Pros and cons of Practical Geostatistics

Dr Isobel Clark is the author of Practical Geostatistics. What she did in this 1979 textbook took me by surprise. It would have baffled many a thinker who has never bothered to read works of others. She deserved praise because she had derived the variance of the distance-weighted average. Agterberg in 1974, David in 1977, and Journel in 1978 never took the trouble to derive this variance. It’s a bad omen that the distance-weighted average morphed into a kriged estimate on Matheron’s watch. It was Matheron who had failed to derive the variance of this kriged estimate long before geostatistics was hailed far and wide as a new science.

Dr IC pointed out on the jacket of her textbook, “Geostatistics is the popular name for the application of statistical methods to problems in mining and geology”. She confessed in her Preface that Journel and others at Fontainebleau, France had taught her all she knows about the Theory of Regionalized Variables. Now that’s what got me worried. Matheron’s way of drawing strings of symbols on a blackboard may well have driven his most gifted disciple to routinely assume spatial dependence. All it took is to assume, krige, smooth, a leap of faith, and a lot of luck!

Author 1974 Geomathematics
ex NRCan Emeritus Scientist

Dr Frederik P Agterberg, too, took a leap from applied statistics to geostatistics. He went from serial correlations in 1958 to functions without variances in 1974. In contrast, Dr Isobel Clark derived not only the distance-weighted average grade of a set of measured values but also the variance of the set and the variance of its central value. Much of it is detailed in Figure 4.1 and Table 4.1 of her textbook and a few pages about her data set. She fretted a bit on page 72 whether or not the Central Limit Theorem would hold. But it always does! Surely, David himself in 1977 would not have written about if it were false.

Dr IC’s set of hypothetical uranium concentrations seem to have been measured in hypothetical samples selected at positions with real coordinates in a two-dimensional sample space. I wish she had given the mass of each of her hypothetical samples. She reports that her 95% confidence range for a distance-weighted average hypothetical uranium grade of 400 ppm has a lower limit of 95% CRL=350 ppm and an upper limit of 95% CRU=450 ppm. It doesn’t look too bad, does it? Of course, t0.95;4=2.776 from the t-distribution rather than z0.95=1.96 from the normal distribution should have been used to derive the 95% confidence interval for xbar=400 ppm. But who would split hairs at this stage? Sir R A Fisher was already counting degrees of freedom for small data sets when Dr IC was a cute tot.

Assumes with poise!

The question is not so much whether this distance-weighted average hypothetical grade of 400 ppm is an unbiased estimate for the unknown true grade at 1,244 m Easting and 713 m Northing. Of course, I wish I knew her sampling protocol. The question is whether it is an unbiased estimate for the sample space defined by this set of five (5) hypothetical uranium concentrations. The most effective method to test for spatial dependence is to do what door-to-door sales people have always done. Travel from point to point such that each point is called on but once and the shortest distance is traveled. The next step is to derive the variance of the set and the first variance term of the ordered set. Finally, Fisher’s F-test is applied to the variance of the set and this first variance term. The observed F-value of 2.07 between var(x)=4,480 for the set and var1(x)=2,161 for the ordered set is below the tabulated F-value of 4.53 at 95% probability. Hence, her set of measured values does not display a significant degree of spatial dependence. By implication, the distance-weighted average grade of 400 ppm is not an unbiased estimate for this sample space. Dr IC did what Journel had taught her to do. All she did was assume spatial dependence between measured values in the ordered set. Here’s what happens when spatial dependence is assumed and functionally dependent kriged estimates are added to measured values.

Assume, krige, smooth, and be happy!

Dr Isobel Clark is due to teach a 3-day course at Global InfoMine in Vancouver, BC on May 3-5, 2011. She will take students from "no knowledge of statistics or geostatistics to understanding the mysteries of ordinary kriging and its variants in 30 hours (or less)". I wish I were there to shed light on that shameless practice of converting bogus grades and barren rock into a phantom gold resource. But then, why rush the good stuff!