Efficient probabilistic inversion of geophysical data

Estimation of uncertainties is critical for subsequent decision making in all applications of geosciences such as geological hazard analysis and risk mitigation, management and exploitation of subsurface resources, and environmental waste disposal. More efficient probabilistic inversion methods in geosciences are vital to making rapid and improved predictions of geological hazards and estimation of subsurface resources from geophysical data, and estimation of associated uncertainties. While this thesis focuses on seismic data inversion for the estimation of geological properties, the methods developed may find a wide variety of applications in all fields of research that involve spatial data analysis. New concepts, models and methods are developed to perform more efficient probabilistic inversion by making use of the latest developments in machine learning and Bayesian inverse theory to solve geophysical inverse problems. The major contribution of this thesis is the development of efficient geostatistical inversion methods for approximate inference for structured inverse problems where probabilistic dependence between unknown model parameters may be expressed as a Markov random field (MRF). These methods are many orders of magnitude faster than the corresponding sampling based methods in such types of inverse problems. Further, some of the commonly used but avoidable assumptions in conventional geostatistical inversion methods are progressively relaxed and finally removed in this research. The faster inversion methods allow more complex models to be evaluated for more accurate predictions and improved estimation of uncertainty for given compute power and time. Most existing geostatistical inversion methods are based on the localized likelihoods assumption, whereby the seismic data at a location are assumed to depend on the geology only at that location. Such an assumption is unrealistic because of imperfect seismic data acquisition and processing, and fundamental limitations of seismic imaging methods. It is also assumed in most such previous research that the data are completely free of any correlated noise or errors. Although these requirements are almost never met in reality, existing methods use these assumptions to make solutions computationally tractable. Both of these assumptions are progressively removed in this thesis while still allowing computationally tractable solutions to be found for suitably structured problems. The class of problems considered here spans a broad range of spatial data analysis and geosciences, where geology at a location is assumed to depend directly only on the geology within some pre-specified neighbourhood of that location – the so called Markovian assumption – which is the core assumption across the entire literature of geostatistics and has been proven to be valid for all practical purposes. Exact Bayesian inference is intractable in most models of practical interest because it requires normalization of the posterior distribution by integrating model parameters over a very high dimensional space. Therefore, approximate inference is used in practice. Stochastic sampling (e.g., by using Markov-chain Monte Carlo – McMC) is the most commonly used approximate inference method but is computationally expensive and detection of its convergence is often based on subjective criteria and hence is unreliable. New Bayesian inversion methods are introduced that estimate the spatial distribution of geological properties from attributes of seismic data, by showing how the usual probabilistic inverse problem can be solved using an optimization framework while still providing full probabilistic results – the so called variational inference approach. The intractable posterior distribution is replaced by a tractable approximation in the variational approach. Inference can then be performed using the approximate distribution in an optimization framework, thus circumventing the need for sampling, while still providing probabilistic results. The methods developed in this thesis infer the post-inversion (posterior) probability density of the unknown model parameters from seismic data and geological prior information. These methods are shown to be robust against weak prior information and correlated noise in the data. The methods are computationally efficient, and are expected to be applicable to 3D models of realistic size on modern computers without incurring any significant computational limitations.