[Ct+1 e- ik (l+l).d l in .NET

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[Ct+1 e- ik (l+l).d l
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+ DZ+l eik(l+l).dl ]
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with A o = RTE , Bo = 1, At = 0, B t = TTE, Co = R , Do = 1, Ct = 0, and D t = TTM. The wave amplitudes can now be determined using the propagating matrices. The reflection coefficients RTE and RTM are dual of each other and
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can be calculated by the recurrence relation of (5.2.12). However, because we are deriving the electric dyadic Green's function, the amplitudes Gl, Dl, and TTM are not dual of A l , B l , and TTE. We define
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R _ kl z - k(l+l)z l(l+l) - k k lz (l+l)z
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(5.2.22a) (5.2.22b)
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El+1 k lz - Elk(l+l)z 8 l (l+1) = ------'---'El+1 k lz El k(l+l)z
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Rl(l+l) and 8 l (l+1) are the reflection coefficients for TE and TM waves, respectively, between regions I and 1+ 1. The wave amplitudes in regions 1+ 1 and I are related by the TE and TM propagating matrices.
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V (l+l) l
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is called the TE forward propagation matrix and is given by -2
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V(l+l)l =
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1 + -k-(l+l)z
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kl Z
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e-ik(l+1)% (d'+1 -d,) [ R(l+l) l e ik (l+1).(d 1 1-d,) +
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R(l+l)l e-ik(l+1)%(dl+1-dl)]
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e1'k (l+1). (d 1+1 -
dI )
and, similarly,
V (l+1) l =
2 k(l+l)
1 + -El- k(l+l)z
k lz
e-ik(l+1).(dl+1-dl) +1-d1) [ 8(l+ 1) l eik (l+1).(d1
8(l+1) l e- ik (l+1).(d 1+ 1-dz)]
eik (l+1)' (d1+1-d,)
Using reciprocity and k1.. --t -k1.. and the property that e( -k x , -ky , k z ) = -e(k x , k y , k z ) and h( -k x , -ky , k z ) = h(k x , k y , -k z ), we have
COl(1', r') = GOt (1', 1")
C~oCr', 1') = 8~2
i k .r 90zCk1.., z')e-ik.L.r'.L
= C:O(1", 1') =
eik.r 9olk J.., z')e-ik.L.r'.L (5.2.26)
2.2 Dyadic Green'8 Function for Stratified Medium
gozCkl.., z') = e(k z ) [Ale l (_klz)eiklZZ'
+ Blel(klZ)e-iklzZ']
+ h(k z ) [Czht(-ktz)eikIZZ' + Dtht(ktz)e-ikIZZ']
gotCkl.., z') = e(kz)et(ktz)e-iktzZ'TTE + h(kz)ht(ktz)e-iktzZ'TTM (5.2.28)
Numerical solution of the Green's functions of layered media for arbitrary field point can be done by performing numerical integration of Sommerfield integral. In the past, this was usually done by computing the mixed potentials in the spatial domain [Mosig, 1989; Michalski and Mosig, 1997]. Recently, it is shown that the electric field dyadic Green's function of layered media can be computed in the spatial domain by using the Sommerfield integral with extractions [Tsang et al. 2000] For remote sensing, the observation point r in.-:.egion 0 is in the far-field. It is useful to have a far-field approximation for GOl(r, r') with r r'. We may evaluate GOl(r, r') by the stationary-phase method. The exponent is
kxx + kyY + (k 2
Then the stationary point is at
k~ - k;)~z
(5.2.29a) (5.2.29b) (5.2.29c) (5.2.30a) (5.2.30b) (5.2.30c)
kx = k sin ecos <p
k y = k sin esin <p k z = kcose
x = r sin ecos <p Y = r sin esin <p z = rcose
This gives the asymptotic result of = - -') eikr G01 (r, r = =
_ .- -, = (k l.., z ') e -zk,L r ,L gOI
(5.2.31a) (5.2.31b)
eikr _ .- -, Got(r,r') = -4 gOt(kl..,z')e-zk,L.r,L
where the value of kl.. = kxx + kyY in gOI(kl.., z') and got(kl.., z') is to be evaluated at the stationary phase point given by (5.2.29).
2.3 Brightness Temperatures for a Stratified Medium with Temperature Distribution
Using the result of fluctuation dissipation theorem in 3, we have
TBV(So)] l' ~) 1m 167f211r; ~ [ TBh ( So = To-+OO A 0 cos 00 L
dxdy d"( Z )rJ"1 (Z ) ZWEI .11 (5.2.32)
[~(so) GOl(ro,r) . Gzt(ro,r) '~(So)]
h(so) . GOl(ro, 1') . GOl(ro,r) . h(so)
where GOI is the dyadic Green's function for stratified medium with SOurce point in region l. Using the asymptotic formula of (5.2.31), we have
211 167f ; Aocos
fff dx dy dz WE1'(Z)1l (z)G Ol (ro,r) . G~;(ro,1')
ko = -0-
dz--1}(z)gOI(k.L, z) . gOI(k.L, z)
-*t -
Hence, using (5.2.27) and (5.2.28) in (5.2.33) gives
k TBh(k,w) = -0cos 0
L:L j-dl-1dz'TI(Z')
1=1 Eo -d l
IAI l( -klz ) eiklzZI
+ BI el(kIZ)e-iklzzI12
k TBv(k, w) = -0cos 0
L :L j-dl-1dz' 7Hz')
1=1 Eo
-d l
leI hl(-kl z) eiklZZI
+ D I hl(klz)e-iklzzl
Carrying out the integrations in (5.2.34) and (5.2.35), we find the brightness temperature as observed from a radiometer at an angle 00 to be
2.3 Brightness Temperatures for Stratified Medium
for horizontal polarization, and
for vertical polarization where kx = ksinO o . In the derivation of (5.2.36), we made use of the identities 2k~zk~~ = w2p,E~ and Ik tz l 2 + k~ = w2p,(E~k~z + E~/k~~)/k~z' The procedure for evaluating these expressions is as follows: (1) Both reflection coefficients R TE and R are evaluated by the recurrence relation method as given in (5.2.12). (2) The propagation matrix formalism of (5.2.23) and (5.2.24) are used to calculate the upward and downward wave amplitudes Al' El' GI, and Dl in each layer, as well as the transmitted wave amplitudes in the bottom layer TTE and TTM. That is, A o = RTE, B o = 1, Co = R , and Do = 1 are known from step (1), we can use (5.2.23a) and (5.2.23b) to calculate AI, B I , CI and D I and then A 2, B 2, C 2 and D 2, and so on. (3) The temperature Tl and permittivity El in each layer are used to perform the summation with the wave amplitudes previously obtained, as done in (5.2.36a) and (5.2.36b). When the medium is of constant temperature T, then the brightness temperatures are given simply by