Page 126 - Statistics and Data Analysis in Geology

Spa tia I Analysis

spacing. Figure 5-2 b is an equivalent graph for a search pattern consisting of a
square grid of lines.
If the shape of the target is specified, the probabilities of intersection can be
graphed for different patterns of search. Figure 5-3 a, for example, shows the prob-
ability of intersecting a circular target with search patterns ranging from a square
grid, through rectangular grid patterns, to a parallel-line search. Figure 5-3 b is
the equivalent graph for a line-shaped target. Between the two graphs, all possible
shapes of elliptical targets and all possible patterns of search along two perpendic-
ular sets of parallel lines are encompassed.

Distribution of Points

Geologists often are interested in the manner in which points are distributed on a
two-dimensional surface or a map. The points may represent sample localities, oil
wells, control points, or poles and projections on a stereonet. We may be concerned
about the uniformity of control-point coverage, the distribution of point density,
or the relation of one point to another. These are questions of intense interest to
geographers as well as geologists, and the burgeoning field of locational analysis
is devoted to these and similar problems. Although much of the attention of the
geographer is focused on the distribution of shopping malls or public facilities, the
methodologies are directly applicable to the study of natural phenomena as well.
The patterns of points on maps may be conveniently classified into three
categories: regular, random, and aggregated or clustered. Examples of point dis-
tributions are shown in Figure 5-4 and range from the most uniform possible (the
face-centered hexagonal lattice in Fig. 5-4a, where every point is equidistant from
its six nearest neighbors) to a highly clustered pattern composed of randomly lo-
cated centers around which the probability of occurrence of a point decreases
exponentially with distance (Fig. 5-4f). Of course most maps will have patterns
intermediate between these extremes, and the problem becomes one of determin-
ing where the observed pattern lies within the spectrum of possible distributions.
For example, most people would intuitively regard the distribution of points in
Figure 5-4 c as random. However, intuition is wrong, because the map was created
by dividing the map area into a 4 x 4 array of regular cells and then placing four
points at random within each cell (except in the shortened bottom row, which re-
ceived only two points per cell). The distribution therefore has both random and
regular aspects and is more uniform in density than a purely random arrangement
such as Figure 5-4d.
The pattern of points on a map is said to be uniform if the density of points
in any subarea is equal to the density of points in all other subareas of the same
size and shape. The pattern is regular if the spacings between points repeat, as
on a grid. That is, the distance between a point i and another point j lying in some
specified direction from i is the same for all pairs of points i and j on the map.
Obviously, a regular pattern also will be uniform, but the converse is not necessarily
true. A random pattern can be created if any subarea is as likely to contain a point
as any other subarea of the same size, regardless of the subarea’s location, and the
placement of a point has no influence on the placement of any other point. In an
aggregated or clustered pattern, the probability of occurrence of a point varies in
some inverse manner with distances to preexisting points.

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