The Centre de Géosciences/Géostatistique deserves praise for posting to its Online Library a treasure trove of writings. A great deal came from the seminal work of Professor Dr Georges Matheron (1930-2000). Most of it merits long overdue scrutiny and review. The problem is not so much that Matheron put a few spurious findings on paper but that his students took it for doctrine. The Online Library has made it easy to pinpoint what Matheron did wrong and when he did so.
Matheron derived the length-weighted average grade of a set of metal grades determined in core samples with variable lengths. He did so in his Rectificatif of January 13, 1955, to Formule des Minerais Connexes of November 25, 1954 (see Note Statistique No 1). What he didn’t derive was the variance of this length-weighted average grade. Neither did he show how to test for spatial dependence between grades of ordered core samples by applying analysis of variance. He didn’t report primary data sets because of his penchant for working with symbols rather than with real measured values.
Matheron concocted the honorific eponym krigeage in his 1960 Krigeage d’un Panneau Rectangulaire par sa Périphérie. In this Note géostatistique No 28, Matheron derived k*, his “estimateur”, and a precursor to kriged estimate or kriged estimator. In real statistics, Matheron’s k* is in fact the length-weighted average grade of a single block. In this case, too, he didn’t derive var(k*), the variance of his “estimateur”. Sadly, kriging became a curse of sorts for Professor D G Krige.
Matheron’s Stationary Random Function seemed not to have troubled those who were at the first geostatistics colloquium in the USA in 1970. Matheron even called on Brownian motion to infer by hook or by crook the continuity of his Riemann integral. He didn’t explain what Brownian motion and mineral deposits have in common. Matheron, unlike John von Neumann in 1941 and Anders Hald in 1952, didn’t work ever in his life with Riemann sums. On the contrary, he would rather infer spatial dependence than apply Fisher’s F-test to the variance of a set and the first variance term of the ordered set.
It is to Matheron’s credit that it was not him who lost variances of all weighted averages. It was Dr Frederik P Agterberg who failed to derive the variance of his distance-weighted average. He did derive the distance-weighted average grade of a set of five (5) points at positions with different coordinates but failed to derive the variance of this central value. What he didn’t point out was that as few as two such points define an infinite set of distance-weighted averages. He fumbled the variance of his central value for the first time in his 1970 colloquium paper and once again in his 1974 Geomathematics.
Matheron’s length-weighted average grade was reborn as an honorific kriged estimate or estimator. But then Agterberg’s distance-weighted average grade was honored in the same way! And here’s the clincher! An infinite set of Agterberg's zero-dimensional point grades fits along any borehole, and within any ore block, sampling unit or sample space. That’s why distance-weighted average point grades without variances became the heart and soul of geostatistics. Matheron’s seminal work merely set the stage for Agterberg’s giant step into the abyss of mineral reserve and resource estimation with confidence but without confidence intervals and ranges.
The above figure is a facsimile of Fig. 203 on page 286 of David's 1977 Geostatistical Ore Reserve Estimation. It shows the infinite set of "estimated" values within B derived from the same set of nine (9) holes.
The more geostatistocrats tinkered with real statistics, the more flawed geostatistics grew. It’s a scientific fraud to derive confidence limits from pseudo kriging variances. It’s as silly to talk about confidence without limits as it is to infer spatial dependence within or between boreholes. To discount degrees of freedom would make no sense at all in real statistics. To count degrees of freedom makes no sense in geostatistics. That’s the very reason why geostatistics does not give unbiased confidence limits for metal contents and grades of mineral reserves or mineral resources.
Professor Dr Roussos Dimitrakopoulos is a catch of sorts for the Department of Mining, Metals and Materials Engineering at McGill University. I don’t know why! I told him in 1993 that weighted averages have variances because one-to-one correspondence between functions and variances is sine qua non in statistics. This basic rule is still beyond his grasp in 2008. All the same, he is Editor-in-Chief, Journal of Mathematical Geosciences. Agterberg, President, International Association for Mathematical Geosciences, left his fingerprints when he failed to derive the variance of his distance-weighted average point grade. Dimitrakopoulos talks about “gazillion types” of probabilistic models. What he doesn’t talk about is that the odds to select the least biased subset of some infinite set of kriged estimates are immeasurable. The problem is not so much he himself believes it but the world’s mining industry believes it. The more so because he does all of that with voodoo variances.
Mining engineers, mine geologists, resource analysts, and project managers were invited to a course on Applied Risk Assessment for Ore Reserves and Mine Planning at McGill University. The same course deals with Strategic Risk Quantification and Management for Ore Reserves and Mine Planning and with Conditional Simulation for the Mining Industry. That’s a lot of buzz for a bundle of bucks! Too bad that voodoo variances underpin all that risk assessment and quantification stuff! That’s why one should come with a buddy. For it’s more difficult to baffle a few birds of a feather than a single sitting duck. Dimitrakopoulos should explain why Agterberg’s distance-weighted average point grade aborted its variance during its rebirth as an honorific kriged estimate on Matheron’s watch.
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