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(5.2.26)
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(5.2.27) nl We calculate Green's functio~ of the upper medium on the coarse grid. These are represented by CL a f3 and baf3 .
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2.4 Bistatic Scattering Coefficient and Emissivity
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where ~x is the coarse grid sampling, ~x = n1~x, and ~x is the dense grid sampling. Thus we use this averaging for the second terms of both (5.2.20) and (5.2.21). Equation (5.2.13) becomes
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(5.2.30)
In (5.2.30) we use subscript "intp" to represent linear interpolation. Note in (5.2.30) that L::~g1 a:nnu(xn) has N dg values of m = 1,2, ... , N dg , while L:~=1 a~!il(x(3) only has N values ofa = 1,2, ... , N. Thus we first compute
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a~(3u(x(3) = d(x a )
(5.2.30a)
for a = 1,2, , N on the coarse grid. To find d(x m ) on the dense grid of X m , m = 1,2, ,Ndg , we use linear interpolation of d(xa)'s to get d(xm)'s. We further use BMIAjCAG to solve matrix equation. We divide nonnear-field interactions into two regions which are separated by Td. We now have three distance ranges for the upper medium Green's function, 0 :S T :S Tj, Tj < T :S Td, and T > Td with different operations. For 0 :S T :S Tj known as very near-field region, we use direct matrix and column vector product on the dense grid. For Tj < T :S Td known as near-field region, we use direct matrix and column vector product on the coarse grid and interpolation as in (5.2.30). For T > Td known as non-near-field region, we expand aa(3 and ba (3 in Taylor series as in the BMIAjCAG so that the FFTs can be used to compute this part of the matrix-vector multiplication. The Taylor series expansion is as in Section 1
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(5.2.31)
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2.4 Bistatic Scattering Coefficient and Emissivity
After the matrix equation is solved, the surface field can be calculated. The bistatic scattering coefficient ()( Bs , B ) for the spectral domain tapered wave i is given in (4.1.57). We next illustrate the numerical simulation results of wave scattering from a rough lossy dielectric surface for both TE and TM waves [Tsang and
5 FAST METHODS FOR ROUGH SURFACE SCATTERING
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Figure 5.2.1 Comparison of the bistatic scattering coefficients between the single dense grid of 30 points per wavelength and the single coarse grid of 10 points per wavelength. TE wave, rms h = 0.5>", correlation length of I = 0.6>", dielectric constant of Er = 25 + i, surface length of L = 100>.., and tapering parameter of 9 = L/4 at incidence angle of (Ji. (a) One realization, (b) 20 realizations.
Li, 1997; Li et al. 1999]. Simulations are based on Gaussian random rough surfaces with Gaussian correlation functions. First, we show the comparisons of bistatic scattering coefficients and surface fields based on a single dense grid (SDG) and a single coarse grid (SCG) with a complex dielectric constant of 25 + i, surface length of 100 wavelengths, and at an incidence angle of 30 . The results show that the dense grid is required for the case with large dielectric constant. We shall regard the SDG results to be correct. Next, we compare the results based on PBTG-BMIAICAG with that of SDG. Then we use the PBTG-BMIAICAG method to calculate the cases with large surface length and compare with SDG. The tapering parameter was taken to be L I 4 for the case of surface length of 100 wavelengths and L 18 for the case of surface length of 500 wavelengths at near-grazing incidence. The critical distance r f that defines the very near field is fixed at 1 wavelength. The cases with a surface length of 100 wavelengths were run on a SPARC 20 workstation, and the cases with a surface length of 500 wavelengths were run on a Pentium-Pro Personal Computer with a clock rate of 200 MHz.