Application of finite difference methods to the inverse problem of wave propagation

Abstract

Two methods of determining the wave velocity in an inhomogeneous medium from the reflection coefficient are presented. The medium, which is assumed to be bounded on one side by a perfectly reflecting surface, is divided into homogeneous layers with equal travel times, and the wave equation is solved within each layer. By matching the solutions at the boundaries, a nonlinear difference equation for the ratio of the amplitudes of the in- to the out-going waves in adjacent layers is obtained. This equation which relates the variable wave velocity to the phase shift can be solved by continued fraction expansion for the direct or the inverse problem. An alternative method consists of transforming the one-dimensional wave equation into a similar equation by replacing the spatial coordinate by the travel time coordinate. Converting this transformed equation into a linear difference equation with variable coefficient, the direct problem is reduced to that of calculating two phase shifts, one a discrete form of the JWKB phase and the other the phase shift obtained from the Schrödinger equation. This latter phase can be found conveniently from a discrete form of the variable phase equation. The inverse problem in this case becomes the same as the determination of a "potential" from the phase shift in quantum scattering theory.