Geoscience Reference
In-Depth Information
4 and 16 cells with concentration values as shown in Fig.
10.21
. The maximum
element concentration value after
k
subdivisions is (1 +
d
)
k
and the minimum value
is (1
d
)
k
;
k
is kept even in 2-D applications in order to preserve mass but the
frequency distribution of all concentration cannot be distinguished from that arising
in 1-D or 3-D applications of this multiplicative cascade model.
In a random cascade, larger and smaller values are assigned to cells using a
discrete random variable. Multifractal patterns generated by a random cascade have
more than a single maximum. The frequency distribution of the element concen-
trations at any stage of this process is called “logbinomial” because logarithmically
transformed concentration values satisfy a binomial distribution. The logbinomial
converges to a lognormal distribution although its upper and lower value tails do
remain weaker than those of the lognormal (Agterberg
2007
). Notation can be
simplified by using indices that are powers of (1 +
d
) and (1
d
), respectively; for
the example of Fig.
10.21
,(1+
d
)
3
(1
d
) is written as 31 in the 16-cell matrix on
the left in the next row. If at each stage of subdivision, the location of higher and
lower concentration cells is determined by a Bernoulli-type random variable, the
arrangement of cells may become as shown in the 16-cell matrix on the right.
Because of its property of self-similarity, the model of de Wijs was recognized to be
a multifractal by Mandelbrot (
1983
,
1989
) who adopted this approach for applica-
tions to the Earth's crust.
Figure
10.22
shows a 2-D logbinomial pattern for
d
14. Increasing
the number of subdivisions for the model of de Wijs (as in Fig.
10.21
) to 14 resulted
in the 128
¼
0.4 and
k
¼
128 pattern shown in Fig.
10.22
in which values greater than 4 were
truncated to more clearly display most of the spatial variability. The frequency
distribution of all 214 values is logbinomial and approximately lognormal except in
the highest-value and lowest-value tails that are thinner than lognormal (also see
Sect.
12.4
). When the number of subdivisions becomes large, the end product cannot
be distinguished from that of multiplicative cascade models in which the dispersion
index
D
is modeled as a continuous random variable with mathematical expectation
equal to 1 instead of the Bernoulli variable allowing the values +
d
and -
d
only
(Sect.
12.5
). The lognormal model often provides good first approximations for
regional background distributions of trace elements.
10.4.2 The Model of Turcotte
Figure
10.23
shows Turcotte's variant of the model of de Wijs: After each subdi-
vision, only the half with larger concentration in further subdivided into halves with
concentration values equal to (1 +
d
)
. This simplifies the process
as illustrated for 16 cells in 2-D space. At each stage of this process the concentra-
tion values have a Pareto-type frequency distribution. In analogy with Turcotte's
(
1997
) derivation for blocks in 3-D space, it can be shown that a fractal dimension
equal to
D
ʾ
and (1
d
)
ʾ
log
2
(1 +
d
) can be defined for this process. The final element
concentration map has only one maximum value contrary to patterns generated by
¼
2
Search WWH ::
Custom Search