Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 411

ordered values along the logarithmic scale is equal to ln ʷ . It can be rewritten in the

form ( cf . Matheron 1962 , p. 309):

ln V
v

2

˃ ¼ α

where v represents the volume for which the concentration value is determined; V is
a larger volume in which v occupies a random position. In one-dimensional
applications, the volumes v and V are reduced to line segments, and V / v 2 n . The ¼

constant α satisfies:

1
4ln2 ln

2

α ¼ ½ ʷ :

According to the De Moivre-Laplace theorem (Bickel and Doksum 2001 ,
p. 470), the frequency distribution of ln converges to normal form when n

increases. Frequency density values in the upper tail of the logbinomial are less
than those of the lognormal. The logbinomial would become lognormal when n
representing the number of subdivisions of blocks is increased indefinitely.
Paradoxically, its variance then also would become infinitely large. In practical
applications, it is often seen that the upper tail of a frequency density function of
element concentration values is not thinner but thicker and extending further than a
lognormal tail. Several cascade models (see, e.g., Schertzer and Lovejoy 1991 ;
Veneziano and Furcolo 2003 ; Veneziano and Langousis 2005 ) result in frequency
distributions that resemble the lognormal but have Pareto tails.

ʷ

Box 11.1: Self-Similarity and Power Laws

Korvin ( 1992 ) offers a heuristic exposition of the idea that self-similarity
necessarily leads to power-law type relations. A more rigorous approach can
be found in Acz´l and Dhombres ( 1989 ). Arguing along the same lines as
Korvin, suppose that ʼ 1 , ʼ 2 , and ʼ 3 are the measures of a fractal set with
“singularity”

E 3 ,
respectively. Self-similarity would imply that the ratio of the measures for
two cells depends on the ratio of their sizes only, or: ʼ 1

α in three cells with different sizes labeled

E 1 , E 2 and

; ʼ 2 ;

E 1
E 2

E 2
E 3

ʼ 2 ¼

f

ʼ 3 ¼

f

.Hence: f

¼

. The function f is such that it

ʼ 1
ʼ 3 ¼

E 1
E 3

E 1
E 3

E 1
E 2

E 2
E 3

f f f

satisfies a relation of the type f ( ab ) f ( a )· f ( b )where a and b are constants.

Almost 200 years ago, the French mathematician A.L. Cauchy had shown that
this implies that f must be power-law type with f ( x )

¼

x p where p is a constant. ¼

Other types of functions cannot be of the type f ( ab ) ¼ f ( a )· f ( b ). It follows that

E α where c and the singularity for the measure on the fractal set: ʼ α ¼ c α are

constants.