Aspects of the theory of inversion as applied to geophysical problems.

Abstract

Inverse theory provides an important tool that the geophysicist can use to explore the structure of the Earth. This thesis examines several new approaches to the inverse problem, and suggests ways of improving the conventional least-squares technique. Non least-squares inversion was applied to borehole temperature data from South Africa, and when the norm of the inversion was controlled by the statistics of the misfit, It reduced by over 50% the number of iterations required for the inversion to converge upon a solution. Various damping schemes were also examined, and the use of the misfit in controlling the damping is shown to provide the best solution of those studied (Cooper and Jones, in press). Improvements to the efficiency of the inverse process were also achieved by the fitting of parabolic forms to portions of the misfit surface, using both the misfit value and the gradient of the surface. for gravity data. The presence of nearby minima other than the one that the inversion has just converged to can also be detected in this manner. The set of initial models that converged to a particular solution using leastsquares inversion was studied for magnetic data, and it was noted to have a fractal nature. The fractal dimension of the set was found to be inversely proportional to the damping of the inverse problem. The inverse process was pushed into a chaotic state by the modification of the least-squares inversion equation. The chaotic state was studied, and exploited to