Comparison of various approximation theories for randomly rough surface scattering

Comparisons of several approximation solutions to rough surface scattering are conducted for randomly topographies with an analytical description of the scaled wavenumber ka and the scaled rms height h/a (where k is the wavenumber, h is the surface rms height, and a is the surface correlation length). These approximations include the Kirchhoff approximation, Taylor expansion-based perturbation theory, Rytov phase approximation, and Born series method. The regions of validity of the approximation solutions are examined using both Gaussian and exponential roughness surfaces as benchmark models in comparison with the full-waveform numerical simulations obtained by boundary element method. The Kirchhoff approximation is applicable for large-scale roughness components with smooth surfaces under ka > 6 and shares similar variations with the full-waveform solution for other values of ka  . Rytov phase approximation improves the Kirchhoff approximation to large-scale roughness components in both amplitude and phase for all values of kaka. Perturbation theory is suited for small-scale roughness components under ha<0.2 and ka < 1.9, with rough surfaces characterized by short-range, small-amplitude fluctuations. The medium-scale roughness components, determined by excluding the roughness regions applicable by both the Kirchhoff approximation and perturbation theory, can be generally studied by the second-order or high-order Born series approximations that account for multiple scattering between surface points. Finally, these approximation methods apply to a topographic profile of the path from the seismic event of June 2, 1990 in the Tibet region by separating its surface roughness into the large-, medium-, and small-scale components satisfied by the ranges of validity of the Kirchhoff approximation, the Born series approximation, and the perturbation method, respectively.