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

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.

[1]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[2]  J. Laurie Snell,et al.  Markov Random Fields and Their Applications , 1980 .

[3]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  N. Cressie,et al.  Statistics for Spatial Data. , 1992 .

[5]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[6]  Noel A. C. Cressie,et al.  Statistics for Spatial Data: Cressie/Statistics , 1993 .

[7]  Robijn Bruinsma,et al.  Soft order in physical systems , 1994 .

[8]  C. Geyer,et al.  Annealing Markov chain Monte Carlo with applications to ancestral inference , 1995 .

[9]  Dani Gamerman,et al.  Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference , 1997 .

[10]  Håkon Tjelmeland,et al.  Markov Random Fields with Higher‐order Interactions , 1998 .

[11]  N. Cressie,et al.  Image analysis with partially ordered Markov models , 1998 .

[12]  D. Cox,et al.  Complex stochastic systems , 2000 .

[13]  Hans Kiinsch,et al.  State Space and Hidden Markov Models , 2000 .

[14]  Gerhard Tutz,et al.  State Space and Hidden Markov Models , 2001 .

[15]  Stan Z. Li,et al.  Markov Random Field Modeling in Image Analysis , 2001, Computer Science Workbench.

[16]  S. L. Scott.  Bayesian Methods for Hidden Markov Models , 2002 .

[17]  Francesco Bartolucci,et al.  A recursive algorithm for Markov random fields , 2002 .

[18]  Gerhard Winkler,et al.  Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction , 2002 .

[19]  Sebastien Strebelle,et al.  Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics , 2002 .

[20]  Håvard Rue,et al.  A Tutorial on Image Analysis , 2003 .

[21]  Jesper Møller,et al.  Spatial statistics and computational methods , 2003 .

[22]  Anthony N. Pettitt,et al.  Efficient calculation of the normalizing constant of the autologistic and related models on the cylinder and lattice , 2003 .

[23]  Tapan Mukerji,et al.  Stochastic reservoir characterization using prestack seismic data , 2004 .

[24]  A. Journel,et al.  The Necessity of a Multiple-Point Prior Model , 2007 .

[25]  Håvard Rue,et al.  Recursive computing and simulation-free inference for general factorizable models , 2007 .

[26]  Tapan Mukerji,et al.  Seismic inversion combining rock physics and multiple-point geostatistics , 2008 .

[27]  Anthony N. Pettitt,et al.  Bayesian Inference in Hidden Markov Random Fields for Binary Data Defined on Large Lattices , 2009 .

[28]  Stan Z. Li.  Markov Random Field Modeling in Image Analysis , 2009, Advances in Pattern Recognition.

[29]  Henning Omre,et al.  Improved resolution in Bayesian lithology/fluid inversion from prestack seismic data and well observations: Part 1 — Methodology , 2010 .

[30]  Marita Stien,et al.  Facies Modeling Using a Markov Mesh Model Specification , 2011 .

[31]  Andrew Gelman,et al.  Handbook of Markov Chain Monte Carlo , 2011 .

[32]  H. Tjelmeland,et al.  An approximate forward-backward algorithm applied to binary Markov random fields , 2011 .

[33]  Peter Clifford,et al.  Markov Random Fields in Statistics , 2012 .

[34]  Snehamoy Chatterjee,et al.  Dimensional Reduction of Pattern-Based Simulation Using Wavelet Analysis , 2012, Mathematical Geosciences.

[35]  Tuanfeng Zhang,et al.  Memory-Efficient Categorical Multi-point Statistics Algorithms Based on Compact Search Trees , 2012, Mathematical Geosciences.

[36]  Håkon Tjelmeland,et al.  Exact and Approximate Recursive Calculations for Binary Markov Random Fields Defined on Graphs , 2012 .