Markov chains and embedded Markov chains in geology

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.

[1]  T. W. Anderson,et al.  Statistical Inference about Markov Chains , 1957 .

[2]  R. F. Blakely,et al.  Random Processes and Lithologic Transitions , 1968, The Journal of Geology.

[3]  Adrian E. Scheidegger,et al.  Stochastic branching processes and the law of stream orders , 1966 .

[4]  W. C. Krumbein,et al.  Stratigraphy and sedimentation , 1963 .

[5]  R. Pyke.  Markov renewal processes: Definitions and preliminary properties , 1961 .

[6]  J. Smart,et al.  Statistical Properties of Stream Lengths , 1968 .

[7]  John Warvelle Harbaugh,et al.  Mathematical simulation of marine sedimentation : with IBM 7090/7094 computers , 1966 .

[8]  A Comparative Study of Upper Chester and Lower Pennsylvanian Stratigraphic Variability , 1955, The Journal of Geology.

[9]  The Use of Multivariate Markov Schemes in Petrology , 1966, The Journal of Geology.

[10]  P. Gingerich.  Markov analysis of cyclic alluvial sediments , 1969 .

[11]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[12]  R. Pyke.  Markov Renewal Processes with Finitely Many States , 1961 .

[13]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[14]  W. C. Krumbein,et al.  Frequency Distributions of Stream Link Lengths , 1969, The Journal of Geology.

[15]  Luna Bergere Leopold,et al.  The concept of entropy in landscape evolution , 1962 .

[16]  W. Schwarzacher.  An Application of Statistical Time-Series Analysis of a Limestone-Shale Sequence , 1964, The Journal of Geology.

[17]  N. Matalas.  Mathematical assessment of synthetic hydrology , 1967 .

[18]  W. B. Langbein,et al.  Probability concepts in geomorphology , 1966 .

[19]  P. Billingsley,et al.  Statistical Methods in Markov Chains , 1961 .

[20]  Robert F. Blakely,et al.  Generation of a Synthetic Vertical Profile of a Fluvial Sandstone Body , 1967 .

[21]  P. E. Potter,et al.  Stratigraphic Sections, Bedding Sequences, and Random Processes , 1966, Science.