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
