Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 396

One of the basic assumptions in geochemical abundance models (Brinck 1974 ;
Harris 1984 ; Garrett 1986 ) is that trace elements are lognormally distributed.
Originally, Ahrens ( 1953 ) postulated lognormality as the first law of geochemistry.
In general, it cannot be assumed that the element concentration values for very
small blocks of rock collected from a very large environment satisfy a single
lognormal frequency distribution model. However, the lognormal model often
provides a valid first approximation especially for trace elements. Reasons why
the lognormal model may not be applicable include the following: Concentration
values for all constituents form a closed number system and this prevents major
constituents from being lognormally distributed. Also, discrete boundaries (con-
tacts) between different rock types commonly exist in regional applications and
mixtures of two or more lognormals would occur if rock types have lognormals
with different parameters. Three types of generating mechanisms or explanations
have been suggested to explain lognormality. The first two were previously
discussed in Chap. 3 . Aitchison and Brown ( 1957 ) already had reviewed processes
in which random increases of value are proportional to value do result in lognormal
distributions, in the same way that processes subject to conditions underlying the
central limit theorem of mathematical statistics lead to normal distributions. The
second type of explanation was advocated by Vistelius ( 1960 ): mixtures of
populations with mean values that are proportional to standard deviations tend to
result in positively skewed distributions that resemble lognormal distributions even
if the original populations are normal. Thirdly, multiplicative cascade models such
as the model of de Wijs can help to explain lognormality ( cf . Agterberg 2001 , 2007 ;
Sect. 12.5.2 ).
All ` gre and Lewin ( 1995 ) provided an overview of geochemical distributions
that are either lognormal or Pareto. A relatively simple generalization of the
lognormal model is to assume that the concentration values for a chemical element
in a large region or 3-D environment originate from two different populations
representing background and anomalies, respectively. The largest concentration
values then primarily represent anomalies. This type of modeling either uses
lognormal Q - Q plots (Sinclair 1991 ), or use is made of concentration-area (C-A)
log-log plots (Sect. 10.3.1 ) to distinguish between two or more separate
populations. Often it can be assumed (Agterberg 2007 ) that (a) the relatively
small concentration values (background) represent a mixture of different
populations, and (b) the largest values satisfy a Pareto distribution with a tail that
is thicker than lognormal.

10.4.1 The Model of de Wijs

The simplest multiplicative cascade model in 1-D, 2-D or 3-D is the model of
de Wijs ( 1951 ) . This model is graphically illustrated in Figs. 10.21 and 10.22 .
In the original model of de Wijs, any block of rock is divided into two equal parts
( cf . Sect. 6.2 ). The concentration value (

ʾ ) of a chemical element in the block then