Radio Science, Volume 16, Number 6, pages 1021-1024, November-December 1981 ScaRering from a rough interface J. A. DeSanto Naval Research Laboratory, Washington, D.C. 20375 (ReceivedNovember 3, 1980;accepted February 18, 1981.) This paper treats plane wave scattering from a rough interface separating two semi-infinite fluids of different density occupyingregions Vl and Va. The interface is initially consideredto be deterministic. A coordinate-space integral equation is derived for the surface value of the Green's function using Green's theorem in both regions and continuity conditions at the interface. For a source in Vl , Green's theorem in V2 whenevaluated at a field pointin Vi is a nonlocal impedance-type boundary condition. It is also called an extended or extinction boundary condition. Since there is only one free-space Green's function, the results of Green's theorem in both regions combine to yield a single integral equation. An integral relation is then used to f'md thescattered or transmitted field values. In Fourier transform space the T-matrix for the scattered field satisfies a Lippmann- Schwinger (LS) integral equation familiar from quantum potential scattering theory. Here the 'potential' is noncentral and complex. Several examples and a discussion of the generalization to the electromagnetic problemare listed. When the surfaceis considered to be a homogeneously distributed Gaussian random variable, this LS equation, Feynman diagram methods, and cluster decomposition methods from statistical mechanics are usedto derive an integralequation (Dyson equation) satisfied by the coherent part of T. This is a one-dimensional equation, and an approximation to it, whose Born term is the result due to Ament, is solved numerically. The result is a multiple scatter theory for coherent specularreturn which, when compared with experimental measurements, yields good results even for large roughness. DETERMINISTIC ROUGH INTERFACE We treat scalar scatteringfrom the deterministic roughinterface (in general,nonperiodic) illustrated in Figure 1. It separates two semi-infinite media (regions V•, j = 1, 2) of different density 0i with P = P2/P•. A point in spaceis denotedby x = (x•_, z) with x•_ = (x, y), and a point on the surface by x s= (x•_, h) wherez = h (x•_) denotes the surface. Mitzner [1966] has previously treated this problem but only for media of greatly different density. The Green's functions Giin regions V•. satisfy (C•mC• m '4- kl2)Gj(X, X tt) '- --•J(X- X n) (1) wherex E V•, kl is the wave number, and O m = (0/OX, 0/Oy, 0/OZ) where repeated subscripts are summed. They also satisfycontinuity conditions at the interface Ol (x•, x") = p(•: (x•, x") Sl (Xs, X tt) -- S 2 (Xs, x 't) where N•(%, x") = n m (X•_) Om Gj(Xs, nit) and nm(X•_ ) This paper is not subject to U.S. copyright. Published in 1981 by the American Geophysical Union. ---- •m3 -- O m_L h(x•) is a vector in the direction of the surface normal. We next sketch the method of finding the Lipp- mann-Schwinger equation. A more thoroughreview is in the works by DeSanto [ 1980, 1981]. Using the free-space Green's function G Oand Green's theoremin V1 (with the sourcepoint in V•) yields a standard integral relation between field and sur- face values of G• and N•. Green's theorem in V 2 (with both source and field points in V•) yields' a nonlocal impedance-type boundary condition,also called an extended boundary condition [ Waterman, 1975] or extinction coefficient [Pattanayak and Wolf, 1976]. Combining these two representations using the continuity conditions and transforming to Fourier transform space, it can be shown that the scattering amplitude F satisfies a Lippman- Schwinger integral equation [Newton, 1966] r(k', k") = V(k', k")A (k'- k") + V(k', k) ß A (k' - k) Gø(k)F(k, k" dk (2) with a noncentral and complex 'potential' term VA composedof a 'kinematical' vertex function V Paper number IS0280. 1021