Michael F. Dacey

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.

The syntax of a triangle and some other figures

Abstract An approach suggested by Kirsch is used to construct sets of rules for two-dimensional languages that generate arrangements of symbols that form polygons. The syntactic structure of these languages is analyzed and it is shown that a mathematical group summarizes the structure holding between languages constructed for polygons that are related by proper and improper rotations.

Stokes’ Settling and Chemical Reactivity of Suspended Particles in Natural Waters

Equations are given for the Stokes settling velocities of the following particle shapes: the sphere, oblate spheroid, prolate spheroid, circular cylinder, elliptic cylinder, disc, and hemispherical cap. Dissolution of calcareous and silicate particles settling through ocean water, based on literature data, is analyzed in terms of a model for dissolution rate independent of the particle surface area, and a model for dissolution rate dependent on a surface reaction. The settling of dissolving particles in the presence of a countercurrent of upwelling water may lead to formation of thin nepheloid layers. Settling of calcite crystals through a stratified water column is treated as a case of variable nucleation (production) rates, dissolution and agglomeration of crystals en route to the bottom. A stochastic model presented in the paper gives a reasonably simple method for treating transient transport of particles in a physically heterogeneous water column.

Models of bed formation

The model for bed formation by Kolmogorov consists of an unending sequence of alternating periods of deposition and erosion of sediments, with the amounts of deposition and erosion being independent random variables. This paper examines this model in relation to recent mathematical studies that are relevant to, and simplify, the analysis of the thickness of the bed that remains after the sequence is operated for a long time. This thickness obeys the exponential probability law when the amounts of deposition and erosion also obey the exponential law distributed. For the discrete version formulated by Schwarzacher, the thickness obeys the geometric probability law when the amounts of deposition and erosion obey the geometric law.

Markovian models in stratigraphic analysis

A stratigraphic section may be divided into lithologic units which in turn may be divided into beds. This paper gives a mathematical formulation of stratigraphic sections that takes these two levels into account and uses bed properties to yield the thickness and number of beds in lithologic units. The model is a semiMarkov chain in which the succession of lithologic bed types forms a Markov chain and is an independent random variable. The model is tested against stratigraphic data obtained from micrologs. There is close agreement between the observed and calculated thicknesses of lithologic units. Tests for the degree of agreement between observed and calculated numbers of beds in lithologic units are hampered by inability to observe thin beds on micrologs. Some implications of this limitation to stratigraphic analysis are noted.

Poly: A two dimensional language for a class of polygons

Abstract Poly is a two dimensional language that produces line pictures of polygons that may be decomposed into 45° right triangles and rectangles. The elements of the language are described and an example illustrates use of the language to produce a picture.

Models of breakage and selection for particle size distributions

It is generally agreed that particle size distributions of sediments tend ideally to approximate the form of the lognormal probability law, but there is no single widely accepted explanation of how sedimentary processes generate the form of this law. Conceptually, and in its simplest form, sediment genesis involves the transformation of a parent rock mass into a particulate end product by processes that include size reduction and selection during weathering, transportation, and deposition. The many variables that operate simultaneously during this transformation can be shown to produce a distribution of particle sizes that approaches asymptotically the lognormal form when the effect of the variables is multiplicative. This was first shown by Kolmogorov (1941). Currently available models combine breakage and selection in differing degrees, but are similar in treating the processes as having multiplicative effects on particle sizes. The present paper, based on careful specification of the initial state, the nth breakage rule and the nth selection rule, leads to two stochastic models for particle breakage, and for both models the probability distributions of particle sizes are obtained. No attempt is made to apply these models to real world sedimentary processes, although this topic is touched upon in the closing remarks.

Sediment Growth and Aging as Markov Chains

Preservation of sediments as a function of their geologic age can be described by stochastic models for the following cases: (1) fixed sediment mass with constant probabilities of erosion and preservation; (2) fixed sediment mass with erosion and preservation probabilities that vary with geologic age; and (3) growing sediment mass with constant probabilities of erosion, preservation, and removal from the sedimentary cycle. For geologically long periods of time, the stochastic models can be transformed into such models of sediment preservation as the exponential or compound decay of a fixed sediment mass and a continuous growth of the sediment mass. Several of the statistical distributions arising out of the stochastic models- the geometric, exponential, compound decay, and inverse hypergeometric-produce mass-age distribution curves for the Phanerozoic and Late Proterozoic sediments that agree reasonably well with the actual masses reported in the literature. A general characteristic of the stochastic models is that for any mass-age distribution of sediments when the mass decreases with an increasing geologic age, there always exist sets of preservation and erosion probabilities that describe the process of sediment aging.

Catastrophe theory: Application to the Permian mass extinction

Catastrophe theory, the recent development of Rene Thorn, permits mathematical descriptions to be obtained for physical processes in which the dependent parameter has abrupt changes in value and, over some range of values of the independent parameters, assumes one of several values for a given value of the independent parameters. Catastrophe theory is briefly discussed and potential applications to geologic processes are mentioned. The theory is applied to the Permian extinction of marine invertebrates; our conclusion is that a reduction in oceanic salinity was a more significant factor in the extinction of marine invertebrates than was a reduction in the area of shallow seas.

Topological properties of disjoint channel networks within enclosed regions

Enclosure of some portion of one or more natural stream-drainage basins by superposition of a rectangle on a map of drainage network results in fragmentation of the natural basins into a set of disjoint channel networks. Each of these may have some channel links and forks of the natural network plus truncated links intersected by the enclosure boundary. The topological properties of the network elements in the enclosure are used to set up a model of random network patterns, in which the number of disjoint channel networks is expressed as a function of the number of links and forks in the enclosures. This function is shown to be a multiplicative constant times the square root of the number of links or forks. Empirical data on square and rectangular enclosures of several sizes from the Inez (Kentucky)Quadrangle map showed that the predicted multiplicative constants do not agree with observation, but that the square-root relation seems to hold at least to a first approximation. The models thus can be used as a norm against which deviations of real-world enclosures from network pattern randomness can be studied.