Bayesian Inversion of Seismic Attributes for Geological Facies using a Hidden Markov Model

Markov chain Monte-Carlo (McMC) sampling generates correlated random samples such that their distribution would converge to the true distribution only as the number of samples tends to infinity. In practice, McMC is found to be slow to converge, convergence is not guaranteed to be achieved in finite time, and detection of convergence requires the use of subjective criteria. Although McMC has been used for decades as the algorithm of choice for inference in complex probability distributions, there is a need to seek alternative approaches, particularly in high dimensional problems. Walker & Curtis (2014) developed a method for Bayesian inversion of two-dimensional spatial data using an exact sampling alternative to McMC which always draws independent samples of the target distribution. Their method thus obviates the need for convergence and removes the concomitant bias exhibited by finite sample sets. Their algorithm is nevertheless computationally intensive and requires large memory. We propose a more efficient method for Bayesian inversion of categorical variables, such as geological facies that requires no sampling at all. The method is based on a 2D Hidden Markov Model (2D-HMM) over a grid of cells where observations represent localized data constraining each cell. The data in our example application are seismic attributes such as P- and S-wave impedances and rock density; our categorical variables are the hidden states and represent the geological rock types in each cell – facies of distinct subsets of lithology and fluid combinations such as shale, brine-sand and gas-sand. The observations at each location are assumed to be generated from a random function of the hidden state (facies) at that location, and to be distributed according to a certain probability distribution that is independent of hidden states at other locations – an assumption referred to as localized likelihoods. The hidden state (facies) at a location cannot be determined solely by the observation at that location as it also depends on prior information concerning the spatial distribution of other hidden states elsewhere. The prior information is included in the inversion in the form of a training image which represents a conceptual depiction of local geologies that might be expected, but other forms of prior information can be used in the method as desired. The method provides direct estimates of posterior marginal probability distributions over each variable, so these do not need to be estimated from samples such as in McMC. Nevertheless, in case samples are desired, these can be generated. On a 2-dimensional test example the method is shown to outperform previous methods significantly, at a fraction of the computational cost. In many foreseeable applications there are therefore no serious impediments to extending the method to 3-dimensional cases.