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Fig. 10.23 Turcotte's
variant of model of de Wijs
as shown in Fig. 10.21
(Source: Agterberg 2007 ,
Fig. 2b)
(1+ d )
(1- d )
(1+ d ) 2
(1- d )
(1- d )
(1+ d )(1- d )
(1+ d ) 4
(1+ d ) 2 (1- d )
(1+ d ) 2 (1- d )
(1- d )
(1- d )
(1+ d ) 3 (1- d )
(1- d )
(1- d )
(1+ d )(1- d )
(1+ d )(1- d )
(1+ d )(1- d )
(1+ d )(1- d )
(1- d )
(1- d )
(1- d )
(1- d )
11
11
11
11
01
01
01
01
40
31
11
11
21
21
01
01
01
01
01
01
01
01
11
11
01 01
01 01
40
31
21
21
the model of de Wijs that have many local maxima. Figure 10.23 is a mosaic of four
patterns resulting from Turcotte “fractal” cascades with d
12; vertical
scale is logarithmic (base 10). Contrary to the multimodal logbinomial patterns, the
Turcotte fractal cascade develops a single peak. However, the same Turcotte
cascade could have been operative in different parts of a study area. If the index
of dispersion ( d ) remained the same for all separate cascades, the combined
frequency distribution after many subdivisions for each cascade would satisfy a
single Pareto distribution plotting as a straight- line with slope determined by d .
Turcotte's cascade model is a modification of the original multifractal-
generating cascade (Sect. 10.4.1 ). Only cells with largest concentration value
during a previous subdivision are further subdivided into parts with different
element concentration values. It is assumed that the same type of cascade was
operational at n different random locations generating a pattern with n maxima. The
frequency distribution of the concentration values then would remain the same
except for enlargement of all frequencies by the factor n . Element concentrations
generated by Turcotte's cascade satisfy a Pareto distribution which is associated
with a fractal instead of a multifractal. The slope
¼
0.4 and k
¼
β
of the straight-line representing
this Pareto distribution on log-log paper satisfies
β ¼
1/log 2 (1 + d ). Figure 10.25
is a C-A diagram for the Turcotte model with d
¼0.4 and k
¼14. Consequently,
β ¼ 2.060.
10.4.3 Computer Simulation Experiments
The following computer simulation experiments (from Agterberg 2007 ) illustrate
that an unbiased estimate of the Pareto parameter can be obtained in the hypothet-
ical situation of a study area where background satisfies the model of de Wijs with
overall average concentration value set equal to 0.1 and d
¼
0.3. Suppose that one or
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