Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 384

Fig. 10.11 Example of
point pattern for 295 gold
deposits in the vicinity of
Timmins and Kirkland
Lake, Abitibi volcanic belt
on Canadian Shield. Circles
are gold mines (mainly past
producers); triangles
represent small, unmined
deposits (Source: Agterberg
2013 , Fig. 2)

80°30' 79°30'W

further events within distance r from an arbitrary event point. The relation between
ʻ 2 ( r ) and K ( r )is denotes proportionality and K 0 ( r ) is the first

derivative of K ( r ) with respect to r . Application of this statistical point process
technique to fractal cluster theory yields

K 0 ( r )/ r where ʻ 2 ( r ) / /

r Dc 2 and K ( r ) r Dc where ʻ 2 ( r ) / / /

again denotes proportionality.
For rectangular areas such as that of Fig. 10.11 ,the K ( r ) function can be estimated
readily by the method originally developed by Ripley ( 1976 ) according to which edge
effects due to the boundaries of the study area are avoided. Diggle ( 1983 , p. 72) has
published explicit formulae for unbiased estimation of K ( r ) for event points in a
rectangular study area. Application of this method to the pattern of Fig. 10.11 resulted
in the pattern of solid circles in Fig. 10.12a . The slope of the straight line fitted
to these points gives the estimate D c ¼1.493 mentioned before.
Results of two experiments to study the effects of lacunarity and anisotropy are
also shown in Fig. 10.12 . The largest holes in the point pattern of Fig. 10.11 occur in
the northern half of this study area. For this reason, a new study area consisting only
of the southern half of Fig. 10.11 was defined. It contains most of the gold deposits
in Fig. 10.11 and has three holes (lacunae) that are smaller than the two largest holes
in Fig. 10.11 . Application of the previous method to the point pattern for the
southern half of Fig. 10.11 resulted in the pattern of solid circles in Fig. 10.12b
with D c ¼ 1.493. The fact that these two estimates are nearly

equal to one another illustrates that lacunarity does not significantly affect the
estimation of D c .
The effect of anisotropy was studied as follows. A so-called affine transforma-
tion was applied to the pattern of Fig. 10.11 by changing the north-south coordi-
nates of all event points. The north-south coordinate was measured by taking
distance from the southern boundary of the study area and these distances were

1.467 instead of D c ¼