# Markov chains and embedded Markov chains in geology

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## 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.

## Keywords

Markov Chain Markov Process Diagonal Element Nonzero Element Thickness Distribution## Preview

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## References

- Allegre, C., 1964, Vers une logique mathematique des series sedimentaires: Bull. Soc. Geol. de France, Series 7, Tome 6, p. 214–218.Google Scholar
- Anderson, T. W., and Goodman, L. A., 1957, Statistical inference about Markov chains: Ann. Math. Stat., v. 28, p. 89–110.Google Scholar
- Billingsley, P., 1961, Statistical methods in Markov chains: Ann. Math. Stat., v. 32, p. 12–40.Google Scholar
- Carr, D. D., and others, 1966, Stratigraphic sections, bedding sequences, and random processes: Science, v. 154, no. 3753, p. 1162–1164.Google Scholar
- Feller, W., 1968, An introduction to probability theory and its applications (3rd ed.): John Wiley & Sons, New York, 509 p.Google Scholar
- Gingerich, P. D., 1969, Markov analysis of cyclic alluvial sediments: Jour. Sed. Pet., v. 39, no. 1, p. 330–332.Google Scholar
- Karlin, S., 1966, A first course in stochastic processes: Academic Press, New York, 502 p.Google Scholar
- Kolmogorov, A. N., 1951, Solution of a problem in probability theory connected with the problem of the mechanism of stratification: Am. Math. Society Trans., no. 53, 8 p.Google Scholar
- Krumbein, W. C., 1967,fortran iv computer programs for Markov chain experiments in geology: Kansas Geol. Survey Computer Contr. 13, 38 p.Google Scholar
- Krumbein, W. C., 1968
*a*, Computer simulation of transgressive and regressive deposits with a discrete-state, continuous-time Markov model,*in*Computer applications in the earth sciences: Colloquium on simulation, D. F. Merriam, ed.: Kansas Geol. Survey Computer Contr. 22, p. 11–18.Google Scholar - Krumbein, W. C., 1968
*b*,fortran iv computer program for simulation of transgression and regression with continuous-time Markov models: Kansas Geol. Survey Computer Contr. 26, 38 p.Google Scholar - Krumbein, W. C., and Sloss, L. L., 1963, Stratigraphy and sedimentation (2nd ed.): W. H. Freeman and Co., San Francisco, 660 p.Google Scholar
- Pettijohn, F. P., 1957, Sedimentary rocks: Harper and Bros., New York, 718 p.Google Scholar
- Potter, P. E., and Blakely, R. F., 1967, Generation of a synthetic vertical profile of a fluvial sandstone body: Jour. Soc. Petroleum Engineers, v. 6, p. 243–251.Google Scholar
- Potter, P. E., and Blakely, R. F., 1968, Random processes and lithologic transitions: Jour. Geology, v. 76, no. 2, p. 154–170.Google Scholar
- Potter, P. E., and Siever, R., 1955, A comparative study of Upper Chester and Lower Pennsylvanian stratigraphic variability: Jour. Geology, v. 63, no. 5, p. 429–451.Google Scholar
- Pyke, R., 1961
*a*, Markov renewal processes: definitions and preliminary properties: Ann. Math. Stat., v. 32, p. 1231–1242.Google Scholar - Pyke, R., 1961
*b*, Markov renewal processes with finitely many states: Ann. Math. Stat., v. 32, p. 1243–1259.Google Scholar - Scherer, W., 1968, Applications of Markov chains to cyclical sedimentation in the Oficina Formation, eastern Venezuela: Unpubl. master's thesis, Northwestern Univ., 93 p.Google Scholar
- Schwarzacher, W., 1964, An application of statistical time-series analysis of a limestone-shale sequence: Jour. Geology, v. 72, no. 2, p. 195–213.Google Scholar
- Schwarzacher, W., 1967, Some experiments to simulate the Pennsylvanian rock sequence of Kansas,
*in*Computer applications in the earth sciences: Colloquium on simulation, D. F. Merriam, ed.: Kansas Geol. Survey Computer Contr. 18, p. 5–14.Google Scholar - Smart, J. S., 1968, Statistical properties of stream lengths: Water Resources Res., v. 4, p. 1001–1014.Google Scholar
- Vistelius, A. B., 1949, On the question of the mechanism of the formation of strata: Doklady Akad. Nauk SSSR, v. 65, p. 191–194.Google Scholar
- Zeller, E. J., 1964, Cycles and psychology,
*in*Symposium on cyclic sedimentation: Kansas Geol. Survey Bull. 169, p. 631–636.Google Scholar

## Additional references

- Agterberg, F. P., 1966
*a*, The use of multivariate Markov schemes in geology: Jour. Geology, v. 74, no. 5, pt. 2, p. 764–785.Google Scholar - Agterberg, F. P., 1966
*b*, Markov schemes for multivariate well data: Min. Ind. Experiment Sta., Pennsylvania State Univ. Sp. Publ. 2–65, p. Y1-Y18.Google Scholar - Conover, W. J., and Matalas, N. C., in press, A statistical model of turbulence applicable to sediment laden streams: ASCE, Jour. Engineering Mechanics.Google Scholar
- Graf, D. L., Blyth, C. R., and Stemmler, R. S., 1967, One-dimensional disorder in carbonates: Illinois Geol. Survey Circ. 408, 61 p.Google Scholar
- Griffiths, J. C., 1966, Future trends in geomathematics: Pennsylvania State Univ., Mineral Industries, v. 35, p. 1–8.Google Scholar
- Harbaugh, J. W., 1966, Mathematical simulation of marine sedimentation with IBM 7090/7094 computers: Kansas Geol. Survey Computer Contr. 1, 52 p.Google Scholar
- James, W. R., and Krumbein, W. C., 1969, Frequency distributions of stream link lengths: Jour. Geology, v. 77, no. 5, p. 544–565.Google Scholar
- Kemeny, J. G., and Snell, J. L., 1960, Finite Markov chains: D. Van Nostrand Co., Inc., Princeton, New Jersey, 210 p.Google Scholar
- Leopold, L. B., and Langbein, W. B., 1962, The concept of entropy in landscape evolution: U.S. Geol. Survey Prof. Paper 500-A, p. A1–A20.Google Scholar
- Matalas, N. C., 1967
*a*, Mathematical assessment of synthetic hydrology: Water Resources Res., v. 3, p. 937–946.Google Scholar - Matalas, N. C., 1967
*b*, Some distribution problems in time series simulation,*in*Computer applications in the earth sciences: Colloquium on time-series analysis, D. F. Merriam, ed.: Kansas Geol. Survey Computer Contr. 18, p. 37–40.Google Scholar - Scheidegger, A. E., 1966, Stochastic branching processes and the law of stream orders: Water Resources Res., v. 2, p. 199–203.Google Scholar
- Scheidegger, A. E., and Langbein, W. B., 1966, Probability concepts in geomorphology: U.S. Geol. Survey Prof. Paper 500-C, p. C1–C14.Google Scholar
- 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.Google Scholar - 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.Google Scholar
- 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.Google Scholar
- Vistelius, A. B., and Feigel'son, T. S., 1965, On the theory of bed formation: Doklady Akad. Nauk SSSR, v. 164, p. 158–160.Google Scholar
- 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.Google Scholar