Multiparameter inversion in anisotropic elastic media

Abstract

In this paper, we formalize the linearized inverse scattering problem for a general anisotropic, elastic medium and describe two approaches to the construction of a stable inversion procedure. The first uses generalized Radon inversion and requires extra information limiting the independent variation of material paramters. The second uses a stationary base approximation and requires extra information to the effect that the medium is everywhere locally stratified with known dip. The point of common departure is the single-scattering or Born approximation to the scattered field in perfectly elastic media. The formalism is simple to outline: the medium being modelled (or reconstructed) is thought of as a perturbation of a simpler, known, background medium. We are to find the unknown medium perturbation, given the scattered field, which is defined as the difference between the actual (total) field and the background medium. If the background medium is sufficiently smooth, the background fields can be well approximated within the framework of ray theory for elastic waves. If the background medium is sufficiently close to the actual medium, the scattered field can be well approximated by an integral involving the background field and linear in the medium perturbation (the Born approximation). Within this regime we show how to find which combinations of parameters can be determined for a given arrangement of sources and receivers (aquisition geometry). Many expressions simplify when sources and receivers coincide (zero-offset), but then only one parameter may be reconstructed. At several points we use 'generalized' linear inversion, implemented through a singular value decomposition, which enables us to find and rank the vest-determined linear combinations of the unkown parameters. Our first method of inversion depends upon the inversion formula for the generalized Radon transform (GRT) to leading order asymptotically for high spatial frequencies. Our second method benefits from the use of further information to the effect that the medium has a locally stratified microstructure within which the medium properties vary rapidly in the direction normal to the local layering, and, assuming that the orientation of the layering can be estimated separately, the scattering problem becomes locally 1-D and the inversion procedure reduces to an amplitude versus (scattering) angle (AVA) analysis.