An examination of the “full‐wave” method for rough surface scattering in the case of small roughness

Abstract

We examine the accuracy of Bahar's “full‐wave” method for rough surface scattering in the case of small roughness. The specific problem examined is scattering from one‐dimensional rough surfaces assuming the Dirichlet boundary condition. This corresponds to scattering of horizontally polarized electromagnetic waves from perfectly conducting one‐dimensional surfaces. By one dimensional surfaces we mean surfaces with constant height in the direction perpendicular to the incident wave direction. Our main result is that full‐wave predictions for the normalized bistatic radar cross section are found to disagree substantially with first‐order perturbation theory when the latter is known to be accurate through comparison with exact integral equation studies. This disagrees with Bahar's conclusions. The full‐wave predictions for the normalized radar cross section were computed using a Monte Carlo method and also by averaging formally; the two methods show excellent agreement. For the examples discussed in the paper, the full‐wave prediction agrees closely with the Kirchhoff approximation result, though the Kirchhoff approximation is not accurate away from the specular direction for these cases. In fact, for the special case of backscattering with shadowing neglected, the full‐wave predictions are shown to agree exactly with the Kirchhoff approximation results, independent of surface roughness properties. We examine Bahar's analytic argument for the reduction of full‐wave theory to perturbation theory and show that it is based on an inconsistent expansion in small surface height. When the small height expansion is done consistently, the full‐wave prediction does not reduce to first‐order perturbation theory. Finally, we consider the computational procedure given by Bahar to reduce the numerical effort in evaluating full‐wave theory. We find that numerical results obtained via this procedure approximately coincide with numerical results from first‐order perturbation theory for sufficiently small surface heights and slopes. This may at first seem confusing in light of our other findings. However, we show that certain simplifying assumptions in Bahar's computational procedure change the numerical results significantly from those computed from full‐wave theory without such assumptions. Thus these assumptions are not acceptable approximations for full‐wave theory, and results based on them should not be considered valid results of full‐wave theory.