Markov Random Fields and Random Walks

Markov Random FieldsMarkov Random Fields, which are spatial generalizations of Markov ChainsMarkov Chain in time, are mainly used in geophysical inverse problemsInverse problems and geologic image processingImage processing (Table 10.1). Discrete-time and continuous-time Random Walk modelsRandom Walk Models (Sect. 10.2, and Table 10.2) are among the most popular petrophysical simulation tools. Three important applications are discussed in details: anomalous diffusionAnomalous diffusion, the NMR response of fluids in porous rocksNMR response of fluids in porous rocks, and an[aut]Chatzis, I. attempt[aut]Kwiecen, M.J. (of Ioannidis[aut]Ioannidis, M.A. et al. 1997) to derive the formation factorFormation factor F in Archie’s equationArchie equation. Mathematical details (such as: Derivation of the Gibbs DistributionGibbs distribution; Proof of the Hammersley-Clifford TheoremHammersley-Clifford theorem; Polya’s TheoremPolya’s Theorem and its proof; and Partial Differential Equations governing Continuous Time Random WalksContinuous Time Random Walks) are discussed in appendices.