## Methods for Bayesian inversion of seismic data

##### Abstract

The purpose of Bayesian seismic inversion is to combine information derived from
seismic data and prior geological knowledge to determine a posterior probability
distribution over parameters describing the elastic and geological properties of the
subsurface. Typically the subsurface is modelled by a cellular grid model containing
thousands or millions of cells within which these parameters are to be determined.
Thus such inversions are computationally expensive due to the size of the parameter
space (being proportional to the number of grid cells) over which the posterior is to
be determined. Therefore, in practice approximations to Bayesian seismic inversion
must be considered. A particular, existing approximate workflow is described in
this thesis: the so-called two-stage inversion method explicitly splits the inversion
problem into elastic and geological inversion stages. These two stages sequentially
estimate the elastic parameters given the seismic data, and then the geological parameters given the elastic parameter estimates, respectively. In this thesis a number
of methodologies are developed which enhance the accuracy of this approximate
workflow.
To reduce computational cost, existing elastic inversion methods often incorporate only simplified prior information about the elastic parameters. Thus a method
is introduced which transforms such results, obtained using prior information specified using only two-point geostatistics, into new estimates containing sophisticated
multi-point geostatistical prior information. The method uses a so-called deep neural network, trained using only synthetic instances (or `examples') of these two estimates, to apply this transformation. The method is shown to improve the resolution
and accuracy (by comparison to well measurements) of elastic parameter estimates
determined for a real hydrocarbon reservoir.
It has been shown previously that so-called mixture density network (MDN) inversion can be used to solve geological inversion analytically (and thus very rapidly and efficiently) but only under certain assumptions about the geological prior distribution. A so-called prior replacement operation is developed here, which can be
used to relax these requirements. It permits the efficient MDN method to be incorporated into general stochastic geological inversion methods which are free from the
restrictive assumptions. Such methods rely on the use of Markov-chain Monte-Carlo
(MCMC) sampling, which estimate the posterior (over the geological parameters) by
producing a correlated chain of samples from it. It is shown that this approach can
yield biased estimates of the posterior. Thus an alternative method which obtains
a set of non-correlated samples from the posterior is developed, avoiding the possibility of bias in the estimate. The new method was tested on a synthetic geological
inversion problem; its results compared favourably to those of Gibbs sampling (a
MCMC method) on the same problem, which exhibited very significant bias.
The geological prior information used in seismic inversion can be derived from real
images which bear similarity to the geology anticipated within the target region of the
subsurface. Such so-called training images are not always available from which this
information (in the form of geostatistics) may be extracted. In this case appropriate
training images may be generated by geological experts. However, this process can
be costly and difficult. Thus an elicitation method (based on a genetic algorithm)
is developed here which obtains the appropriate geostatistics reliably and directly
from a geological expert, without the need for training images. 12 experts were asked
to use the algorithm (individually) to determine the appropriate geostatistics for a
physical (target) geological image. The majority of the experts were able to obtain
a set of geostatistics which were consistent with the true (measured) statistics of the
target image.