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Figure 5.1.3 Comparison of Monte-Carlo simulations of small and large surface lengths with (a) L = 40>- with 4000 realizations, (b) L = 200>- with 800 realizations, (c) L = 800>with 200 realizations, and (d) periodic boundary condition method with L = 40>- with 100 realizations. The parameters are h = 0.5>-, I = 1.0>-, and ()i = 10 .
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The numbers of realizations are 4000 for L = 40)" 800 for L = 200)" 200 for L = 800)" and 100 for the periodic case with P = 40).. The results of all four cases are similar upon averaging over realizations. It is to be noted that in Fig. 5.1.3d of PBe, the data points for angles beyond -78 are not available since the number of angular data points are limited by the Floquet modes.
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Example 3. Close to grazing and moderate RMS height and correlation length
In Fig. 5.1.4a, we illustrate the results for an incidence angle = 85 , rms height h = 0.5). and l = 1.0). and averaged over 50 realizations. We compare three cases of (L = 200)" 9 = L/4), (L = 800)" 9 = L/4), and (L = 800)" 9 = L/8). The results for the cases agree except for the forward specular direction (Fig. 5.1.4b) and the vicinity of the backscattering direction (Fig. 5.1.4c). The difference in the forward direction is due to the
1.5 Results of Composite Surfaces and Grazing Angle Problems
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Figure 5.1.4 (a) Comparison of bistatic scattering coefficients for various surface lengths and 9 for 50 realizations with h = 0.5)", l = 1.0)." and ()i = 85 . For L = 200)" the bandwidth is b = 200(20),,), and for L = 800)" it is b = 400(40),,). (b) Comparison of bistatic scattering coefficients of (a) near-specular direction. (c) Comparison of bistatic scattering coefficients of (a) near backscattering direction. The flat surface result is for L
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fact that the forward scattering peak of the coherent wave depends strongly on the surface lengths. The bistatic scattering coefficient from the largest surface length and the smallest tapering gives the smallest scattering level for angles beyond -85 . In Fig. 5.1.4c, an additional simulation is performed with a flat surface to illustrate the edge diffraction contribution to backscattering. Note that the backscattering level is at least two orders of magnitude larger than the edge diffraction. For this particular example, we have achieved convergence up to -87 .
Example 4. Close to grazing angle and composite surface
Next we examine the case of scattering from a composite random rough surface at a near-grazing incidence angle of ()i = 85 . The composite surface has a small-scale roughness (hI = 0.1). and II = 0.3),) superimposed on a larger scale roughness (h 2 = 0.5). and [2 = 5),). Figure 5.1.5a shows the bistatic scattering coefficient for one realization for the surface length of 2500 wavelengths with 25,000 surface unknowns. From Table 5.1.3 we can see that in order to perform Monte Carlo simulations with the 25,000 unknowns case, other methods which require the storage of a full matrix would be impossible on a workstation. However, using the BMIA/CAG, we were able to compute the solution. For a problem of this size, it requires 6 CPU hours on a SPARCIO workstation with 75 Mbytes of memory. In Fig. 5.1.5b, the bistatic scattering coefficient for 50 realizations is shown. In Fig. 5.1.5c, we compare the four cases of L = 200)" L = 400)" L = 800)" and L = 2500), near the backscattering angle. Clearly, the L = 200), surface is not large enough for the parameters used for this example. On the other hand, the results for L = 400)" L = 800)" and L = 2500), agree near the backscattering angle of -85 and up to -88 . This shows convergence of the bistatic coefficient with respect to the surface length for the scattering angles up to -88 . Next in Fig. 5.1.5d, the bistatic scattering coefficient from the 2500 wavelengths composite surface is compared with the result from the PBC with P = 40).. Since the periodic surface has the periodicity of 40 wavelengths, the angular resolution is only 6 near the backscattering angle of -85 . This is illustrated in Fig. 5.1.6 for one realization with P = 40).. In fact, for the 40). surface, the maximum backscattering angle for PBC is -78 . If we use a period of 200), for PBC, The matrix building time for this corresponding 2000 x 2000 matrix can be large.