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involving a good understanding of the underlying physical and chemical processes
with detailed knowledge of the ages of rock units and processes. As discussed in
Agterberg (
2013
), geologists have had a long history of interaction with mathemat-
ical statisticians. These collaborations continue to be fruitful. During his long and
distinguished career, William Krumbein regularly consulted with John Tukey.
Krumbein also wrote one of the first geomathematical textbooks together with the
mathematical statistician Franklin Graybill (Krumbein and Graybill
1965
). These
authors distinguished between three types of models in geology: (1) scale-models;
(2) conceptual models; and (3) mathematical models. Traditionally, geologists have
been concerned mainly with scale-models and conceptual models.
John Griffiths (
1967
), who was one of the other pioneers of geomathematics,
based much of his work on advanced statistical sampling techniques, especially as
they had been developed by Ronald Fisher. In the Soviet Union, Andrew Vistelius
(
1967
) worked closely with Andrey Kolmogorov, already when he was preparing
his PhD thesis in the 1940s. Felix Chayes (
1956
,
1971
) developed modal analysis in
petrography and took to heart Karl Pearson's admonishment regarding spurious
correlations that could result from closed-number systems. This later led John
Aitchison (
1986
) to develop compositional data analysis, which continues to be
an important research topic (Egozcue et al.
2003
).
In turn, geoscientists have inspired statisticians to pursue new research direc-
tions. Georges Matheron (
1962
,
1965
) introduced the idea of regionalized random
variables and this has helped to found the important field of spatial statistics
(Cressie
1991
). Ronald Fisher (
1953
) became interested in statistics of directional
features when a geophysics student at Cambridge University asked him for help
in dealing with greatly dispersed paleomagnetic measurements. It led to the estab-
lishment of the cone of confidence for unit vectors. Geoffrey Watson (
1966
) was
instrumental in developing statistical significance tests for directional features,
which are similar to those that were in existence for ordinary data analysis. Geo-
scientists often work with very large data sets to be subjected to exploratory data
analysis with use of jackknife and bootstrap techniques for uncertainty estimation.
John Tukey (
1962
,
1970
,
1977
) pioneered these approaches and advised mathe-
matical geologists to use them. GIS (Bonham-Carter
1994
) and its 3-D extensions,
for which credit is owed to engineers and computer scientists, are widely used
in geological map modeling and image analysis. Promising new developments
include projection pursuit (Friedman and Tukey
1974
; Xiao and Chen
2012
),
boosting (Freund and Shapire
1997
), radial basis function theory (Buhmann
2003
) and bi-dimensional empirical mode decomposition (Huang et al.
2010
).
1.2 Geological Data, Concepts and Maps
The first geological map was published in 1815. This feat has been well
documented by Winchester (
2001
) in his topic:
The Map that Changed the
World - William Smith and the Birth of Modern Geology
. According to the concept
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