Equations for the description of wave scattering by multi-valued random surfaces

We consider a scattering theory for multi-valued rough surfaces which cannot be described by the conventional equation of the type z = (x,y). Both Dirichlet and Neumann problems are analyzed. Starting with Green's theorem we obtain a representation of the 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(u1,u2), y = y(u1,u2), and z = z(u1,u2), where u1 and u2 are the parameters describing the surface. We introduce two scattering amplitudes S +/- for describing the scattered wave above and below the surface. The extinction theorem, if formulated in terms of S-, allows us to determine S- for an arbitrary multi-valued surface and after this it becomes possible to derive a simple integral equation for surface sources. Knowledge of the surface sources allows us to find S+ by integration.