Page 134 - Statistics and Data Analysis in Geology

Spatial Analysis

The t-test compares the ratio between m/T and s2, which should be equal to 1.0 if
the two statistics are the same
-
(F) 1.0
t= (5.18)
Se
The test has T - 1 degrees of freedom.
For the eastern Permian Basin area, the variance in number of wells per tract is

The standard error of the mean number of wells per tract can be estimated as

se = & 0.112
=
The t-statistic for the test of equivalence of the mean and variance is
(1.05/1.46) - 1.0
t= = -8.86
0.112
At a significance level of o( = 0.05 and 159 degrees of freedom, the critical
value of t for a two-tailed test is 1.96; the computed statistic far exceeds this and
so we may conclude as we did in the x2 test that the spatial distribution is not
random. Since the variance is significantly greater than the mean, we must also
conclude that discovery wells are areally clustered.

CI ustered patterns
Many naturally occurring spatial distributions show a pronounced tendency toward
clustering. This is especially true of certain biological variables, such as presence
of specific organisms or occurrences of an infectious disease. The descendants of
a sedentary parent, perhaps a coral or a tree, tend to grow nearby, leading to devel-
opment of densely populated areas surrounded by areas that are relatively barren.
Clustered patterns of points can be modeled by many theoretical distributions,
most of which can be regarded as combinations of two or more simpler distribu-
tions. One of the distributions describes the locations of the centers of clusters,
while the other describes the pattern of individual points around the centers of the
clusters.
The negative binomial distribution can be used to model the occurrence of
clustered points in space in a manner equivalent to the use of the Poisson to model
randomly arranged points. An extensive discussion with citations to studies in
many fields is given by Ripley (1981). Griffiths (1962, 1966) advocated the use of
the negative binomial as an appropriate model for the occurrence of oil fields and
ore bodies.
One derivation of the negative binomial is as a compound Poisson and loga-
rithmic distribution with clusters of points randomly located within a region; indi-
vidual points within a cluster follow a logarithmic distribution. In the formulation
appropriate for describing spatial patterns, the negative binomial is

k
(5.19)

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