Computer Simulations of Stochastic L-Systems and Input Files in .NET

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2.3 Computer Simulations of Stochastic L-Systems and Input Files
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Figure 13.2.2 Tree-like scattering object generated using L-systems.
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Input file B
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Tree-like structure with a big stem and binary branching d1 160.00 ao 40 lm 0.04 W m 0.01 * (9 lb 0.04 wb 0.004 width 100
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#def ine maxgen 5 #define #def ine #define #define #define #def ine #define
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+ rand(2))
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/ * divergence angle * / / * branching angle * / / * 1/100 of average stem length * / / * stem width * / / * 1/10 of average branch length * / / * 1/10 of average branch width * /
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START: !(wm)F(lm
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PI :
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* (9 + rand(2)))/(180 + rand(180))A [&(ao + rand(10))!(wb * (9 + rand(2))) F(lb * (9 + rand(2)))]/(d 1 + rand(40))
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[&(ao + rand(10))!(wb * (9 + rand(2))) F(lb * (9 + rand(2)))JI(d i + rand(40))!(wm) F(lm * (9 + rand(2)))A
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13 ELECTROMAGNETIC WAVES SCATTERING BY VEGETATION
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Figure 13.2.3 Configuration of the tree-like scattering object generated by L-systems.
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tribution between 0.04 * (1 - 10%) and 0.04 * (1 + 10%). The branching angles have a mean value of 45 with a uniform distribution between 40 and 50 . Apex A also grows the main stem. The growth part of the main stem has a mean length of 0.4 with the length uniformly distributed in [0.4 * (1 - 10%),0.4 * (1 + 10%)). The divergence angles have a mean value of 180 with a uniform distribution between 160 and 200 . The number of growth stages is maxgen = 5. After 5 generations, the total number of branches, including the main branch, is 11. Figure 13.2.3 shows one of the generated simple trees with binary branching structure.
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Scattering from Trees Generated by L-Systems Based on Coherent Addition Approximation
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In this section we study wave scattering from trees without leaves by using coherent addition approximation. A cylindrical model is employed to represent the tree trunks, stems, and branches. The scattering amplitude of each branch is calculated by using the infinite cylinder approximation discussed in 1, Section 6.2 of Volume 1. The correlations of scattering by different branches are included by using their relative positions, which can be obtained after decoding the growth procedure.
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3.1 Single Scattering by a Particle in the Presence of Reflective Boundary
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We first consider in Section 3.1 coherent single scattering by a particle in the presence of a reflective boundary. Scattering by trees generated with L-systems is discussed in Section 3.2.
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Single Scattering by a Particle in the Presence of Reflective Boundary
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3.1.1 Electric Field and Dyadic Green's Function
Consider a particle located above a reflective boundary. The boundary is at z = 0 which separates region 0 and region 1. From Eqs. (5.2.15a-c) of Volume I, the dyadic Green's function is, for z > z',
G (r, r')
J ik~.(p-pl)eikzze(kz) + 8~2 J L
= 8~2
dkJ.. :z dkJ.. k = kxx K = kxx kJ.. =
[e(kz)e-ikzZI
+ RTE(kz)e( _kz)eikzZI] + R
(kz)h( _kz)eikzZI]
(13.3.1)
ikdp-pl)eikzz h(k z ) [h(kz)e-ikzZ'
where, based on the notations from 5 of Volume I,
+ kyY + kzz + k/iJ - kzz kxx + kyY
2 kx 2 ky
(13.3.2a) (13.3.2b) (13.3.2c) (13.3.2d) (13.3.2e) (13.3.2J)
k z =.V k 2 /
xx + YY p' = x'x + y'y
For a far-field observation point r in the
direction, where (13.3.3)
ks = sin Os cos cPsx + sin Os sin cPsY + cos Osz
the far-field Green's function is
G (r, r')
= ::: e(k sz ) [e(ksz)e-ikszZI + R TE (ksz)e( -ksz)eikszZI] e- ikd .p'
+ ::~ h(ksz ) [h(ksz)e-ikszZI + R
where
(ksz)h( -ksz)eikszZI]
e-iks~ pl
(13.3.4)
k sx = k sin es cos is
(13.3.5a)
13 ELECTROMAGNETIC WAVES SCATTERING BY VEGETATION
k sy = k sin es sin s ksz=kcose s ksJ.. = ksxx + ksyY
(13.3.5b) (13.3.5c)
(13.3.5d)
Let the incident wave be a plane wave with the following wavevector components
kix = k sin ei cos <Pi kiy = k sin ei sin <Pi kiz = k cos ei kiJ.. = kixX
(13.3.6a)
(13.3.6b)
(13.3.6c)
+ kiyY
(13.3.6d)
The incident wave is downward going in the direction (7f - i , i) while the reflected wave is upward going in the direction (e i , <Pi)' Thus the total field in region 0 is
E = [ETEe( -kiz ) + ETMh( -kiZ )] /Ki r
+ [ETEe(kiZ)RTE(kiZ) + E
RTM (kiZ)h(k iZ )]
i ki r
(13.3.7)
where ETE and E TM are the amplitudes of the TE and TM components of the incident wave. The TE and TM Fresnel reflection coefficients are denoted by R T E and R T M, respectively.
3.1.2 Scattering by a Single Particle
Consider a particle of permittivity Ep and relative permittivity the volume of Vp and is centered at rp . The scattered field is
occupying
E s = k21 dr' (E r
1) G (r, r') . E(r')
(13.3.8)
Then in the far field and in the ks-direction, the scattered field becomes
= -e(k. z )k 2 1 dr' (E r e
47fr
[e(ksz)e-ikszZ'
+ RTE(ksz)e(-ksz)eikszZ']
e-iks.L p'. E(r')
47fr
h(k sz )k 2
r dr' (E
(ksz)h( _ksz)eikszZ'] e- iksd/ . E(r') (13.3.9)
[h(ksz)e-ikszZ'
+ R
In the Born approximation, we approximate E(r') by the electric field of (13.3.7) given in the previous section. Thus
3.1 Single Scattering by a Particle in the Presence of Reflective Boundary
ikr E s = -e(ksz )k 2 e
47fr
(lOr -
[e( k sz )e-ikszz'
+ R TE (ksz)e( -ksz )eikszz']
e-iksi- .p' .
{ [ETEe( -kiz ) + ETMh( -kiZ )] iKi';P'