Bayesian inversion of convolved hidden Markov models with applications in reservoir prediction
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Efficient assessment of convolved hidden Markov models is discussed. The bottom-layer is defined as an unobservable categorical first-order Markov chain, while the middle-layer is assumed to be a Gaussian spatial variable conditional on the bottom-layer. Hence, this layer appear as a Gaussian mixture spatial variable unconditionally. We observe the top-layer as a convolution of the middle-layer with Gaussian errors. Focus is on assessment of the categorical and Gaussian mixture variables given the observations, and we operate in a Bayesian inversion framework. The model is defined to make inversion of subsurface seismic AVO data into lithology/fluid classes and to assess the associated elastic 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 algorithms converges far too slow 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 Markov chain Monte Carlo algorithm, to assess the correct posterior model. A set of synthetic realistic examples are presented. The proposed approximations provides efficient proposal densities which results in acceptance probabilities in the range 0.10-0.50 in the Markov chain Monte Carlo algorithm. A case study of lithology/fluid seismic inversion is presented. The lithology/fluid classes and the elastic material properties can be reliably predicted.
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