In many geophysical inverse problems, smoothness assumptions on the underlying geologic model are utilized to mitigate the effects of poor data coverage and observational noise and to improve the quality of the inferred model parameters. In the context of Bayesian inference, these assumptions take the form of a prior distribution on the model parameters. Conventionally, the regularization parameters defining these assumptions are fixed independently from the data or tuned in an ad hoc manner. However, it is often the case that the smoothness properties of the true earth model are not known a priori, and furthermore, these properties may vary spatially. In the seismic imaging problem, for example, where the objective is to estimate the earth's reflectivity, the reflectivity model is often smooth along a particular reflector but exhibits a sharp contrast in the direction orthogonal to the reflector. In such cases, defining a prior using predefined smoothness assumptions may result in posterior estimates of the model that fail to correctly image these reflectors. In this thesis, we explore the application of Bayesian inference to different geophysical inverse problems and seek to address issues related to regularization and smoothing by appealing to the hierarchical Bayesian framework.
In the first part of this talk, we explore a hierarchical Bayesian extension of the seismic imaging problem. We capture the smoothness properties of the prior distribution on the model by defining a Markov random field (MRF) on the set of model parameters and assigning weights to the edges of the underlying graph; we refer to these parameters as the edge strengths of the MRF. Formulating the seismic imaging problem within the hierarchical Bayesian setting, we provide a framework for computing the marginal MAP estimate of the edge strengths by application of the expectation-maximization (E-M) algorithm. Our methodology is validated on synthetic datasets arising from 2D models, where the images we obtain after inferring the edge strengths exhibit the desired spatially-varying smoothness properties, yielding sharper, more coherent reflectors.
In the second part of this talk, we shift our focus and consider the problem of time-lapse seismic processing, where the objective is to detect changes in the subsurface over a period of time using repeated seismic surveys. We focus on the realistic case where the surveys are taken with differing acquisition geometries. In such situations, conventional methods for processing time-lapse data often perform poorly as they do not correctly account for differing model uncertainty between the surveys due to differences in illumination and observational noise. Applying the machinery explored in the previous problem, we formulate the time-lapse processing problem within the hierarchical Bayesian setting and present a framework for computing the marginal MAP estimate of the time-lapse change model. The results of our inference framework are validated on synthetic data from a 2D time-lapse seismic imaging example, where the hierarchical Bayesian estimates significantly outperform conventional time-lapse inversion results.
Thesis Supervisor: Dr. Michael C. Fehler
Earth Resources Laboratory