Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 495

previously shown in Fig. 3.11 for gold values in Merriespruit Mine, Witwatersrand
goldfield, South Africa. It is noted, however, that values in the lower value tail
cannot be less than a minimum value that is greater than zero.
12.6 Trends, Multifractals and White Noise
Regional geology generally can be explained as the result of deterministic pro-
cesses that took place millions of years ago. How does the well-documented and
explained mosaic constituting the upper part of the Earth's crust fit in with the
concept of fractals and multifractals?
Brinck's ( 1974 ) model constituted an early application of the model of de Wijs
( cf . Sect. 11.1.2 ). Estimation of parameters in this model including d could be
improved by adopting the multifractal modeling approach explained in Sect. 12.4 .
At first glance, the Brinck approach seems to run counter to the fact that mineral
deposits are of many different types and result from different genetic processes.
However, Mandelbrot ( 1983 ) has shown that, for example, mountainous landscapes
can be modeled as fractals ( cf . Sect. 10.1 ). Smoothed contours of elevation on such
maps continue to exhibit similar shapes when the scale is enlarged, as in Krige's
( 1966 ) example for Klerksdorp gold contours (Sect. 11.1.1 ). Lovejoy and Schertzer
( 2007 ) argued convincingly that the Earth's topography can be modeled as a
multifractal, both on continents and ocean floors in accordance with power-law
relations originally established by Vening Meinesz ( 1964 ) as explained in Sect.
10.1.1 . These broad-scale approaches to the nature of topography also seem to run
counter to the fact that landscapes are of many different types and result from
different genetic processes. Nevertheless, it can be assumed as Brinck did that
chemical elements within the Earth's crust or within smaller, better defined envi-
ronments like the Witwatersrand goldfields can be modeled as multifractals.
In early applications of mathematical statistics to geoscience, it often was
assumed that regional features can be modeled by using deterministic functions
(e.g., in trend surface analysis, Chap. 4 ) . Residuals from the trends were assumed to
be white noise with the properties of uncorrelated (iid) random variables. It
gradually became clear that residuals often are better modeled as stationary random
functions with a spatial covariance function or semi-variogram. Universal kriging
(cf. Sect. 7.2 ) is an approach that embodies the three components consisting of
deterministic trends (or drifts), stationary random functions and white noise (nugget
effect). A frequently used geostatistical model is that the semivariogram shows
nugget effect at the origin, a range that can be modeled for use is kriging and a sill
related to regional mean. Nested designs of superimposed models of this type also
are frequently used. Multifractal modeling can help to refine the nugget effect.
Based on the concept of self-similarity (or scale independence), the spatial auto-
correlation function can be extrapolated to very short distances. Use of the Chen
algorithm (Sect. 11.6 ) resulted in identification of white noise components that are
much smaller than previously suggested by semivariogram or correlogram. It was