Autocorrelation and Geostatistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 209

Fig. 6.14 Estimated autocorrelation coefficients for original data ( diamonds ) and logarithmically
transformed Pulacayo zinc values ( squares ), shown together with best-fitting negative exponential
autocorrelation functions. The curves for original and transformed data coincide approximately
illustrating that logarithmic transformation of the original data does not significantly affect
autocorrelation in this application (Source: Agterberg 2012 , Fig. 5)

satisfied as demonstrated in Fig. 6.14 . Approximate equality of results shown in
Fig. 6.14 applies to both the estimated autocorrelation coefficients and negative
exponential functions fitted by non-linear least squares to data points with h 0.

Consequently, variograms of zinc values and logarithmically transformed zinc
values also are approximately the same. Later the variogram of logarithmically
transformed zinc values will be used. Substituting fitted values from Fig. 6.14 into
ʳ

>

2 (1

ˁ h ) yields a variogram (Table 6.1 , see later) that is close to estimates
originally obtained by Matheron ( 1962 ).

( h ) ¼ ˃

Box 6.6: Whittle's Space-Time Model

(x, t ) that adopts a value at every point of a
space with Cartesian co-ordinates x

ʾ

Whittle ( 1962 ) considered a variable

¼

( x ,

y , z )forime t with

∂ ʾ

2

∂ t þ ʱʾ ¼ ½∇

ʾ þ E . This is the standard diffusion equation “driven” by

2
ʾ

E ¼ E

representing the spreading of, for example, a chemical
element through the medium and

(x, t )with

∇

ʱʾ as a spatial “trend” term. The solution is:

(continued)