Page 93 - Geochemical Anomaly and Mineral Prospectivity Mapping in GIS

92 Chapter 4

Multifractality of geochemical landscapes
The two line segments fitted to the log-log plot of the concentration-area relationship
(Fig. 4-3) indicate that there are at least two sets of fractal patterns in the soil Fe data.
This suggests that geochemical landscapes can be multifractals. A multifractal is
considered to be spatially intertwined sets of monofractals (Feder, 1988; Stanley and
Meakin, 1988). Whereas monofractals are characterised by one fractal dimension and
thus apply to binary patterns, multifractals have different fractal dimensions and thus
apply to patterns with continuous spatial variability (Agterberg, 1994, 2001). As the
subset of “high” Fe value can be characterised by a fractal dimension, so can the subset
of “low” Fe values, although this is not to say that either subset is a monofractal because
they both have continuous spatial variability. Thus, the multifractality of geochemical
landscapes can be related to the probability density distributions and spatial distributions
of geochemical data (Cheng and Agterberg, 1996; Gonçalves, 2001; Wei and Pengda,
2002; Panahi and Cheng, 2004; Xie and Bao, 2004; Shen and Cohen, 2005), which are
influenced by various processes that have occurred throughout geological time at various
rates and at various scales (e.g., Rantitsch, 2001). If that is the case, then the
concentration-area relation introduced earlier is appropriately a multifractal model,
which can be used to separate geochemical anomalies from background as proposed
originally by Cheng et al. (1994). The concentration-area fractal method has been
demonstrated by several workers to map significant anomalies using various
geochemical sampling media (e.g., Cheng et al., 1996, 1997, 2000; Cheng, 1999b; Sim
et al., 1999; Gonçalves et al., 2001; Panahi et al., 2004) and is reviewed and further
demonstrated here.

THE CONCENTRATION-AREA METHOD FOR THRESHOLD RECOGNITION

The following discussion of the concentration-area method for separation of
geochemical anomalies from background is adapted from Cheng et al. (1994). For a
series of contours of uni-element concentrations, the concentration contours v and the
areas of uni-element concentrations equal to or greater than v or the areas enclosed by
each contour [i.e., A(•v)] satisfy the following power-law relation if they have
multifractal properties:

A( ≥v) ∝ v − α (4.5)

where ∝ denotes proportionality and the exponent α represents the slope of a straight
line fitted by least squares through a log-log plot of the relation. If, on the one hand, the
concentration-area relation represents a fractal model, then the log-log plot can be fitted
by one straight line and thus by one value of α corresponding to the whole range of v,
representing a group of similarly-shaped concentration contours. If, on the other hand,
the concentration-area relation represents a multifractal model, then the log-log plot can