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,ed on the first and second order vector radiative transfer theory, IEEE Tmn8 Ge08ci Remote Sens . 29, 242--253 T~mng L and J A Kong (1977), Thermal microwave emission from a random homogeneous layer over a homogeneous mediulll using the method of invariant imbedding, Radio Sc i, 12, 185195 Tsang, L and J 11. Kong (1982), Effective propagation wnstants for coherent electromagnetic wave propagation in media embedded with dielectric scatterers, / Appl Phys, 53,7162-7173 Tsang, L, 1 A Kong, Z Chen, K Pak, and C Hsu (1995), Theory of microwave scattering from vegetation ha;;ed on the e-olleetive se-aLtering effects of dise-rete scatterers, Pa"sive Micmwa'll(~ lle-mote Scn,inlJ of Land-At1l/0sphe1"l~ Jnt,emetion, B J Choudhury, Y H Kerr E G Njoku, and P Palllpaloni, Eds, 117-15/1 Tsang, L, J A Kong, and R T Shin (1985), 1'heo'I~/j of Micmlllave Remote Sen"ing, WileyInterscience, New York.



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Twersky, V (1977), Coherent scalar field in pair-correlated random distributions of aligned scatterer;;, / Math Phys . 18, 2468--2486 Yueh, S n, J A Kong J K lao, R T Shin, and T Ie Toan (1992), Branching model for vegetation, JEEE Tmn, Geosci Remote Sens, 30, 390-402..

Scattering of Electromagnetic Waves: Advanced Topics. Leung Tsang, Jin Au Kong. Copyright 2001 John Wiley & Sons, Inc. ISBNs: 0-471-38801-7 (Hardback); 0-471-22427-8 (Electronic)

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4

1.1 1.2

Dyson's Equation for the Mean Field Bilocal Approximation Nonlinear Approximation Second Moment of the Field





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2.1 2.2

Random Medium with Spherically Symmetric Correlation Function Very Low Frequency Effective Permittivity Effective Permittivity Under tk Bilocal Approximation Backscattering Coefficients Results of Effective Permittivity and Bistatic Coefficients

3.1 3.2

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In this chapter we will use the diagrammatic approach to study multiple scattering in random media. The diagrammatic approach leads to Dyson's equation for the mean field and the Bethe-Salpeter equation for the covariance of the field. Both Dyson's equation and Bethe-Salpeter equation are exact. Approximations are made on the Dyson's equation. The bilocal and nonlinear approximations will be examined. For the Bethe-Salpeter equation, the ladder approximation is used. We also study the case when the variance of the permittivity fluctuation is appreciable, leading to the analytic theory of strong permittivity fluctuations.

Consider an unbounded random medium with permittivity E(1') = (E) +E j(1'). The dyadic Green's function G(1', 1'0) in the medium satisfies the vector wave equation

\7 x \7 x G(1',1'o) - W2Il(E)G(1', 1'0) = Q(r)G(1',1'o) + 18(1' - 1'0) (4.1.1) 2fl,Ej(r). In (4.1.1). we have put the (E) part on the left-hand with Q(1') = W side of the equation while putting the E j part on the right-hand side. The solution to (4.1.1) can be expressed as an integral equation by using

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