Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 400

Fig. 10.24 Four 2-D
realizations of Turcotte's
fractal model (Fig. 10.23 )
for d ¼ 0.4 and N ¼ 12.
Total area was subdivided
into four quadrants. Vertical
scale is for 5 + log 10 (Value)
(Source: Agterberg 2007 ,
Fig. 4)

5.4

5.2

100

5

4.8

50

50

100

more Turcotte cascades with overall average value equal 0.4

( cf . Fig. 10.24 ) are superimposed on this background. Figure 10.26 is a C-A
diagram for this hypothetical situation. The pattern in this diagram is approximately
linear with slope of approximately

to 1.0 and d ¼

2. It was obtained by random sampling of
concentration values resulting from de Wijs and Turcotte cascades combining the
resulting values with one another.
Suppose now that the Turcotte cascades were operational in only 25 % of the area.
Use of Cheng's ( 2003 ) method of piecemeal fitting of successive straight-line
segments in a C-A diagram indicates that the largest concentration values approxi-
mately fall on a straight line with slope equal to

2asinFigs. 10.25 and 10.26 .
Consequently, the estimated value of d is 0.4. These experiments (Figs. 10.27 and
10.28 ) illustrate that unbiased estimates of this type can be obtained irrespective of how
many Turcotte cascades were operative in the area or how much of the study area
consists of approximately lognormal background without anomalies. Figure 10.28 is a
plot of the frequency density values for the second experiment. The smaller peak on the
right corresponds to the line-segment for largest concentration values in Fig. 10.27 .
A practical example of dispersion index ( d ) estimation for Turcotte's model is as
follows. Suppose a measure ʼ of the amount of a chemical element in a square cell

E ʱ where c is a constant, and

measuring is the

singularity also known as H¨ lder exponent (Mandelbrot 1989 ; Evertsz and Man-
delbrot 1992 ); then

km on a side satisfies ʼ ¼ c · ʱ

E

ʱ can be estimated by measuring the slope of the straight line in

E ʱ2
represents average element concentration value in the cell. If element concentration
values for samples taken at the surface of a study area are realizations of a stationary
random variable with constant population mean,

2 where

a log-log plot of ʼ against . In this 2-D application, ʼ ¼ ʾ · ʾ ¼ c ·

∈ E

then ʱ ¼ 2 represents

non-singularity. “Singular” locations (where ʱ< 2) may indicate anomalous