Discrete inverse methods for elastic waves in layered media

Abstract

The seismic inverse problem for waves at normal incidence on horizontally layered media is discussed. The emphasis is theoretical rather than practical, but some long‐standing questions concerning the general applicability of the often taught Goupillaud inverse method are answered. The main purpose is to demonstrate in detail the equivalence between the Goupillaud method of inversion for the wave equation and the Marchenko integral equation (inverse scattering) method for the Schroedinger equation. We show that the very simple method of solution due to Goupillaud for a specialized model (layers of equal traveltime) actually has a much wider significance. If seismic data are smoothed before sampling using a type of antialiasing filter, the Goupillaud method gives a valid approximate inversion for models with arbitrary layer thicknesses (or continuous impedance variation) when the “reflection coefficients” are appropriately reinterpreted. In all, three inverse methods are considered: (1) the Goupillaud method for the wave equation and both (2) continuous and (3) discrete inverse scattering methods for the Schroedinger equation. A computationally fast algorithm for solving the inverse scattering formulas is deduced from the equivalent Goupillaud method. By comparing the continuous and discrete formalisms in the continuum limit, a preferred form is found within the class of symmetric tridiagonal discretizations of the Schroedinger equation. For the elastic wave inverse problem, two cases are distinguished: (1) If the impedance is continuous, we show that both the Goupillaud method and the discrete inverse scattering method converge to the impedance when the equal‐traveltime layer thickness goes to zero; and (2) if the impedance has a finite number of discontinuities, we show that the inverse scattering method assigns the arithmetic average across the discontinuity at the point of discontinuity, while the Goupillaud method assigns the value of the right‐hand (spatially deeper) limit. Thus, in the continuum limit, both methods will reconstruct the same impedance except (possibly) for the values at a finite number of jump points in any finite span of traveltime.