Voodoo statistics at IAMG

Acronyms serve to make long tags short. Ranking high among the world’s most famous acronyms are USA and IBM. Laser and taser are well-known objects that have but rhyme in common. EMF stands for Eclipse Modeling Framework. ASTM, DIN and ISO are familiar to those who develop and work with national and international standard methods. IAMG stood for the International Association for Mathematical Geology from 1968 to 2007. IAMG’s Council in January 2008 called it the International Association for Mathematical Geosciences. What IAMG's Council never did was set up an ISO Technical Committee on Reserve and Resource Estimation.

Professor Dr Georges Matheron may well have thought that he was peerless. In way too many ways he was indeed without peers. It was a blessing of sorts in disguise. All of his work is so richly embellished with symbols that tallied up to a tangle of formulas. All of it fell far short of a clear and concise text. He made up all sort of terms if and when required. But what he didn’t do was provide primary data sets. So, his work does not make an easy read even in French let alone in English. What does matter is that CdG's website has made Matheron's work accessible to the world.

So it came about in 1970 that Matheron’s new science of geostatistics got all geared up to do more with less. That was the very year it made its way to the University of Kansas, Lawrence. D F Merriam, Chief of Geologic Research, Kansas Geological Research, and IAMG Historian, called it a colloquium. It was a thoughtful touch that he dedicated the proceedings to ‘all geostatisticians and statistical geologists’. Matheron had come all the way from his Centre de Morphologie Mathematique to talk about Random Functions and their Application in Geology. His tour de force was to somehow force Brownian motion along a straight line. He didn’t spell out what Brownian motion and ore deposits could possibly have in common. What did matter most was that his so-called random functions are continuous along intervals between measured values in ordered sets.

Matheron was not the only geostatistical scholar from his Centre de Morphologie Mathematique. A Marechal and J Serra had come along to talk about Random Kriging. What captured my attention was M&S’s Figure 10. It turned out to be a dead ringer for Figure 203 in David’s 1977 textbook. Both figures showed how to derive a set of sixteen (16) distance-weighted averages from the same set of nine (9) holes. It may look like the miracle at the wedding of Cana in Galileo. But that’s what geostatistics is all about. Agterberg derived but a single distant-weighted average point grade from a set of five (5) measured values. Marchal, Serra and David derived a set of sixteen (16) distance-weighted averages. Each and every so-called kriged estimate is a zero-dimensional and variance-deprived weighted average point grade. It turned into the heart and soul of Matheronian geostatistics.

Infinite set of distance-weighted average point grades

When Matheron's new science of geostatistics struck the University of Kansas, Lawrence in June 1970 it didn’t hit any raw nerves. At that time, IAMG stood for International Association for Mathematical Geology. And Matheronian geostatistics kept coming along by hook and by crook.

Assume, krige, smooth, be happy!

IAMG’s News Letter No 38 reported that all members of its discipline belong to one of three schools of thought: Those who practice and strongly advocate geostatistics, those who are violently (and vocally) opposed to geostatistics, and the silent majority, who wonder what all of the shouting is about. The same newsletter shows Michel David accept the Krumbein Medal from IAMG’s President John Davis. News Letter No 38 did put into perspective why our paper on Precision Estimates for Ore Reserves troubled David as much as it did. He didn’t know how to derive unbiased confidence limits for the mass of metal in a volume of in-situ ore. Why then did David expect Merks & Merks to refer to twenty years of geostatistical literature? Many questions and but few answers. Stay tuned for sound statistics!

Rebranding Professor Dr Georges Matheron

Dr Frederik P Agterberg tried to do so when he sang the praises of Professor Dr Georges Matheron and called him the Founder of Spatial Statistics. The keepers of Matheron’s magnum opus at his own Centre de Géostatistique didn’t quite see eye to eye with Agterberg’s rebranding. Matheron’s disciples were taught to hold him in the highest regard as the Creator of Geostatistics. It was Matheron himself who called geostatistics a new science in the early 1960s. Here’s in a nutshell what had inspired Matheron so much in his most creative of days. He taught that, “geologists stress structure and statisticians stress randomness”. I liked that a lot. I would have liked it even more had Matheron shown how to test for absence or presence of structure. All it would have taken is to apply Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set. He would have had to count the number of degrees of freedom for each set. That was a bit of a problem. Matheron and his following never got around to counting degrees of freedom.

On a positive note, Matheron did test for associative dependence between lead and silver grades of drill core samples. This test may well have been the very reason why Matheron thought he was a statistician. His 1954 Formule des Minerais Connexes is indeed marked Note Statistique No 1. In his Rectificatif à la Note Statistique No 1, Matheron derived weighted average lead and silver grades. What he failed to derive were variances of weighted average lead and silver grades. So, I am quite pleased that the Centre de Géostatistique has posted so much of Matheron’s work. On the negative side, its webmaster saw fit to predate the evolution of Matheron’s new science of geostatistics. That’s why his very first paper did end up as Note Géostatistique No 1. Providentially, his 1954 Formule des Minerais Connexes and its Rectificatif are still marked Note Statistique No 1.

So it was that Matheron didn’t take to working with the Central Limit Theorem. David did recall the famous Central Limit Theorem in his 1977 Geostatistical Ore Reserve Estimation. He didn’t much work with it either. A critical subject that failed to make Matheron’s list of things to teach is one-to-one correspondence between functions and variances. Yet, it is a condition sine qua non in mathematical statistics. It is no wonder then that the properties of variances are beyond the grasp of the geostatistical fraternity. I have never thought much of Professor Dr Georges Matheron’s thinking. Whenever I do think of Matheron, I remember him as a self-made wizard of odd statistics.

Professor Dr Michel David took a shine to Matheron’s new science of geostatistics. David did so while he was teaching at l’École Polytechnique, University de Montréal, Québec, Canada. And he did predict that ‘statisticians would find many unqualified statements’ in his 1977 Geostatistical Ore Reserve Estimation. He didn’t predict he couldn’t care less if someone pointed out what was wrong and why. Some twenty years ago I did but few cared. So, I’ll just keep doing it again and again! Chapter 10 The Practice of Kriging shows how to do more with fewer boreholes by paying no attention at all to the rules of mathematical statistics.

Fig. 203. Pattern showing all the points within B,

which are estimated from the same nine holes.

David borrowed the above figure from Maréchal and Serra’s 1970 Random Kriging. Both were scholars at Matheron’s Centre de Morphology Mathématique. Here’s word for word what I have come to call David’s test for geostatistical acuity. “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics”. I have pointed out that a good test to find out whether one really understands mathematical statistics is to count the number of degrees of freedom for David’s system of equations. The correct count is zero! That’s how Matheron and his timid minions took reserve and resource estimation into a dead-end street.

But even more bad science pops up in Chapter 12 Orebody Modelling. In Section 12.2 Conditional Simulation, David wrote about the infinite set of simulated values. He wonders how to make infinite sets smaller and get models closer to reality. In Section 12.2.1 Using a Simulated Model, he wrote about some pudding proof and a posteriori proved simulations. But nobody cries out loud in the face of such blatant nonsense. What are the odds to win when playing 649?

So, why then did Agterberg try to rebrand Matheron the Founder of Spatial Statistics after he had passed away? Now that’s a long story. The short of it is that there are many more geoscientists than geologists on our little planet. Remember global warming? And how to assume spatial dependence between measured values in ordered sets? That’s what way too many geoscientists are taught. Stay tuned for real statistics. And tune out to surreal geostatistics.

One more message to CIM's President

CIM stands for Canadian Institute of Mining, Metallurgy and Petroleum. Once upon a time I was a proud CIM Member. Today I am the accidental CIM Life Member. My first message to CIM’s President was snail mailed on March 20, 1992. CIM’s President was William E Stanley of The Coopers & Lybrand Group in Vancouver. He was the first of many whom I had told why geostatistics is an invalid variant of applied statistics. We met, he listened to my story, and I wrote him a letter. CIM Bulletin of March 1989 had published Armstrong and Champigny’s A Study on kriging Small Blocks. Both authors were geostatistical scholars at the Centre de Géostatistique, France. They thought up the study since, “The kriging variance rises up to a maximum and then drops off.” What they also found out is that “…mine planners are often tempted to kriging very small blocks.” How about that? Smoothing is good but smoothing very small blocks is bad. That sort of a pass-the-buck study did pass David’s peer review with red flags blazing.

Early in 1990 we found out that Precision Estimates for Ore Reserves was rejected. Our paper showed how to test for spatial dependence between gold grades of ordered rounds in a drift. David’s 1977 textbook didn’t show how to test for spatial dependence, or how to count degrees of freedom. Neither did his work show how to derive unbiased confidence limits for metal contents and grades of in-situ ores. So, I put Geostatistics or Voodoo Science on paper, and The Northern Miner printed it on March 20, 1992. Champigny was no longer a geostatistical scholar at the Centre de Géostatistique in France but a Senior Consultant with The Coopers & Lybrand Group in Toronto. He never lost his passion for kriging and smoothing. As a matter of fact, he rounded up a team of anonymous ore reserve practitioners to stand on guard against the rise and fall of kriging variances. What he and his team did prove was that the properties of variances were far beyond their grasp. The Northern Miner put Champigny’s rambling tale in print on May 18, 1992. Armstrong went beyond the pale and lectured on scientific integrity in De Geostatisticis of July 1992.

Following is the text of my emessage of January 13, 2010, to Michael J Allen, CIM’s President, Vice President, Engineering, with Teck Corporation, Member of APEGBC and SME, and a CIM Fellow:

About twenty years ago I reported to CIM that geostatistics is an invalid variant of applied statistics. Geostatistocrats with CIM Bulletin promptly put up a spirited battle to salvage the new science of geostatistics. And a fine job they did! Matheron's madness of surreal geostatistics even survived the Bre-X fraud. Statistics turned into geostatistics under the guidance of Professor Dr Georges Matheron, a French probabilist who became a self-made wizard of odd statistics in the 1950s. A brief history of my 20-year campaign against the geostatocracy and its army of degrees of freedom fighters is chronicled on my website.

Dr Frederik P Agterberg, Past President, International Association for Mathematical Geosciences formerly know as International Association for Mathematical Geology, called Matheron (1930-2000) the Founder of Spatial Statistics. Agterberg ranked Matheron on a par with giants of real statistics such as Sir Ronald A Fisher (1890-1962) and Professor Dr J W Tukey (1915-2000). Agterberg was wrong! Matheron fumbled the variance of the length-weighted average in 1954. Agterberg himself fumbled the variance of the distance-weighted average first in his 1970 Autocorrelation Functions in Geology and once more in his 1974 Geomathematics. Agterberg is Emeritus Scientist with Natural Resources Canada. He ought to but has yet to explain why his distance-weighted average point grade does not have a variance. After all, Gemcom's geostatistical software converted Bre-X's bogus grades and Busang's barren rock into a massive phantom gold resource. I applied Fisher's F-test to prove that the intrinsic variance of Bre-X's phantom gold resource was statistically identical to zero. Duplicate test results for gold by cyanide leaching determined in a few boreholes would have been enough to unravel the Bre-X fraud in a timely manner.

I make a clear and concise case for real statistics. Test for spatial dependence by applying Fisher's F-test to the variance of a set of measured values and the first variance term of the ordered set. Chart a sampling variogram to show where spatial dependence in a sample space (or in a sampling unit) dissipates into randomness. We applied Fisher's F-test in Precision Estimates for Ore Reserves. And we did it again in our APCOM 2009 paper entitled Metrology in Mineral Exploration.

Geostatisticians assume spatial dependence between measured values in ordered sets, interpolate by kriging, smooth some kind of least biased subset of an infinite set of Agterberg's zero- dimensional and variance-deprived distance-weighted average point grades AKA kriged estimates or kriged estimators, and rig the rules of real statistics with reckless abandon. I urge CIM to investigate whether or not geostatistics is a scientific fraud. I do so as a CIM Life Member. Please do not assume that CIM need not resolve this matter.

To strip or not to strip?

CIM Bulletin approved Abuse of Statistics for publication. Dr Frits Agterberg wanted to know when and where Wells spoke so highly about statistical thinking. I wasn’t about when Wells said what he did. What I do know is that Darrell Huff said Wells did. That’s good enough for me. Huff did so in his 1954 How to Lie with Statistics. It was the very same year that young Matheron didn’t know how to test for spatial dependence between metal grades of ordered core samples, how to derive the variance of the set of metal grades, and how to derive the variance of the central value of the set. Huff never found out what Matheron did wrong. But then, neither did Matheron himself! Agterberg, Armstrong, David, Journel and scores of geostatistocrats never broke rank with Matheron.

I want to move fast forward to the present. Michael J Allan, CIM President in 2010, writes under President’s Notes about A time of renewal. Let’s see what else he wrote. “Our work in providing standard reserve and resource definitions that are used by the country’s securities regulators is an example of the ongoing technical contributions CIM makes to the industry at large”. For heaven’s sake! Geostatistics is as alive and flawed as it was in the 1970s. So it seems that CIM is not about to kill the incredible kriging machine. Surely, infinite sets of kriged estimates and zero kriging variances set the stage for boundless krige and smooth fests. APEGBC ‘s Code of Ethics is not written to rule against scientific fraud. What will kill the kriging machine is the study of climate dynamics on our little planet. No ifs and buts!

Whatever happened to Setting New Standards

The Bre-X fraud brought about an orgy of hand wringing but not even a token search for the truth, the whole truth, and nothing but the truth. The Ontario Securities Commission and the Toronto Stock Exchange set up a Mining Standards Task Force. Morley P Carscallen, OSC’s Commissioner, and John W Carson, TSE’s Senior Vice President, called on Canadian mining experts to set new standards. Of course, the old standards were dreadfully wrong. All it took was to assume gold between salted boreholes. That’s how Bre-X’s bogus grades and Busang’s barren rock added up to a phantom gold resource! So what did the Mining Standards Task Force do? It wrote a lot but little else. Here's why!

Hardcore krigers and cocksure smoothers were silent after Bre-X had gone bust. So much so that none served on the task force. They would have had a tough time to explain why kriging variances rise first and then fall. Or to prove why spatial dependence may be assumed without proof. Without genuine geostatisticians on board the task force was in limbo. The more so since I had proved that Bre-X was a salting scam. My son and I had shown in 1992 how to verify spatial dependence by applying Fisher’s F-test to the variance of test results for gold determined in bulk samples taken from a set of rounds in a drift, and the first variance term of the ordered set. Stanford’s Journel wrote to Professor Dr R Ehrlich, Editor, Journal of Mathematics Geology, (in those days!) that I am, “… too encumbered with Fischerian (sic) statistics.” I confess to have worked with Fisher’s F-test most of my life. So what?

The Mining Standard Task Force was put to work in July 1997. MSTF released its Interim Report in June 1998, and published its Final Report in January 1999. MSTF’s Final Report is high on verbiage but low on sound sampling practices and proven statistical methods.


It took a while to find out that Setting New Standards had done nothing to improve sampling practices in mineral exploration. The task force could have but did not show how to derive unbiased confidence limits for metal contents and grades of mineral inventories. Sadly, geostatistics was very much alive when I looked at CIM’s website under APCOM 2009. The program for this event set the stage for another krige-and-smooth bash. But this time the stage was set on my home turf. The scientific fraud behind the Bre-X fraud turned out to be alive ten years after MSTF’s Final Report had been released. It is as much alive as it was on Journel’s watch in 1992. So much for setting new standards!

I dug into my data base and retrieved test results for gold and silver determined in pairs of interleaved bulk samples taken from 1 m³ volumes of crushed gossan ore mined from a vertical pit. I had designed this sampling program to test for spatial dependence, to derive confidence limits for gold and silver contents and grades, and to estimate the intrinsic variances of gold and silver. The same test proved that the intrinsic variance of gold in Bre-X’s gold resource was statistically identical to zero. My son and I submitted to APCOM 2009 for review a paper on Metrology in Mineral Exploration. It was accepted as “a highly specialized topic reserved for the advanced geostatistician.” How about that!

My coauthor was talking about EMF in Europe. His presentation was also of interest at L’Ecole des Mines in Nantes. So, his mom and my partner for life listened to my APCOM 2009 talk in Vancouver, BC. I asked again why the variance of Agterberg’s distance-weighted average point grade had gone missing. The question was met with solemn silence. My spouse got some kind of revised textbook on a CD. Long ago I had bought a copy of the original edition. What it taught me was not to mess around with sloppy semi-variograms. That's why I took a systematic walk, tested for spatial dependence between hypothetical uranium concentrations, and counted degrees of freedom properly.

NRCan’s Emeritus Scientist is loath to bring back the long-lost variance of his distance-weighted average point grade. But then, how could JMG’s Editor-in-Chief possibly do what Rio Tinto wants him to do if each and every weighted average point grade were to have its own variance? He may need but a few boreholes. But what he does need most of all are infinite sets of distance-weighted average point grades to play with by hook or by crook. I really don’t give a fiddle about JMG’s Editor-in-Chief and his models. What I want is a world free of Matheron’s mad science of geostatistics.


I agree with H G Wells. I like statistical thinking. And I like to write about it. A good grasp of statistics is needed to bridge the gap between sampling theory and sampling practice. I have written a great deal about spatial dependence in sample spaces and sampling units. I want to write much more. My website gets a load of traffic. I blog for fun and play mind games when I do. I found out in 2007 that geostatistics plays a role in the study of climate change. It was some Canadian hockey stick that struck a panic button around the world. The study of climate change is much more relevant to the world than unbiased mineral inventories are to mining investors. Securities commissions ought to set rules and regulations that protect the public at large against all sorts of scientific frauds. The kriging machine will be shredded as soon as the ugly factoids are clear to investors. Surely, geoscientists should apply classical statistics when they study climate change. After all, functions without variances are as dead as dodos. CRIRSCO does not think so but I know!

What if our world were free of geostatistics

A world free of surreal geostatistics is long past due. Geostatistics was called a new science in the 1960s but it turned out to be an insidious scientific fraud. Real statistics would have nipped the infamous Bre-X fraud in the bud but CIM and IAMG ruled in favor of surreal geostatistics in the early 1990s. Matheron’s so-called new science of geostatistics did make a mess of the study of climate change. That’s why our world ought to get rid of surreal geostatistics. And fast! Come frost bites or sun burns!
Thanks to all those who read my blogs. More than two million have done so. But I got fewer than ten comments. So, what’s the matter? Is it the way I write? All I do is put in plain words why geostatistics is a scientific fraud. Here’s what I have been writing for more than twenty years. Each weighted average has its own variance. Could I have put it any other way? It is a truism in real statistics. The Central Limit Theorem is bound to stand the test of time. Why then was the variance of the weighted average done away with in Matheron's novel science? It was G Matheron in the early 1960s who called the weighted average "a kriged estimate" to honor D G Krige. Matheron never derived the variance of his own kriged estimate. Neither did any of his docile disciples.
What happened in the 1970s defies common sense and sound science. Was it Matheron himself or one of his disciples who thought that every one set of kriged estimates ought to have its own kriging variance? Stanford’s Journel was Matheron’s most astute student. He figured out that an infinite set of kriged estimates gives a zero kriging variance. Wow! Here’s what he taught Stanford's neophytes in a nutshell. Assume spatial dependence between measured values in ordered sets, interpolate by kriging, and smooth a little but not a lot. Stanford’s finest geostatistical mind never took to testing for spatial dependence, or to counting degrees of freedom.
Some readers may want to study the odd opus on geostatistics. I suggest a paper on kriging small blocks. It was put together by genuine geostatisticians from the Centre de Géostatistique in France. Professor Dr Margaret Armstrong and Normand Champigny were the first scholars who cautioned against reckless over-smoothing by careless mine planners.

I messed up my own copy of Armstrong and Champigny’s A Study on Kriging Small Blocks. The Canadian Institute of Mining, Metallurgy, and Petroleum has not yet posted this study on its website. It would have passed David’s review at CIM Bulletin with flying colors. Elsevier in 1988 published Professor Dr Michel David’s 1988 Handbook of Applied Advanced Geostatistical Ore Reserve Estimation. It’s by far the worst textbook I’ve ever read. Yet, universities all over the world have added this work of geostatistical fiction to their libraries.
It was early in October 1989 when Precision Estimates for Ore Reserves ended up on David’s desk. That's when we found out that geostatistical peer review is a shamelessly self-serving sham. Too many geoscientists do not know that measured values do give degrees of freedom, and that functionally dependent values (calculated values!) do have variances. If the difference between calculated and measured is a bit of a mystery, buy Moroney’s Facts from Figures, read Abuse of Statistics, or take Statistics 101.
So, who’s to blame for the rise of Matheron’s new science of surreal geostatistics? What comes to mind first and most of all is the Canadian Institute of Mining, Metallurgy, and Petroleum and its APCOM appendix. The International Association for Mathematical Geosciences and a score of institutions of higher learning such as McGill, Stanford, UBC, and scores of others, are close seconds.
Thank goodness I still have plenty of geostats and stats stuff to write about. Every night I fall sleep in my straight-thoughts jacket and figure out what to do next. Tonight it’s full moon in Vancouver. I feel really good about real statistics!

Matrix report worth its weight in gold

Same time thirteen years ago some of Bre-X’s test results for gold landed on my desk. I had not asked for Bre-X’s data. But I had agreed to and signed a three-year confidentiality agreement with Barrick Gold Corporation. I did so on December 16, 1996. It was the very same confidentiality agreement that Barrick Gold Corporation and Bre-X Minerals had signed a few days earlier. The first set of Bre-X data were transmitted by facsimile on December 17, 1996. I didn’t know then that my life would never be the same. Bre-X‘s infamous phantom gold resource is but part of a tangled tale with as many twists and turns as Matheron took to create his new science of geostatistics. It’s a tale that taught me a lot more about the mining industry than I cared to know.
I sorted out the Bre-X fraud faster than Bre-X’s salting squad took to cook it up. I think Barrick liked what I did. At least Barrick did when I applied statistics to prove that Bre-X was a salting scam. So much so that I signed on July 4, 1997 a Consulting Services Agreement with Barrick Goldstrike Mines Inc. I submitted on August 18, 1997 my report on Statistical Quality and Grade Control . Geostatisticians on Barrick’s staff didn’t think much of it. I had applied Fisher’s F-test to verify spatial dependence between gold grades of ordered core sections from a single borehole by applyingit to the variance of the set and the first variance term of the ordered set. I had done the same thing with Bre-X’s salted boreholes. Stanford’s Journel would have assumed rather than verified spatial dependence. But then, Matheron’s most gifted disciple never signed a Consulting Services Agreement with Barrick Goldstrike Mines Inc.
When I was working with Bre-X’s test results my closest contact was a staff mining engineer at Barrick Gold Corporation in Toronto. We got along great because he knew plenty about sampling and assaying. So, he knew why Bre-X’s bogus grades and Busang’s barren rock added up to a geostatistically engineered gold resource. He also knew how to test for spatial dependence, and why geostatistics should not be applied in reserve and resource estimation. And he asked me whether I wanted to take a look at a large set of borehole data for a real gold deposit. Guess what? So, I did agree to and signed on October 22, 1997 a confidentiality agreement with Barrick Gold Corporation. I submitted my report on Confidence Limits for Gold Contents and Grades on February 9, 1998. When I called my contact to find out what he thought of my report, he said, “It’s worth its weight in gold”. I didn’t ask him to put it in writing. His word was good enough for me!

Worth its Weight in Gold

Geologists, mining engineers and mineral process engineers rarely agree on metal grades of in-situ ores, mined ores and mill feed. I witnessed many such rituals. Top brass wants high mineral inventories in glossy annual report and geostatisticians always deliver. Barrick’s geologists may find confidence limits for gold contents and grades of mineral inventories a bit much of a commitment. Shareholders do want a measure for risk.
Another year passed by, Christmas 1999 came along, and the Confidentiality Agreement between Barrick Gold Corporation and Bre-X Minerals expired. I liked to talk about the Bre-X fraud. Barrick engaged lawyers who wanted to come to Vancouver and tell me not to talk. I called on a friend and the visit to Vancouver was cancelled. All I have done since Christmas 1996 is show why geostatistics is a scientific fraud.
What Barrick asked me ten year later blew my mind. Barrick wanted consulting services. I’m not about to describe the required services but it had nothing to do with confidence limits for gold contents and grades of in-situ ore. I agreed to and signed on March 20, 2007 a Consulting Services Agreement for services to be provided at Barrick Technology Centre, Vancouver, BC. My contact had a lot of practical experience but stood to gain from a touch of real statistics. Before we could get going he was needed at Barrick’s Bulyanhulu gold mine in northwest Tanzania. Long before Barrick acquired Placer-Dome and its former Bulyanhulu gold deposit I knew Placer-Dome had born geostatisticians on board.
A Munk Debates on scientific fraud makes no sense whatsoever. Who would dare make a case for scientific fraud? Yet, a scientific fraud underpins the geostatistical practice of reserve and resource estimation all over the world. Blatantly biased, shameless self-serving peer review is all it took. But that’s another story. I have called it Behind Bre-X, The Whistleblower’s Story.

Who wants more Munk Debates

Who wouldn’t! Debates beat apathy. The Munk Debates is cool. The more so since climate change was the theme for the Fourth Munk Debates. Climate change, just like continental drift, has been around for a few billion years. It took geologists from 1915 to 1950 to slow down to continental drift and call it plate tectonics. So, it’s about time to debate climate change. And why not call it weather dynamics? I work with metrology, the science of measurement. I took a crack at testing whether or not annual temperatures at several locations in Canada have changed significantly as a function of time. The average temperature of 6.57 centigrade in 2007 at Ottawa International Airport was significantly higher than the average temperature of 4.79 centigrade in 1939. Similarly, the average temperature of 8.30 centigrade in 2007 at Toronto International Airport was significantly higher than the average temperature of 6.04 centigrade in 1939. Average temperatures didn't change at international airports in Calgary, Vancouver and Victoria. Neither did the average temperatures in Coral Harbour and Iqaluit change significantly during the test period under examination.


Some grasp of statistics is required to apply Fisher’s F-test and verify spatial dependence between annual temperatures in ordered sets. Weather dynamics do change from day to day, from week to week, and from month to month. Such short-term changes in temperatures do not merit a Munk Debates. What does merit a Munk Debates is the question whether or not geostatistics is a scientific fraud.
Here’s in a nutshell my take on the Fourth Munk Debates. Elizabeth May is Leader of the Green Party of Canada. She is a gifted and confident speaker. She knows a lot of environmental stuff. She doesn’t know much about temperatures recorded by Environment Canada. Given that the Leader of the Green Party does speak a lot in public, she should know where temperatures went up or down, since when, and by how much.
George Monbiot was her partner in the Fourth Munk Debates. He is a superb scribe with the Guardian newspaper where his penchant for hyperboles runs rampant. How to measure climate change as a function of space and time is far beyond his grasp. Monbiot says cool things such as, “Canada is a cultured, peaceful nation, which every so often allows a band of Neanderthals to trample over it.” He doesn’t know Sir Ronald A Fisher ‘s work is trampled over by a tribe of statistically dysfunctional geoscientists bred in France, Great Britain, and elsewhere on this planet. The May/Monbiot side debated The Case For Climate Change.
Lord Nigel Lawson and Bjorn Lomborg debated The Case Against Climate Change. Lord Lawson is in a class apart when it comes to a life of public service in the United Kingdom of Great Britain. His work has done much to cool down global warming to climate change. He is the author of An Appeal to Reason, A Cool Look at Global Warming. He is the Chairman of Oxford Investment Partners, and of Central Europe Trust. As such, he knows all about mining conglomerates and mineral inventories in annual reports. He is bound to remember the Bre-X fraud. He may be unaware that geostatistical software converted Bre-X’s bogus grades and Busang’s barren rock into a huge phantom gold resource. Neither may Lord Lawson remember the cast of characters behind the Bre-X fraud.
Bjorn Lomborg’s claim to fame is based on The Skeptical Environmentalist and on Cool It. He is adjunct professor at the Copenhagen Business School. He also set up the Copenhagen Consensus Center to bring together those who set priorities for the world. I had brought to his attention in August 2008 that junk statistics underpins Matheron’s new science of geostatistics. I wanted to know whether he applies geostatistical data analysis. Environment Canada points to geostatistical data analysis in its handbook for inspectors. The skeptical environmentalist did not respond to my message.
The Merks and Merks team wants to debate The Case Against Geostatistics. Dr Frits P Agterberg, Emeritus Scientist with Natural Resources Canada, and Dr Roussos Dimitrakopoulos, Professor with McGill University, are highly qualified to debate The Case For Geostatistics. Both are serving in key positions with IAMG (International Association for Mathematical Geosciences). Once upon a time, IAMG stood for International Association for Mathematical Geology. Nowadays, our world needs more mathematical statistics.

Chatting with NRCan's Emeritus Scientist

Dr Frits P Agterberg is Emeritus Scientist with Natural Resources Canada. He wrote a textbook on Geomathematics and scores of papers on a wide range of geological topics. He is the nimblest of geostatistical minds on this planet. His gift to assume spatial dependence between measured values in ordered sets is second to none but Stanford’s Journel. I called him on November 4, 2009, at NRCan in Ottawa but he was away from his Office. I caught him at home when I called his residence at 09:10 AM PDST. I asked him to explain why his zero-dimensional distance-weighted average point grade does not have a variance.

He hummed and huffed but didn’t speak to the matter of the missing variance. All I wanted to know is why the variance of his distance-weighted average went missing. I pointed out that the Central Limit Theorem pops up if all of his measured points are equidistant to his selected point. NRCan’s Emeritus Scientist beats around the bush with the best. His textbook does refer to the Central Limit Theorem in Chapter 6 Probability and Statistics and Chapter 7 Frequency Distributions of Independent Random Variables but not in Chapter 10 Stationary Random Variables and Kriging. NRCan’s Emeritus Scientist has yet to give a clear and concise explanation why the Central Limit Theorem doesn’t apply to his distance-weighted average point grade.

I included Agterberg’s problems in my talk about Metrology in Mineral Exploration. I wanted to make a case at APCOM 2009 that distance-weighted average point grades do have variances. Nobody was ready for my show-and-tell but I got a gift. It was Clark’s Practical Geostatistics 2000. I found out that semi-variograms are still alive and below par. Here’s Clark’s problem. Her set of five (5) hypothetical uranium data doesn’t display a significant degree of spatial dependence. Thus, the concentration at the selected coordinates is not necessarily an unbiased estimate. Let’s find out what happens when coordinates are selected beyond her sample space.

Who expects the distance-weighted average point grade to converge on zero? And who expects it to converge on the arithmetic mean? It's a good test to find who is geostatistically gifted and who is not. I would rather test for spatial dependence between measured values in ordered sets and chart sampling variograms that show where spatial dependence dissipates into randomness. Come hell, high water, global cooling, polar warming, or another Bre-X.

My first APCOM affair was just as cluttered with geostat drivel as are all of IAMG’s shindigs. McGill’s Professor Dr Roussos Dimitrakopoulos sought to shed light on stochastic mine planning optimization. He is Editor-in-Chief, Journal for Mathematical Geosciences. That’s why all his work passes his own litmus test for scientific integrity with flying colors. Somehow, it may have slipped his mind how geostatistical software converted Bre-X’s bogus grades and Busang’s barren rock so smoothly into a massive phantom gold resource. But then, the geostatocracy has worked long and hard to ensure mining professionals never get a grasp of classical statistics.

It brings me back to my chat with NRCan’s Emeritus Scientist. I brought to his attention that a good test to verify McGill's stochastic mine planning optimization would be to apply it to Bre-X’s data. Agterberg saw it differently because Bre-X's data was “no real data”. No real data? But mining investors thought Bre-X was real! Didn't Gemcom’s software convert Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource? And wasn't the battle to take over Bre-X Minerals a really bizarre affair?

This was my second chat with NRCan’s Emeritus Scientist after we had found out in 1989 that geostatistics is a scientific fraud. It brought back an odd dialogue in 1992 with Dr W D Sinclair, Editor, CIM Bulletin, and Dr F P Agterberg, Associate Editor. We talked about a technical brief on Abuse of Statistics. I'll keep that tangled tale for some other place and time!

Spatial dependence in mineral exploration

Some twenty years ago my son and I submitted to CIM Bulletin a paper on Precision Estimates for Ore Reserves. David, CIM Bulletin's reviewer, blew a fuse because we didn’t refer to “twenty years of geostatistical literature”. We did study David’s 1977 Geostatistical Ore Reserve Estimation and Clark’s 1979 Practical Geostatistics. Neither author showed how to test for spatial dependence. So, we showed how to test for spatial dependence between gold assays determined in bulk samples taken from twelve (12) rounds in a drift. CIM Bulletin was but one of several journals to reject our paper. Yet, the very same paper was praised by and published in Erzmetall 44, October 1991. We could not show how to estimate the intrinsic variance of gold because but a single bulk sample was taken from each round.
It was easy to estimate the intrinsic variance of gold in Bre-X’s phantom resource. Bre-X’s quality control program was based on selecting and testing duplicate test portions of every tenth crushed and salted core sample. The set of duplicate gold assays for Bre-X’s bonanza borehole BSSE198 gave enough degrees of freedom to estimate the analytical variance with a high degree of precision. Fisher’s F-test proved that the analytical variance and the first variance term of the ordered set are statistically identical. Hence, the intrinsic variance of gold in BSSE198 was statistically identical to zero. Plenty of placer gold was present in crushed and salted core samples but Bre-X’s bonanza borehole BSSE198 was barren.


When APCOM 2009 asked for abstracts, I talked to my son about presenting one more paper on our home turf. His talk about EMF at some school of mines in Nantes, France, took him too far away from Vancouver to attend APCOM 2009. Our abstract was based on a bulk sampling program at the Cerattepe project in Turkey where core recovery was poor. So, I advised my client to implement an interleaved bulk sampling program in order to derive unbiased confidence limits for in-situ gold and silver. Our abstract was accepted and Metrology in Mineral Exploration was approved.
I spoke to a small group on Thursday, October 8, 2009, at 15:30. I showed how to unscramble the Bre-X fraud, and how to derive the statistics for Cerattepe's bulk sampling program.

Spatial dependence is significant at 99.9% probability
Lag of 4.30 m at 95% probability is defined for gold

Spatial dependence is significant at 99.9% probability
Lag of 4.09 m at 95% probability is defined for silver

I explained how to correct those sampling variogram for the extraneous measurement variance estimated from pairs of interleaved primary samples, and how to derive 95% confidence limits for in-situ masses of gold and silver.

I asked my audience why the variance of Agterberg’s distance-weighted average point grade is still missing. I didn't get any response. Not a single question was asked. There was but a pinch of polite applause. My soul mate got an anonymous note together with the second coming of Clark’s 1979 Practical Geostatistics on DVD. Which APCOM 2009 sponsor ignored my question but did hand my spouse that anonymous note? Was it Gemcom? Or did Geovariances do it?
I was tickled pink with that priceless gift. In her first coming Clark cooked up a semi-variogram, berated those who "sloppily" call it a variogram. Yet, Clark praised Journal and his buddies for teaching her all she knows about “the theory of the Theory of Regionalized Variables.” Journel may well have taught Clark how to assume spatial dependence between measured values in ordered sets. He might even have cautioned Clark, too, not to become “too encumbered with Fischerian [sic!] statistics”. But what did Professor Dr William V Harper teach Dr Isobel Clark between 1979 and 2000? Sadly, Clark’s learning curve simply flat lined! She still doesn’t test for spatial dependence in sampling units and sample spaces. She still scolds those who work with variograms rather than with her own sacred semi-variogram. There's still no progress!
Statistics or geostatistics? Sampling error or nugget effect? Clark talked about those questions at WCSB4 in Cape Town on 21-23 October 2009. Sampling error adds a nice touch of Gy’ological thinking to Clark’s repertoire. Testing for spatial dependence failed to make her grade. Why did she take the factor two (2) out of degrees of freedom for ordered sets. Why does she deem too sloppy sampling variograms that show where orderliness in sample spaces or sampling units dissipates into randomness. Clark and Harper are ready to take Gy's sampling theory to sampling practices in mineral exploration, mining, processing, smelting and refining? Why does Harper not recognize that geostatistics is a scientific fraud? Strip the variance of the distance-weighted average, assume spatial dependence between measured values, interpolate by kriging, smooth the least biased subset of some infinite set of distance-weighted averages, and rig the rules of real statistics with impunity.

Who wrote bogus stats, when, where, and why

Professor Dr Roussos Dimitrakopoulos came up all the way from Down Under to chair a Forum on Geostatistics for the Next Century at McGill University on June 3-5, 1993. His task was to honor Professor Dr Michel David for writing the very first textbook on Matheron’s new science of geostatistics. David didn’t know how to test for spatial dependence and how to count degrees of freedom. He wrote his first textbook against all odds since he didn’t even know that functions do have variances. I have written quite a bit about the properties of variances. So, I send by registered mail an abstract to that futuristic forum at McGill University. Some person at McGill’s Conference Office encouraged me in an unsigned letter of March 31, 1993, to submit my abstract to another event. I'll have to dig up more bits and pieces about genuine variances.
Dimitrakopoulos likes McGill a lot. In fact, he settled down in La Belle Province after the Bre-X fraud was no longer on his mind. In a candid interview with the National Post on August 15, 2005, he clarified the intricacies behind his valuations of mining projects. Here’s what he said, “You drill a few holes, you think you understand something but what you know is very, very little. Uncertainty means probabilistic models, and there are a gazillion types of them.” How about that? Some mining investors might wonder how RD selects the least biased probabilistic model. Peter Ravenscroft, a senior executive with Rio Tinto and an expert at geostatistics, thinks what RD does is kind of cool and gave him a stack of dough.
Professor Dr Roussos Dimitrakopoulos was present at APCOM 2009 in Vancouver, British Columbia. The first line of his abstract reads, “Conventional approaches to estimating reserves and optimizing mine planning and production forecasting result in single, often biased forecasts.” I wonder what would have happened if Stochastic Mine Planning Optimization: New Concepts, Applications, and Monetary Value in an Ever Uncertain Market, had been applied to Bre-X’s exploration data. I also wonder why regulators and financial institutions do not insist the International Organization for Standardization set up a Technical Committee on Reserve and Resource Estimation. It's long past due! Matheron thought he was a statistician in 1954. Yet, his Note Statistique No 1 shows he didn't know how to test for spatial dependence between metal grades in ordered core samples. Neither did he know how to derive variances of length-weighted average lead and silver grades determined in core samples of variable lengths. So much for Matheron's new science of geostatistics!

Dr Frederik P Agterberg wrote in 2000 that Matheron was the Founder of Spatial Statistics. Matheron thought he was a statistician in 1954 when he wrote his Note Statistique No 1. He didn't write about spatial dependence between metal grades of ordered core sections with variable length. He did derive length-weighted average lead and silver grades but didn't derive the variances of these central values. In 1907 he stirred up "Brownian motion on a straight line." He did so because he liked Riemann integrals better than Riemann sums. He wrote in his 1978 Foreword to Mining Geostatistics why he proposed the name geostatistics in the 1960s. Professor Georges Matheron would have been shocked had he read in his obituary that he was the Founder of Spatial Statistics. Agterberg invited me on October 1, 2004, to present my views at the next IAMG annual meeting in Toronto. I happen to know a lot about IAMG events where geostatistocrats talk bafflegab. I would rather make my case against bogus stats at APCOM 2009.

Dr Michel David wrote a few words of caution in his 1977 Geostatistical Ore Reserve Estimation. First, he wrote, "...statisticians will find many unqualified statements..." Then, he blew the sales of his work by writing, "This is not a book for professional statisticians." But he was indeed right. David did prove it when he wrote his test for geostatistical proficiency. He took M&S's set of nine (9) measured values and "estimated" the same set of sixteen (16) what he came to call "...points..." He wrote on page 286 of his textbook, "Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics." Why did the author of the very first textbook on geostatistics fail to derive the variance of each of this sixteen (16) functionally dependent values? Why didn't he count degrees of freedom? If M&S's set of nine (9) measured values were evenly spaced, the set and the ordered set would give df=n-1=8 and dfo=2(n-1)=16 respectively. Why is the geostatocracy still asleep at the switch? Why is Bre-X's massive phantom gold resource all but forgotten?

A Marechal and J Serra wrote Random kriging in 1970 to celebrate the first krige and smooth bash in North America. M&S toiled under Matheron's tutelage at his Center de Morphology Mathematique, Fontainebleau, France. So, why did M&S set out to simplify Matheron's kriging equations with their own random kriging procedure? Under Punctual Kriging in Random kriging they show how to get a set of sixteen (16) functionally dependent values from a set of nine (9) measured values. M&S didn't show how to derive a variance of a functionally dependent value. Neither did they show how to test for spatial dependence by applying Fisher's F-test to the variance of the set of measured values and the first variance term of the ordered set. What Matheron never taught M&S was how to count degrees of freedom. In his own 1970 Random functions and their applications in geology Matheron wrote, "Let us denote a Brownian motion on a straight line." In Matheron's mind it somehow seemed to replace Riemann sums with Riemann integrals. Matheron never explained what Brownian motion and ore deposits have in common. M&S put Random kriging "within the geostatistical framework of the French school." Go figure why!

Dr Isobel Clark is the author of Practical Geostatistics. She wrote on the first page of Chapter 5 Kriging, "It would seem sensible to use a weighted average of the sample values, with the 'closer' sample values having more weight." On the same page she wrote, "The arithmetic mean is simply a special case where all the weights are identical." She wrote in her Preface that Journel and others at Fontainebleau taught her all she knows about the theory of the Theory of Regionalized Variables." She transposed for "mathematical convenience" the factor two (2) in dfo=2(n-1), the number of degrees of freedom for an ordered set of n measured values. That's how Clark's semi-variogram was born. Why did Fisher's F-test for spatial dependence between hypothetical uranium data fail to make Clark's grade in her 1979 Practical Geostatistics? And why does nobody care?

Statistically dysfunctional geoscientists write all sorts of things that are bound to hound them in time. Read what Stanford’s Journel wrote to the Editor of the Journal for Mathematical Geology. What he did was set the stage for conditional simulation on Stanford stationary. Take note of when he wrote it. And read what JMG’s Editor wrote to me. So, my feeling that geostatistics is invalid might be correct. How about that? He also wrote that different “flavors” of geostatistics may fail at different times. Now that’s kind of cool. I do know which flavor failed in the Bre-X fraud. It was the flavor of assuming continued gold mineralization between salted boreholes. The odd geostatistician might be taught how to test for spatial dependence and how to count degrees of freedom. Most are doomed to assume, krige, smooth, and rig the rules of statistics.

Degrees of freedom fighters struck at Stanford

It’s a strange but annual ritual of sorts. Degrees of freedom fighters assume, krige, smooth, and rig the rules of statistics. Today's fighters call that mathematical statistics. This year the stage was set at Stanford Campus on 23-28 August. Once upon a time IAMG stood for International Association for Mathematical Geology. A few years ago IAMG morphed into International Association for Mathematical Geosciences. Its present mission is to promote, worldwide, the advancement of mathematics, statistics and informatics in the Geosciences. This latest variant of IAMG talks about statistics without degrees of freedom.

The famous feud between Pearson (1857-1936) and Fisher (1890-1962) came about because of degrees of freedom. Fisher added degrees of freedom to Pearson’s chi-square distribution, and was knighted in 1952. Fisher's F-test is applied to verify spatial dependence in sampling units and sample spaces alike. Pearson’s coefficient of variation, too, stood the test of time. Meanwhile in Algiers, young Matheron didn’t count degrees of freedom. In fact, he didn’t have a clue what degrees of freedom were all about. IAMG’s most advanced thinkers still do not count degrees of freedom.

The very first textbook about Matheron’s new science of geostatistics was David’s 1977 Geostatistical Ore Reserve Estimation. Table 1.IV Copper grades Prince Lyell in Chapter 1 Elementary Statistical Theory and Applications gives a chi-square distribution with 13 degrees of freedom. David’s Index lists neither Chi-square distribution nor Degrees of freedom. What the author did list are Best linear unbiased estimator, Brownian motion, and Bull’s eye shot.

Figure 203 on page 286 of David’s first textbook takes the cake for boldness. The same figure saw the light as Figure 10 in Marechal and Sierra’s 1970 Random Kriging. It is printed in Proceedings of a Colloquium on Geostatistics held on campus at the University of Kansas, Lawrence on 7-9 June 1970.

Fig. 203. Pattern showing all the point within B,

which are estimated from the same nine holes.

David derived the covariances of his set of sixteen "samples", each of which was "estimated" from the same nine holes.What he didn't do was count degrees of freedom. His set of nine (9) holes gives df=n-1=9-1=8 degrees of freedom. The ordered set gives dfo=2(n-1)=2(9-1)=16 degrees of freedom. The number of degrees of freedom is a positive integer for evenly spaced holes but becomes a positive irrational for unevenly spaced holes.

A set of sixteen (16) functionally dependent values does not give a single degree of freedom. What David did not know either is that every functionally dependent value does have its own variance. He did know that his set of nine (9) holes gives an infinite set of functionally dependent values. David called them simulated values but statistically dysfunctional thinkers call them kriged estimates. The question is then why kriging variances of sets of kriged estimates became the building blocks of Matheronian geostatstics.

Dr Jef Caers chairs IAMG 2009. He is Associate Professor, Energy Resources Engineering, with Stanford University. His 1993 MS in Mining Engineering and Geophysics and his 1997 PhD in Engineering were obtained with the Katholieke Universiteit, Leuven, Belgium. He speaks French fluently. This is why he should belatedly review Matheron’s 1954 Note Statistique No 1 to assess if anything else but degrees of freedom and primary data went missing. Some scholar at Stanford Earth Sciences should know all about associative dependence, functional dependence and spatial dependence. I think Dr Jef Caers may be that scholar!

Casting dice and tossing coins at Stanford

Behind Stanford’s motto “The wind of freedom blows” is a rich history. It was President Gerhard Casper on October 5, 1995 who put a score of fine points to it. Who could possibly object to the freedom to teach and be taught sound sciences? When President Casper spoke in 1995 the freedom to assume spatial dependence between measured values in ordered sets had been entrenched in geostatistics since 1978. Herbert Hoover, Thirty-First President and Stanford’s very first mining engineer, would have been shocked. Who would put a mine stope together by casting dice? How could geostatistics have converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource?

Here’s what I have been trying to bring to the attention of Dr J L Hennessy, Stanford’s President. Geostatistics ignores the concept of degrees of freedom and violates one-to-one correspondence between functions and variances. Agterberg’s distance-weighted average does not have a variance. Neither does David’s distance-weighted average. I pointed out that it took the Papacy 300 years to right a wrong. I did so the last time I wrote to Stanford’s President on February 13, 2008. I wrote that I thought Stanford could right a wrong much faster. He could have asked a Stanford statistician whether or not the geostatocracy has the freedom to assume spatial dependence between measured values in ordered sets. What I wrote in 2008 didn’t hit Dr Hennessy’s list of things to do.

I do appreciate my own freedom and am a stickler for degrees of freedom. So, I looked at Stanford’s statistical scholars and warmed to what I read about Professor Dr Persi Diaconis. He looked like the kind of scholar who would take seriously my crusade against the geostatocracy and its army of degrees of freedom fighters. Stanford Report of June 7, 2004, pointed out, “Persi Diaconis has spent much of his life turning scams inside out.” Now there’s a professional scam buster of sorts. It became even better than I thought it would be when I read what Professor Dr Persi Warren Diaconis had done. He left home at 14, hit the road with Dai Vernon, the famous Ottawa-born slight-of-hand magician, and got Vernon’s magic touch.

When I was searching Stanford’s website for a genuine statistician, I found out that Dr Diaconis doesn’t respond to email. I took a chance and did send him an email anyway on February 23, 2009. That was more than year after my last email to Stanford’s President. Diaconis is indeed true to his word and did not respond to my email. I had suggested that Stanford should give real statistics a fighting chance. So, I decided to call Diaconis but nobody picked up the phone. I called between March 26 and April 22, 2009, and did so between 13:00 and 16:00 PST. I called sixteen times and the line was busy twice. I could have but decided not leave a message.

Diaconis knows how to toss a coin. So much so that he can make the same side of a coin come up ten times in a row. He designed a mechanical coin tossing contraption that gives the same odds. What he did do was defy the Central Limit Theorem. Coins and dice played cameo roles when I taught sampling theory and practice in places are far apart as Greenland and Tasmania, and as Finland and the Philippines. I put in plain words how to tamper with the outcomes of tossing coins and casting dice. What I didn’t show is how to test for bias. A Stanford student should cast the same die often enough to infer absence of bias within acceptable bias detection limits. The catch-22 is that abrasion is bound to cause a bias before acceptable bias limits are obtained.

I taught sampling theory and practice on the basis of a binomial sampling unit that consists of 90% white beans and 10% of the same but red-dyed beans.

Each participant would take a small increment and a large increment, and count white and red beans in each. This simple sampling experiment made it easy to explain Visman’s sampling theory and practice, and his composition and distribution components of the sampling variance. Visman’s work proved that the most effective method to estimate the variance of the stochastic variable of interest in a sampling unit or a sampling space is to partition the set of primary increments into a pair of interleaved subsets. Of course, one pair of subsets gives but one degree of freedom. That’s why SQC programs should be implemented on a routine basis. The interleaved sampling protocol has been incorporated in several ISO standards.

The wind of freedom blows at Stanford University. What geostatistocrats have blown is the concept of degrees of freedom. Agterberg blew the variance of the distance-weighted average. Journel blew Fisher’s F-test for spatial dependence. Once upon a time Herbert Hoover wrote, “It should be stated at the outset that it is utterly impossible to accurately value any mine, owing to the many speculative factors involved. The best that can be done is to state that the value lies between certain limits, and that the various stages above the minimum given represent various degrees of risk.” Hoover’s 1909 Principles of Mining Valuation, Organization and Administration still make sense. Why then is the world’s mining industry hooked on assuming, kriging, smoothing, and rigging the rules of real statistics?

Teaching junk statistics at Stanford

Stanford University is Professor Dr Andre G Journel’s world. He has put down deep roots at Stanford since 1978. Journel teaches the same flaky stats that Professor Dr Georges Matheron taught him between 1969 and 1978. Journel was Matheron’s most gifted student. Matheron taught him all of the ins and outs of his novel science of geostatistics. Matheron may not have told Journel that he thought in 1954 he was a statistician. It took almost ten years to teach Journel how to assume, krige, and smooth with a lot of confidence and pride. Journel was Mining Project Engineer at the Centre de Morphology Mathematique from 1969 to 1973, and Maitre de Recherches at the Centre de Geostatistique from 1973 to 1978. Not surprisingly, he worked as profusely with symbols as Matheron did in his magnum opus. What Matheron failed to show his star disciple is how to test for spatial dependence between ordered sets of measured values in sample spaces and sampling units. Matheron and Journel never found the lost variance of Agterberg's distance-weighted average point grade.

Journel is the lead author of Mining Geostatistics. When the ink had dried in 1978 he took his book to Stanford’s students and taught them all about assuming, kriging and smoothing. My copy is a “1981 reprint with corrections.” Matheron’s Foreword makes a deeply dense read. In contrast, Dr Isobel Clark’s Preface to her 1979 Practical Geostatistics makes an easy read. Her cradle once rocked on the side of the Channel where Sir R A Fisher was knighted. Clark confessed it was Journel who taught her all she knows about the Theory of Regionalized Variables. Clark messed up degrees of freedom for ordered sets of measured values. She slashed for "mathematical convenience" the factor 2 in df₀=2(n-1) degrees of freedom for ordered sets, cooked up her silly semi- variogram, and scolded the poor souls who “sloppily call it a variogram”. Clearly, Clark and Journel disagreed about semi-variograms and variograms. Neither knew how to test for spatial dependence, how to chart sampling variograms, or how to count degrees of freedom.
Matheron’s 1978 Foreword to Mining Geostatistics went off on a tangent just as much as did his 1954 Note statistique No 1. He beat around the bush about geologists who “stress structure” and statisticians who “stress randomness.” Matheron’s point of view flies in the face of Visman’s sampling theory with its composition and distribution variances. Matheron predicted, “The user of Mining Geostatistics will come across nothing more than variances and covariances, vectors and matrices”. Matrices and vectors do indeed abound from cover to cover but so do pseudo variances and pseudo covariances. What all those so called “variances” and “covariances” in Mining Geostatistics do have in common with genuine variances and covariances are squared dimensions. The concept of degrees of freedom, too, failed to make the grade in Matheronian geostatistics. And that’s what will kill the kriging game!
I came across a genuine variance in a numerical example on page 63 of Mining Geostatistics. The authors divided a stope into four equal units, and assigned to each unit a grade equal to the outcome of a cast of “an unbiased six-sided die.” Now that does indeed give a genuine variance. Casting an unbiased die a large number of times gives a uniform probability distribution with a population mean of μ=3.5 and a population variance of σ²=2.917. The authors deserve praise for giving correct values, and for pointing out that the die ought to be unbiased. Surely, Stanford’s students ought to be taught how to measure the risk of playing all sorts of games of chance.

No real data in 1954 - Casting dice in 1978

The set of three (3) stopes is given on the same page. Each set of four units within its stope was put together with a six-sided unbiased die such that each unit has the same mean of 3.5. That sort of applied research is time-consuming but of critical importance when teaching all of the intricacies of geostatistics. A touch of classical statistics is required to test whether or not a given die is unbiased. The question of whether Journel's die was biased may have been solved by assuming it was unbiased. Fisher’s F-test shows that the variances of the sets and the first variance terms of ordered sets are statistically identical. Read what Journel said about “Fischerian (sic) statistics” in October 1992. How’s that for creative thinking and writing?
The zero kriging variance of σ²_k=0 is given on page 308, Chapter V The Estimation of in situ resources in Mining Geostatistics. Another unique feature of Matheronian geostatistics is one-to-one correspondence between zero kriging variances and infinite sets of kriged estimates. Even the OCS might find it a bit of a stretch to report a 95% confidence interval of zero ounces of gold for a mineral inventory with 9.9 million ounces.
Armstrong and Champigny solved this Catch-22 with a strict caution against over-smoothing. They did so in A Study on Kriging Small Blocks, CIM Bulletin, March 1989. The study implies that requirement of functional independence may be violated a little but not a lot All that geostatistical gobbledycook is cooked up because one-to-one correspondence between distance-weighted averages and variances became null and void in Agterberg's 1974 Geomathematics.
On a positive note, Dr John L Hennessy, Stanford’s President, is but one of the few leaders at institutes of higher learning who did bother to respond to my letters.

On August 23-28, 2009, IAMG’s Annual Conference will be held at Stanford University. What a wonderful opportunity for Stanford's President to peek around the corner and ask why the variance of Agterberg’s distance-weighted average point grade is still missing. Or he might ask Professor Dr Persi Diaconis to pose a few questions on his behalf. Diaconis is Stanford's Mary V Sunseri Professor of Statistics and Mathematics. He’ll know all about the Central Limit Theorem and its role in sampling theory and practice.