Autocorrelation and Geostatistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 215

6.2.4 Geometrical Probability Modeling

In principle, Matheron ( 1962 )'s semivariogram · log e h also can be

applied to untransformed data. However, the following applications of geometrical
probability are for logarithmically transformed distance.
Returning to the geometry of channel sampling (Fig. 2.9 ) : suppose that AA 0 BB 0
represents a rectangle with sides AA 0 ¼

ʳ ( h ) ¼ 3 ʱ

BB 0 ¼ A 0 B 0 ¼ L / h .

If the concentration value for a small volume at a point is taken to be the concen-
tration value of another volume of rock that either contains the small volume or is
located elsewhere, this results in uncertainty expressed by means of the “extension
variance”. In Matheron ( 1962 , Sect. 39) or Agterberg ( 1974 , Sect. 10.11) it is
discussed in detail that the variogram value of parallel line segments of length L
that are distance h apart along a straight line can be interpreted as an extension
variance

h ,AB ¼ L and tan ʸ ¼

˃ h 2

¼ ʲ

￿ f (

ʸ ʲ ¼

6 A and:

) with

h 2
L 2 ln

L
L 2

2 h
L

tan 1 L

h
L 2

p p

f ðÞ ¼ ln þ

h þ

h 2 h 2

þ þ

Table 6.1 shows the first ten Pulacayo Mine variogram values as estimated by
Matheron ( 1962 , p. 180) applying this equation to log-transformed (base e ) zinc
values. For comparison, theoretical variogram values for the exponential model
(derived from autocorrelation model graphically shown in Fig. 2.10 ) are listed as
well, illustrating that this model with a sill also provides a good fit. For other
theoretical autocorrelation functions fitted to the Pulacayo zinc values, see
Sect. 6.2.1 and Chen et al. ( 2007 ).
Use of Matheron's original variogram model resulted in multiple estimates of

ʲ ( h ) for different lag distances ( h ) in Table 6.1 . A better estimate is obtained by

using constrained least squares estimation as follows. The theoretical variogram
values in the second last column of Table 6.1 are based on a single estimate
(

0.0988) representing the slope of a line of best fit (Fig. 6.17 ) forced through
the point where f (

ʲ ¼

0. This additional point receives relatively strong
weight in the linear regression because it is distant from the cluster of the other ten
points used. The constraint can be used because, for decreasing h :

ʸ ) ¼ 0 and h ¼

( ) ( )

L
L 2

2 h
L

tan 1 L
h

h
L 2

p p

lim

h! 0 f

ðÞ ¼ lim

h! 0

ln þ lim

h! 0

þ lim

h! 0

¼ 0

h 2 h 2 þ þ

0.0165 not only produces
theoretical variogram values, which are nearly equal to the estimates based on the
logarithmically transformed zinc values, it also is nearly equal to

The new estimate of absolute dispersion A ¼ ʲ /6 ¼

ʱ ¼

0.015 previ-
ously derived from the logarithmic variance in the previous section, confirming the
applicability of Matheron's original method within a neighbourhood extending
from about 2 to 400 m.