Adding and subtracting (8.2.100) and (8.2.101) gives in .NET

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Adding and subtracting (8.2.100) and (8.2.101) gives
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aJi l- =A.l+ aJi l+=W.lwhere A = -~e
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(8.2.110) (8.2.111) (8.2.112) (8.2.113) (8.2.114)
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+ F . a+ B . a = -~e + F . (i - B . a
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Combining (8.2.110) and (8.2.111) gives the eigenvalue problem
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(/i-I. W . /i-I. A - ( 2 )1+ = 0
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Thus, if a is an eigenvalue, so is -a. Equation (8.2.114) has 2n eigenvalues a~, a~, ... ,a~n and 2n eigenvectors 1+1 ,1+2 , ... ,1+2n' The solution is written in the following form
1+ = L {Pl1+l eQIZ + p- ll +l e-QI(Z+d)}
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(8.2.115)
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2.4 Discrete Ordinate Method for Passive Remote Sensing
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The constants PI and P-I are to be determined from the boundary conditions. Also, 1_ is determined from (8.2.110). Let superscript H denote homogeneous solution. Then,
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= - L....J
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1 Pl (1 + - j--1 . = i A)
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1+1 eO<IZ
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+ P-l(1- ~JL-1 . A) 1+1e-O<I(Z+d)}
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(8.2.116)
1 L....J 1 i I d = - ~ { Pl(1- - j--1 . = 1+1 eO<IZ A) 2 1=1 al
+ P-l(1 + ~JL-1
. A) .1+1 e-O<I(Z+d)}
(8.2.117)
To determine the particular solution, we note that
Jo1r dO' sin 0'
[(v(O),v(O'))
+ (v(O),h(O'))] = "'sv(O) = "'ev(O) + (h(O),h(O'))]
Jo1r dO' sinO'
[(h(O),v(O'))
= "'sh(O) = "'eh(O) - "'a
The particular solution to (8.2.94) and (8.2.95) is, with superscript P denoting particular solution,
= -I d p = CT
(8.2.118)
where T is a 2n x 1 column matrix with each element equal to the temperature T. The total solution is the sum of the homogeneous solution of (8.2.116) and (8.2.117) and the particular solution of (8.2.118). The 4n unknown constants PI and P-I are to be determined from the boundary conditions. Suppose the scattering medium is bounded by dielectric interfaces at z = 0 and at z = -d characterized by Fresnel reflectivity and transmissivity matrices (Fig. 8.2.6). Then the boundary conditions are, in matrix form,
= 0) = 1'10 . Iu(z = 0) 1u (z = -d) = 1'12 .1d (z = -d) + t12 . CT 2
Id(z
where
rlO =
(8.2.119) (8.2.120)
diag [rv10" rVlO2' r12 = diag [r v12" r v12 2,
, rv10 n , rhlO, ,rhlO2, ... , rhlOJ (8.2.121) , r v12 n , rh12" rh12 2,"', rh12J (8.2.122)
8 SOLUTION TECHNIQUES OF RADIATIVE TRANSFER THEORY
Region 0
--------1'------- z = 0
Region 1
scattering layer
- - - - - - - - - - - - - - - z =-d
Region 2
Figure 8.2.6 Thermal emission of a scattering layer lying above a homogeneous dielectric half-space.
(8.2.123)
T2 = -
[~~]
(8.2.124)
T2 T2 is the temperature for the dielectric half-space below z = -d, r alO is the Fresnel reflectivity for the interface at z = 0 for 0: polarizations, and r a 12 and t a 12 are, respectively, Fresnel reflectivity and transmissivity for polarization 0: at the interface z = -d. Equations (8.2.119) and (8.2.120) provide 4n equations for the 4n unknowns PI and P-I, l = 1,2, ... , 2n. After the 4n unknown constants are determined, the brightness temperature T B is given by
with
= ~ tlOIu(z = 0) = ~ tlO {I~ (z = 0) + CT}
tlO = 1-1"10
(8.2.125)
(8.2.126)
and 1 as the 2n x 2n unit matrix. Thus, tlO is the transmissivity matrix for the n directions and the two polarizations at the boundary z = O. In the following two sections, we illustrate the brightness temperature solutions for random media with three-dimensional variations and discrete spherical scatterers.
2.5 Passive Remote Sensing of a 3D Random Medium
2.5 Passive Remote Sensing of a Three-Dimensional Random Medium
The phase matrix of a three-dimensional medium with correlation function
(Elf(r )Elf(r )) = c5E 1m exp _, _II
[Iz' - z"l I
(x' - x")2 + (y' _ y")2] z2
(8.2.127) has been derived in 7. The coupling coefficients of passive remote sensing (v, v'), (v, h'), (h, v'), and (h, h') can be calculated by integration of corresponding phase matrix elements over dell. They are
(v, v') = Q( e, e') e- w { [sin 2 e sin 2 e'
+ ~ cos2 e cos 2 e'] I o ( w)
+ 2 sin e sin e' cos e cos e'It (w) + ~ cos2 e cos2 e'12 (w)} (8.2.128)
( v, h') , e = Q(e, e) e- cos -[Io (W) - h(w)] 2
(8.2.129) (8.2.130) (8.2.131)
(h, v') = Q(e, e') e- W
cos2 e'
-[Io (w) - 12 (w)]
(h, h') = Q(e, e') e- w ~[Io(w)
where ,
+ h(w)]
lz ] 1 + k~~J~(cose _ cose')2
Q(e, e) =
8k~~I~ [
.exp [- k~~l~ (sine-sinO,)2]
(8.2.132) (8.2.133)
k,2 12
sin 0 sin 0'
The scattering coefficients are
~sv (e) =
~sh(e)
de' sin e' [(v', v)
+ (h', v)] + (h', h)]
(8.2.134) (8.2.135)
de' sine' [(v', h)
and the integration can be carried out using Gaussian-Legendre quadrature. The extinction coefficient is a summation of the absorption coefficient of the background medium and the scattering coefficient. (8.2.136)