Page 130 - Statistics and Data Analysis in Geology

Spat ia I An a I ysis

or expected numbers of quadrats can be used in a x2 procedure to test whether the
points are distributed at random within the area.
As an application, we can determine if oil discoveries in a basin occur at ran-
dom or are distributed in some other fashion. It is not intuitively obvious that the
Poisson distribution can be expressed in a form appropriate for this problem, so
we will work through its development.
Assume a basin has an area, a, in which m discovery wells are randomly lo-
cated. The density of discovery wells in the basin is designated A, and is simply

A=- 712 (5.10)
a
The basin may be divided into small lease tracts, each of area A (here the term “tract”
is equivalent to “quadrat”). In turn, each tract may be divided into n extremely
small, equal-sized subareas which we might regard as potential drilling sites. The
probability that any one of these extremely small subareas contains a discovery
well tends toward zero as n becomes infinitely large.
The area of each drilling site is Aln. The probability that a site contains a
discoverv well is

and the probability that it does not contain a discovery well is
( 3
1-p= 1-A-

We wish to investigate the probability that Y of the n drilling sites within a
tract contain discovery wells, and n - Y drilling sites do not. The probability of a
specific combination of discovery and nondiscovery well sites within a tract is

P = (A;)r (1 - A;).-.

However, within a tract, there are (:) combinations of the n drilling sites, of which

Y contain discovery wells and all are equally probable. The probability that a tract
will contain exactly Y discovery wells is therefore
P (Y) = (;) (A:)r (1 - A:).-.

Note that this is simply the binomial probability of Y discovery wells on n drilling
sites.
The combinations can be expanded into factorials,
n(n - 1) (n- 2) *. . (n-Y + 1) (AA)’ AA
P(Y) =
r ! nr
Rearranging and canceling terms yields

(1
AA
AA
-
P (Y) = (1 - i) f) ... (1 - G) (1 - q) -‘ [(I - --) 71 (5.11)
(AA)‘
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