Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 394

Fig. 10.19 Lognormal Q - Q
plot of gold values (in ppb)
of stream sediment samples
at locations shown in
Fig. 10.18 . Secondary peak
at zero in (a) is for gold
values below detection limit
(Source: Cheng et al. 1996 ,
Fig. 9b)

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Standard Normal Quantile

Fig. 10.20 Concentration-
Area (C-A) model for
Au. Areas were measured
on contour map of gold
values generated by Cheng
et al. ( 1996 ). Notice that
values fall on two distinct
straight-line segments, and
are interpreted as
background and anomaly
populations (Source: Cheng
et al. 1996 , Fig. 11)

10.4 Cascade Models

In a paper concerned with multifractal measures for the geoscientist, assuming that
ore tonnage is equivalent to volume, Mandelbrot ( 1989 ) uses the model of de Wijs
( 1951 ) as a starting point for spatial distribution of metals in the Earth and
concludes that the zinc ore “curdled” into a multifractal. The model of de Wijs
will be taken as a starting point in this section. In general, it leads to lognormal
distribution of metal concentrations without Pareto tails and modifications of the
approach will be needed to explain separate straight-line segments commonly
resulting from applications of the C-A method.
The model of de Wijs is a simple example of a binomial multiplicative cascade
( cf .Mandelbrot 1989 ). The theory of cascades has been developed extensively
over the past 25 years by Lovejoy and Schertzer ( 1991 ) and Schertzer et al. ( 1997 )
and other geophysicists, particularly in connection with cloud formation and rainfall
(Schertzer and Lovejoy 1987 ; Over and Gupta 1996 ; Veneziano and Furcolo 2003 ;
Veneziano and Langousis 2005 ). More recently, multifractal modeling of solid-Earth