Generalization of the geometrical optics scattering limit for a rough conducting surface

Scattering cross sections for randomly rough surfaces such as the ocean are often computed using geometrical optics for near-normal incidence or the composite surface model for higher incidence angles. Several obstacles limit the applicability of these methods. Since geometrical optics is exact only in the infinite frequency limit, it breaks down for surfaces with roughness on scales much smaller than a wavelength for which the correlation function is not analytic at the origin. Physically, surface features which are much smaller than the electromagnetic wavelength cannot influence the backscatter, but a rigorous understanding of the cutoff in sensitivity to surface features has lacked. For surfaces having roughness at all scales, the choice of separation parameter required to implement the composite model has been unclear. It has also been difficult to match composite model cross sections to geometrical optics at the transition from near-normal to mid-range incidence angles. In this paper, the authors overcome these difficulties by extending the geometrical optics limit to finite frequencies and deriving the composite model as an approximation to the finite frequency generalization. In the infinite frequency limit, the scattering cross section in the physical optics approximation can be evaluated asymptotically to yield the usual geometrical optics result with Gaussian dependence on incidence angle. At finite frequencies, this dependence generalizes to an a-stable distribution, where the parameter a is determined by the asymptotic decay of the surface height power spectral density. This stable distribution is insensitive to surface features smaller than a rigorously derivable cutoff size. The composite model arises as an approximation to the tails of the stable distribution, and a smooth transition from near-normal to mid-range incidence angles is obtained