Metrology in mining and metallurgy

Trans Tech Publication printed Volume 4 in its series on Bulk Materials Handling in 1985. It was called Sampling and Weighing of Bulk Solids. An unauthorized translation into Mandarin surfaced in November 1989. We do have a Canadian copyright on Metrology in Mining and Metallurgy. This text will also deal in detail with mineral exploration. It will do so because the Bre-X fraud was by far the worst salting scam I have ever unscrambled. I did it for Barrick Gold Corporation several months before Bre-X’s boss salter wound up in the Kalimantan jungle. That’s but one reason why I have registered the Canadian copyright for Metrology in Mining and Metallurgy.  

What has put my work on the map was the interleaved sampling protocol for mineral concentrates. The same protocol underpins the design of a mechanical sampling system to determine trace elements in cathode copper. I know how to derive 95% confidence limits for metal grades and contents of reserves and of proven parts of resources. Page 120 of my textbook in Section 4.5 Propagation of Variances gives the variance of a general function as defined in probability theory. One would expect a scholar with a PhD in epidemiology and biostatistics to be familiar with the properties of variances and the concept of degrees of freedom. I had given a pair of copies of Sampling and Weighing of Bulk Solids to Dr Martha Piper and she gave both to Professor Dr Alastair J Sinclair, PEng, PGeo. Dr Piper could have but did not give a copy to Dr M Klawe, her Dean of Science in those days. UBC’s library in 2011 finally put a copy of my book on one of its shelves. Why it took much too long would make a story in itself at this stage.

UBC’s Department of Geological Sciences took a liking to Matheron’s new science of geostatistics. It came about when Professor Dr Alastair J Sinclair, PEng, PGeo thought that his students stood to benefit more from Matheronian geostatistics than from applied statistics. Dr Piper might have been aware that one-to-one correspondence between functions and variances is sine qua non in applied statistics. All it would have taken in those days was a brief call to Professor Dr Nathan Divinsky.

A peculiar event took place at the Department of Geological Sciences on November 22, 1989. That’s when Dr A J Sinclair greeted those who took my short course on Sampling Precious Metal Deposits, Metrology - A New Look.  CIM Bulletin had earlier entrusted Sinclair with the review of Precision Estimates for Ore Reserves. He had initialed his review with AJS:131 on November 15, 1989. What may have troubled Professor Dr Alastair J Sinclair, PEng, PGeo was that Matheron’s science of geostatistics had left us cold. It may explain why he hopped in and out of Room 330A like a jack-in-the-box. But he could have asked the odd question during my talk. For Sir R A Fisher’s sake!

One-to-one correspondence between functions and variances was as far beyond the grasps of David and of Sinclair in 1989 as it was beyond Matheron’s grasp in 1952. The question is why variances of distance-weighted averages are still missing in 2012. It is true that the distance-weighted average itself was never lost but had merely morphed into a kriged estimate. But its variance had vanished when Matheron and his disciples had cooked up geostatistics.

The attachment to my letter of November 30, 1994 to Mr John Drury, CIM Ad Hoc Reserve Definitions Committee, shows how to derive the variance of a mass of metal in crushed ore or insitu ore.

ISO/DIS 13543-Determination of Mass of Contained Metal in the Lot

 

Borehole statistics with spreadsheet software
SME Volume 308, Transactions 2000

It had come about that the new science of geostatistics called for a mind-numbing step. Matheron and his minions stripped the variance off the distance-weighted average and called what was left a kriged estimate. The miracle of that stripped variance was embraced at UBC with as much zeal as it was at Stanford. Professor Dr A J Journel was asked why Fisher’s F-test was not applied to test for spatial dependence between measured values in ordered sets. His reply has graced my website since 2003. Journel seems to encourage those who assume, krige, smooth and rig the rules of applied statistics with impunity!

From human error to scientific fraud

Such reads the caption that these days graces my website. A few changes have been made since it was posted in 2003. What pleased me most was that loads of facsimiles and scores of snail mails could be whittled down to links. It didn’t take Merks and Merks long to figure out why geostatistics is an invalid variant of applied statistics. All it took was a close look at geostatistics when CIM Bulletin did reject Precision Estimates for Ore Reserves. We did so since it was praised by and published in Erzmetall 44 (1991) Nr 10. It was easy  to find out what was wrong with geostatistics. It matters not at all that the distance-weighted average is called a kriged estimate. What does matter is that it did somehow shed its variance.  Geostatistocrats have not yet put into plain words why each and every kriged estimate has lost its variance.  

Matheron’s new science of geostatistics has made landfall on this continent in 1970. A geostatistics colloquium in North America took place on campus at The University of Kansas, Lawrence on 7-9 June 1970. Its proceedings were recorded by Daniel F Merriam and published by Plenum Press, New York-London, 1970. A Maréchal and J Serra had graduated at the Centre de Morphologie Mathématique at Fontainebleau, France. They had come to shed light on Random Kriging. The authors point to Punctual Kriging in Figure 10. It shows how to derive a set of sixteen (16) grades from a set of nine (9) grades. It looked a bit of a slight of hand but it seemed to make sense to Professor Dr Michel David. So he posted  Maréchal and Serra’s Figure 10 on page 286 in Chapter 10 The Practice of Kriging of his 1977 textbook.

Figure 10 – Grades of n samples belonging to
nine rectangles P of pattern surrounding x

Figure 203 – Pattern showing all points within B,

which are estimated from the same nine holes

Why geostatistics is but a bogus variant of applied statistics is simple comme bonjour! Functions do have variances. No ifs or buts! That’s why one-to-one correspondence between functions and variances is sine qua non in applied statistics. Degrees of freedom are positive integers when all measured values in the set have the same weight. Degrees of freedom are positive irrationals when all measured values in the set have variable weights.

The power of applied statistics has served me well throughout my career. It did because so much of applied statistics is intuitive. For example, any set of measured values has a central value, a variance, a standard deviation and a coefficient of variation. The central value is either its arithmetic mean or some weighted average. Numbers of measured values in sets define confidence limits for central values. Testing for spatial dependence between measured values in ordered sets shows where orderliness in sample spaces or sampling units dissipates into randomness. Never did it make any sense in my work to assume spatial dependence between measured values in ordered sets.  What does make sense is testing for spatial dependence, skewness and kurtosis.

The central limit theorem defines the relationship between a set of measured values and its central value. Even David did refer to “the famous central limit theorem”. Yet, he didn’t deem it famous enough to add to his Index. Testing for spatial dependence between measured values in sample spaces and sampling units plays a key role in scores of applications in a wide range of disciplines. Participation in several standard committees served to make applied statistics indispensable in so many ways. I do have but a few simple questions at this stage. Why did Professor Dr Georges Matheron (1930-2000) cook up such a silly variant of applied statistics? Why was Matheron’s work deemed beyond peer review! Why didn’t anybody point out to him that all functions do have variances? Why doesn’t the mining industry care about unbiased confidence limits for metal contents and grades of reserves and resources?

Today I woke up as a certified octogenarian. I took a ride on my stationary bike and got nowhere. Yet I felt good. But I am still sick and tired of those who play games with other people’s money.  All I want to do at this stage of my life is show how to work with sound statistics and how to get rid of bogus science.