Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 376

of similarly shaped sets, there exist power-law relationships between any two
measures of volume or area and perimeters. For example, two areas A i and A j
enclosed by contours i and j are related to their perimeters L i and L j as follows:

D AL = 2

L j ðÞ A j ðÞ

ðÞ ¼

L i A i ðÞ

ʴ

where is the common yardstick used for measuring both areas and perimeters. The

fractal dimension D AL satisfies D AL ¼ 2 D L / D A , where D L and D A denote fractal

dimensions of perimeter and area, respectively. The contours for Au and Cu in
Fig. 10.5 show similar shapes suggesting that the current perimeter-area modeling
approach may be valid. Experimental results for Au, Cu and As are shown on
log-log plots in Fig. 10.7a, c, and e using perimeters and areas for different contours
with concentration values above the thresholds previously defined using the ele-
ment concentration-area method (Fig. 10.6 ). Figure 10.7b, d and f show the
relationships between estimated lengths of perimeters of contours of concentration
values for yardsticks of different lengths. There appear to be two different fractal
dimensions in these diagrams on the right side of Fig. 10.7 . The first of these ( D 1 )
for yardsticks less than 300 m is equivalent to the so-called “textural” fractal
dimension (Kaye 1989 , p. 27). It is close to 1 for the elements considered and can
be explained as the result of smoothing during interpolation. The other estimate of
D L ( D 2 ) ranges from 1.14 to 1.33 and is probably representative of the true
geometry of anomalous areas for the elements considered. It is noted that the
least squares estimates of D 1 and D 2 in Fig. 10.7 are subject to considerable
uncertainty because they are based on relatively few points.

Box 10.1: Fractal Perimeter-Area Relation

The following perimeter-area relationship was introduced by Mandelbrot
( 1983 ) for similarly shaped sets: L

p where C is a con-
stant. A modification of this equation was introduced by Cheng ( 1994 )as
follows ( cf . Cheng et al. 1994 ). For yardstick

A

ð 1 D Þ

ðÞ ¼ C ʴ ʴ

, the estimated length and area

δ

D ) .
Similarly shaped patterns at different scales can be derived from one
another by changing the scale. Suppose that r i represent the ratio to
create the i -th geometrical pattern from the k -th one. Estimates of L and A
for these two geometries then can be obtained from: L k

(1 D ) ; A ( (2

of a pattern in 2-D can be expressed as: L ( ʴ ) ¼ L 0 ʴ ʴ ) ¼ A 0 ʴ

ð 1 D L Þ ;

ðÞ ¼ L 0 ʴ

L 0 r i ðÞ

D L

A 0 r i ðÞ

D A

ð 2 D A Þ 2 . Consequently,

L i ðÞ ¼ ʴ ; A k ðÞ ¼ A 0 ʴ A i ðÞ ¼ ʴ

;

D AL = 2 D AL = 2

L i ðÞ
L k

A i ðÞ
A k

A i ðÞ
ʴ

1 D AL A i D AL = 2 , and DAL

ðÞ ¼

L i ðÞ ¼ ʴ ¼ ʴ ½ ðÞ ¼

;

2

ðÞ

2 DL / DA .