Wave scattering by multi-valued random surfaces

We consider a scattering theory for multi-valued and multi-connected rough surfaces, which cannot be described by the conventional equation of the type z = zeta(x, y). Both Dirichlet and Neumann problems are analyzed. Starting with the Green's theorem we obtained representation of scattered field, the surface integral equation, and the extinction theorem for such surfaces. In contrast to conventional theory, these equations contain three random functions x = x(mu(1), mu(2)), y = y(mu(1), mu(2)), and z = z(mu(1), mu(2)), where mu(1) and mu(2) are the parameters describing the surface. The Kirchhoff approximation and the first Born approximation are derived and a general relation between these approximation was found. All final results are presented in the form of surface integrals, which axe independent of choosing parameterization of surface.