Efficient Earthquake Inversion using the Finite Element Method

Abstract

A vital component in the management of seismic hazard is the study of past seismic events. Classically, this has been the domain of seismology, which studies the dynamic manifestations of the event to infer properties such as epicenter and moment magnitude. More recently it has become possible to perform similar analyses on the basis of the static consequences of a seismic event, as satellite borne Synthetic Aperture Radar (SAR) data allows us to compare the local surface geometries before and aftera seismic event. The locality of the deformation data promises reconstructions with greater detail and subject to fewer model uncertainties.
With current technology, it is not possible to use SAR to their full potential. The non-linearity of the static dislocation problem that links faulting mechanisms to observed deformations causes any inverse method to require many evaluations of the forward model. This poses limits on the permissible cost of solving the dislocation problem, restricting most approaches to simplified model assumptions such as material homogeneity and absence of topography. In situations where more accurate information is available, this presents a clear opportunity for improvement by accelerating the computational methods instead.
This thesis presents the Weakly-enforced Slip Method (WSM), a modification of the Finite Element Method (FEM), as a fast approach for solving static dislocation problems. While the computational cost of the WSM is similar to that of the FEM for single dislocations, the WSM is significantly faster when many different dislocation geometries are considered, owing to the reuse of computationally expensive components such as matrix factors. This property makes the method ideally suited for inverse settings, opening the way to incorporating all available in situ data in a forward model that is simultaneously flexible and cheaply evaluable. Moreover, we prove that the WSM retains the essential convergence properties of the FEM.
A limitation of the WSM is that it produces continuous displacement fields, which implies a large error local to the dislocation. We show that this error decreases rapidly with distance, and that in a typical scenario the majority of deformation data has a discretization error that is smaller than observational noise, particularly when a fault is buried. In the case of shallow or rupturing faults, neighbouring data needs to be discarded from the analysis to avoid disruption. With this measure in place, we show via Bayesian inference of synthesized datasets that the discretization errors of the WSM do not significantly affect the inverse problem.