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The Kirchhoff approximation (KA) for elastic wave scattering from two-dimensional (2D) and three-dimensional (3D) rough surfaces is critically examined using finite-element (FE) simulations capable of extracting highly accurate data while retaining a fine-scale rough surface. The FE approach efficiently couples a time domain FE solver with a boundary integration method to compute the scattered signals from specific realizations of rough surfaces. Multiple random rough surfaces whose profiles have Gaussian statistics are studied by both Kirchhoff and FE models and the results are compared; Monte Carlo simulations are used to assess the comparison statistically. The comparison focuses on the averaged peak amplitude of the scattered signals, as it is an important characteristic measured in experiments. Comparisons, in both two dimensions and three dimensions, determine the accuracy of Kirchhoff theory in terms of an empirically estimated parameter σ2/λ0 (σ is the RMS value, and λ0 is the correlation length, of the roughness), being considered accurate when this is less than some upper bound c, (σ2/λ0<c). The incidence and scattering angles also play important roles in the validity of the Kirchhoff theory and it is found that for modest incidence angles of less than 30°, the accuracy of the KA is improved even when σ2/λ0>c. In addition, the evaluation results are compared using 3D isotropic rough surfaces and 2D surfaces with the same surface parameters.

The validity of Kirchhoff theory for scattering of elastic waves from rough surfaces