To have or not to have true variances

It all depends on who applies what! Statisticians apply true variances but geostatisticians work with false variances. The problem is that geostatistocrats call theirs kriging variances. The matter of true variances versus kriging variances came up at a seminar sponsored by the PDAC (Prospectors and Developers Association of Canada). The PDAC had set the stage at the Royal York Hotel in Toronto, Ontario, on Saturday, March 23, 1991. It was organized by H E (Buzz) Neal, PEng, William A Roscoe, PhD, PEng, Henrik Thalenhorst, PhD, and Lorne A Wrigglesworth. I had called my talk Sampling in Exploration, Theory and Practice. I was slated first to speak. As luck would have it, I would give the same talk at Mount Isa, Queensland, Australia, on November 3-7, 1992. That’s where I also presented the Conference Dinner address. But that’s one more part of my story!

During my talk Professor Dr Michel David was sitting sort of face to face with me on the first row. A few of his buddies were close by. David himself had put on paper the very first work on Matheron’s new science of geostatistics. He had simply called it Geostatistical Ore Reserve Estimation. Elsevier Scientific Publishing Company had printed in 1977. David himself had predicted in this book that it was not for professional statisticians. He also predicted that statisticians would find many unqualified statements. And he did get that right too! What David did not predict is that he would blow a fuse if and when he was to review a paper that was short on references to geostatistical literature. But that’s exactly what he did as a reviewer for CIM Bulletin. David did so when he reviewed in September 1989 our paper on Precision Estimates for Ore Reserves.

We had decided not to point out what was wrong with geostatistics but to show what made sense in applied statistics. We had tested for spatial dependence between gold grades of bulk samples taken from a set of ordered rounds in a drift. We had done so by applying Fisher’s F-test to the variance of the set and the first variance of the ordered set. We pointed out that each function does have its own variance in applied statistics and that variances of gold contents are additive. What we didn’t do was estimate the intrinsic variance of gold. It would have required that a pair of interleaved primary samples be taken from every crushed round. We mentioned that extraneous variances such as those for dividing whole core sections into halves, and for selecting and assaying test portions of test samples may be subtracted before deriving unbiased confidence limits for contained gold. We were tickled pink that Precision Estimates for Ore Reserves was praised by and published in Erzmetall, October 1991.

David has made peer review at CIM Bulletin a shameful sham. Read what he wrote about our paper: “The authors present their own method for calculating precision estimates for ore reserves without a single reference to 20 years of work in geostatistical ore reserve estimation (see attached references)”. What he had missed were references to Dagbert & Myers, to himself, and to Journel & Huijbregts. In his 1977 Geostatistical Ore Reserve Estimation he did praise “the famous Central Limit Theorem”. What he didn’t show was how to test for spatial dependence between measured values in ordered sets. Neither did he show how to derive unbiased confidence limits for masses of contained metals.

David may have reviewed A study on Kriging Small Blocks. Its authors called attention to the fact that mine planners are often tempted to over-smooth small blocks. Armstrong and Champigny failed to show how to smooth both small and large blocks to perfection. Good grief! That sort of bogus science was approved by and published in CIM Bulletin of March 1988. Nowadays, mineral analysts are blamed when geostatistically predicted grades mess up metallurgical balances in mineral processing plants. It’s all a huge game of chance for mining investors!

Marechal and Serra showed in 1974 how to derive a set of sixteen (16) distance-weighted averages from a set of nine (9) boreholes. David shows the same set on page 286 of his book. Each distance-weighted average is a function of the same set of nine (9) holes. As such, each is blessed with its own variance in applied statistics. Here’s where statistics went missing in geostatistics. The variance-deprived distance-weighted average morphed into a kriged estimate. What’s more, geostatisticians never took to counting degrees of freedom.

Infinite set of kriged estimates within B

David got into calling a kriged estimate a simulated value. Here’s literally what he wrote on page 324 of his 1977 work, “The criticism to this model is obvious. The simulation is not reality. There is only one answer: The proof of the pudding is …! So far the few simulations made which it has been possible to check have a posteriori proved to be adequate”. Nobody knows all of the nonsense I've had to put up with!