Autocorrelation and Geostatistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 217

Fig. 6.18 Relationship between normalized extension variance ( ˃ E 2 ) and h / L (Source: Agterberg
2012 , Fig. 9)

applications of multifractal modeling to the Pulacayo Mine (Cheng and Agterberg
1996 ; Chen et al. 2007 ; Lovejoy and Schertzer 2007 ), it is assumed that the
zinc concentration values can be converted into measures of amounts of zinc in
adjoining 2-m wide samples along a line parallel to the drift on the 446-level.
It implies that every zinc concentration value for a channel sample at a point along
this line is taken as representative for a width of 2 m. Associated uncertainty then is
given by the extension variance

E . Figure 6.18 shows that normalized extension

˃

E /3 A depends on h / L . From our estimate A

variance ˃ ¼ 0.0165, it follows that

E

˃ 50 cm long. It probably signif-

icantly overestimates true value because absolute dispersion is less than 0.0165 over
very short distances due to the nugget effect (see later). If A

¼ 0.0622 for h ¼ 2 m wide samples that are L ¼

< 0.0165, the normal-

E

ized extension variance is greater than 0.0622 as derived for the same value of

h / L from the curve in Fig. 6.18 , that is for A

˃ ¼

¼ 0.0165.

Box 6.7: Geostatistics of Mean Block Values

Suppose that f ( x ) represents concentration of a chemical element at a point x ,
which may be a 3-D vector ( x 1 , x 2 , x 3 ). The average m for any volume v is
m

Z

1

¼

v

fx ðÞ dx . Suppose M is an assemblage average for a large volume V ,

Z

v

X m i where the m i are for smaller volumes v i .On
the basis of V and f ( x ) the intrinsic functions can be defined for the covariance

1

then M ¼

V

fx ðÞ dx ¼

V