Perturbation of the Plane‐Wave Reflectivity of A Depth‐Dependent Elastic Medium By Weak Inhomogeneities

Abstract

The first‐order effects produced by plane inhomogeneities on the plane‐wave response of a vertically heterogeneous elastic medium are derived. the derivation is based on a perturbation analysis of the wave equations which yields the Fréchet derivatives of the displacement fields in terms of the Green's functions for the unperturbed medium. the accuracy and limitations of the first‐order solutions are checked numerically by comparing τ‐p seismograms calculated from the analytical expressions of the Fréchet derivatives with a complete solution obtained by introducing discrete perturbations in the medium properties. Calculations performed with a realistic marine model show that the first‐order perturbation theory is remarkably accurate at all angles of incidence, and even in the evanescent regime. Furthermore, it is shown that the formulae obtained allow us to model density or velocity perturbations of up to 10 per cent. the expressions of the Fréchet derivatives are directly applicable to linear and non‐linear inversion algorithms of plane‐wave seismograms using gradient techniques. In addition, they may be useful in examining the sensitivity of the plane‐wave response of complex geological sequences to various combinations of elastic parameters. Copyright © 1990, Wiley Blackwell. All rights reserved