2.2 Numerical Procedure of Discrete Ordinate Method
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metry relations exist for phase functions such that (8.2.35) and (8.2.36) Thus (8.2.37) denotes the forward scattering phase function matrix and (8.2.38) denotes the backward-scattering phase function matrix. We also define diagonal matrices Ii and ~ containing the j.li'S and the ai's. They are both N x N diagonal matrices.
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0 (8.2.39)
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0 (8.2.40)
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Also, let f3 uUl f3ud' !JdUl and !Jdd represent the source terms on the right hand sides of (8.2.19a)2 and (8.2.19b). They are N x 1 column vectors
Po(Jll,l) r(l)R Jll 0
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f3 uu =
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(8.2.41)
8 SOLUTION TECHNIQUES OF RADIATIVE TRANSFER THEORY
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po(1l1,-I) F.
PO(1l2,-I) F. 112 0
/3ud
(8.2.42)
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(8.2.43)
(8.2.44)
pO(-IlN,-I) F. fiN 0
Also the reflectivity matrix is N x N and is
r(JLI)
(8.2.45)
r(JLN)
Then in matrix notation, (8.2.27) and (8.2.28) become
dI;;T) = _ Ii-I Iu(T)
+ Ii-I pUu aIu(T) + Ii-I pUd aId(T)
+ /3ud + f3dd
+ /3uu
_ dXd(T) dT
T e-(T+2 d)
(8.2.46)
Ii-I
Id(T)
+ /i-I plu a Iu(T) + /l-I pdd a Id(T)
+ f3du
e-(T+2 Td)
(8.2.47)
(8.2.48)
and the boundary conditions in matrix form are
2.2 Numerical Procedure of Discrete Ordinate Method
I u (7 = -7d) = 'if. I u (7 = -7d)
(8.2.49)
Solutions of (8.2.46) and (8.2.47) consist of homogeneous solutions and particular solutions. For homogeneous solution, we try
I u = I ua eaT I d = Ida eaT
(8.2.50a) (8.2.50b)
Substituting (8.2.50a) and (8.2.50b) into (8.2.46) and (8.2.47) gives
+aIua + Ii-II ua = /i-I. F a' I ua + /i-I. B . a Ida (8.2.51a) -aIda + 1i-IIda = /i-I. B . a' I ua + /i-I. F . a Ida (8.2.51b)
Adding (8.2.51a) and (8.2.51b) gives
a/i (fua - Ida) = A (I ua + Ida)
where
(8.2.52) (8.2.53)
A=-I+F a+B a
where 1 is N x N unit matrix. Similarly, taking the difference of (8.2.51a) and (8.2.51b) gives
ali (I ua + Ida)
where
W . (I ua
- Ida)
(8.2.54)
(8.2.55)
W=-I+F a-B a
From (8.2.52) and (8.2.54), we have
a/i (Iua + Ida)
so that
W ./i-I . A . (Iua + Ida)
(/i-I. W . /i-I. A - a 2 . (I ua + Ida) )
2 2 2 al,a2"" ,aN
(8.2.56)
Thus we have eigenvalue problem for a 2 . It is an N x N eigenvalue problem for a 2 so that there are N values of a 2 Let I be the eigenvector associated with a , that is,
[/i-I. W . /i-I. A - a~] . I al
(8.2.57)
l = 1,2, ... , N. Then al and -al will both be eigenvalues of (8.2.51a)(8.2.51b) with corresponding eigenvectors
(1 + ~l Ji-
A) . I
(8.2.58)
8 SOLUTION TECHNIQUES OF RADIATIVE TRANSFER THEORY
1( Ida = 2 1 -
1 =-1 A (};l J.L
=) .I
(8.2.59)
(};l,
I 1 ua = 2
(};l
( 1--11. A 1 =-1 r
1 ( 1 =-1 Ida. = 2 1 + (};l J.L A
=) I =) -
(8.2.60) (8.2.61)
(};l
-(};l.
We also have 2N arbitrary constants with PI corresponding to eigenvalue and with P- 1 corresponding to eigenvalue -(};/. The homogeneous solution is a linear combination of these 2N eigenvectors.
1= 1 ~ [ ( 1 + (};l,T 1 . A . I PI
=) -
al e
(1 - ~l/i-l A) .1 't: ~ ~t [11 (1- ~,li-I A) .'to, + (1 + ~ -1 .
+ P-l
. P-l
al e-a!(T+Td)]
(8.2.62a)
eO" (8.2.62b)
A) . I
al e-al(T+Td)]
We next determine particular solutions of (8.2.36)-(8.2.37). The first set is
1 uu
e-(T+ 2Td)
(8.2.63a) (8.2.63b)
I d = Idu e-(T+2Td)
Putting (8.2.63a) and (8.2.63b) into (8.2.46) and (8.2.47) give the equations
- Iuu =
du =
_/i-I. I uu + /i-I. F . a I uu + /i-I. B . a' I du + lJuu _Ji-l .I du + /i-I. B . a' I uu + /i-I. F a' Ldu + lJdu
I u = Iud e I d = I dd e T T
(8.2.64a) (8.2.64b) (8.2.65a) (8.2.65b)
The second set is
where
obey the equations
Iud =
-Idd
-1-1=--1=--Ji- . Iud + Ji- . F a' Iud + Ji- . B . a' Idd + f3ud (8.2.66a) -1 === -Ji- .- + TC1 B . a - + - 1 F . a' - + -f3dd (8.2.66b) Idd . Iud Ji- . Idd
2.3 Active Remote Sensing: Oblique Incidence
Thus the total solution is
I u = I~ + I uu e-(T+2 T + Iud eT d) (8.2.67a) I d= + I du e-(T+2Td) + I dd eT (8.2.67b) To determine p" and P-I, l = 1, ... , N, we impose the boundary conditions (8.2.48)-(8.2.49) at 7 = 0 and 7 = -7d. For 7 = 0, these give
1Jl =) '1 {(1- a1 --1 A Icx1+P-I (1 --1 A . 2~ PI t:7' l+ i1 =) ICXle-CX1Td } al
and at
+ I du e 1'0 + I dd = 0 = -7d, the boundary condition gives
- 2Td
(8.2.68a)
1;-'1'2 ;r { PI (1 + al Jl+ I uu e- Td
=) - + P-I (1 - . =) -CX1 } Icx/e-CX1Td Ji.
1 - al
+ Iud e- Td
[~t {p, (1- ~, /i-I
+ P_ I
(1 + ~l
jl-1 .
A) T eA) .Icx,}+ Idu e-Td + Idd e-Td ]
a, a ",
(8.2.68b)
Equations (8.2.68a) and (8.2.68b) provide 2N equations for the 2N unknowns
P1 , P2, ... , PN , P- 1 , P- 2, ... , P-N.
2.3 Active Remote Sensing: Oblique Incidence
For the case of an incident wave that is obliquely incident on the layer of scatterers, the radiative transfer equations become
jL ~; (7, jL, 1 = - I u (7, jL, 1
2 djL'
d1>' [Pn (jL, 1>; jL', 1>')Iu (7, jL', 1>')
(8.2.69a)
+ Pn(jL, 1>; -jL', 1>')Id(7, jL', q/)] -jL
~: (7, jL, 1 = -
I d(7, jL, 1
2 djL'
d1>' [Pn( -jL, 1>; jL', 1>')Iu(7, jL', 1>')
(8.2.69b) (8.2.70a)
+ Pn(-jL, 1>; -jL', 1>')Id(7, jL', 1>')]
