Page 122 - Statistics and Data Analysis in Geology

I
Spat ia Ana I ysis
the problems of data-point distribution, lack of fit, computational “blowup,” and
inappropriate applications. Statistical tests are available for trend surfaces if they
are to be used as multiple regressions; we will consider these tests and the assump-
tions prerequisite to their application.
The exchange between Earth scientists and statisticians has been mostly one
way, with the notable exception of the expansion of the theory of regionalized vari-
ables. This theory, developed originally by Georges Matheron, a French mining
engineer, describes the statistical behavior of spatial properties that are interme-
diate between purely random and completely deterministic phenomena. The most
familiar application of the theory is in kriging, an estimation procedure important
in mine evaluation, mapping, and other applications where values of a property
must be estimated at specific geographic locations.
Two-dimensional methods are, for the most part, direct extensions of tech-
niques discussed in Chapter 4. Trend-surface analysis is an offshoot of statistical
regression; kriging is related to time-series analysis; contouring is an extension
of interpolation procedures. We have simply enlarged the dimensionality of the
subjects of our inquiries by considering a second (and in some cases a third) spa-
tial variable. Of course, there are some applications and some analytical meth-
ods that are unique to map analysis. Other methods are a subset of more general
multidimensional procedures. It is an indication of the importance of one- and
two-dimensional problems in the Earth sciences that they have been included in
individual chapters.
Systematic Patterns of Search

Most geologists devote their professional careers to the process of searching for
something hidden. Usually the object of the search is an undiscovered oil field
or an ore body, but for some it may be a flaw in a casting, a primate fossil in an
excavation, or a thermal spring on the ocean’s floor. Too often the search has
been conducted haphazardly-the geologist wanders at random across the area of
investigation like an old-time prospector following his burro. Increasingly, how-
ever, geologists and other Earth scientists are using systematic procedures to
search, particularly when they must rely on instruments to detect their targets.
Most systematic searches are conducted along one or more sets of parallel lines.
Ore bodies that are distinctively radioactive or magnetic are sought using airborne
instruments carried along equally spaced parallel flight lines. Seismic surveys are
laid out in regular sets of traverses. Satellite reconnaissance, by its very nature,
consists of parallel orbital tracks.
The probabilities that targets will be detected by a search along a set of lines
can be determined by geometrical considerations. Basically, the probability of dis-
covery is related to the relative size of the target as compared to the spacing of
the search pattern. The shape of the target and the arrangement of the lines of
search also influence the probability. If the target is assumed to be elliptical and
the search consists of parallel lines, the probability that a line will intersect a hidden
target of specified size, regardless of where it occurs within the search area, can
be calculated. These assumptions do not seem unreasonable for many exploratory
surveys. Note that the probabilities relate only to intersecting a target with a line,
and do not consider the problem of recognizing a target when it is hit.
McCammon (1977) gives the derivation of the geometric probabilities for cir-
cular and linear targets and parallel-line searches. His work is based mostly on the

295