Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 385

a b

2

2.5

2

1.5

1.5

1

1

0.5

0.5

0

0

− 0.5 − 0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Log r Log r

Fig. 10.12 Log-Log plots of K ( r ) function versus distance r (in km) to estimate cluster fractal
dimension D c (estimated from slopes of best-fitting straight lines). (a) Point pattern for gold
deposits in Fig. 10.11 ;(b) southern half of point pattern shown in Fig. 10.11 . Solid circles show
results for original (anisotropic) point patterns; open circles show results for transformed (approx-
imately isotropic) point pattern. Estimates of D c for solid circles are (a) 1.493 and (b) 1.498;
estimates of D c for open circles are (a) 1.506 and (b) 1.524. Solid circles in Fig. 10.12b are based
on fewer points because edge correction formula used requires r 2 where h is height of map

area (measured in north-south direction) in km. (Source: Agterberg 2013 , Fig. 3)

< h

multiplied by 2.5 to provide a more isotropic pattern. Results of application of the
previous methods are shown as open circles in Fig. 10.12 . The resulting estimates
are D c ¼

1.506 (Fig. 10.12b ). Like the two other
estimates these two estimates are approximately equal to 1.5 illustrating that, like
lacunarity, anisotropy does not significantly affect the estimation of D c .
Results of the box-counting fractal method applied to the point pattern of
Fig. 10.11 were tabulated in Agterberg et al. ( 1993 ) and here are graphically
shown in Fig. 10.12 . As already mentioned in Sect. 10.1.3 , the box-counting fractal
method is less accurate and less precise than cluster density estimation for point
patterns. Because there are only 295 points in Fig. 10.11 , D b approaches 0 for
decreasing box size ( E ! 0). For any very small value of E , the number of boxes
containing points becomes N Є ¼Log 10 295¼2.47. In Fig. 10.13 the “roll-off”
pattern of measurements asymptotically approaches this value that is almost
reached for boxes measuring 1 km on a side, because very few of these contain
more than a single point. This source of bias can be seen on Fig. 10.13 for values
less than N Є . Larger boxes are not subject to this type of bias but are relatively
imprecise because their frequency approaches 0 for increasing box size. The lack of
precision is shown on Fig. 10.13 as increased scatter of measurements along the
straight line pattern (broken line in Fig. 10.13 ). This line with slope equal to

1.498 (Fig. 10.12a ) and D c ¼

1.53

was fitted to the values with 10 Log

E

0.6 only. It would result in the estimate

D b ¼ 1.5 obtained in the

previous section. Although other interpretations remain possible ( cf . Raines 2008 ,
p. 289), the most reasonable conclusion is that there is no significant difference
between D b and D c

1.53, which is comparable with the estimates of D c

1.5 for the spatial distribution of gold deposits in the Kirkland

Lake ar ea.