I. Seismic Ray-Tracing in Piecewise Homogeneous Media. II. Analysis of Optimal Step Size Selection in Homotopy and Continuation Methods

Author: Perozzi, David J.

Year: 1980

Degree: Dissertation (Ph.D.)

Advisor: Keller, Herbert Bishop

Committee Member: Unknown, Unknown

Option: Applied Mathematics

DOI: 10.7907/djxb-gg89

Abstract

Part I:

 The general problem of the inversion of seismograms usually involves the solution of a nonlinear least squares system. The major component of any such system is the solution of the direct problem. That is, the tracing of a ray between two given end points, where all velocities and interface shapes are specified.

 This problem is studied for piecewise constant velocities and fairly arbitrary interface shapes. An efficient computer code is developed for the solution of this problem yielding travel times, amplitudes, ray paths, phase shifts, and caustic locations.

 The results are then extended to a wider class of velocity distributions. Conditions are given for the class of velocity distributions for which the problem may be studied completely algebraically.

 A standard nonlinear least squares technique is then applied to invert for hypocenters, interface shapes, and elastic constants.

Part II:

 A brief historical survey of continuation methods is given, with particular emphasis on contributions after 1950.

 The problem of selecting an "optimal" step is studied. Optimality here refers to work and storage required for the computation of the solution. The problem is first cast in its most general setting and a couple of trivial theorems are presented.
 
 The problem is then dissected into its component parts, each of which is studied separately. Several combinations of components are also examined. For several specific iterative methods, theorems are presented which optimize an upper bound on the work.

 Several computational procedures naturally arising from this theory are suggested.

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