Analytical vs. Numerical Methods for Rough Surface Scattering

The integral equation method can be used to obtain exact solutions for scattering from one-dimensional surfaces. Scattering cross sections for randomly rough surfaces can then be found by averaging scattered intensities for many surface realizations. The accuracy of analytical methods for rough surface scattering can then be examined. Traditionally, the perturbation and Kirchhoff approximations have been the most commonly used analytical methods—the former for surfaces with small roughness and the latter for surfaces with smooth roughness. These approaches can be extended systematically to obtain the perturbation series and the multiple scattering series, respectively. The first few terms in the perturbation series can be formally averaged, extending the range of analytic perturbation theory. The terms in the multiple scattering series beyond the lowest order (the Kirchhoff approximation) have not yet been formally averaged. The multiple scattering series is thus implemented as an approximate numerical method. Numerical results will be presented showing the accuracy and limitations of these two series approaches. These results will also clarify how the two complementary series apply when the surfaces have both small and smooth roughness. The relationship of the multiple scattering series to shadowing phenomena and a major shortcoming with the multiple scattering approach will also be addressed. Recently, several new approximations to rough surface scattering have been developed which reduce properly to the perturbation and Kirchhoff approximation limits. These new approaches will be briefly reviewed and their accuracy discussed.