Construction of Binary Multi-grid Markov Random Field Prior Models from Training Images

Abstract

Bayesian modeling requires the specification of prior and likelihood models. In reservoir characterization, it is common practice to estimate the prior from a training image. This paper considers a multi-grid approach for the construction of prior models for binary variables. On each grid level we adopt a Markov random field (MRF) conditioned on values in previous levels. Parameter estimation in MRFs is complicated by a computationally intractable normalizing constant. To cope with this problem, we generate a partially ordered Markov model (POMM) approximation to the MRF and use this in the model fitting procedure. Approximate unconditional simulation from the fitted model can easily be done by again adopting the POMM approximation to the fitted MRF. Approximate conditional simulation, for a given and easy to compute likelihood function, can also be performed either by the Metropolis–Hastings algorithm based on an approximation to the fitted MRF or by constructing a new POMM approximation to this approximate conditional distribution. The proposed methods are illustrated using three frequently used binary training images.

Appendix:  Additional Plots

In this Appendix, the results of the simulation from $p_{\theta}^{\star}(x)$ and from the conditional distribution $\tilde{p}_{\theta}(x|z)$ are presented, when z is the observations of 11 vertical traces in the training image. In Fig. 12, we show realizations from the fitted $p_{\theta}^{\star}(x)$, and in Fig. 13 we show Box and Whisker plots of the corresponding standardized descriptive statistics. In Fig. 14, we show realizations from the POMM approximation of the fitted $\tilde{p}_{\theta}(x|z)$, when z is 11 vertical traces taken from the training image, and corresponding marginal probabilities are shown in Fig. 15. Box and Whisker plots of the corresponding standardized descriptive statistics are shown in Fig. 16.

Fig. 12

For each training image, three realizations from the fitted $p^{\star}_{\theta}(x)$

Fig. 13

Box and Whisker plots of the standardized descriptive statistics for unconditional simulations from the fitted model $p^{\star}_{\theta}(x)$

Fig. 14

For each training image, three realizations from the POMM approximation of the fitted $\widetilde{p}_{\theta}(x|z)$ when z contains exact observations of eleven columns in the training image

Fig. 15

Estimated marginal probabilities for the POMM approximation of the fitted $\widetilde{p}_{\theta}(x|z)$ when z is exact observations of eleven columns in the training image

Fig. 16

Box and Whisker plots of the standardized descriptive statistics for the conditional simulation when conditioning on eleven columns