Geoscience Reference
In-Depth Information
The Brinck approach to modeling metal occurrence in the upper part of the
Earth's crust has not been adopted widely. Economic geologists know that
orebodies for uranium and other metals are of many different types with character-
istic features that differ from type to type. Also, the genetic processes resulting in
different types of orebodies were very different. These facts and concepts do not
seem to fit in with the simplicity of Brinck's approach. Nevertheless, comparison of
sizes and grades of ore deposits for uranium and other metals (
cf
. Brinck
1971
)fitin
remarkably well with diagrams such as Fig.
11.3
. Approximate multifractal distri-
bution of some metals in the Earth's crust or within smaller blocks provides an
interesting alternative approach for regional mineral resource evaluation studies.
This different kind of approach is equivalent to what was discussed in the previous
chapter: the Earth's topography can be modeled as a fractal although, obviously, the
genetic processes that have caused differences in elevation on Earth were very
different. It is also equivalent to the remarkable fact that fractal modeling can be
used to model the geographical distribution of mineral deposits within worldwide
permissive tracts (Sect.
10.2.3
).
11.2 The Multifractal Spectrum
Multifractals arose in physics and chemistry as a generalization of (mono-)fractals
(Meneveau and Sreenivasan
1987
; Feder
1988
; Lovejoy and Schertzer
1991
).
Multifractals can be regarded as spatially intertwined monofractals (Stanley and
Meakin
1988
). Mandelbrot (see, e.g., Evertsz and Mandelbrot
1992
) has empha-
sized that multifractals apply to continuous spatial variability patterns, whereas
monofractals are for binary Yes-No type patterns. The relation between mono-
fractals and multifractals also was considered by Herzfeld et al. (
1995
) who showed
that the ocean floor could not be modeled as a monofractal. Better results were
obtained by a multifractal model after incorporation of a non-stationary component
(Herzfeld and Overbeck
1999
). The multifractal spectrum is the tool
par excellence
in the study of multifractals. In this spectrum a monofractal plots as a single spike
because it has only one singularity associated with it. Examples of multifractal
spectra for geoscience data include Gon¸alves et al. (
2001
), Pahani and Cheng
(
2004
) and Arias et al. (
2011
,
2012
).
A 1-D example of a multifractal is as follows. Suppose that
ʼ
¼
x
·
represents
E
total amount of a metal for a line segment of length
and
x
is the metal's
concentration value. In the multifractal model it is assumed that (1)
E
ʼ /
E
α
where
/
is the singularity exponent corresponding to
concentration value
x
; and (2)
N
α
(
denotes proportionality, and
α
/
E
f
(
α
)
)
represents total number of line
E
segments of length
with concentration value
x
, and
f
(
α
) is the fractal dimension
E
of these line segments.
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