Convergence of difference methods for one-dimensional inverse problems

We consider the one-dimensional inverse problem of reflection seismology. An impulsive plane wave in pressure is applied to toe surface of a stratified elastic half space and the resulting particle velocity is measured at the surface. The characteristic impedance of the medium is to be recovered as a function of travel time. We study two methods for the numerical solution of the inverse problem. One method la a discretization of the Gopinath-Sondhi integral equation. The other is a downward continuation method. Both methods are exact for so-called Goupilllaud-tayered media. Both use second-order difference schemes to reconstruct a solution of the system of partial differential equations governing motion in the medium. A formally second-order accurate equation based on the theory of propagation of singularities is used to recover the impedance. We prove that these two methods are second-order convergent for smoothly stratified media by analyzing the discretization of the Gopinath-Sondhi integral equation.