h . 0 Mean 0.064 0.056 0.018 0.001Max 0.134 0.246 0.056 0.005
h , 0 Mean 1.751 0.622 0.014 0.002Max 4.926 3.967 0.053 0.015
z 5 100 km h , 0 Mean 0.845 0.137 0.019 0.003Max 1.637 0.474 0.045 0.012
0.55 µm, and phase function i 5 51 as shown in Fig. 1.The Lambertian surface withLs 5 0.8 is taken here inorder to have some amount of reflected radiation.In this case the peak in the intensity at the surfacelevel disappears because of high optical thicknessvalues from 0 to 2 km 1.222 and the low absorption inthis layer 1the single scattering albedo is .12. Theradiation diffuses almost uniformly in all directionsthrough the fog layer because of the high number ofscattering events. The fog layer reflects most of theincident solar radiation back to the upper atmosphere.The mean deviations of the intensity values are listedin Table 2. Here and in the following the columns ofthe tables in which all the errors are lower than 1024
are omitted. Furthermore, for this case 1and also forthe case in Subsection 4.D2 the mean and the maxi-mum deviations for h , 0 at the surface level are thesame, since the reflected radiation does not depend onthe polar angle, so they are presented in one line.An average error lower than 1023 is guaranteed withN $ 8.
D. Stratus@Stratocumulus CloudsThe boundary layer from 0.66 to 2 km is characterizedby the presence of stratocumulus clouds 1l 5 0.55µm2. For clouds and rain the Henyey–Greensteinapproximation for the phase function is used in theLOWTRAN program and the asymmetry factor is, in thiscase, equal to 0.8535. The surface is Lambertianwith Ls 5 0.7. For the intensity computation thesame considerations that were made for advective foghold true, except that the inaccuracy is generallylower 1Table 32. This is due to the fact that the opticalthickness between 0 and 2 km is higher 146 comparedwith 222 and the peak of the phase function is not sopronounced. The error is correspondingly lower than1023 withN 5 4 in all the considered cases.
E. Extinction by Rural Aerosol and by Volcanic Particles
The distinctive features of this case are extinction byrural aerosol in the boundary layer 10–2 km2 andextreme volcanic profile with extinction by freshvolcanic particles in the stratospheric layer 110–30
Table 3. Mean and Maximum Deviations of the FEM Intensity Values forDifferent Numbers of Grid Points for Stratocumulus Clouds with l 5 0.55
mm, LS 5 0.7, and gS 5 0.7
Layer Level Angle Deviation
Grid Points
N 5 2 N 5 4 N 5 8
z 5 0 km h . 0 Mean 0.039 0.005 0.000Max 0.067 0.007 0.001
h , 0 Mean and max 0.019 0.003 0.000z 5 2 km h . 0 Mean 0.256 0.019 0.002
Max 0.803 0.073 0.005h , 0 Mean 0.036 0.009 0.002
Max 0.140 0.016 0.002z 5 100 km h , 0 Mean 0.356 0.037 0.009
Max 0.748 0.071 0.029
km2 with l 5 0.4 µm, gs 5 0.7, and Ls 5 0.7. Thephase function 1Fig. 1, i 5 552 has the strongest for-ward peak of all the considered cases. The computa-tion of the errors for downgoing intensity just below10 km shows thatN 5 32 grid points are necessary foran average error of 0.1% 1Table 42.
F. Rural Aerosol and Cirrus Clouds
These results are characterized by extinction by ruralaerosol in the boundary layer 10–2 km2 and cirrusclouds from 9 to 10 km with l 5 0.4 µm, gs 5 0.8, andLs 5 0.8. The phase function of the cloud layer is aHenyey-Greenstein approximation with an asymme-try factor equal to 0.6707. The characteristics of theintensity estimates are quite similar to those inSubsection 4.E, but the peak in the downgoing radia-tion is lower both at 0 and at 10 km and, consequently,the error is also lower 1Table 52.For most of the considered cases eight grid points
are enough to obtain, on average, an accuracy to three
Table 4. Mean and Maximum Deviations of the FEM Intensity Values forDifferent Numbers of Grid Points for Rural Aerosol and Extreme Volcanic
Profile with l 5 0.4 mm, LS 5 0.7, and gS 5 0.7
LayerLevel Angle Deviation
Grid Points
N 5 2 N 5 4 N 5 8 N 5 16 N 5 32
z 5 0 h . 0 Mean 15.708 10.079 2.842 0.825 0.041km Max 59.180 31.749 16.927 4.935 0.386
h , 0 Mean 0.183 0.012 0.000 0.000 0.000Max 0.233 0.036 0.001 0.000 0.000
z 5 10 h . 0 Mean 39.698 27.002 7.066 1.239 0.049km Max 90.909 63.881 24.177 7.915 0.384
h , 0 Mean 0.997 0.158 0.004 0.000 0.000Max 1.823 0.538 0.014 0.001 0.000
z 5 100 h , 0 Mean 2.287 0.644 0.012 0.002 0.000km Max 1.191 4.660 0.003 0.015 0.001
Table 5. Mean and Maximum Deviations of the FEM Intensity Values forDifferent Numbers of Grid Points for Rural Aerosol and Cirrus Clouds with
l 5 0.4 mm, LS 5 0.8, and gS 5 0.8
Layer Level Angle Deviation
Grid Points
N 5 2 N 5 4 N 5 8
z 5 0 km h . 0 Mean 1.252 0.089 0.011Max 2.139 0.171 0.037
h , 0 Mean 0.101 0.003 0.000Max 0.144 0.015 0.000
z 5 10 km h . 0 Mean 0.663 0.145 0.026Max 1.647 0.271 0.067
h , 0 Mean 1.114 0.044 0.004Max 2.061 0.176 0.018
z 5 100 km h , 0 Mean 0.751 0.026 0.003Max 1.440 0.142 0.014
Table 6. CPU Times on a VAX 7000-610 for Computation of One Harmonicfor Different Values of N with Two Interpolation
decimal places. The most critical cases are the esti-mates of the downgoing radiation at the bottomboundary of a layer characterized by a phase functionwith a strong forward peak, a high value of the singlescattering albedo 1.12, and small optical thickness.In these cases it is necessary to use a higher numberof grid points 1N 5 32, 642 to obtain high accuracy.Nevertheless, no nonphysical oscillations appear inthe solution and, if larger errors, those that occurmainly in the directions close to the Sun, can betolerated, a good approximation of the solution canalready be obtained withN 5 8, 16.The analysis would not be complete without some
consideration of the computer time. It is not easy togive a priori an exact estimate of the CPU timeneeded for the intensity computation because of itsvariability with respect to the input parameters.The CPU time for each harmonic depends mostly onthe number of grid points, but it also depends on thenumber of interpolation points and levels. The num-ber of harmonics that one needs in order to have thedesired accuracy depends on the characteristics of thesolution itself at the level and direction under consid-eration. Nevertheless the following results are in-tended to give an idea of the CPU time required forthe cases outlined in Section 4. In Table 6 the CPUtime on a Vax 7000-610 that is needed for eachharmonic with different values of N is shown for twointerpolation polar angles and t levels. The CPUtime increases, but comparatively slowly, with thenumber of interpolation angles. For example, withN 5 16, it varies from 4 to 9 s when the number ofinterpolation points varies from 2 to 20. The maxi-mum value of each harmonic 3Im1t, h2 in Eq. 1124,measured over all the interpolation angles h andlevels t for some of the atmospheric models consideredin Section 4, is presented in Table 7 as a function ofharmonic numberm. Since the slowest convergenceusually occurs at the points at which the solution hasa maximum, the data in Table 7 allow one to estimatethe necessary number of harmonics that should beincluded in the sum in order to obtain the givenaccuracy of the intensity.
5. Conclusions
An extension of the FEM algorithm for the intensitycomputation in a plane-parallel vertically inhomoge-
Table 7. Maximum Value of Each Harmonic for Some AtmosphericModels that Were Measured over All the Interpolation Angles and Levels
neous multiple scattering atmosphere has been de-scribed. The numerical stability that is due to fluxconservation,1 which does not depend on the shape ofthe phase function and the number of grid points,makes this method especially suitable for dealingwith phase functions with a large forward peak,which is one of the major difficulties in the numericaltreatment of radiative transfer in the atmosphere.Tested on realistic atmospheric models, the algo-
rithm shows an average error of 0.1% for most of theconsidered cases with only eight grid points coveringthe half-interval of the angular variable. In particu-larly critical cases 1phase functions with strong for-ward peaks2, a higher number of grid points1N 5 32, 642 is needed to obtain the same accuracy.The accuracy of the FEM has been shown to besignificantly higher than that obtained with the DOMwith the same number of grid points. The firstversion of the FORTRAN code with documentationexists and is available upon request.
This study was carried out at the Polytechnic ofTurin, Turin, Italy, as part of a cooperation betweenthis institution and the RussianAcademy of Sciences.Laura Roberti acknowledges the financial supportfrom Fondazione SANPAOLO di Torino as part of theGlobal Climate and Environment project.
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