Thursday, September 06, 2012

To krige or not to krige?

Not only is it a verb with a touch of a noun but it is also a true eponym. Matheron had written in 1960 what he himself had called Krigeage d’un panneau rectangulaire par sa périphérie. Nowadays it is posted as Note géostatistique No 28. An anthology of Matheron’s life and time, and of his creation of geostatistics, is posted on a massive website. Danie G Krige had put together a Preface to David’s 1977 Geostatistical Ore Reserve Estimation. References to Krige pop up on many pages. Journel’s 1978 Mining Geostatistics, too, refers not only to D G Krige but also to the zero kriging variance. 

Geostatistical software made Bre-X’s bogus grades and Busang’s barren rock look like a massive gold resource. So why had geostatistics been hailed as a new science in the 1970s. The Bre-X scam was well on its way when geostatisticians got together to praise David’s 1977 Geostatistical Ore Reserve Estimation. He was praised at a celebration called Geostatistics for the Next Century at Montreal on June 3-5, 1993. My take on The Properties of Variances clashed with the celebrations at McGill University. What applied statistics did do is prove that the intrinsic variability of Bre-X’s gold was statistically identical to zero. How about that? The geostatocracy is still poised in 2012 to assume, krige, smooth, and rig the rules of applied statistics with impunity.  

David’s 1977 textbook displayed his tenuous grasp of applied statistics. The author points on page 33 of Chapter 2 to what he calls “the famous central limit theorem”. On page 286 in Figure 203 he shows how to derive a set of sixteen (16) “famous central limit theorems” from the same set of nine (9) holes. Next, he points out on this page, “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics”. Counting degrees of freedom would have  shown that the author of the first textbook on geostatistics did grasp applied statistics.

It is simple to verify spatial dependence between measured values in an ordered set by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. The F-test requires that degrees of freedom be counted. Stanford’s Journel claims that spatial dependence between measured values may be assumed. For crying out loud! He did so in his letter to JMG’s Editor. Now how’s that for a nouveau science! Surely, spatial dependence in sample spaces should be proved beyond reasonable doubt. It took but two steps to go from goofy geostatistics to a genuine fraud. The first step was to strip the variance off the distance-weighted average. The second step was to call a kriged estimate what had once been a distance-weighted average with a variance. Now that’s simple comme bonjour, n’est ce pas? Kriging is a stacked game of chance. Thou shall not krige when scientific integrity matters!

Mineral Inventory Studies of Precious Metal Deposits in British Columbia is one work of geostatistical fiction that I have kept on file. The study that peeked my interest most of all was Ordinary Block Kriging with Geological Control, A Practical Approach to Estimating Mineral Inventory, Nickel Plate Mine, Hedley, British Columbia. I did so simply because primary data are given. The authors of this study were A J Sinclair et al. It hit the spotlight on June 3-5, 1993 when “Geostatistics for the Next Century” was hailed for no reason whatsoever!   

Dr A J Sinclair, Professor Emeritus (Geological Engineering), was 2000-2001 recipient of a distinguished lecturer award. Sinclair talked about “Geology and data analysis: essential components of high quality resource/reserve estimation”.  He talked across the country in both official languages. His paper on Ordinary Block Kriging with Geological Control, A Practical Approach to Estimating Mineral Inventory, Nickel Plate Mine, Hedley, BC was presented when David was praised at McGill in June 1993. I applied Fisher F-test to test for spatial dependence.

Fisher's F-test for spatial dependence
The set of production data didn’t display a significant degree of spatial dependence. Neither did the set of ordinary block kriging data. Bartlett’s chi-squared test would have shown significant discrepancies not only between variances of sets but also between first variance terms of ordered sets. 
95% Confidence limits for arithmetic means

The central values in this table are arithmetic means. Confidence intervals and ranges are derived in Excel spreadsheet files. Shortly, a link to both files will be be posted.
The Society for Mining, Metallurgy, and Exploration published in Volume 308 Transactions 2000 a reviewed paper entitled Borehole statistics with spreadsheet software. The paper shows how to fingerprint boreholes. Its reviewer expected it would “stir up a hornets’ nest” but it never did! This paper underpins a report in which confidence limits for a large gold reserve had been derived. It was submitted to Barrick Gold early in 1998. 

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