Coherent Addition Approximation with Attenuation in .NET

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Coherent Addition Approximation with Attenuation
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For sparse media like vegetation, Foldy's approximation can be employed to account for the effects of absorption and scattering on the coherent wave caused by the inhomogeneities of the random medium. Rigorous derivation of the Foldy's approximation is treated in Volume III. As discussed in 7, Section 2.2 of Volume I, the propagation of coherent wave is governed by
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13 ELECTROMAGNETIC WAVES SCATTERING BY VEGETATION
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: independent -.-.-.-.-.-.-. : tree-independent
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Figure 13.3.9 Comparison of backscattering coefficient !Jvh by coherent addition approximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 0.45 GHz. The fractional volume is f = fvzfa = 0.12%. Cs = (11+i4)co. The scattering layer has a thickness of 2.46A, and the underlying half-space is flat and has a permittivity Esoil = (16 + i4)E o . The number of branches of each tree is 324.
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. independent -.-.-.-.-.-.-. : tree-independent
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Figure 13.3.10 Comparison of backscattering coefficient !J vv by coherent addition approximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 0.45 GHz. The fractional volume is f = fvzfa = 0.12%. E = (11+i4)co. s The scattering layer has a thickness of 2.46'\, and the underlying half-space is flat and has a permittivity Esoil = (16 + i4)Eo. The number of branches of each tree is 324.
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5 Scattering from Plants Generated by L-systems
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the following equation under Foldy's approximation
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= (zk o + Mhh)E h + MhvEv
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(13.4.1) (13.4.2)
d"; = MvhE h + (zk o + Mvv)Ev
where Eh and E v are the horizontal and vertical components of the electric field, s is the distance along the propagation direction (0, ), and i27rn o M jl = ~(Jjl(O, i 0, )) (13.4.3) with j, I = v, h. Here no is the number density of scatterers, ko is the wavenumber of the background medium, ijl is the scattering amplitude matrix element, and the angular brackets denote the configuration average. If the canopy structure exhibits statistically azimuthal symmetry, there will be no coupling between horizontal and vertical components of the coherent field. Thus
Ahv =
M vh
(13.4.4)
The effective propagation constants of horizontally and vertically polarized coherent waves are given respectively by
kh = k o
iMhh
(13.4.5) (13.4.6)
where k o is the wavenumber of free space. The attenuation of coherent wave is accounted for by the real parts of M hh and M vv . These two equations of coherent wave propagation can be incorporated to take into account the attenuation and phase shift before and after the wave is scattered by a particular vegetative element.
Scattering from Plants Generated by L-Systems Based on Discrete Dipole Approximation
In this section we use the discrete dipole approximation (DDA) method to calculate the scattering from trees generated by stochastic L-systems. The discrete dipole approximation is a volume integral approach as described in 2. The advantage of this approach is that the mutual interactions between the branches are included. The scattering from a layer of trees overlaying ground is calculated by assuming each tree scatters independently. As shown in Section 3, this assumption has compared well with the coherent addition approximation through the C-band, the L-band, and the P-band.
13 ELECTROMAGNETIC WAVES SCATTERING BY VEGETATION
5.1 Formulation Method
of Discrete
Dipole
Approximation
(DDA)
Using the DDA equations from (2.3.48) of 2, we have
E~iV = E:~
where
nc -
E:~'
j=l,#i
L (d~V)Aij' Eiv
E' (
(13.5.1)
is the permittivity of the background medium, and
ai =
f3ix
(XixXX
+ (XiyYY + C\;izZZ =
:2 - 1 7Ji
(13.5.2) (13.5.3)
(13.5.4) (13.5.5)
EL\V
f3iy
1+ 1+ 1+
(E:,
-1) (Lx - D x k 2 )
EL\V
(3iz
(E:, (E:,
1) (L y
Dy k 2 ) Dz k 2 )
EL\V
1 -1) (L z
and L\ V = dxdyd z . In the case of cells of circular cylindrical shape of radius a and length l, the corresponding results of Lx, L y , L z and D x , D y and D z are
L - L l_--cx - y - 2(4a2 +[2)1/2
l L = 1 -2 - - z (4a +[2)1/2
D =D =ika2l+i{./[2+4a2_l}+a2ln(l+V[2+4a2) x y 6 8 V 4 2a
(13.5.6a) (13.5.6b)
(1356) .. c (1:3.5.6d)
2 2 ika l (l a D =--+-In z 6 2
v[2 + 4a2 )
(1"i
and L\V = 7ra 2 l. The equation for A(1",1"') is given in (2.3.8) of 2. The value of
A(1",1"'), for 1"
= 1"i and 1'" = 1"j,
f= 1"j)
is expressed as
(13.5.7)
However, these may not be accurate enough when 1"i and 1"j are in the neighborhood of each other. Accuracy can be improved by numerical integration over the cell Vj centered at 1"j. Thus we can define a neighborhood distance