Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 395

processes has been advanced by Qiuming Cheng and colleagues (Cheng 1994 ; Cheng
and Agterberg 1996 ; Cheng 1999 ;Cheng 2005 ). In applications concerned with
turbulence, the original binomial model of deWijs has become known as the binomial
p -model (Schertzer et al. 1997 ). It is noted several advanced cascade models in
meteorology (Schertzer and Lovejoy 1991 ; Veneziano and Langousis 2005 )result
in frequency distributions that resemble the lognormal but have Pareto tails. An
example of a cascade model resulting in a frequency distribution with Pareto tail
will be given in Sect. 12.7 .
The Pareto is characterized by a single parameter that can be related to a fractal
dimension. The lognormal has two parameters ( cf . Sect. 4.4 ) . A basic difference
between the two models is that Pareto frequency density approaches infinity in its
low-value tail whereas lognormal frequency density at zero-value is zero. During
the last 25 years it became increasingly clear that the Pareto often performs better
than the lognormal in modeling the upper-value tails of frequency distributions in
geochemistry and resource analysis. Turcotte ( 1997 , 2002 ) has developed a variant
of the model of de Wijs that results in a Pareto distribution, which is truncated in its
lower tail. In practical applications, frequency density at zero-value always is
observed to be zero. Thus the lognormal generally provides a more realistic
model for modeling low-value tails. This is probably one reason why the lognormal
has been preferred to the Pareto in the past. The other reason was that, in practical
applications, the largest values in the upper tails of frequency distributions become
increasingly rare when value goes to infinity. In goodness-of-fit tests the largest
values generally are combined into a single class so that it is not possible to
distinguish between lognormal and Pareto (Agterberg 1995a ). In Sect. 4.4 a new
method of intercept analysis was applied according to which the Pareto performs
better than the lognormal for large copper deposits in the Abitibi area on the
Canadian Shield.
This section is concerned with combining data from large regions. The problem of
interest is how to explain and model a regional frequency distribution of element
concentration values that resembles the lognormal but displays a power-law tail. Two
ways to solve this problem are (1) to adopt a single-process model with an end
product that is lognormal except in its upper tail that is Pareto, and (2) to consider the
end product to be a mixture of two or more separate processes resulting in lognormal
and Pareto distributions, respectively. Single-process models include the previously
mentioned meteorological models (e.g., the beta-lognormal cascades of Veneziano
and Langousis 2005 ). Already in the 1980s, Schertzer and Lovejoy ( 1985 )had
pointed out that the binomial p- model can be regarded as a “micro-canonical” version
of their -model in which the strict condition of local preservation of mass is replaced

by the more general condition of preservation of mass within larger neighborhoods
(preservation of ensemble averages). Cascades of this type can result in pure lognor-
mals or in lognormals with Pareto tails. The applicability of such single cascade
approaches to geological processes that took place within the Earth's crust remains to
be investigated. With respect to mixtures of two separate cascades, a promising
approach to be discussed at the end of this section consists of superimposing
Turcotte's Pareto-type models on a lognormal background.

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