Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 454

0.1

0.08

0.06

0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

-0.1

0 50 100 150 200 250

Distance, m

Fig. 11.34 Residuals from straight line in Fig. 11.33 display white noise pattern with variance
equal to 0.00118 (Source: Agterberg 2012a , Fig. 23)

algorithm (also see Chen et al. 2007 , Table 1). However, the minimum and maximum
singularity estimates obtained by singularity analysis differ significantly from esti-
mates based on the multifractal spectrum previously obtained by the methods of
moments (see Figs. 11.7 and 11.11 ) as will be explained in the next paragraph.
Evertsz and Mandelbrot ( 1992 , p. 941) show that the mass exponents satisfy:

log 2 [ p q +(1

p ) q ]. Cheng and Agterberg ( 1996 ) illustrated that, for ˄

( q )

¼

q ranging from 35 to 50, the method of moments provides the multifractal

spectrum of Fig. 11.7 . For small values of q , mass exponent estimates are excellent;
e.g., ˄ (2) ¼0.979 0.038 (Fig. 11.11 ). From ˄ (2) ¼0.979 it would follow that

d ¼ 0.121, α min ¼ 0.835, and α max ¼ 1.186. The latter estimate differs not only

from α max ¼ 1.719 derived in this section, it also is less than the estimate

α max ¼ 1.402 on the right side of the multifractal spectrum shown in Fig. 11.7 .

The estimate d ¼ 0.121 obviously is much too small. Use of the relations

p ¼ 0.5(1 d ), α min ¼ log 2 (1 p ) ¼ 0.369 and α max ¼ log 2 p ¼ 1.719, produces

the much larger estimates d ¼ 0.369 and d ¼ 0.392, respectively. Clearly, the

binomial/ p model has limited range of applicability although α max ¼ 1.402 on the

right side of the multifractal spectrumwould yield d 0.243, which is closer to 0.392

than 0.121. The preceding inconsistencies suggest that a more flexible model with
additional parameters should be used. Two possible explanations are as follows.
A “universal multifractal model” with three parameters was initially developed
during the late 1990s by Schertzer and Lovejoy ( 1991 ). Lovejoy and Schertzer
( 2007 ) have successfully applied this model to the 118 Pulacayo zinc values as will
be discussed Sect. 12.6 . The other possible explanation is that, in general,

¼

α min and