An improved formulation of coherent forward scatter from random rough surfaces

The operator expansion method is known to give accurate numerical results for scattering from individual surfaces that are too complicated for other methods. It is less widely appreciated that the method can be applied to random surfaces as well. The simplest application is the modelling of mean forward scatter from a homogeneous Gaussian ensemble of surfaces. To lending order in the admittance operator, the formula for the scalar Dirichlet boundary includes an exponential form in the roughness correlation function. The scalar Neumann boundary adds terms involving the gradients of the exponential form. These factors modestly alter the magnitude and advance the phase of the coherent scatter relative to the conventional one-point (Kirchhoff) approximation when the significant surface correlation scales are comparable to the radiation wavelength. Narrow troughs in the surface undulations 'repel' the radiation and effectively elevate and flatten the mean surface. These results are reliable over a wide range of surface amplitudes and correlation scales, provided the slope times Rayleigh height (Dirichlet problem) and slope (Neumann problem) are not large.