The effect of stochastic rough interfaces on coupled-mode elastic waves

Abstract

The effect of stochastic fluctuations of the interface boundaries has been incorporated into the elastic coupled-mode equations for 2-D range-dependent media. We assume the medium to be characterized by some deterministic range-dependent layered structure superposed by stochastic boundary fluctuations. The deterministic range-dependent structure defines the reference structure, and the reference structure with superposed stochastic fluctuations defines the true layer boundaries in the medium. The boundary conditions on the true boundary are expanded about the reference boundary, and first-order perturbation theory is applied to the boundary conditions, as well as to the equation of motion. The O(1) system and the O(ε) system represent wave propagation in the deterministic model and in the stochastic model, respectively, with a first-order perturbation of roughness. The solution of the O(1) system is referred to as the primary field. The solutions to the O(ε) system relate the coherent field and the scattered field, which are represented as the superposition of local modes multiplied by the stochastic modal amplitudes. Propagation of coupled-mode elastic waves in a 2-D deterministic range-dependent medium is represented by a unitary coupled-mode propagator. The evolution equation for the stochastic medium and the stochastic coupling matrices, which acts to convert the coherent field to the scattered field, is derived. Enforcing energy conservation on the O(ε) system leads to the requirement that the propagator for the stochastic medium must satisfy a Lippman-Schwinger-type integral equation, whose solution can be represented by a formal perturbation series for the multiply scattered wavefields. The integral equations for the mean field and the covariance of the field are also presented. These formal theoretical results are valid to all orders of multiple scattering. In the second part of the paper the Born approximation to the Lippmann-Schwinger integral equation is used to extract information about attenuation due to rough surface scattering and in the design of an inverse problem for the boundary roughness variance and correlation length. By defining the modal scattering cross-section, the formula for the scattering Q(s)-1 from the Born approximation for the scattered field is derived. The modal scattering cross-section and scattering Q(s) for a range-dependent model with stochastic roughness are computed for both exponential and Gaussian correlation functions. Finally, a formula for the power spectrum of the coherent field is derived from the Born approximation, and an inversion for the roughness variance and correlation length is designed by power spectrum fitting.