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where r is gamma function. The parameter f{2 is related to the mode size a c . The mode radius a c is such that n(a c ) is a maximum in the distribution. The mode radius a c can be found by setting the derivative of (7.7.8) with respect to a equal to zero. Then
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Instead of using J{l, J{2, P, and Q to characterize the distribution, we will use fiot, a c , P, and Q.
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Figure 7.7.1 Comparison of brightness temperature of vertical V polarization (TM) and horizontal H polarization (TE) as a function of angle Bo as computed by conventional radiative transfer theory and dense media radiative transfer theory. The input physical parameters are for a half-space medium with One particle species with frequency = 18 GHz, h = 0.3, E1 = (3.2+iO.016) 0, al = 0.175 cm, and T = 272 K. The background medium permittivity is o. The calculated mixing formula effective permittivity f~~t) is (1.49+iO.0029) 0. Parameters represent dry snow.
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Figure 7.7.2 Comparison of snow field brightness temperature experimental measurements [Shiue et a!. 1978; Kong et a!. 1979] with theoretical results of dense media radiative transfer equations using the input physical parameters of Fig. 7.7.1. V stands for vertical polarization and H stands for horizontal polarization.
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Figure 7.7.3 Brightness temperature of vertical polarization (solid line) and horizontal polarization (dashed line) as a function of observation angle Ou at frequency = 18 GHz, for a half-space medium with a gamma size distribution of particles with P = 6, Q = 2, a c = 0.075 cm, ftut = 0.3, Es = (3.2 + iO.002)E u and T = 272 K. The computed values are: effective propagation conHtant J( = (4.60 + iO.2865 x 10- 2 ) cm- 1 , "e = 0.00573 cm- 1 , and W = 0.7573.
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Figure 7.7.4 Brightness temperature of vertical pohtrization (solid line) and horizontal polarization (dashed line) as a function of observation angle Ou at frequency = 18 GHz, for a half-space medium with a gamma size distribution of particles with P = 6, Q = 2, a c = 0.1 cm. hut = 0.3. f s = (3.2 + iO.O(2)f o and T = 272 K. The computed valucH arc: effective propagation constant J( = (4.60 + iO.584 x 10- 2 ) cm- 1 , "e = 0.0117 cm- 1 , and W = 0.881.
7 Numerical Illustrations
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Figure 7.7.5 Comparison of dense medium radiative transfer theory with backscattering data from snow. The input parameters are frequency = 17 GHz, lOs = (3.2 + iO.002)f o , d = layer thickness = 27 em, 102 = permittivity of homogeneous half-space below scattering layer = (6 + iO.6)f o . The gamma size distribution parameters are P = 6, Q = 2, a c = 0.125 em, and ftot = 0.2. Based on these input parameters, the computed values are K = (4.064 + iO.00941) em-I, K e = 0.0188 em-I, and w= 0.961.
In Figs. 7.7.3 and 7.7.4 we plot the brightness temperatures for a halfspace medium with particles obeying a gamma size distribution for two different mode radii of 0.075 cm and 0.1 cm, respectively. The permittivity of the particles is f s = (3.2+iO.002)fo ' From the figures, it can be seen that the larger particles which correspond to the tail of the gamma size distribution can contribute significantly to scattering and scattering-induced decreasing of brightness temperatures. Figure 7.7.4 has a lower brightness temperature than Fig. 7.7.3 because of a larger mode radius. In Fig. 7.7.5 we compare the result of backscattering coefficient in active remote sensing with experimental data at 17 GHz [Stiles and Ulaby, 1980]. The parameters are a two-layer medium with the uppermost layer being air. The middle layer is a slab of scatterers with f s = (3.2 + iO.002)f o of thickness d = 27 cm overlying the lowermost layer of a homogeneous half-space of f2 = (6 + iO.6)fo . The particles in the scattering layer obey gamma size distribution with P = 6, Q= 2, and a c = 0.125 cm. The backscattering coefficient is calculated by solving (7.4.5) subject to boundary conditions at the top and the lower boundary. The backscattering coefficient (J = 41f cos ()oi . I o(()oi,1f + cPoi) for HH polarization is illustrated in Fig. 7.7.5 as a function