Page 98 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 4

that state j will be the next state to occur, given that the present state is i. [We here
introduce the unconventional but equivalent notation, p (i - j), which can be read
as the probability that state i will be followed by state j. This alternative notation
will be useful later.]
to Row
A B C D Totals
A 0.78 0 0.22 0 1 .oo
B 0 0.71 0.29 0 1 .oo
from
C 0.18 0.07 0.64 0.11 1.00
D 0 0 0.60 0.40 1 .oo
Here, for example, we see that if we are in state C at one point, the probability is
64% that the lithology 1 ft up will also be state C. The probability is 18% that the
lithology will be state A, 7% that it will be state B, and 11% that it will be state D.
Since the four states are mutually exclusive and exhaustive, the lithology must be
one of the four and so their sum, given as the row total, is 100%.
If we divide the row totals of the transition frequency matrix by the total num-
ber of transitions, we obtain the relative proportions of the four lithologies that are
present in the section. This is called the marginal (or fixed) probability vector:
D F1

C 0.44
0.08

You will recall from Chapter 2 (Eq. 2.7) that the joint probability of two events,
A and B, is
p(A,B) = p(BIA)p(A)
rearranging ,

So, the probability that state B will follow, or overlie, state A is the probability that
both state A and B occur, divided by the probability that state A occurs. If the
occurrence of states A and B are independent, or unconditional,

and

That is, the probability that state B will follow state A is simply the probability that
state B occurs in the section, which is given by the appropriate element in the fixed
probability vector. If the occurrences of all the states in the section are independent,
the same relationship holds for all possible transitions; so, for example,

This allows us to predict what the transition probability matrix should look like if
the occurrence of a lithologic state at one point in the stratigraphic interval were

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