An investigation of the small slope approximation for scattering from rough surfaces. Part II. Numerical studies

The small slope approximation (SSA) of Voronovich [Sov. Phys. JETP 62, 65–70 (1985)] is a promising method for modeling wave scattering from rough surfaces. The SSA T-matrix series, which can be interpreted as an expansion in a generalized surface slope, satisfies the appropriate reciprocity condition at each order and reduces to the standard perturbation series for small surface roughness. When the SSA T matrix is found to second order in generalized slope, it reduces to that of the Kirchhoff approximation as the frequency is increased. In an earlier paper [E. I. Thorsos and S. L. Broschat, J. Acoust. Soc. Am. 97, 2082–2093 (1995)] the derivation of the SSA for surfaces subject to the Dirichlet boundary condition was examined in detail. In this paper the accuracy of the SSA for the Dirichlet problem is investigated through comparison with exact results. Expressions for the first three terms of the SSA incoherent bistatic scattering cross-section series are presented, followed by numerical results for one-dimensional surfaces with Gaussian statistics and a Gaussian roughness spectrum. Surfaces with rms slope angles up to 45° are considered. It is found that, for the numerous cases studied, the SSA results agree well with the exact results over a broad range of scattering angles. When the lowest-order results are inaccurate, successive addition of each higher-order term generally yields improvement. The range of scattering angles over which the SSA results are accurate depends on both the rms slope angle and the surface correlation length, as well as on the angle of incidence. A simple rule of thumb, however, is that for an incident angle of 45°, the highest-order SSA scattering cross section examined here is accurate to within ±1 dB from backscatter to a forward grazing angle of 5° for rms slope angles less than about 30°. When the surface roughness is such that perturbation theory is accurate, the SSA is accurate over the full range of scattering angles for small to moderate slopes.