Variational Bayesian inversion (VBI) of quasi-localized seismic attributes for the spatial distribution of geological facies. Issue 2 (26th April 2018) - Nawaz, Muhammad Atif; Curtis, Andrew
SUMMARY: We introduce a new Bayesian inversion method that estimates the spatial distribution of geological facies 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. Our mathematical model consists of seismic attributes as observed data, which are assumed to have been generated by the geological facies. The method infers the post-inversion (posterior) probability density of the facies plus some other unknown model parameters, from both the seismic attributes and geological prior information. Most previous research in this domain is based on the localized likelihoods assumption, whereby the seismic attributes at a location are assumed to depend on the facies only at that location. Such an assumption is unrealistic because of imperfect seismic data acquisition and processing, and fundamental limitations of seismic imaging methods. In this paper, we relax this assumption: we allow probabilistic dependence between seismic attributes at a location and the facies in any neighbourhood of that location through a spatial filter. We term such likelihoods quasi-localized . Exact Bayesian inference is impractical because it requires normalization of the posterior distribution which is intractable for large models and must be approximated. Stochastic sampling (e.g. by using Markov chain Monte Carlo) is the most commonly used approximate inference method but it is computationally expensive and detection of its convergence is often subjective and unreliable. We use the variational Bayes method which is a more efficient alternative that offers reliable detection of convergence. It achieves this by replacing the intractable posterior distribution by a tractable approximation. Inference can then be performed using the approximate distribution in an optimization framework, thus circumventing the need for sampling, while still providing probabilistic results. We show in a noisy synthetic example that the new method recovered the coefficients of the spatial filter with reasonable accuracy, and recovered the correct facies distribution. We also show that our method is robust against weak prior information and non-localized likelihoods, and that it outperforms previous methods which require likelihoods to be localized. Our method is computationally efficient, and is expected to be applicable to 3-D models of realistic size on modern computers without incurring any significant computational limitations.