Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 388

Table 10.1 Comparison
of fractal cluster model
(power-law) variance with
sample variances s 2 ( x ) for
different (square) cell sizes
(Source: Agterberg 1993 ,
Table 1)

S 2 (x) Cell size (km) Power-law estimate

5 1.48 1.450 (0.000 54)

10 13.42 13.257 (0.006 20)

20 128.42 130.903 (0.070 87)

40 111 680 1,296.609 (0.809 56)

Study area 508,777.7 (920.82)

relatively small cells, the variance can also be estimated directly from the observed
data (Table 10.1 ). For large areas, this variance cannot be estimated directly but the
power-law model can be used for this purpose. Table 10.1 (from Agterberg 1993 )
shows variances computed from observed data in comparison with variances based
on the fractal power-law model. Standard deviations of the power-law variances are
also shown. The model can be used for extrapolation to areas of any size including
very large areas. Modeling of uncertainties of this type is useful in regional mineral
resource potential studies.

10.2.3 Worldwide Permissive Tract Examples

Singer and Menzie ( 2010 ) have developed a three-part method for regional quan-
titative mineral resource assessments. The three parts are (1) delineation of per-
missive tracts for selected types of mineral deposits; (2) grade-and-tonnage models
for these deposits; and (3) estimating the number of deposits for each type. They
have compiled a table of worldwide mineral deposit density control areas for
mainly (1) podiform chromite; (2) volcanogenic (Cyprus +Kuroko) massive sul-
phide and (3) porphyry copper deposits (Singer and Menzie 2010 , Table 4.1). For
deposit type per permissive area, these authors listed (a) area (km 2 ), (b) number of
deposits, and (c) median and total tons of metal. Figure 10.15 (based on Singer and
Menzie 2010 , Table 4.1) shows log-log plots for the relations between deposit
density and permissive area for the three selected deposit types. In a separate study,
Singer and Menzie ( 2008 ) have studied the problem of map scale effects on
estimating number of undiscovered deposits within permissive tracts.
If the deposits would be randomly distributed across each permissive tract;
e.g., according to a Poisson model, one would expect that the three best-fitting
straight lines for point density in Fig. 10.15 would be approximately horizontal
because number of deposits per tract then would be proportional to tract area for
each deposit type. Instead of this, the three best-fitting lines in the log density versus
log permissive plots have negative slopes equal to 0.61 for

chromite, volcanogenic massive sulphide and porphyry copper, respectively. From
the relatively large R 2 values of these least squares fits it can be concluded that these
three slopes are significantly greater than zero.
Although the permissive tract approach differs from cluster density estimation, it
also can be used for fractal cluster estimation. From the slopes of the lines of best fit

0.53, 0.62 and