Complexity of the Geological Framework and Use of Mathematics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 2

Keywords Geomathematics • 3-D map-making • Geological cross-sections
• Geometry of folds • Faults • Minkowski operations • Edge effects • Central Baffin
Island supracrustal rocks • Bathurst acidic volcanics • Leduc reef complex
1.1 Use of Mathematics in Geology
As pointed out previously in Agterberg ( 1974 ), geomathematics, in its broadest
sense, includes all applications of mathematics in the studies of the Earth's crust. In
this topic, the emphasis is on use of mathematics to solve 3-D prediction problems
with applications to actual examples in selected case history studies. Of course,
mathematics is widely used in geophysics, traditionally by formulating problems
in terms of differential equations for deterministic processes but during the past
20 years also by means of non-linear process modeling. However, geologists also
need a variety of geomathematical techniques, especially in the following fields of
activity:
1. Data acquisition and processing. Systematic recording, ordering and compari-
son of data, and methods for graphical display of results.
2. Data analysis . Identification of trends, clusters and simple or complex correla-
tions for which geological explanations are needed.
3. Sampling . Design of statistical procedures for data acquisition.
4. Hypothesis testing . Verification of concepts or models of processes believed to
explain the origin and provenance of specific phenomena (includes computer
simulation models).
5. Quantitative prediction in applied geology . Provision of solutions to specific
problems such as estimating probabilities of occurrence of specific types of
mineral deposits, volcanic eruptions, earthquakes and landslides.
The advent of digital computers catalyzed a quantitative revolution in geology,
because it became possible to apply mathematical and statistical models to large
volumes of data (Merriam 1981 , 2004 ). When mathematics is used in geology,
parameters must be defined in a manner sufficiently rigorous to permit nontrivial
derivations. The initial hurdle is to choose parameters that are substantially mean-
ingful. During model design it is important to keep in mind Chamberlin's ( 1899 )
warning: “The fascinating impressiveness of rigorous mathematical analysis, with
its atmosphere of precision and elegance, should not blind us to the defects of the
premises that condition the whole process. There is, perhaps, no beguilement more
insidious and dangerous than an elaborate and elegant mathematical process built
upon unfortified premises.”
The geosciences continually benefit from the use of mathematics. An important
objective of geology is to construct three-dimensional maps of the upper part of the
Earth's crust with hypothetical delineation of various rock units including ore and
hydrocarbon deposits. Uncertainties in predictive geology are very large but need to
be quantified nevertheless. Use of geological concepts is essential in this endeavor