Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 458

60

55

y = x

50

45

40

35

30

25

20

20 25 30 35 40 45 50 55 60

Cu (ppm) 100 iterations

Fig. 11.39 Moving average shown in Fig. 6.22 (Location value 1,000) plotted against c k

obtained after 100 iterations. Because both variables are nearly continuous, their relation plots
as a nearly continuous line as well. The y ¼ x line shown for comparison suggests that both
variables are equal on the average (Source: Agterberg 2012b , Fig. 16)

>

1, 5, 100 and 1,000, respectively.
A procedure to be explained in more detail in Sect. 12.6 can be used to help decide
which pattern is best. Figures 11.36 , 11.37 and 11.38 are plots of log 10 Cu versus
estimated singularities for k

Figure 11.35 shows the coefficients c k for k ¼

¼

1, 100 and 1,000, respectively. The regression line in
Fig. 11.35 provides a poor fit because estimates of the singularities for k

1 cannot
be considered to be free of error. Residual variance with respect to this line is
0.01636. The regression lines in Figs. 11.37 and 11.38 for k

¼

1,000
provide good fits. However, the result in Fig. 11.36 is to be preferred because it's
residual variance (

¼ 100 and k ¼

0.0092)
estimated in the preceding paragraph. Residual variance for the line in Fig. 11.37
amounts to 0.00451 and is about two times too small.
The pattern of c k for k

¼ 0.00823) is close to the white noise variance ( ¼

¼ 100 closely resembles the 101-point moving average

(for sample numbers 1,000) shown in Fig. 6.12 . This relationship is illustrated in

Fig. 11.38 . The relatively good fit of the y

>

x line in this diagram suggests that both
variables are subject to approximately the same uncertainty. Figure 11.39 shows the
corresponding “final” relation between log 10 Cu and singularity (as obtained for
k

¼

1 in Fig. 11.39
(for 500 sampling points only). The minimum and maximum singularity for k

¼ 100). This result in compared with singularities estimated for k ¼

100
are only slightly less and greater than the result obtained after a single iteration.

¼