EM Algorithms and Convex-Envelope Approximations in Joint Facies/Elastic Inversion for AVO and FWI

Seismic inverse problems are always ill-posed because of missing frequencies or limited acquisition. Bayesian-prior rock-property models can compensate for much of this missing information, and are most powerful when furnished in lithology-dependent form. Since most seismic energy comes from facies boundaries, joint inversion for facies and elastic parameters is desirable; with discrete facies variables, the expectation-maximisation (EM) algorithm is then the tool of choice.However, joint inversions with facies variables introduce additional kinds of nonconvexity, so inversion will greatly benefit from starting at solutions of tightest-possible convex relaxations.

The Bayesian prior in this study is a hierarchical model, with a Markov random field for the facies label distribution, and a facies-conditional multivariate loading model for elastic parameters. For the facies mixture distributions which are a consequence of this model, the tight relaxation emerges from building a convex-envelope of the prior model mixture distribution. Since the lithological model has significant facies-proportions and spatial variation, computation of this convex-envelope must be extremely efficient. Accelerated PDE methods are much faster than quickhull for this task. The efficacy of the convex-envelope relaxation as a starting point for the EM algorithm is illustrated for example problems in both AVO inversion and a small-scale FWI instance.