A discrete-time formulation for the variable wave speed scattering problem in two dimensions

Abstract

Motivated by electromagnetic wave propagation in media where permittivity varies in two dimensions, we address the problem of wave scattering for two-dimensional (2D) media having variable speed. Wave speed variations are shown to produce scattering which can be represented in terms of a Schrödinger scattering potential. The wave equation problem is thus reformulated as a Schrödinger equation inverse potential problem, with a variable wave speed. Throughout it is assumed that wave speed varies smoothly and slowly such that a finite-difference approximation is valid, defining a discrete inverse scattering problem. For this discrete problem, we define an equivalent medium on a variable-mesh grid for which the wave speed is constant throughout, yet the equivalent medium has the same scattering response as the actual variable wave speed medium. Going from actual to equivalent medium entails spatially warping the medium, while going from equivalent to actual entails spatial dewarping. The discrete-time forward and inverse scattering problems are then formulated and solved using the equivalent medium. A numerical example illustrating the introduced concepts is presented.