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L e -2Ok ,zZj -2ok szZj 2 sinc [ (k 27r 2Z
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Pj = XjX + y/iJ k d = k(k i - ks )
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Figure 12.2.2 Four scattering contributions for single scattering of the cylinder in the presence of a reflective boundaryo
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The phase dependence can be directly traced from the figure. The phase dependence e-ikizzj-ikszzj is due to the vertical Zj position of the cylinder. The phase dependence of ikdpopj is due to the horizontal position
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incident and scattered wave vectors in the horizontal direction. The amplitude dependence is as illustrated by the conical pattern of the cylinder that was discussed in 1, Section 6.2 of Volume 1. (b) Reflection by boundary followed by volume scattering by cylinder j, which has a dependence of
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LJo e'o-k dpOPj e'Ok iz (Zj +2d) -,ok sZZJ -sinc [(kiZ-kqz)LJo] .
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The phase factor exp( ikiz2d) is a result of the incident wave traveling through the canopy.
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(c) Volume scattering followed by reflection that has a dependence of o ok Zj+ -2 inc [(k ~z - k sz )Lo] e~k dpOPje-~Ok izZje+~ sz (2d)L J s 21f 2 (d) Reflection-volume-reflection scattering. Reflection followed by volume scattering that is further followed by reflection. The dependence is ikdpopj eikiZ(zj+2d) eiksz(zj+2d) L j sinc [(kiz 21f
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In this case both incident waves and scattered waves traveled through the canopyo Thus
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E(s)(l) (7')
ikr e ~ LJo - sinc LJo] = --:;:- ~ 21fe2O-kdpOP] { [(k iz + k sz )2 -fv(k s , ki )e- Ok SzZje-~ok izZj
+ sinc
[(k 2Z - k sz ) L j ] -fvr (k s, k)eikiZZj-ikszzj+2ik,zd 2 2
+ sinc[(k
-ksz )Lj]-frv (k S, k)e-iko,zzj-ikiZZje2iksz(zJ+d) 2 2 A
2 + SinC[(k o +ksz )Lj]-frvr (k s, kAo)eikszZjeikiZZJe2i(k,z+ksz)d}
where lv, lvr' lrv' and lrvr are field vectors that depend on cylinder radius aj and is polarization dependent on the scattering characteristics of the cylinders and the reflection by the half-space boundary. They will be derived rigorously in Section 2.2. The subscripts v and r denote volume and reflection, respectively. In the following, we shall assume that (1) the position of Zj is Zj = L j /2 - d (that is, the cylinder is attached to the boundary of the dielectric half-space) and (2) the length of the cylinder is Gaussian distribution with mean L o and standard deviation O'L. The probability density function (pdf) is
p (L) =
exp -
(L - Lo)2]
The probability distribution of length will smooth out side lobes that may exist in the conical scattering pattern. The radius of the cylinder is equal to constant a for all the cylinders. The pdf and joint pdf of horizontal positions are independent of lengths of cylinders. Thus, letting Zj = L j /2-d in (1202.6)
gives E S (l) (7') = _e_p(l)
L ikdP'Pi!(Lj)
is the first-order scattering amplitude of the canopy, and
7(T, j) ~ ~~ eiik .,+k,,)d { sinc [(I'i' + k." /:; ] f,c -iik +k,,) I., /2
+ sinc + sinc
[(ko~z - ksz )L j ] -fvr ei(kiZ-ksz)LJ/2 2 [(k ~z - ksz )L j ] -f rv ei(ksz-kiz)LJ/2 2
/:; ] Nffcilk...+k" )I., /2 }
+ sinc [( ki' + k."
The pdf and joint pdf of horizontal positions of cylinders are as follows. Let the cylinders be in clusters with N s cylinders per clw,ter. Hence the number of clusters is N c with
N (12.2.10) Ns Note that Nand N c are large numbers while N s may not be a large number. Let the cluster center be at (x a , Ya) with 0: = 1, ... , N c . Each cluster lies within a radius R c . Thus the pdf and joint pdf of clusters are Nc
p(xa , Ya) = A
where A = LxL y is the area under observation and Lx A, L y joint pdf of clusters is
P2 ( Pa , Pe
A. The
- ) - 9pCPm (5(3)
(12212) ..
where 9p is the pair distribution function. We also disallow interpenetration of clw,ters so that (12.2.13) where de is the minimum separation of the centers of two clusters. Within each cluster 0:, the positions of the secondary scatterers are at Pa + Paj,