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α max cannot be determined by the method of moments because they are would be
almost exclusively determined by the largest and smallest observed values. If this
second explanation is valid, mass exponents and multifractal spectrum derived by
the method of moments such as those shown in Fig. 11.7 (and Fig. 11.11 ) are valid
only for the smaller values of q . For very large values ( q
!1
) the multifractal
spectrum develops long narrow tails with f (
α
α
α min and
α max
) intersection the
-axis at
as estimated by singularity analysis.
11.6.2 KTB Copper Example
The second example is for the long series consisting of 2,796 copper (X-Ray
Fluorescence Spectroscopy) concentration values for cutting samples taken at 2 m
intervals along the Main KTB borehole, on the Bohemian Massif in southeastern
Germany. These data (Fig. 6.12 ) were previously analysed in Chap. 6 . It was
assumed that 101-point average copper values representing consecutive 202-m
long segments of drill-core captured deterministic trends in this series reflecting
systematic changes in rock compositions. The data set was divided into three series
(1, 2 and 3) with 1,000, 1,000 and 796 values, respectively. Mean copper values for
these three series are 37.8, 33.7 and 39.9 ppm Cu, and corresponding standard
deviations are 20.3, 11.0 and 20.6 ppm Cu, respectively. Correlograms for the three
series are significantly different (Fig. 6.13 ), although the patterns for series 2 and 3
resemble one another. Figure 6.14 showed correlograms of the three series using the
differences between the copper values and their moving average. Each correlogram
is approximately semi-exponential with equation
ah ) . The slope
coefficients ( a ) of the three curves are nearly equal to one another (0.40, 0.38 and
0.41 for series 1, 2 and 3, respectively). Series 2 and 3 also have similar nugget
effect at the origin and were combined to form a single series to be used here for
local singularity analysis with white noise variance equal to 37.7 %. Series 1 was
not included in this new data set because its nugget effect is markedly different. In
Agterberg ( 2012a ) it was pointed out that quantitative modeling of the nugget effect
in KTB copper determinations yielded better results than could be obtained for
several examples from mineral deposits including the Pulacayo zinc orebody. This
is not only because the series of KTB chemical determinations is much longer but
also because the nugget effect remains clearly visible over lag distances between
2m(
ˁ h ¼
c ￿ exp(
original sampling interval) and 10 m.
Mean and variance of the new series are 36 ppm Cu and 268.4, respectively. The
variance of logarithmically (base 10) copper values is 0.024447. The coefficient of
variation of original data amounts to 0.45. Because this is less than 0.5, the white
noise component probably also provides an approximation for the logarithmically
transformed copper values. Approximate white noise variance then would be
(0.377
¼
) 0.0092. This estimate is probably slightly too large because
the logarithmic variance of 0.24447 does not account for the deterministic trends.
However, it provides a crude estimate that can be used for comparison in the
following application of the Chen algorithm.
0.024447
¼
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