Non-Gaussian Surfaces in .NET

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5.4 Non-Gaussian Surfaces
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BMIA PBTG-BMIA PBTG-FMM
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Figure 5.5.3 Comparisons of CPU time per iteration in the conjugate gradient method required by the PBTG-FMM, PBTG-BMIA, and BMIA. N is the number of surface unknowns.
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The dense grid is fixed at 30 points per wavelength and we change the surface length. It is shown that the PBTG-BMIA and PBTG-MLSDFMM have similar performance for the cases we compute. The first algorithm is an O(N log N) algorithm and the latter is a linear algorithm with N. Both of them take less CPU than the BMIA with the dense grid.
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5.4 Non-Gaussian Surfaces
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There are two types of correlation functions often used [Chen and Ishimaru, 1990], Gaussian and exponential correlation functions. The spectral densities of the Gaussian and exponential are given, respectively, by
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where h is rms height, Z is correlation length, and k is surface wavenumber. It has been found that the surfaces with Gaussian spectral density are far away from real natural rough surfaces such as soil and ocean whereas the surfaces with exponential correlation function are without rms slope, which is required for numerical simulations of wave scattering from random rough
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5 FAST METHODS FOR ROUGH SURFACE SCATTERING
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surfaces. The third type of surface roughness spectrum, power-law spectral density, is proposed as the following [Chen and Ishimaru, 1990; Kuga et al. 1993].
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where (2n - 2)!! = 2 x 4 x ... x (2n - 2), (2n - 3)!! = 1 x 3 x ... x (2n - 3), and (-I)!! = 1. The above spectrum becomes a Gaussian spectrum when the power index of n goes to infinity and is very similar to the spectrum with an exponential correlation function when n is one. The parameters hand l are supposed to be the rms height and correlation length in the above spectrum, respectively. But if we compare the power-law spectrum with a power index of one with the spectrum of the exponential correlation function, we find that the real correlation length of power-law spectrum is actually yl7flj2 . This can be seen by rewriting the power-law spectrum with the power of 1 as:
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Thus, a coefficient varying with the power index is needed and introduced to overcome this problem. The modified power-law spectrum is the following:
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where ap = r(p - 0.5)jf(p) and f is the Gamma function and b1 = yl7fj2, b2 = 0.95, b3 = 0.97, b4 = 0.98, ... , and boo = 1.0 are determined numerically, and h is the rms height and l is the correlation length. The modified powerlaw spectrum becomes a Gaussian spectrum when the power index n goes to infinity, and is the spectrum of an exponential correlation function when n is one. The important feature of the proposed spectrum is that it gives various spectra but with fixed rms height and correlation length, which are physical parameters usually used to describe the rough surfaces. We next show some numerical results of the bistatic scattering coefficients and the brightness temperatures from wet soil with the power law spectrum. The rms height and correlation length are fixed at 0.3 and 1.0 wavelength, respectively. The surface length is 64 wavelengths and the dense grid is 30 points per wavelength. The simulation was performed by the PBTGMLSDFMM. In Fig. 5.5.4, the comparisons of the bistatic scattering coefficients between surfaces with a power law spectrum with different power indices are
5.4 Non-Gaussian Surfaces
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Figure 5.5.4 Comparisons of the bistatic scattering coefficients from various spectra but with fixed rms height of 0.3 wavelength and correlation length of 1.0 wavelength at a angle of incidence of 30 degrees. The relative permittivity is 17.7 + i2.26. (a) TE wave; (b) TM wave.