Variation of Enhance:ment Angular Width with Layer Thickness

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To illustrate the angular width for the backscattering peak, Fig. 12.2.7 compares the bistatic scattering coefficient of the first-order scattering result in linear scale with two different average lengths L 1 = 60 cm and L 2 = 90 cm and standard deviations (JL 1 = 6 cm and (JL 2 = 9 cm, respectively. For the case of a longer cylinder, the peak for backscattering enhancement appears higher while the angular width becomes narrower. The angular widths for the 50-cm case and the 90-cm case are, respectively, 28 and 18 . Thus the enhancement peak is of the order of a wavelength divided by the layer

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2.3 Results of Monte Carlo Simulations

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BistCltic Scattering Coefficient

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Figure 12.2.7 Comparison of HH bistatic scattering coefficients (linear scale) between the case of cylinders with L = 90 em and (TL = 6 em and the case of cylinders with L = 90 em and IJL = 9 cm (50 realizations). The parameters are E = (20+i3)EO' E1 = (15+i5)Eo, N = p 100, VA = 560.5 em, a = 1 em, A = 20 cm, ei = 45.55 , and cPi = 0 .

thickness. This angular width is substantially larger than the purely volume scattering case, which is of the order of a wavelength divided by the mean transport path (see 8, Volume III).

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Second-Order Theory for Uniformly Random Case

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In Fig. 12.2.8 we compare the results of second-order theory, first-order theory, and first-order independent scattering. However, there is only very little difference between the results of first-order and second-order solutions for the uniformly random distributions. This is because of the small optical thickness of the layer.

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Second-Order Theory for Clustered Distribution Case

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We next illustrate the clustering distribution case in Figs. 12.2.9-12.2.11. The local fractional volume is equal to Ns'Tr~2 and is equal to 5% in Figs. 12.2.9 'Tra e and 12.2.10, and 25% in Fig. 12.2.11. The second-order solution becomes important for the clustered distribution cases. We keep the same total fractional volume of 0.1 %; however, the cylinders are placed in clusters with the cluster radius a c = 10 em in Fig. 12.2.9 (VV) and Fig. 12.2.10 (HH). We note that the clustering case gives much larger second-order scattering than

12 MULTIPLE SCATTERING BY CYLINDERS

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Bistatic Scattering Coefficient

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Figure 12.2.8 Comparison of HH bistatic scattering coefficients. Results of first-order, second-order, and independent scattering are shown (50 realizations). The parameters are E = (20 + i3)EQ, E1 = (15 + i5)EQ, N = 100, VA = 560.5 em, a = 1 em, L = 60 em, IJ"L = p 1 em, A = 20 em, e-i = 45.55 , and cP-i = 0 .

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the case of uniformly random distribution. This can be attributed to the fact that the scatterers are in the vicinity of each other in a cluster. The second-order scattering of vertical polarization is more important than that of horizontal polarization. That is because vertical cylinders scatter vertically polarized waves more than horizontally polarized waves and create more coherent wave interaction among the cylinders in a cluster. For second-order radiative transfer theory, the conditional probability of a second order given a first order depends on the optical thickness, which is small in this case. Thus radiative transfer theory cannot account for the enhancement of scattering due to clustering effects. In Fig. 12.2.11 we have more cylinders in each cluster with higher fractional volume of 25%. There are N s = 10 cylinders that are randomly distributed in each of N c = 10 clusters with the cluster radius a c = 6.32 em. The other parameters are the same as in Fig. 12.2.6. In comparison with Fig. 12.2.6, significant enhancements are observed in both backscattering and forward-scattering directions. Since the local density is higher than Figs. 12.2.9 and 12.2.10, the second-order effects are stronger. Also, the clustered scattering predicts much larger second-order scattering effects than in the case of uniformly random distribution. This is due to the coherent mutual wave interactions between scatterers that are in close proximity with each other.