A Tour of Optimisation Methods for Facies Estimation in AVO Seismic Inversion Using Markov Random Fields

We overview some joint lithology-fluid-class/rock-properties inversion techniques for AVO/quantitative-interpretation work from true-amplitude imaged seismic data, using a Bayesian hierarchical model. We use a Markov random-field (MRF) model for facies labels, and a facies-conditional Normal model for elastic rock properties. The MRF forces spatial smoothness in categories, and forbids non-hydrodynamical fluid-class transitions. The rock properties model captures loading effects and mutual correlation between elastic variables within a facies. We study the optimisation problem rather than sampling. The optimisation is a mixed integer-continuous problem, provably NP-hard, but susceptible to good heuristics in certain regimes we discuss. A natural approach to the problem is the EM algorithm, which provides fast local solutions using loopy-belief propagation for the E-step and conjugate-gradient least-squares for the M-step. This yields soft/marginalised estimates of facies labels. Industry-standard Bayesian classifications are related to single cycles of this EM algorithm, but without spatial coupling. A comparable method to the EM algorithm is "relaxed joint inversion'' with continuous "facies indicator'' variables confined to a simplex. The absence of middle frequencies and the multimodality of the objective make gradient methods, including EM and relaxation, vulnerable to local minima. Two globalising approaches, deterministic annealing, and integer-optimisation, are discussed.