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Отправлено: 22.07.08 10:40. Заголовок: Markov chains and embedded Markov chains in geology
Markov chains and embedded Markov chains in geology Mathematical Geology Springer Netherlands ISSN 0882-8121 (Print) 1573-8868 (Online) Volume 1, Number 1 / Март 1969 г. DOI 10.1007/BF02047072 Страницы 79-96 Markov chains and embedded Markov chains in geology W. C. Krumbein1 and Michael F. Dacey2 (1) Department of Geology, Northwestern University, USA (2) Department of Geography, Northwestern University, USA Received: 30 June 1969 Abstract Geological data are structured as first-order, discrete-state discrete-time Markov chains in two main ways. In one, observations are spaced equally in time or space to yield transition probability matrices with nonzero elements in the main diagonal; in the other, only state transitions are recorded, to yield matrices with diagonal elements exactly equal to zero. The mathematical differences in these two approaches are reviewed here, using stratigraphic data as an example. Simulations from chains with diagonal elements greater than zero always yield geometric distributions of lithologic unit thickness, and their use is recommended only if the input data have the same distribution. For thickness distributions lognormally or otherwise distributed, the embedded chain is preferable. The mathematical portions of this paper are well known, but are not readily available in publications normally used by geologists. One purpose of this paper is to provide an explicit treatment of the mathematical foundations on which applications of Markov processes in geology depend. -------------------------------------------------------------------------------- References Allegre, C., 1964, Vers une logique mathematique des series sedimentaires: Bull. Soc. Geol. de France, Series 7, Tome 6, p. 214–218. Anderson, T. W., and Goodman, L. A., 1957, Statistical inference about Markov chains: Ann. Math. Stat., v. 28, p. 89–110. 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E., 1966, Stochastic branching processes and the law of stream orders: Water Resources Res., v. 2, p. 199–203. Scheidegger, A. E., and Langbein, W. B., 1966, Probability concepts in geomorphology: U.S. Geol. Survey Prof. Paper 500-C, p. C1–C14. Schwarzacher, W., 1968, Experiments with variable sedimentation rates,in Computer applications in the earth sciences: Colloquium on simulation, D. F. Merriam, ed.: Kansas Geol. Survey Computer Contr. 22, p. 19–21. Vistelius, A. B., 1966, Genesis of the Mt. Belaya granodiorite, Kamchatka (an experiment in stochastic modeling): Doklady Akad. Nauk SSSR, v. 167, p. 1115–1118. Vistelius, A. B., and Faas, A. V., 1965, On the character of the alternation of strata in certain sedimentary rock masses: Doklady Akad. Nauk SSSR, v. 164, p. 629–632. Vistelius, A. B., and Feigel'son, T. S., 1965, On the theory of bed formation: Doklady Akad. Nauk SSSR, v. 164, p. 158–160. Wickman, F. E., 1966, Repose period patterns of volcanoes, V. General discussion and a tentative stochastic model: Arkiv for Mineralogi och Geologi, v. 4, p. 351–367.
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