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We use the theory for a mixture of three constituents to illustrate the application to wet snow. In Figs. 4.3.4, 4.3.5, and 4.3.6, we plot, respectively, the real part of the effective dielectric constant, the imaginary part of the effective dielectric constant, and backscattering coefficients as a function of fractional volume of scatterer 2 from 0% to 10%. The permittivity of scatterer 2 is taken to be that of water and is given by (4.3.47). The fractional volume of the background medium is fixed at 80%. Therefore, as f.s2 increases from o to 10%, fsl decreases from 20% to 10%. From Fig. 4.3.6 we note that the backscattering coefficients decrease as the wetness of the snow is increased. The radii of both scatterers 1 and 2 are chosen to be 0.5 mm, which falls within the grain size of snow. In Fig. 4.3.6 the theoretical result is also used to match experimental data taken from wet snow [Stiles and Ulaby, 1980]. The frequency of operation for the data was 17 GHz and was for HH polarization. The data were plotted as a function of volumetric snow wetness, M v , which we have equated with fs2' The incident angles were 50 and 70 , and the snow depth d was 30 cm with water equivalent W equal to 5 em. Since ice has a smaller density than water, the quantity obtained by dividing the water equivalent W by snow depth d is generally about 5% to 20% smaller than the fractional

i=12.

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volume of ice. Thus in Fig. 4.3.6 we take the fractional volume of air to be 80%. There is good match between theory and experimental data. In recent years, extensive studies have been made on strong permittivity fluctuation theory with extensions and improvements. Applications have been made to dry and wet snow. Relations were also established between the dense medium discrete scatterer model and the strong permittivity fluctuation model [Veysoglu, 1994]. Interested readers can refer to the references.

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Chuah, H T and H S Tan (19S9), A high order renormalizationmethod for radar backscatter from a random medium, IEEE Trans Gcosci Remote Sens, 27 79-85 Frisch, V (196S), Wave propagation in random medium, Pmbabilistic Methods in Applied Mathematics, 1, Bhamch-Reid, Ed, Academic Press Fung, A K (1979), Scattering from a vegetation layer, IEEE Trans Geosci Electronics, 17, 1-5 Fung, A K and H S Fung (1979), Application of first-order renormalization method to scattering from a vegetation-like half-space, IEEE Trans Geosci Elce!,,imics, 15, 189195 Ishimaru, A (1978), Wave Propagation and Scatte1'inq in Random Media, 1 and 2 Academic Press New York Tin, Y Q (19S9), The radiative transfer pquation for strongly-fluctuation continuous random media, 1 Quant Speetmsc Radiat Transfer, Vol 42, 529 537 lin, Y Q (19S5), Strong fluctuation theory for a bOUJl(jpd layer of random discrete scatterers, PhD.

thesis, lVlassachusetts Institute of Technology Jin, Y Q and J A Kong (1984), Strong fluctuation theory for electromagnetic wave scattering by a layer of random discretp scatters 7 Appl Phys, 551364-1369, Tin, Y Q and J A Kong (1 9S5) Strong fluctuation theory for scattering, attenuation, and transmission of microwave through snowfall, IEEE Trans Geosci Remote Sens, 23 754-760 Karal, F C aud 1 B Keller (19GO), Elastic eleetromaglletie and other waves in random medium, 7 Math Phys, 5 5:37-547 Keller, 1 B (1964), Stochastie equations and WavP propagation in random media, P1'Oc Symp Appl Math, 16 145 170 (Am l\lath Soe, Providence, RI) Keller T B and F C Kantl (196G), Effective diPlectric eonstant, permeability and conductivity of a random mediulll and the velocity and attpnuation coefficient of coherent wavps, 7 Math Phys . 7, GGl--G70 Lim, H H.

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, M E Veysoglu, S H Yueh, R T Shin and T A Kong (1994), Random medillln model approach to scattering from a mndom eollection of diserete seatterers, l Elect1'Omag Waves and Appl, 8, SOl-S17 l\liitzler, C (1996), Microwave permittivity of dry snow, IEEE 7i an~ Geosci Remote Sens, 34(2), 573-5S1 Miitzler, C (1998) Improved Born approximation for scattering of radiation in a granular medium, 7 Appl Phys . 83(11)6111-6117 Nghiem S V (1991), Electromagnetie wave models for polarimetrie remote sensing of geophysical media, PhD thesis, l\I<l~sachusetts Institute of Technology Nghiem, S V, R Kwok 1 A Kong, R T Shin (1993), A nlOdel with ellipsoidal scatterers for polarimetrie remote sensing of anisotropic layereclmeclia, Radio Sci, 28(5) 6S7-703 Nghiem, S V, T Le Toan J A Kong, H C Han.

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