Probability and Statistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 55

Fig. 2.3 Theoretical frequency distribution for shale and sandstone distributions for lithological
components in Oficina Formation, eastern Venezuela (After Krumbein and Dacey 1969 ).
All distributions are geometric by close approximation (Source: Agterberg 1974 , Fig. 35)

For each rock type, the frequencies for sequences of successive letters can be
determined and plotted in a diagram. For example, one counts how many times the
sequences A, AA, AAA, occur in the total series. The results are shown in

Fig. 2.3 . Krumbein and Dacey ( 1969 ) fitted geometric distributions to these data
and tested the degree of fit by means of chi-square tests for goodness of fit.
The theoretical distribution for shale is shown on the left-hand side of Fig. 2.3 .
The good fit of the geometric distributions indicates that the following stochastic
process was controlling the sedimentation.
Suppose that for a lithology, say sandstone, p (A) denotes the probability that at a
distance of 2 ft., another lithology (not sandstone) will occur. Obviously, q (A) ¼
1

...

p (A) then represents the probability that the same rock type (sandstone) will
occur. If for every lithology p (and q ) remained constant during deposition of the
entire series, then the probability that a sequence for any lithology is k letters long,
satisfies the negative binomial form. Consequently, r

1 because each sequence is
terminated at the point where it is replaced by another lithology. The result is a
geometric frequency distribution for each lithology. It is possible to divide the
probability p for each lithology into three parts, one for each other lithology.
All probabilities can be arranged into the following transition matrix:

¼

2
4

3
5 ¼

2
4

3
5

qA ðÞ pAB ðÞ pAC ðÞ pAD ðÞ 0 : 787 0 : 071 0 : 075 0 : 067

pBA ðÞ qB ðÞ pBC ðÞ pBD ðÞ 0 : 048 0 : 788 0 : 061 0 : 103

pCA ðÞ pCB ðÞ qC ðÞ pCD ðÞ 0 : 105 0 : 316 0 : 430 0 : 149

pDA ðÞ pDB ðÞ pDC ðÞ qC ðÞ 0 : 182 0 : 388 0 : 132 0 : 298

This matrix is the transition matrix of a Markov chain of the first order as
demonstrated by Krumbein and Dacey ( 1969 ). It illustrates the close connection
between geometric frequency distributions and first order Markov chains. Doveton
( 2008 ) discusses how Markov mean first-passage time statistics can be obtained for
sedimentary successions.