Scattering from a random rough surface: Linear equations for coefficients of Wiener‐Hermite expansion of the wave field

This paper describes an exact formulation of random surface scattering by a probabilistic theory. No Rayleigh hypothesis and no approximate boundary condition are used in this formulation. Using the translation invariance property of a homogeneous random surface, we first determine a form of the scattered wave, which is written as a product of unknown homogeneous random function and an exponential phase factor. Regarding the scattered wave as a stochastic functional of the Gaussian random surface, we then develop the scattered wave in terms of orthogonal stochastic functionals (Wiener‐Hermite expansion) with unknown coefficients. As a result, the problem of finding the scattered wave is mathematically reduced to determining the coefficients. In the external region above the highest excursion of the random surface, each stochastic functional is physically represented by a linear combination of outgoing plane waves (and evanescent waves). Then Green's theorem and the ergodic theorem relate the surface field to the coefficients to get two sets of new equations; a set of equations determines the surface field, whereas the other set of equations gives the coefficients in terms of the surface field. Suggestions for solving these equations are described.