Title

NONLINEAR INVERSE METHODS APPLIED TO INTERPRETING GRAVITY ANOMALIES PRODUCED BY MULTI-INTERFACED GEOLOGIC BODIES

Date of Completion

January 1983

Keywords

Geophysics

Degree

Ph.D.

Abstract

Nonlinear inverse methods can be used to determine shapes of source bodies directly from observed gravity anomalies when density distributions are known. Two nonlinear inverse methods are applied in this study, the nonlinear optimization and the least-squares methods, which are here grouped under optimization methods. For an initially assumed model of a geological structure, optimization methods obtain values for source parameters by iterative adjustments that minimize a nonlinear objective function. This function is the sum of the squares of residuals of observed anomaly minus calculated value at each station. Whether optimization methods can be used to delineate a particular structure depends primarily on whether an analytical formula can be developed to represent the gravity anomaly.^ The performance of ten different nonlinear, unconstrained and constrained, optimization methods and one least-squares method (see Table 1) are tested and compared in their abilities to resolve the shape of a given two-dimensional sedimentary basin model. These eleven methods provide quite adequate resolution in finding a solution to the specified problem, particularly Powell's method, the variable metric method, and Marquardt's method.^ Five hypothetical and one field structure (the Aleutian trench) are analyzed by Powell's method to illustrate the versatility of that method for simulating two-dimensional multi-interfaced earth structures. Other optimization methods can be developed to obtain similar analyses. The behavior of an objective function may be studied visually by means of two-dimensional cross-sections in an n-dimensional space. Contoured values of an objective function can indicate locations of solutions for acceptable structures. Contour patterns can also provide ranges of possible solutions. Multi-modality is the main characteristic of the objective function associated with a complex structure and is the consequence of nonuniqueness in inverse problems.^ Optimization methods applied in analyzing gravity anomalies produced by multi-interfaced structures can (1) provide high resolution for simple structures or, at least, find gross models for very complex structures in short computer time, (2) simulate various geological models where the corresponding objective function can be defined, and (3) indicate available constraints placed on some parameters. Other types of geophysical inverse problems can be similarly analyzed by optimization methods. ^