Radio Science, Volume 21, Number 5,Pages 771-786, September-October 1986 Vector solution forthe mean electromagnetic fields ina layer ofrandom particles R. H. Lan•] Department of Electrical Engineering and Computer Science, Geor#e Washington University, Washington, D.C. S.S. Seker Department of Electrical En#ineerin#, Bo•aziqi University, Istanbul, Turkey D. M. LeVine Goddard Space Fli•7ht Center, Greenbelt, Maryland (Received August 13,1985; revised April 11,1986; accepted April 11,1986.) The mean electromagnetic fields arefound in a layer of randomly oriented particles lying over a half space. A matrix-dyadic formulation of Maxwell's equations is employed in conjunction with the Foldy- Lax approximation to obtain equations for themean fields. A twovariable perturbation procedure, validin thelimit of small fractional volume, is then used to derive uncoupled equations for theslowly varying amplitudes of the mean wave. These equations aresolved to obtain explicit expressions for the mean electromagnetic fields in the slab region in thegeneral case of arbitrarily oriented particles and arbitrary polarization of the incident radiation. The results include special cases previously treated in the literature on propagation through the atmosphere. Numerical examples are given for the appli- cationto remote sensing of vegetation. 1. INTRODUCTION Many of the scenes of interest in remote sensing of the environment can be conveniently modeled as a collection of randomly orienteddiscrete objects. Ex- amples are rain, the leaves of trees or cropssuchas soybeans, ice crystals formed in cold cirrus clouds, etc.In many applications, especially in remote sens- ing from space at microwave frequencies, the col- lection can be regarded as existing in a layer above a ground plane(e.g., the earth). It is the purpose of the present paperto deriveexpressions for the mean elec- tric field which propagates in such a layer whenthe layer is composed of sparsely distributeddiscrete particles whichhave arbitraryshape, size, and orien- tation. The knowledge of the mean field is useful for com- putingattenuation constants for the layer, depolar- ization effects and backscattering effects near nadir. In addition,it is hoped that many of the sameap- proaches usedsuccessfully on the mean wave prob- lems can,in turn, be applied to the fieldcorrelation Copyright 1986 by theAmerican Geophysical Union. Paper number 6S0198. 0048-6604/86/006S-0198508.00 calculation. This in turn can lead to a better under- standingof the relationship of transporttheory to Maxwell'sequations. The starting point for the calculation is theFoldy- Lax mean equation. This equationhas beenderived by Foldy [1945-1 for scalar wavesand dipoles scat- terers by assuming that the meanfield is the incident field on the particle. It has been extended to scat- terers having an arbitrary size with respect to a wavelength by Lax [1951]. The equation in its scalar version has been rederived by Twersky [1964], Keller [1964-1, Frisch [1968-1, Furutsu [1975-1 and Ishi- maru [1975]. The corresponding Foldy-Lax mean equation for vector waves subsequently hasbeen ob- tainedby Lang [1981•. The mean equation in both the scalarand vector cases is an integrodifferential equation which for con- stant density p and unbounded regions can be alge- bratized by a three-dimensional Fourier transform. For cases in which p is not constant or only piece- wise constant, such as layers, the three-dimensional Fourier transformdoes not completely algebratize the mean equationand thus it does not provide a simple way to obtain a solution. An alternative method which does not require p to 771