Asymptotic Wave Methods in Heterogeneous Media

Wafik B. Beydoun

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on May 2, 1985 in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Geophysics

Abstract

The limitations of asymptotic wave theory and its geometrical manifestations are newly formalized and scrutinized in Chapter II. Necessary and sufficient conditions for the existence of acoustic and seismic rays and beams in general inhomogeneous media are expressed in terms of new physical parameters: the threshold frequency w0associated with the P/S decoupling condition, the cut-off frequency wc associated with the radiation-zone condition, the total curvature of the wavefront and the Fresnel-zone radius. The analysis is facilitated with the introduction of a new ancillary functional_the hypereikonal which is capable of representing ordinary as well as evanescent waves. The hypereikonal is the natural extension of the eikonal theory. With the aid of the above parameters, simple conditions are obtained for the decoupled far field, the decoupled near field, two point dynamic ray tracing, paraxial wave fields and Gaussian beams.

Chapter III deals with a canonical problem. The Green's function, in a constant gradient medium, is presented, for an explosive point source, in frequency and time domains. The analytical dynamic ray tracing (DRT) solution is re-derived with conditions stated in Chapter II. The Gaussian beam (GB) solution is investigated and new beam parameters are defined. Comparisons between exact and approximate solutions are made; for both methods, DRT and GB, conditions of validity are explicit and quantitative. An accuracy criterion is defined in the time domain, and measures a global relative error. The range of validity is expressed in the form of two inequalities for the dynamic ray tracing method and of five inequalities for the Gaussian Beam method. Results remain accurate at ray turning points. For the type of medium considered, the breakdown of the dynamic ray tracing method is smoother and better behaved than that of the Gaussian beams. As examples, a vertical seismic profiling configuration and a shallow earthquake are modeled using Gaussian beams.

Chapter IV describes the paraxial ray method and its uses in modeling seismic waves. It is a flexible and fast method for computing asymptotic Green»s functions. The method is an extension of the standard ray method and a degenerate case of Gaussian beams. Accuracy is controllable, within ray and paraxial conditions. Comparison of results with finite difference and discrete wavenumber are very satisfactory. Examples for different heterogeneous media are shown.

A full-waveform inversion is then presented in Chapter V. A new approach, using tensor algebra formalism, is presented. Combined data sets (e.g. VSP and surface reflection data) with prior information are simultaneously handled. The forward model is generated by the paraxial ray method. The inversion is performed in the frequency domain, for interface and layer parameters. Sensitivity analysis is studied for each parameter. Data generated by finite difference is inverted and obtained estimates are accurate. VSP field data is inverted to estimate local geologic structure.