Page 100 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 4

The critical value of x2 for nine degrees of freedom and a 5% level of significance
is 16.92; the test value comfortably exceeds this, so we may conclude that the hy-
pothesis of independence of successive states is not correct. There is a statistically
significant tendency for certain states to be preferentially followed by certain other
states.
A sequence in which the state at one point is partially dependent, in a prob-
abilistic sense, on the preceding state is called a Markov chain (named after the
Russian statistician, A.A. Markov). A sequence having the Markov property is inter-
mediate between deterministic sequences and completely random sequences. Our
stratigraphic section exhibits first-order Markov properties; that is, the statistical
dependency exists between points and their immediate predecessors. Higher order
Markov properties can exist as well. For example, a second-order Markov sequence
exhibits a significant conditional relationship between points that are two steps
apart.
From the transition probability matrix we can estimate what the lithology will
be 2 ft (that is, two observations) above a given point. Suppose we start in limestone
(state B). The following probabilities estimate the lithology to be encountered at
the next point upward:
State A (sandstone) 0%
State B (limestone) 71%
State C (shale) 2 9%
State D (coal) 0%
Suppose the next point actually falls in a shale; we can then determine the probable
lithology of the following point:
State A (sandstone) 18%
State B (limestone) 7%
State C (shale) 64%
State D (coal) 11%
So, the probability that the lithologic sequence will be limestone -* shale - limestone
is
p(B - C) x p(C - B) = 29% x 7% = 2%
However, there is another way to reach the limestone state in two steps. The se-
quence limestone - limestone - limestone is also possible. The probability attached
to this sequence is

p(B -, B) X p(B + B) = 71%~ 71%= 50%

Since the other transitions limestone - sandstone and limestone - coal have
zero probability, these two sequences are the only possible ones which lead from
limestone and back again in two steps. The probability that the lithology two steps
above a limestone will also be a limestone, regardless of the intervening lithology,
is the sum of all possibilities. That is,

p(B-A-B)= 0%
p(B - B - B) = 50%
p(B-C-B) = 2%
p(B-D-B)= 0%
Total = 52%

172