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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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