Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 485

Fig. 12.14 Method of moments applied to 64 cell average gold concentration values. A property
of self-similarity is that power moment sums of cell masses raised to powers of q are related to cell
side according to power cells with mass exponents ( q ) given by the slopes of the straight lines.

Because of uncertainties related to gold detection limit, results for q

˄

0 can be used only in this

application (Source: Agterberg 2007a , Fig. 11)

Logbinomial distributions for gold and arsenic are shown in the lognormal Q-Q
plots of Fig. 12.16 . Because the negative binomial is discrete, the preceding
estimates of N were rounded off to the nearest integer. The parameters of the
distributions shown are

ʾ ¼ 1.82 ppb, d ¼ 0.433, and N ¼ 26 for gold (Fig. 12.16a );

and ʾ

were determined by forcing the log-binomials and their lognormal approximations
through the centers of clusters of points on lognormal Q-Q plots for original data.
These results remain approximate but are interesting because they permit shape
comparison of logbinomial and lognormal tails.
The dispersion index of gold (

ʾ ¼ 7.05 ppm, d ¼ 0.069, and N ¼ 28 for arsenic (Fig. 12.16b ). Mean values

¼ 0.433) is much larger than that of arsenic

0.069), but the patterns on Q-Q plots in Fig. 12.16 for these two elements are
similar. Both are approximately lognormal over the ranges of observed values.
Outside these ranges the logbinomial upper (and lower) tails become increasingly
thinner than their lognormal approximations. However, the resulting differences
have relatively little practical significance. For example, the logbinomial would
predict that 1.45 % of all possible till samples have gold concentrations greater than
32 ppb, whereas the corresponding lognormal prediction (

¼

(

¼ 1.56 %) is hardly

different from this.