Elastic-wave reverse-time migration and tomography: A multi-scale approach

Valeriy V Brytik, Purdue University

Abstract

We develop a comprehensive theory and microlocal analysis of reverse-time imaging—also referred to as reverse-time migration or RTM—for the anisotropic elastic wave equation based on the single scattering approximation. We consider a configuration reminiscent of the seismic inverse scattering problem. In this configuration, we have an interior (point) body-force source which generates elastic waves, which scatter off discontinuities in the properties of earth's materials (anisotropic stiffness, density), and are observed at receivers on the earth's surface. The receivers detect all the components of displacement. We introduce (i) an anisotropic elastic-wave RTM inverse scattering transform, and for the case of mode conversions (ii) a microlocally equivalent formulation avoiding knowledge of the source via the introduction of so-called array receiver functions. These allow a seamless integration of passive source and active source approaches to inverse scattering. We discuss decoupling of the isotropic elastic wave equation with coefficients of limited smoothness into modes (ie. P and S constituents). The Lamé parameters have two bounded derivatives in space. We show that the isotropic elastic wave equation can be decoupled into P and S modes and that the "leakage" of modes is noticeable only in a lower order term. Our proof is constructive. Earthquakes, viewed as passive sources, or controlled sources, like explosions, excite seismic body waves in the earth. One detects these waves at seismic stations distributed over the earth's surface. Wave-equation tomography is derived from cross correlating, at each station, data simulated in a reference model with the observed data, for a (large) set of seismic events. The times corresponding with the maxima of these cross correlations replace the notion of residual travel times used as data in traditional tomography. Using first-order perturbation, we develop an analysis of the mapping from a wavespeed contrast (between the "true" and reference models) to these maxima. We develop a construction using curvelets, while establishing a connection with the geodesic X-ray transform. We then introduce the adjoint mapping, which defines the imaging of wavespeed variations from "finite-frequency travel time" residuals. The key underlying component is the construction of the Fréchet derivative of the solution to the seismic Cauchy initial value problem in wavespeed models of limited smoothness. The construction essentially clarifies how a wavespeed model is probed by the method of wave-equation tomography.

Degree

Ph.D.

Advisors

Hoop, Purdue University.

Subject Area

Applied Mathematics|Geophysics|Geophysical engineering

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