Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 424

10

9

8

7

6

5

4

3

2

1

0

0 10 20 30 40 50 60

Distance

Fig. 11.10 Theoretical form of multifractal semivariogram in comparison with three experimen-
tal semivariograms based on 128 rows in patterns similar to the one shown in Fig. 10.22 .
Deviations between experimental semivariograms and continuous curve are relatively large, but
there is probably no significant bias (Source: Agterberg 2001 , Fig. 5)

adjoining blocks along a line. Experimentally, their semivariogram resembles
Matheron's semivariogram for infinitesimally small blocks. Extrapolation of their
spatial covariance function to infinitesimally small blocks would yield infinitely
large variance when h approaches zero.
Figure 11.17 shows three experimental semivariograms each based on 128 rows
of 128 numbers in patterns similar to the 2-D multifractal previously shown in
Fig. 10.22 . The theoretical semivariogram satisfies:

n i

1
2

Þ ˄ ðÞþ 1 Þ ˄ ðÞþ 1

2 k ˄ ðÞþ 1

ʳ k

ðÞ ¼ ʾ 2 ðÞ

1 ð k þ 1 þ ð k 1 g

where k represents distance between successive cells measured in

multiples of E , ʾ 2 ( E ) is the non-centered second-order moment obtained by
dividing the mass-partition function for q

¼ 1, 2, ...

(2) is
the second-order mass exponent. The two parameters used for the theoretical
curve in Fig. 11.10 were in accordance with the results for q

¼ 2 by number of cells, and ˄

2shownin
Figs. 11.4 and 11.5 . The experimental semivariograms are each based on 128 2
individual values. They deviate markedly from the theoretical curve although
on the average they are closer to it. Problems associated with lack of precision
of

¼

experimental semivariograms are well-known in spatial statistics

( cf . Cressie 1991 ).