Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 389

a b

0.5
0.0

0.5

0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0

y=0.5304x+0.2157
R 2 =0.3028

-0.5
-1.0
-1.5
-2.0
-2.5
-3.0

y=-0.087 7x-1.673
R 2 =0.0061
y=-0.6155x-0.5844
R 2 =0.6421

y=-0.3537x+0.475
R 2 =0.0506

1.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

10 Log (area) 10 Log (area)

c

-2.0
-2.5
-3.0
-3.5
-4.0
-4.5
-5.0
-5.5

y=-0.5655x-1.0487
R 2 =0.6215

y=-0.6949x-0.6734
R 2 =0.3518

3.0 3.5 4.0 4.5 5.0 5.5 6.0

10

Log (area)

Fig. 10.15 Log-Log plots of original density versus permissive tract area for three types of
deposits (original data from Singer and Menzie 2010 , Table 4.1). (a) Podiform Cr deposits;
(b) volcanogenic massive sulphide deposits; (c) porphyry copper deposits (Source: Agterberg
2013 , Fig. 6)

it follows that all three deposit types satisfy a fractal cluster model with D c equal to
1.47, 1.38 and 1.39, respectively. These results are not only remarkably similar to
one another but also fairly close to those obtained previously for gold deposits in the
Abitibi Volcanic Belt on the Canadian Shield with D c

1.5. Singer and Menzie
( 2010 ) had already established empirically that deposit density for the three types of
deposits significantly decreases with size of permissive tract but they did not offer a
fractal explanation. A factor probably contributing to the fact that deposit density
clearly decreases linearly with tract size in this type of application is that the
boundaries of the permissive areas have curved shapes determined by local geology.
Roll-off effects should be negligibly small because the tracts are fairly large.
Singer and Menzie ( 2010 , pp. 60-64) also have considered the relation between
total tonnage of deposits in their Table 4.1 with area of permissive tract and deposit
density simultaneously. In the Fig. 10.15 a second set of points is shown for deposits
of each type weighted according to their total metal tonnage. For podiform chromite
and volcanogenic sulphide deposits, the slopes of the lines fitted to the second set of
points are not statistically significant but for the porphyry coppers there is an
indication that size-weighted density decreases with tract area. Figure 10.16 shows
that for each of the three deposit types (podiform chromite, volcanogenic massive
sulphide and porphyry copper) the largest deposits approximately satisfy a Pareto size
frequency distribution. Because the three data sets are not large, Quandt's ( 1966 )
method was used for parameter estimation, because it yields unbiased (consistent)