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emphasized in Chap.
11
that this approach is close to the original Matheron (
1962
)
approach to ore reserve calculations.
The applications of Chap.
11
make use of multifractal spectra that are symmet-
rical. It was assumed that the central part of these spectra with shapes estimated by
the method of moments (Sect.
11.2.1
) is according to the model of de Wijs
(binomial
p
/model) or one of its generalizations, although commonly the minimum
and maximum singularities, with fractal dimensions close to zero, are outside the
range predicted by the model of de Wijs. They represent extreme events in the tails
of the frequency distributions of the phenomena that are being modeled. In practice,
multifractal spectra can be asymmetrical, display negative fractal dimensions in the
tails or suggest the existence of infinitely large or small singularities (
cf
. Mandel-
brot
1999
). In Agterberg (
2001a
) it was shown by means of computer simulations
experiments that 2-D element concentration patterns generated by the model of de
Wijs with superimposed regional trends can result in asymmetrical multifractal
spectra and negative fractal dimensions that are artefacts.
In this section, white noise will be added to a 1-D pattern generated by the model
of de Wijs that seems to show some broad systematic variations resembling regional
trends. It will be shown that the white noise component can be extracted from the
data by means of the Chen algorithm but after a limited number of iterations.
12.6.1 Computer Simulation Experiment
Figure
12.27
shows an artificial series of 250 values generated by using a
one-dimensional version of the model of de Wijs with dispersion index
d
0.4
(
cf
. Agterberg
2012b
). The pattern of Fig.
12.27
was generated in the same way as
that of Fig.
12.17
using different random numbers. These hypothetical element
concentration values were logarithmically (base 10) transformed and a Gaussian
white noise component with zero mean and standard deviation equal to 0.25 was
added. The result is shown in Fig.
12.28
. The antilogs (base 10) of the values shown
in Fig.
12.28
were subjected to local singularity analysis. Patterns of the smoothing
coefficients
c
k
are shown in Fig.
12.29
for
k
¼
1, 5, 10, 120, and 1,000, respectively.
Obviously, degree of smoothing increases when
k
is increased. The purpose of the
iterative algorithm is to optimize local singularity
¼
ʱ
k
rather than
c
k
which in the
limit would become a straight-line pattern with values close to average element
concentration value (
cf
. Agterberg
2012a
,
b
). The result for
k
1 represents ordi-
nary local singularity analysis. The other patterns are for larger values of
k
obtained
by means of the Chen algorithm. Figures
12.30
,
12.31
and
12.32
show logarithmi-
cally transformed input values plotted against singularities for
k
¼
1, 120, and
1,000, respectively. In these three diagrams, the best-fitting straight line obtained
by least squares is also shown. It can be assumed that the logarithmically
transformed input value is the sum of a “true” value that in this application satis-
fies the model of de Wijs for
d
¼
0.4, plus a random error with zero mean and
standard deviation of 0.25 corresponding to the Gaussian white noise component.
¼
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