The Kirchhoff approximation for acoustic scattering from a rough fluid–elastic solid interface

Abstract

The Kirchhoff approximation for wave scattering from a penetrable surface is applied to the case of acoustic scattering from a rough fluid–elastic solid interface. An approximation to a surface integral yields a simpler expression for the scattering amplitude. It turns out to be similar to an approximation previously introduced by Hagfors [J. Geophys. Res. 69, 3779 (1964); J. Geophys. Res. 71, 379 (1966)] to study radar backscattering from the moon. This simpler version produces results that compare favorably with exact numerical results for the case of periodic fluid–solid interfaces. This approximation is well suited to describe acoustic scattering from a randomly rough fluid–solid interface. To illustrate this point, the coherent field reflection coefficient as a function of angle of incidence and the incoherent scattering coefficient as a function of scattering angle were computed and plotted for the case of a model randomly rough fluid–solid interface with acoustical parameters of a water–granite interface and compared with corresponding results from a previously developed renormalized perturbation theory. The geometrical parameters of the randomly rough interface (i.e., the rms roughness and the correlation length) were chosen so that the perturbation results are expected to be accurate in order to provide another test of the reliability of the approximation developed in this article. Overall this approximation is satisfactory, but since it is a single scattering theory it fails to incorporate multiple scattering effects displayed by the renormalized perturbation theory results. These effects, such as those due to Rayleigh angle phenomena, are small in the domain of validity of the Kirchhoff approximation.