Bayesian Inversion of Convolved Hidden Markov Models With Applications in Reservoir Prediction

The efficient assessment of convolved hidden Markov models is discussed. The bottom layer is defined as an unobservable categorical first-order Markov chain, whereas the middle layer is assumed to be a Gaussian spatial variable conditional on the bottom layer. Hence, this layer appears marginally as a Gaussian mixture spatial variable. We observe the top layer as a convolution of the middle layer with Gaussian errors. The focus is on assessing the categorical and Gaussian mixture variables given the observations, and we operate in a Bayesian inversion framework. The model is defined to perform the inversion of subsurface seismic amplitude-versus-offset data into lithology/fluid classes and to assess the associated seismic material properties. Due to the spatial coupling in the likelihood functions, evaluation of the posterior normalizing constant is computationally demanding, and brute-force, single-site updating Markov chain Monte Carlo (MCMC) algorithms converge far too slowly to be useful. We construct two classes of approximate posterior models, which we assess analytically and efficiently using the recursive forward-backward algorithm. These approximate posterior densities are used as proposal densities in an independent proposal MCMC algorithm to determine the correct posterior model. A set of synthetic realistic examples is presented. The proposed approximations provide efficient proposal densities, which results in acceptance probabilities in the range 0.10-0.50 in the MCMC algorithm. A case study of lithology/fluid seismic inversion is presented. The lithology/fluid classes and the seismic material properties can be reliably predicted.