On the solution of discrete-valued inverse problems through guided fuzzy c-means clustering

Abstract

Geophysical inversion is a valuable tool for construction and interpretation of subsurface physical property models. Traditional inverse formulations recover smooth models, hindering interpretation of different units. Constrained inversions present methods to incorporate additional information and restrict the recovered model. Discrete-valued inversion is the only method that guarantees recovery of sharp boundaries and expected physical property values. Strict imposition of discrete values within inversion is computationally expensive in terms of required memory, storage, and computation time. The discrete values also have uncertainties, which if fit exactly, propagate into the recovered model. To maintain the benefits of discrete-valued inversion and improve the computational efficiency, I develop a new discrete-valued inversion by applying guided fuzzy c-means clustering. This method approximates the discrete inversion as a continuous-variable minimization. Clustering inversion aims to balance the goals of both traditional and discrete-valued inversions, i.e., recovering models that fit the geophysical data, are spatially cohesive, and have distinct physical property values, that fit the target values. Within this thesis, I first develop and examine the new method in the context of discrete-valued gravity inversion. I develop an inverse work flow to determine the weighting parameters required to balance the inversion goals. To further expand and understand the new method, I apply the discrete inverse formulation to gravity gradient data and investigate the improvements of added information and constraints for salt imaging. Exploring the general applicability of this formulation, I extend the method to the inversion of induced polarization data and increase the number of clusters. I demonstrate the feasibility and produce a hypothesis testing work flow to explore the model space and uncertainty assessment when no prior physical property information is available. Through diverse examples I demonstrate the strengths of the discrete-valued inversion, not only by recovering sharp boundaries and target physical property values, but in the ability to incorporate generic prior information and obtain additional knowledge from the data. I further show that the discrete-valued inversion can balance the physics of a geophysical response, observed data, prior information, uncertainties, and human intuition, in providing an optimal physical property model for advance interpretation and understanding of the subsurface.