Geoscience Reference
In-Depth Information
Chapter 11
Multifractals and Local Singularity Analysis
Abstract Multifractals are spatially intertwined fractals. For example, a chemical
concentration value obtained from rock samples in a study area may be a fractal
with fractal dimension different from those of other concentration values for the
same element, but together the fractal dimensions may form a multifractal spectrum
f
(
, which depends on the
concentration value. Self-similar patterns produce multifractals of this type. If a
block of rock with chemical concentration value
X
is divided into equal parts, the
halves have concentration values (1 +
d
)·
X
and (1
α
) that is a continuous function of the singularity
α
d
)·
X
where
d
is a constant.
The model of de Wijs assumes that the dispersion index
d
is independent of block
size. This cascade process produces a multifractal. The properties of a multifractal
can be estimated by the method of moments or by the histogram method. The four-
step method of moments has the advantage that the assumption of multifractality is
being tested during its first step because this should produce an array of straight
lines on a log-log plot of the spatial mass-partition function
ˇ
(
E
,
q
) against measure
of block size (
) used. It should be kept in mind, however, that the multifractal
spectrum estimated by the method of moments is primarily determined by the
majority of measurements that are clustered around the mean or median. Very
large or very small observed values are rare; because of this, the low-singularity and
high-singularity tails of a multifractal spectrum generally cannot be estimated with
sufficient precision. Because of strong local autocorrelations effects, singularity
analysis can provide better estimates of singularities associated with the very large
or very small observed values. Practical examples of multifractal modeling include
the distribution of gold in the Mitchell-Sulphurets area, northwestern British
Columbia, uranium resources in the U.S. and worldwide, lengths of surface frac-
tures in the Lac du Bonnet Batholith, eastern Manitoba, geographic distribution of
gold deposits in the Iskut map area, British Columbia. Local singularity mapping is
useful for the detection of geochemical anomalies characterized by local enrich-
ment even if contour maps for representing average variability are not constructed.
Examples include singularity maps based on various element concentration values
from stream sediment samples and their relation to tin deposits in the Gejiu area,
Yunnan Province, as well as Ag and Pb-Zn deposits in northwestern Zhejiang
E
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