Page 94 - Geochemical Anomaly and Mineral Prospectivity Mapping in GIS

Fractal Analysis of Geochemical Anomalies 93

be depicted by at least two straight lines and thus by different values of α corresponding
to different ranges of v, reflecting groups of similarly-shaped concentration contours.
Consequently, the breaks in slopes of the straight lines fitted through the log-log plot of
the relation can be used to distinguish different ranges of v, which intuitively represent
different populations in the probability density distributions and spatial distributions of a
data set of uni-element concentrations.
Suppose a bifractal a geochemical landscape, meaning there is a threshold value (v t)
separating background and anomalous uni-element concentrations. The background uni-
element concentrations v b and the areas occupied by such uni-element concentrations
[i.e., A(v b”v t)] satisfy the following power-law relation:

A( v ≤ v = C b v b − α b (4.6)
)
b
t

where C b is a constant and α b is an exponent associated with the background component.
The anomalous uni-element concentrations v a and the areas occupied by such uni-
element concentrations [i.e., A(v a>v t)] satisfy the following power-law relation:

A( v > v = C a v a − α a (4.7)
)
a
t

where C a is a constant and α a is an exponent associated with the anomalous component.
Thus, if the soil Fe data are considered to represent a bifractal geochemical landscape
and an inflection point (or threshold) at 8.14% Fe is selected for the purpose of
illustration, two straight lines can be fitted through the concentration-area plots for the
soil Fe values have power-law equations shown in Fig. 4-6. The concentration-area
relation for the “low” Fe values (≤8.14% Fe) has α low=0.6939 and C low= 85536, whilst
the concentration-area relation for the “high” Fe values (>8.14% Fe) has α high=14.574
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and C high= 3×10 .
The interpolated soil Fe data do not portray, however, a bifractal geochemical
landscape because by careful inspection of the concentration-area plots shown in either
Fig. 4-3 or 4-6, three inflection points or thresholds (1.6% Fe, 7.2% Fe, 8.6% Fe) can be
distinguished (Fig. 4-7). Accordingly, four straight lines can be fitted through the
concentration-area plots. Each of the three straight lines that fit the plots to the left of
any threshold satisfies the power-law relation in equation (4.6), whilst the straight line
that fits the plots to the right of the rightmost (or highest) threshold satisfies the power-
law relation in equation (4.7). Each of the lines represents a population in the soil Fe
data, which, from lowest to highest, can be classified as “low-background” (or LB),
“moderate-background” (or MB), “high-background” (or HB) and “anomaly” (or A). The
concentration-area relation for the “low-background” class (≤1.6% Fe) has α LB=0.2391
and C LB= 74877. The concentration-area relation for the “moderate-background” class
(1.6-7.2% Fe) has α MB=0.695 and C MB= 93259. The concentration-area relation for the
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“high-background” class (7.2-8.6% Fe) has α HB=4.6729 and C MB= 2×10 . Finally, the