Page 87 - Geochemical Anomaly and Mineral Prospectivity Mapping in GIS

86 Chapter 4

GEOCHEMICAL LANDSCAPES AS FRACTALS

What are fractals?

Mandelbrot (1982, 1983) introduced the term fractal to describe an object or a pattern
consisting of parts (i.e., fractions) that have geometries (e.g., shape or form), except
scale or size, that are more or less similar to the whole object or pattern. Thus, fractals
have the property of being self-similar or self-affine at various scales, meaning they are
scale-invariant or scale-independent entities. As a consequence of this property, it is not
possible to determine the scale of a fractal based on its shape or form alone. A fractal is
strictly self-similar if it can be expressed as a union of objects or patterns, each of which
is a reduced copy of (i.e., geometrically similar to) the full object or pattern. The most
fractal-looking natural objects are not, however, precisely self-similar but are self-affine.
On the one hand, a statistically self-similar fractal is isotropic (Turcotte, 1997), meaning
that patterns with different orientations appear to have similar orientations at the same
scale. On the other hand, a statistically self-affine fractal is anisotropic, meaning that
patterns with different orientations appear to have similar orientations at different scales.
The property of either (statistical) self-similarity or (statistical) self-affinity is an
attribute that can be used to characterise seemingly-disordered natural objects or
phenomena. That is to say, natural systems or patterns resulting from stochastic
processes at various scales are plausibly fractals. For, example, Bölviken et al. (1992)
suggested that geochemical distribution patterns (or geochemical landscapes) plausibly
consist of fractals (background and anomalous patterns), because such patterns were
formed by processes that have occurred throughout geological time at various rates and
at various scales. They tested their hypothesis by applying various methods for
measuring fractal geometry.

Fractal geometry
As originally defined by Mandelbrot (1982, 1983), a fractal has a dimension D f,
known as the Hausdorff-Besicovitch dimension, which exceeds its topological (or
Euclidean) dimension D. A fractal linear feature does not have D=1 as expected from
Euclidean geometry, but has a D f between 1 (D for a line) and 2 (D for an area). An in-
depth review of methods to measure fractal dimension of linear features is given by
Klinkenberg (1994). Similarly, fractal areas or surfaces (e.g., geochemical landscapes)
have values of D f between 2 and 3 (D for volumes), fractal volumes have values of D f
greater than 3, and so on. An in-depth review of methods to measure fractal dimension
of surfaces or landscapes is given by Xu et al. (1993). Further authoritative explanations
of methods for measuring fractal dimensions of geological objects can be found in Carr
(1995, 1997).
To test the fractal dimensions of geochemical landscapes, Bölviken et al. (1992) used
four of the available methods for measuring fractal dimensions of areas or surfaces: (1)
variography; (b) length of contour versus measuring yardstick; (3) the number-area