Formulation of rough-surface scattering theory in terms of phase factors and approximation solutions based on this formulation

This paper presents the formulation of a rough-surface scattering theory in which the bounded phase shift factors, L(r, α) ≡ exp[iαζ(r)], replace the elevation, ζ(r). Both the Dirichlet and the Neumann problems are considered. The integral equations for secondary surface sources are obtained that contain only this phase function in their kernels. The Neumann (iterative) series for the solutions of the integral equations thus derived are functional Taylor series in powers of L(r, α), not in powers of ζ. If we expand L(r, α) in these series in powers of ζ(r), we obtain the standard perturbation theory series. Thus, the new formulation corresponds to the partial summation of the perturbation series. Using the Neumann series, we obtain several uniform (with respect to αζ) approximate solutions that contain, as limiting cases, Bragg scattering, the Kirchhoff approximation, and most known advanced approximations. In the case of random surface z = ζ(r), these new expansions contain the function ζ(r) only in the exponents, and, therefore, the result of averaging can be expressed only in terms of the characteristic functions of the multivariate probability distribution of elevations.