Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 463

Chapter 12
Selected Topics for Further Research
Abstract The case history studies described in the preceding eleven chapters leave
some questions that could not be answered in full. New theoretical approaches in
mathematical statistics and nonlinear physics provide new perspectives for the
analysis of geoscience data. For example, bias due to incomplete information
continues to be one of the most serious problems in 3-D mapping. How methods
such as the jackknife and bootstrap can help to reduce this type of bias is briefly
investigated and illustrated using volcanogenic massive copper deposits in the
Abitibi area on the Canadian Shield. Compositional data analysis offers new
ways to analyze multivariate data sets. Geochemical data from Fort ` la Corne
kimberlites in central Saskatchewan are used to illustrate the isometric logratio
transformation for chemical data that form a closed number system. Three gener-
alizations of the model of de Wijs are: (1) the 3-parameter model with finite number
of iterations; (2) the random cut model in which the dispersion index d is
replaced by a random variable D ; and (3) the accelerated dispersion model in
which d depends on concentration value during the cascade process. Universal
multifractals constitute a useful generalization of multifractal modeling. As illus-
trated on the basis of the Pulacayo zinc values, new tools such as use of the first
order structure function and double trace analysis generalize conventional
variogram-autocorrelation fitting. Measurements on compositions of blocks of
rocks generally depend on block size. For example, at microscopic scale chemical
elements depend on frequencies of abundance of different minerals. On a regional
basis, rock type composition depends on spatial distribution of contacts between
different rock types. Frequency distribution modeling of compositional data can be
useful in ore reserve estimation as well as regional mineral potential studies. During
the 1970, Georges Matheron proposed the theory of permanent frequency distribu-
tions with shapes that are independent of block size. The lognormal is a well-known
geostatistical example. The probnormal distribution is useful for the analysis of
relative amounts of different rock types contained in cells of variable size. It arises
when probits of percentage values are normally distributed. Its Q - Q plot has scales
derived from the normal distribution along both axes. Parameters (mean and
variance) of the probnormal distribution are related to the geometrical covariances