Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 459

1.6

1.4

1.2

1

0.8

0.6

0.4

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500

Sample #

Fig. 11.40 “Final” singularities ( heavy line ) of Fig. 11.37 for KTB copper concentration values at
500 sampling points compared with singularities (thinner line of Fig. 11.39 obtained after single
iteration (Source: Agterberg 2012b , Fig. 17)

However, there are several new relatively low values not detected for k ¼ 1

(Fig. 11.40 ).
The application of the Chen algorithm described in this section differs from the
application to the Pulacayo zinc values described in the previous section. Full
convergence for k

!1

generally results in a subhorizontal straight line. For the
KTB copper values it can be assumed that there exist systematic changes in average
copper content along the borehole and convergence was stopped at a point that the
c k closely approximated the deterministic copper trend pattern. The proof that the
singularities after approximately 100 iterations are realistic is provided by the fact
that the residuals from the trend pattern have nearly the same semi-exponential
autocorrelation function that dies out rapidly with distance. This semi-exponential
curve is equivalent to the semi-exponential close to the origin in Fig. 6.19 .

References

Acz´l J, Dhombres J (1989) Functional equations in several variables. Cambridge University
Press, Cambridge
Agterberg FP (1994a) Fractals, multifractals, and change of support. In: Dimitrakopoulos R
(ed) Geostatistics for the next century. Kluwer, Dordrecht, pp 223-234