Energy Conservation for Rough-Surface Scatter

Abstract

The description of high-frequency scatter from random rough surfaces by the usual surface slope density formulation is internally inconsistent for, in general, energy is not conserved. Nonphysical energy losses are introduced by the neglect of multiple scatter and become more pronounced with increasing rms slope. Nonphysical energy gains arise from the neglect of shadowing considerations and tend to infinity at grazing incidence. These deficiencies in the theory are demonstrated for a perfectly reflecting, one-dimensionally rough surface. It is shown that the infinity at grazing incidence is removed by the extension of slope density to joint probability density of slopes and illumination. However, this modification is only the first term in a ray-trace expansion of the scattering coefficient in the number of surface interactions encountered by an incident ray. A numerical calculation of the shadow-consistent, single- and double-scatter terms demonstrates a significant reduction in the magnitude of the nonconservation discrepancies over the entire angular range, and a virtual elimination of these difficulties over an appreciable angular range about grazing.