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Markov Random Fields and Random Walks

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Statistical Rock Physics

Part of the book series: Earth and Environmental Sciences Library ((EESL))

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Chapter Highlights

Markov Random Fields, which are spatial generalizations of Markov Chains in time, are mainly used in geophysical inverse problems and geologic image processing (Table 10.1). Discrete-time and continuous-time Random 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 diffusion, the NMR response of fluids in porous rocks, and an attempt (of Ioannidis et al. 1997) to derive the formation factor F in Archie’s equation. Mathematical details (such as: Derivation of the Gibbs Distribution; Proof of the Hammersley-Clifford Theorem; Polya’s Theorem and its proof; and Partial Differential Equations governing Continuous Time Random Walks) are discussed in appendices.

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  1. Earth Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Gabor Korvin

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Appendix

Appendix

See Tables 10.1 and 10.2.

Table. 10.1 Some important applications of random Markov fields
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Table. 10.2 Some applications of random walk models
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Korvin, G. (2024). Markov Random Fields and Random Walks. In: Statistical Rock Physics. Earth and Environmental Sciences Library. Springer, Cham. https://doi.org/10.1007/978-3-031-46700-4_10

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