A Geostatistical Primer

Contents

It is common these days for science and engineering students to have a basic course in statistics as part of their degree structure. Such courses typically deal with statistics associated with univariate data and hypothesis testing and may extend into correlation between two variables and linear regression. However in the discussion of correlation between variables it is usually two different attributes that are considered. It is unusual in elementary courses to find a treatment of the correlation between values of a single variable which are measured at different points in space. In the earth sciences, where many measurements are made against a spatial framework, the ability to analyse such data sets is important and the necessity for it may be illustrated easily in the following figures. Figure 1.1 shows the histograms of two data sets taken at the nodes of a regular two-dimensional grid. Visually there is seen to be little difference and a Kolmogorov-Smirnoff test for a comparison of the populations from which these samples come does not reject (with a 5% significance level) a null hypothesis of no difference between the two populations. However when seen in space, in the contour maps of Figure 1.2, there is an apparent difference; the lower map appears busier. Both maps have been produced by the same contouring package so that the difference between the two is a reflection of the more random spatial arrangement of data in the second map. Another way of expressing this would be to say there is less correlation between neighbouring data pairs in the second map. (The problems addressed by time series analysis are similar to those of geostatistics but are rarely treated in elementary statistics courses.)…

The following sections are included:

• The Semivariogram

• Modelling the Semivariogram

The following sections are included:

• Estimation Regimes

• Estimation Variance

• Kriging

Dispersion Variance as a function of the Semivariogram Model

The following sections are included:

• Block Variances

• Estimation Variances

In any situation where there is selection of units above a certain critical limit, (e.g. in ore reserve assessment where there is a cut-off grade where a block is mined if it will recover the costs of mining) the consequences of selection are often discussed assuming that the estimated value is achieved. This is of course not so. Selection is made on the basis of the estimated block value being greater than the critical level; but the true value of this block is what is actually achieved and there can be no guarantee that it will be above the critical level. It is easy to see that there can be large biases introduced by considering the following scatter diagrams relating reality and estimate for the four estimators of the blocks of this data set…

The following sections are included:

• Semivariogram Models

• Estimation Error Results: Geostatistical Theory and Experiment

• Kriging Variance as a function of Semivariogram Parameters

• Block Variance Results: Geostatistical Theory and Experiment

• Grey-scale Maps from Nine Data Sets

The following sections are included:

• Appendix A Numerical values of the data

• Appendix B Estimation in One Dimension

• References