## 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.

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Krumbein, W.C., Dacey, M.F. Markov chains and embedded Markov chains in geology.
*Mathematical Geology* **1, **79–96 (1969). https://doi.org/10.1007/BF02047072

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### Keywords

- Markov Chain
- Markov Process
- Diagonal Element
- Nonzero Element
- Thickness Distribution