63-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
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Figure 6.1.2 Convergence of bistatic scattering coefficient with the number of realizations for a 2-D rough surface. Four cases are shown: 155, 225, 275, and 310 realizations. The cases of 275 and 310 realizations overlap each other.
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--this method (310) ..... Kirchhoff (360)
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Figure 6.1.3 Comparison between the SMFSIA and the second-order Kirchhoff method. The second-order Kirchhoff method result is based on 360 realizations.
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Number of CGM Iterations 27 20 13 7 3 1
Error Norm % 29.31 6.74 2.56 1.04 0.12 0.01
Table 6.1.2 Convergence of the SMFSIA of 2-D surface of a single realization.
1.1 Scalar Wave Scattering
For each order of solution the error norm E(n) can be easily computed as follows. From (6.1.12) and (6.1.15) we obtain
liZ X(n) -
bll =
!!(Z(s)
+ Z(FS))X(n)
_ (b _ Z(w) X(n))11
= Ilb(n) _ b(n+l) II
(6.1.18) Table 6.1.2 shows the convergence of the SMFSIA for a single realization.
1.1.3 Convergence of SMFSIA
In actual implementation of the SMFSIA the iteration stops when the error norm of the original matrix equation has reached the established smallness criterion. In this section we examine the convergence of the SMFSIA. We note that the right-hand side b corresponds to the incident wave and is of the order of 0(1), where 0 stands for the order. The column vector
Z(w) X corresponds to the last term of (6.1.8). It is the original impedance
matrix with the removal of the near field sparse matrix and the flat surface impedance matrix.
_ eXP(ik P)] U(x, y)} JJp>rd 47f(xd+Yd+ z d)2 47fp (6.1.19) where Xd = x - x', Yd = Y - y', Zd = f(x, y) - f(x', y') and P = (x~ + yJ)~. We note that U(x, y) is of the order kO(l) and has a phase that is randomly fluctuating, whereas Zd is of the order of rms height h. Thus in the limit of large r d, with r d h, the first term in the square brackets can be Taylorexpanded. Therefore we have
O(Z(W) X)
dxdy
[eXP[ik(:~ +;J +:Jl~]
o [Z(W) X] = 0
=0 =0
[ff dxdy exp(ikp) (ikZ J ) U(x, y)] JJp>rd 47fp 2p
dpp f21r d exp(ikp) (ikZ J ) U(x, y)] Jo 47fp 2p (6.1.20)
[1~ d/xP~~kP) ikh2 U(x, y)]
By performing integration by parts of (6.1.20), we get (6.1.21) which is much smaller than 0(1) in the limit of large rd.
6 3-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
Electromagnetic Wave Scattering by Perfectly Conducting Surfaces
1.2.1 Surface Integral Equation
Consider an electromagnetic wave impinging upon a rough surface that is perfectly conducting. Then the integral equation for r above the rough surface is
H(r) =
Letting r have
---+
~,dS' [~ x G(r, r') . n' x H(r')] + Hinc(r)
(6.1.22)
r + which is a point infinitesimally above the rough surface, we
nxH(r) = nxHinc(r)+JiI!l nx
r-tr+
r J8' dS' [~x G(r,r'). n' xH(r')]
(6.1.23)
For a constant vector a
~ x G(r, r') . a = ~ x
Using (6.1.24) in (6.1.23), we have
(1 + :2 ~~ )g(r, r') . a
a (6.1.24) (6.1.25)
~g(r, r') x
nxH(r) = nxHinc(r)+ JiI!l nx
r-tr+
r J8' dS' [~g(r,r') x (n' x H(r')]
The integral in (6.1.25) is singular at r = r'. To handle this problem, we perform similarly to the Neumann case of 4, Section 1.5. Let
r = p + On (6.1.26) where p is a point on the rough surface. If 0> 0, r = r +, and if 0 < 0, r = r_. Thus, r + is infinitesimally above the surface while r _ is infinitesimally below
the surface. In (6.1.25) and (6.1.26)
n is the normal to the surface at point
p. Then nxH(p)=nxHinc(p)+ lim nx
<5-tO+
r J8' dS'[(~g(r,r')) __- + r-r
Xn'XH(r')]
(6.1.27) The f8' integration is divided into a circular disk Sa of small radius a about p and the rest which is known as the principal value integral
where
1 1 +1
dS' =
8-8a
dS' =
1 +f
(6.1.28)
represents principal value integral, which is the integration over S
with an infinitesimal circular disk of radius a, Sa, removed from S.
1.2 Electromagnetic Wave Scattering by Perfectly Conducting Surfaces
We next examine
I = lim 71 x
0-+0
f JSa dS' ('Vg(r, r')) r-r
x 71' x H(r') x (71XH(r))]
(6.1.29)
=lim71x
0-+0
[(f dS''Vg(r,r')) - Iss
r=r
where {) > 0 for r = r + and {) < 0 for r = r _. Let (p', ') be the polar coordinates of r' of circular disk of radius a centered about p
fSa dS''Vg(r, r') = r-+r J
)iJ!.l
dp' p'
d ''V
1 1 1 1
411" r - r
1_1 -, I
dp' p'
d ' [-
_ 1 _ 3 (r - r')] 411"lr - r'l r=r
d ' , o pp
d ' [_ (71{) - (p' cos ' X + p' sin ' fJ) ] 411"(p,2 + {)2)3/2
The X and fJ components integrate to zero because of the sin ' and cos ' dependence. Thus
f JS
dS''V (- -')
gr,r
=-n"2
d ' ,
PP(p,2+{)2)3/2
= -71"2 - (p,2 + {)2)1/2
where the
- (a2 + {)2)1/2 1
