Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 450

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1000 iterations

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0 50 100 150 200 250

Distance, m

Fig. 11.29 Results of applying iterative method of Chen et al. ( 2007 ) continuing the iterations
until full convergence is reached. Original zinc values are being smoothed during successive
iterations. Second series ( solid line ) obtained after first iteration resembles “signal” in Fig. 2.10 .
Values after four iterations are shown as diamonds . At the end of the process, after 1,000 iterations,
when convergence has been reached, the result is approximately a straight line with average value
slightly below average zinc content ( ¼ 15.6 % Zn) (Source: Agterberg 2012a , Fig. 18)

four steps of the iterative process exactly duplicated values plotted in Chen et al.
( 2007 )'s Fig. 1 and partially listed in their Table 11.1 except for the first and last
4 values in all successive series because a slightly different end correction was
employed (see before). For k 1, the pattern of c k ( x ) resembles the signal previ-

ously obtained in Fig. 2.10 by eliminating the noise component from the 118 zinc
values. Chen et al. ( 2007 ) selected α *( x ) ¼ α 4 ( x ) because at this point rate of
convergence has slowed down considerably (Figs. 11.29 and 11.30 ).
In Agterberg ( 2012a ) the iterative process was continued until full convergence
was reached in order to obtain more complete information on the nugget effect. In
the limit, after about 1,000 iterations, the final pattern is as shown in Fig. 12.29 with
an average zinc concentration value that is slightly less than 15.61 % Zn
representing the average of the 118 input values. This bias is due to the fact that,
at each step of the iterative process, straight-line fitting is being applied to loga-
rithmically transformed variables followed by the results being converted back to
original data scale. The small bias in this application can be avoided by forcing the
mean to remain equal to 15.61 % during all steps of the iterative process. In other
applications, this type of bias can be much larger and preservation of the mean then
is more important. End product and some intermediate steps this new run are shown
in Fig. 12.30 . In comparison with Fig. 12.29 , the outputs obtained after the first and

¼