Optimized rejection sampling for estimating facies probabilities from seismic data

Abstract

Seismic inversion for facies has nonunique solutions. There are invariably many vertical facies arrays that are consistent with both a data trace and the prior information. Stochastic sampling algorithms set within a Bayesian framework can provide an estimate of the posterior probability distribution of facies arrays by finding the arrays with relatively high posterior probabilities for each data trace. Sample-by-sample facies probabilities can be estimated by measuring the proportions of each facies type at each sample location from the set of posterior facies arrays. To enable the estimation of probabilities of facies mixtures and to obtain high-quality images of facies probability curves, facies must be modeled at high resolution. The facies arrays, or vectors, on which the sampling algorithm operates, must also be long enough to allow for vertical coupling caused by the wavelet. This results in very large sample spaces. The posterior probability distribution is highly nonconvex, which, combined with the large sample space, severely challenges conventional stochastic sampling methods in obtaining convergence of the estimated posterior distribution. The posterior sets of vectors from conventional methods tend to be either correlated or have low predictabilities, resulting in biased or noisy facies probability estimates, respectively. However, accurate estimates of facies probabilities can be obtained from a relatively small number of posterior facies vectors (about 100), provided that they are uncorrelated and have high predictabilities. Full convergence of the posterior distribution is not required. A hybrid algorithm optimized rejection sampling can be designed specifically for the seismic facies probability inversion problem by combining independent sampling of the prior, which ensures posterior vectors are uncorrelated, with an optimization step to obtain high predictabilities. Tests on both real and synthetic data demonstrate better results than conventional rejection sampling and Markov chain Monte Carlo methods.