Comparisons of rms heights. in .NET

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Table 4.5.2 Comparisons of rms heights.
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lated power spectrum with a power law, we have disregarded regions that do not have a linear behavior on a log-log plot. In Table 4.5.2, we list the parameters C and a of the power law. Because the power-law function blows up for K = 0, we have assigned W(O) = 0 in the surface generation. An alternative approach is to taper the power-law spectrum by an exponential function for small values of K n . However, we find that the alternative approach of an exponential tapering did not produce a noticeable difference in the surface profile or the scattering result. It can be seen from Fig. 4.5.4 that the power law gives a good fit for the cases of rock and soil surface spectra. For a rock ~mrface, a power-law fit overestimates the calculated spectrum for spatial frequencies lower than 3/.x. Therefore, a rock surface generated with the fitted spectrum of Fig. 4.5.4a generates surfaces with larger rms surface heights. From Table 4.5.2, we see that the rms height difference between the real-life surfaces and those generated with a power-law spectrum is 38%. For a soil-surface spectrum, the agreement between the calculated spectrum and the best-fit spectrum is good over a wider range than the rock-surface case.
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Figure 4.5.4 Comparison of calculated average spectrum and a best-fit spectrum for (a) rock (0 = 2.5, C = 10- 1 .4), (h) snow (0: = 1.8, C = 10- 2 .85 ), and (c) soil (0 = 2.25, C =
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Rocks (dB)
81 82 83
Snow (dB) -16.0 -14.9 -19.8
Soil (dB) -15.1 -14.8 -19.4
-9.6 -9.1 -4.0
Table 4.5.3 Backscattering level comparison -45 0
This is further evident in the rms height comparison, where the agreement is within 1%. Next, bistatic scattering cross sections from the surfaces of Figs. 4.5.14.5.3 are presented in Figs. 4.5.5a-4.5.5c, respectively. The incident angle is 45 for all simulations. Since the study is a profile testing, all scattering simulations are based on perfect electric conductors. In Table 4.5.3, we list the backscattering (Os = -Od levels in decibel scale; the number of surface realizations is found in Table 4.5.2. All simulations satisfy the power con-
5.3 Numerical Results of the Three Methods
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Figure 4.5.5 Comparison of bistatic scattering coefficient from three types of surfaces: (a) rock surfaces of Fig. 4.5.1, (b) snow surfaces of Fig. 4.5.2, and (c) soil surfaces of Fig. 4.5.3.
servation check to less than 1%. From Figs. 4.5.5a-4.5.5c and Table 4.5.3, it is seen that the backscattering from the calculated spectra (82 surfaces) are within 1.1 dB ofreal-life surfaces (81 ) for all cases. Note that because of the limited number of real-life profiles, the calculated power spectrum W(K n ) from 8 1 still oscillates (Fig. 4.5.4). This W(Kn ) is also used to generate 82 profiles. Nevertheless, the backscattering from 8 2 surfaces all converge to within 1.1 dB of the correct results. From Table 4.5.3, we can see that the backscattering level of the powerlaw spectrum that are used to fit the measured spectrum and the real-life surfaces are not in good agreement. For rocks (Fig. 4.5.5a) the best-fit spectrum model scattering result is 5.6 dB above that of the real-life surface. For snow surfaces, the scattering solutions between the real-life and best-fit spectra have a 4 dB difference. The power-law spectrum of soil provides a better fit of the spectrum than snow or rocks over a wider range of spatial frequencies. Based on this observation, it would seem that the soil-surface case will likely give the best scattering agreement. However, it gave a 4.3-dB difference in backscattering. If we take the point of view of the first-order small perturbation model, the average backscattering is proportional to the spatial frequency component 2k sin Oinc of the spectral density, i.e., W(2k sin Oinc). Thus SPM says that only the spectral component 2k sin Binc in the spectrum matters in backscattering. For the cases considered, 2k sin (hnc is approxi-