Wave motions in stochastic heterogeneous media: a Green's function approach

Abstract

In this work, the authors employ a new type of solution for steady-state wave propagation governed by Helmholtz's equation, which is based on an a priori consideration of the variation of the medium's wave speed with depth given measured values at the surface and at a certain depth. In contrast to the usual integral transformations, this solution is based on an algebraic transformation of the dependent variable (the displacement), which results in a constraint equation for the material properties. This constraint gives closed-form expressions for the wave speed profile and for the fundamental solution (Green's function) involving various constants. Different values for these constants (within a certain range) produce different realistic wave speed profiles. Next, the presence of stochasticity in the material parameters is handled through the perturbation approach. The crucial step here is selection of the appropriate constant in the wave speed profile solution to be treated as a random variable with prescribed mean and variance. This selection subsequently filters into the fundamental solution. Following this first order perturbation approach, closed-form expressions are obtained for the covariance matrix of both wave speed profile and fundamental solution. These results, which are for small variabilities only, are validated against standard Monte Carlo simulations. Finally, the covariance matrices can be used within the context of e efficient boundary element solutions for wave scattering problems.