Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 368

The concept of self-organized criticality (Bak 1996 ) has resulted in experimentally
produced fractal phenomena. One of Bak's physical experiments consisted of
dropping a grain of sand on a pyramid of grains in a “critical” state so that an additional
grain creates a sand avalanche. The number of grains per avalanche then satisfies
a Pareto frequency distribution with a relatively thick high-value tail. Another
application of self-organized criticality is in the study of seismicity: Rundle
et al. ( 2003 ) showed that the Gutenberg-Richter earthquake frequency-magnitude
relation is a combined effect of the geometrical (fractal) structure of fault networks
and the non-linear dynamics of seismicity.
Other successful applications of non-linear modeling are the following.
Most weather-related processes taking place in the atmosphere including cloud
formation and rainfall are multifractal (Lovejoy and Schertzer 2013 ). Other
space-related non-linear processes include “current disruption” and “magnetic
reconnection” scenarios (Sharma 1995 ; Uritsky et al. 2008 ). Within the solid
Earth's crust, processes involving the release of large amounts of energy over
very short intervals of time including earthquakes (Turcotte 1997 ), landslides,
flooding (Gupta et al. 2007 ) and forest fires (Malamud et al. 1998 ) are non-linear
and result in fractals or multifractals.
Increasingly it is realized that various processes that took place millions of
year ago during the geologic past also were non-linear. These include, for example,
the formation of columnar basalt joints (Goehring et al. 2009 ) and self-
organization in geochemical reaction-diffusion systems resulting in banding within
Mississippi Valley-type lead-zinc deposits (Fowler and L'Heureux 1996 ). Several
types of ore deposits resulting from hydrothermal processes display geometric
characteristics that have been known to be fractal or multifractal for a long time
(Blenkinsop 1995 ; Cheng 2008 ). The same consideration applies to oil and gas
pools (Mandelbrot 1995 ; Barton and La Pointe 1995 ). Non-linear process modeling
in these fields has two practical applications. On the one hand it has resulted in new
exploration techniques for the discovery of new mineral deposits; on the other hand,
it allows statistical modeling of the size-frequency distributions of populations
of known deposits.
The relationship between fractal point pattern modeling and statistical methods
of parameter estimation in point-process modeling will be reviewed in Sect. 10.2 .
Statistical estimation of the cluster fractal dimension by using Ripley's ( 1976 )
K -function has advantages in comparison with the more commonly used methods
of box-counting and cluster fractal dimension estimation because it corrects for
edge effects, not only for rectangular study areas but also for study areas with
curved boundaries determined by regional geology. Application of box-counting to
estimate the fractal dimension of point patterns has the other disadvantage that, in
general, it is subject to relatively strong “roll-off” effects for smaller boxes. Point
patterns used for example in this section are mainly for gold deposits in the Abitibi
Volcanic Belt on the Canadian Shield. Additionally, it will be proposed that,
worldwide, the local point patterns of podiform Cr, volcanogenic massive sulphide
and porphyry copper deposits, which are spatially distributed within irregularly
shaped favorable tracts, satisfy a fractal clustering model with similar fractal