## 1. Introduction

[2] In recent years, the scattering of waves from very rough random surfaces has been studied analytically, numerically, and experimentally by many authors [*Chen and Ishimaru*, 1990; *Jin and Lax*, 1990; *Maystre and Dainty*, 1991; *Ishimaru et al.*, 1991, 2002; *Tsang et al.*, 1994; *Ogura and Wang*, 1997; *Torrungrueng et al.*, 2002], with the emphasis on the effect of backscattering enhancement. Very rough surfaces are those with steep slope and RMS roughness characteristics comparable to the wavelength of the incident wave, and the conventional theories based on the Kirchhoff approximation and the perturbation method are not valid for them. It is quite difficult to treat the problem of scattering from very rough surfaces analytically because of the strong multiple scattering effects, and few analytical theories have been proposed for very rough surfaces. The modified second-order Kirchhoff approximation proposed by [*Ishimaru and Chen*, 1990] can give the results that agree well with those obtained by the Monte Carlo numerical simulations and in experiments, using the angular and the propagation shadowing functions introduced in their solutions for accounting for the effects of the higher-order scattering. The small-slope approximation first proposed by [*Voronovich*, 1994] is valid for arbitrary roughness but requiring relatively small slopes.

[3] By means of the stochastic functional approach [*Ogura and Wang*, 1994, 1996; *Wang et al.*, 1995], we have recently proposed a theory of scattering from very rough random surfaces [*Ogura and Wang*, 1997], but merely the first-order Kirchhoff approximation has been considered. For a plane wave incidence, the random wave field is written in the stochastic Floquet form in terms of a random spectral measure and then represented as an integral over a random distributed source through a Green's function or a propagator whose form depends on the scattering structure studied. The consideration on the level shift invariance leads to the extended Voronovich form for the stochastic boundary value solutions, and the use of the boundary condition leads to a stochastic integral equation satisfying by the unknown random distributed source (the boundary value).

[4] The stochastic integral equation can be a Fredholm integral equation of the first or second kind, depending on the boundary value being considered is the field itself or its derivative. For the Fredholm integral equation of the first kind, the unknown surface scattered field in the extended Voronovich form may be further represented as a series expansion based on small values of surface slopes and be determined by a similar procedure in the small-slope approximation method [*Voronovich*, 1994]. For the Fredholm integral equation of the second kind, the inhomogeneous term automatically gives the solution of the Kirchhoff approximation for infinite surface. The iterative substitution of Kirchhoff solution into the integral equation would then give the solution of the “double” (the second-order) Kirchhoff approximation, but the direct solution of the double Kirchhoff approximation leads to a five-dimensional integral for the incoherent scattering distribution, which is difficult to be performed numerically. In the paper, we show that the incoherent scattering angular distribution can be numerically calculated by only the two-dimensional integrals, if we make use of an approximation in which the correlation between the forward and backward scattering process is neglected and use of the propagation shadowing function [*Ishimaru and Chen*, 1990] of a very rough surface to obtain the incoherent scattering distribution from the second-order Kirchhoff terms.