Geoscience Reference
In-Depth Information
Keywords Fractals Fractal dimension Cluster density determination
Concentration-area (C-A) method Cascade models Earth's topography
Mitchell-Sulphurets mineral district Chemical element concentrations Abitibi
lode gold deposits Volcanogenic massive sulphide deposits Iskut River stream
sediment data Model of de Wijs Model of Turcotte
10.1 Fractal Dimension Estimation
Fractals are objects or features characterized by their fractal dimension that is either
greater than or less than the integer Euclidian dimension of the space in which the
fractal is imbedded. The word “fractal” was coined by Mandelbrot (
1975
). On the one
hand, fractals are often closely associated with the random variables studied in
mathematical statistics; on the other hand, they are connected with the concept of
“chaos” that is an outcome of some types of non-linear processes. Evertsz and
Mandelbrot (
1992
) explain that fractals are phenomena measured in terms of their
presence or absence in boxes belonging to arrays superimposed on the domain of
study in 1-D, 2-D, or 3-D space, whereas multifractals apply to “measures”
representing of how much of a feature is present within the boxes used for measure-
ment. Multifractals are either spatially intertwined fractals (
cf
. Stanley and Meakin
1988
) or mixtures of fractals in spatially distinct line segments, areas or volumes that
are combined with one another. During the past 40 years, the fractal geometry of
many natural features in Nature has become widely recognized (see, e.g., Mandelbrot
1983
;Barnsley
1988
;Cheng
1994
; Turcotte
1997
;Raines
2008
; Carranza
2008
;Ford
and Blenkinsop
2009
;Agterberg
2012
;Cheng
2012
). Fractals in geology either
represent the end products of numerous, more or less independent processes (e.g.,
coastlines and topography), or they result from nonlinear processes, many of which
took place long ago within the Earth's crust. Although a great variety of fractals can
be generated by relatively simple algorithms, theory needed to explain fractals of the
second kind generally is not so simple, because previously neglected nonlinear terms
have to be inserted into existing linear, deterministic equations.
Both fractals and multifractals are commonly associated with local self-
similarity or scale-independence, which generally results in power-law relations
that can be represented as straight lines on log-log paper. Frequency distribution
models closely associated with fractals and multifractals include the Pareto, log-
normal and various extreme-value distributions. Computer simulation experiments
can be used to generate artificial fractals. Multiplicative cascade models are a useful
tool for generating artificial multifractals in computer simulation experiments.
Multifractals often result from non-linear process modeling. Lovejoy et al. (
2010
)
have pointed out that non-linear process modeling has made great strides forward in
the geosciences, especially in geophysics, but is not yet as widely accepted as it
should be. These authors briefly describe a number of recent “success stories” in
which non-linear process modeling led to results that could not have been obtained
otherwise. Multifractals will be discussed in more detail in Chap.
11
.
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