Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 480

produces new methods for estimating the parameters of this model. In practical
applications, the frequency distribution of element concentration values for small
rock samples is related to self-similar spatial variability patterns of the element in large
regions or segments of the Earth's crust. The approach will be illustrated by applica-
tion to spatial variability of gold and arsenic in glacial till samples from southern
Saskatchewan. It will be shown that for these two elements the model of de Wijs is
satisfied on a regional scale but degree of dispersion decreases rapidly toward the
local, sample-size scale. Thus the apparent number of subdivisions ( N ) is considerably
less than would be expected if degree of dispersion were to extend from regional to
local scale as generally assumed in the past ( cf .Agterberg 2007a ).

12.4.1 Effective Number of Iterations

Multifractal modeling offers an independent method to verify validity of the model
of de Wijs and to estimate d . In principle, the value of n could be made infinitely
large. However, the logarithmic variance (

2 in the variance equation of de Wijs)
then also becomes infinitely large and the frequency distribution of the element
concentration values would cease to exist. Application of the method of moments in
multifractal analysis results in a multifractal spectrum that is a limiting form for
infinitely large n . The frequency distribution corresponding to this limiting form
cannot exist in reality because it has infinitely large variance. In practice, any set of
element concentration values for very small blocks of rock collected from a very
large environment has a frequency distribution with finite logarithmic variance.
Suppose that the generating process of subdividing blocks under the same
dispersion index ( d ) ceases to be operative for blocks that are larger than the very
small blocks used for chemical analysis. In examples of application to be discussed
in this section, d at the regional scale does not apply at the local scale (for small
blocks used for chemical analysis). Locally, d is either zero or much smaller than
d at the regional scale. Under these conditions, an apparent maximum number of
subdivisions N can be estimated. Self-similarity at scales exceeding a critical lower
limit results in a model of de Wijs with three parameters:

σ

, d , and N .
The multifractal method used for estimating d is similar to the method for
separation of geochemical anomalies from background introduced by Cheng
( 1994 ) and Cheng et al. ( 1994 ). As previously discussed in Agterberg ( 2001a , b ,
2007a ), validity of the model of deWijs for the larger concentration values results in:

ʾ

2

d ˄ ðÞ

dq ¼ ʱ min ¼

q

log 2 ʷ þ

1

lim
q!1

ʷ

In 2-D applications, chemical element measures are formed by multiplying average
cell concentration values by cell areas. Raising these measures to relatively high
powers q filters out the influence of smaller concentration values. Thus our estimate
of d is based on parameters estimated from the relatively large concentration values
of an element. The lower-value tails of the observed frequency distributions could