A major impetus to study the topographic scattering effects is because most of our seismic observations are either at or very near to the earth’s surface. The sensitivity of regional phases to topographic roughness in the crust has been widely observed. Comparisons of several approximation solutions to rough surface scattering are conducted in this study for an analytical description of the close relation of topographic statistics and regional phase attenuation. These approximations include Kirchhoff approximation theory, Taylor expansion-based perturbation theory, two-scale model, Rytov phase approximation, and Born series method, with each valid for a range of roughness scales. Kirchhoff approximation ignores multiple scatterings between any two surface points. In general, it has been considered valid for the large-scale roughness components. Perturbation theory based on Taylor series expansion is valid for the small-scale roughness components where the surface heights deviate from a planar at z = 0 by less than a wavelength. Rytov phase approximation to large-scale topographic roughness is not subject to the stringent restrictions that apply to the Kirchhoff approximation. Tests with the Gaussian and semicircular convex topographies show that the Rytov approximation improves the Kirchhoff approximation in both amplitude and phase. For a two-scale topography that consists of two extreme roughness components (large and small scales), some assumptions are valid to combine the Kirchhoff and perturbation theories for rough surface scattering. The realistic methods for the multiscale surfaces come with the Born series approximation that accounts for multiple scattering between surface points. For instance, the second-order Born approximation might be sufficient to guarantee the accuracy for general rough surfaces without infinite gradients and extremely large surface heights. It must be stressed that the approximation solutions described in this article miss the conversion of energy between SH and P-SV waves that is one of the main features of the crustal wave guide in real situations. Extension to the elastic case must be conducted in the future.

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