Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 504

Log 10 K

Log 10 h

−1.5 −1 −0.5 0.5 1

−1

−2

−3

−4

Fig. 12.36 Double Trace Moment analysis of the Pulacayo zinc values with q ¼ 2 (left), q ¼ 0.5
(right). Slopes yield Levy index ʱ ¼ 1.76, 1.78; and co-dimension C 1 ¼ 0.023, 0.022, respectively
(Source: Lovejoy and Schertzer 2007 , Fig. 27)

Use of the so-called “double trace moment” method ( cf . Lavall´e et al. 1992 )
yielded estimates of

the L´vy index equal to

α ¼

1.76 and

α ¼

1.78, and

codimension C 1 ¼ 0.023, 0.022, respectively (Fig. 12.36 ). In general, a relatively

small value of C 1 with respect to H indicates that the multifractality is so weak that
deviation from conservation ( H ) will be dominant except for quite high moments
(Lovejoy and Schertzer 2007 , p. 491). In the Pulacayo application the estimated
value of H is small so that the two-parameter model of Box 12.3 would be
approximately satisfied. It was shown before that the binomial/ p model produced
inconsistencies between results for very small and very large moments. Universal
multifractal modeling is more flexible and produces realistic zinc concentration
variability. On the other hand, the estimate for the second order moment obtained
by the method of moments

˄ ¼

0.038 produced a realistic autocorrela-
tion function including the nugget effect, which affects the power spectrum for high
frequencies as will be discussed next.
Another important tool in universal multifractal modeling is spectral analysis.
Theoretically, this approach results in a spectrum consisting of a straight line with
slope -

(2) 0.979

ʲ . This parameter can either be estimated directly or indirectly using

ʲ ¼ K 2 +2 H where K 2 representing the “second characteristic function”.

Lovejoy and Schertzer ( 2007 ) estimated K 2 ¼

1

0.05 by double trace moment analysis
(Fig. 12.37 ). With the previously mentioned estimate H

¼ 0.090 this yielded

ʲ 1.12 in good agreement with the experimental spectrum for the 118 zinc values.

Spectral analysis of the 118 logarithmically zinc values was discussed previ-
ously in Sect. 6.2.7 . In Agterberg ( 1974 , Fig. 67) the discrete Fourier transform was
taken of autocorrelation correlation coefficients with lag distances 32 m after

applying a cosine transformation in order to largely eliminate distortions according

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