Autocorrelation and Geostatistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 189

Chapter 6
Autocorrelation and Geostatistics
Abstract Time series analysis has a long history of useful applications in many
branches of science. Estimation of the autocorrelation function and power spectrum
of serial data are important tools that are complementary to one another. In general,
it is assumed that a series is “stationary” with constant mean and variance. Analysis
of glacial varve-thickness data will be presented as an example of establishing
periodicities in temperature related to climate change. The cross-spectrum and
coherence are time series equivalents generalizing the correlation coefficient
between two variables. In geostatistics, the semivariogram is favored as a principal
tool for analysis of space series consisting of mining assays in ore deposits or
chemical element concentration values in rock units. In general, there is a simple
linear relationship between autocorrelation function and semivariogram. The most
frequently used geostatistical semivariograms including the spherical and expon-
ential models allow for a nugget effect at their origin representing non-zero
variance and assume asymptotic convergence to a constant value representing
regional variance at large sampling intervals exceeding a range. However, a
semivariogram model without finite variance for modeling of short-distance auto-
correlation as originally proposed by Matheron (Trait´ de g´ostatistique appliqu´e.
M´m BRGM 14, Paris, 1962, Estimating and choosing, an essay on probability in
practice (trans: Hasover AM). Springer, Heidelberg, 1989) can be more appropriate
in some types of applications. Geometrical considerations related to shapes and
volumes of blocks of rocks are important in geostatistical modeling. Average values
estimated for blocks can be extrapolated into their surroundings with use of their
extension variances. In addition to the time-series applications to glacial varve-
thickness data, practical examples in this chapter are mainly for copper, zinc and
gold concentration values obtained by channel sampling in various ore deposits in
Bolivia, Canada and South Africa, and for copper in chip samples from along the
7-km deep KTB borehole drilled into the Bohemian Massif in southern Germany.
Stationary discrete random variables, for example, those related to successions of
different KTB lithologies also can be modeled geostatistically.