NRCan stuck with geostatistics

Dr Frederik P Agterberg went to work with geostatistics long before NRCan stood short for Natural Resources Canada. He is still Emeritus Scientist with NRCan’s Geological Survey of Canada. He is one of the most gifted geostatisticians in the world. As such, he has a soft spot for Professor Dr Georges Matheron and a penchant for his magnum opus. So much so that he called him the Founder of Spatial Statistics. He did so after Matheron had passed away in 2000. Matheron’s disciples didn’t agree with Agterberg’s view. Matheron taught them how to assume, krige and smooth with infinite confidence. So, they thought of him as the mastermind behind the Centre de Géostatistique and the Centre de Morphology Mathematique. What Matheron taught his disciples was inspired by one or other innovative theme that would call on his most creative thinking. That’s why they thought of him as the Creator of Geostatistics.

Professor Dr Georges Matheron (1930-2000)
Creator of Geostatistics
Founder of Spatial Statistics
Self-made Wizard of Odd Statistics

My son and I checked out what sort of new science Matheron had created. We found out in 1989 that his new science of geostatistics is an invalid variant of applied statistics. As a matter of fact, the author of the very first textbook on geostatistics couldn’t possibly have scored a passing grade on Statistics 101. He praised the “famous Central Limit Theorem” but failed to work with it when he should have. He never tested for spatial dependence between measured values in ordered sets. That’s why it didn’t take us long to get to the bottom of what was wrong. The author had seen fit to shelve degrees of freedom. But it took a long time to unravel what else was shelved, who did it, when, where and why. The power of the internet did make it possible to trace Matheron’s new science to its roots in the early 1950s. In those bygone days young Matheron was a budding geologist who went to work with what he then thought was applied statistics. As such he proved a tenuous grasp of applied statistics early on during his calling.

What struck me as a bad omen for Matheron’s new science of geostatistics was the fact that his grasp of the properties of variances can be traced to the French school of sampling in-situ ores and mined ores. What gets me hopping mad is sloppy sampling and statistics. That’s why I want to put in plain words what Matheron did in the early 1950s. In those days he was a novice geologist with the French Geological Survey (BRGM) in Algeria. His very first paper was called Formule des Minerais Connexes. It was dated November 25, 1954 and marked Note Statistique No 1 straight above its title. It would seem that Matheron saw himself as a statistician of sorts. All the same, CdG’s webmaster early in this century saw fit to mark his very first paper as Note Géostatistique No 1. Perhaps a dash of deception but too little and too late to fool anyone but the odd hardcore kriger.

Matheron’s Note Statistique No 1 was not reviewed by his peers. It took me quite a while to find out that Matheron was without peers. That was just as well since he didn’t quote literature on statistics. What was worse is that he didn’t report any primary data for l’Oued-Kebir. In fact, reporting primary data ranked just as low on his list of things to do as did counting degrees of freedom. Matheron derived population means of μ1=0.45% for lead and μ2=100 g/t for silver, and population variances of σ1=1.82%2 for lead and σ2=1.46 (g/t)² for silver. How he could have done so much with so little is a mystery. A finite set cannot possibly give population means and variances. But that’s what Matheron thought he got! He tested for associative dependence between lead and silver grades and got a correlation coefficient of ρ=0.85. He didn’t point out whether this correlation coefficient was statistically significant at 95%, 99% or 99.9% probability. What put a monkey wrench in Matheron’s first crack at statistics is that he didn’t test for spatial dependence. As luck would have it, Stanford’s Journel and Matheron’s most talented disciple put forward in 1992 that spatial dependence between measured values in ordered sets may be assumed. Testing for spatial dependence was just as trendy in 1992 as it was in 1954.

Matheron may have had a bit of an epiphany when it hit him that l’Oued-Kebir core samples did vary in length. That’s why on January 13, 1955 he tagged on a Rectificatif to his Note Statistique No 1. What he didn’t tag on were the lengths of his core samples. How he derived weighing factors was as clear as drill mud. The same weighting factors should have been applied when he tested for associative dependence between lead and silver. Weighting factors should also have been applied to test for spatial dependence between metal grades determined in ordered core samples of variable length from a single borehole. He didn’t know that degrees of freedom are positive irrationals when core samples vary in length. In other words, he didn’t even know how to fingerprint boreholes. What a shame that peerless Matheron kept on writing more of the same.

No matter what odd statistics Matheron did cook up it would smoothly pass his own peer review. It did make a mockery of statistics that his new science of geostatistics made it all the way to the USA in 1970. Tagged along on the trip were A Marechal and J Serra. They would assist Matheron in making a strong case for his novel science. The stage was set on campus at the University of Kansas for a geostatistics colloquium on 7-9 June 1970. In those days, Matheron, Marechal and Serra were geostatistical scholars at the Centre de Morphology Mathematique at Fontainebleau, France. Matheron himself had thought up Brownian motion along a straight line. It would set the stage for random functions to be continuous between measured values. He did so in his rambling Random Functions and their Application in Geology. What was still beyond Matheron’s grasp is how to verify 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. It did so since counting degrees of freedom had not yet made Matheron’s brief list of significant things to do.

Marechal and Serra in Figure 10 of Random Kriging show how to derive a set of sixteen distance-weighted averages from a set of nine measured values. What M&S didn’t derive was the variance of each and every distance-weighted average. David in 1977 took a long look at M&S’s data but didn’t derive the variance of each and every distance-weighted average either. What David did run into were infinite sets of distance-weighted averages. Journal in 1978 was so taken with David’s infinite set of distance-weighted averages that he took the zero variance with hook, line and sinker. Matheron in 1960 found out about D G Krige’s work at the Witwatersrand gold complex in South Africa and came up all sorts of krige-inspired adjectives and verbs. That’s in a nutshell why Matheron’s new science of geostatistics took on a life of its own for no reason whatsoever.

Agterberg’s 1970 Autocorrelation Functions in Geology was about what he then called “a geologic prediction problem”. He defined a set of measured values at unevenly distributed positions in a sample space. His problem was not so much how to derive the value of the stochastic variable at the selected position. His real problem was that he didn’t derive the variance of his distance-weighted value at the selected position. On a positive note, he didn’t make a point of the fact that his set of measured values didn’t define an infinite set of variance-deprived distance-weighted average values. He didn’t test for spatial dependence by taking a systematic walk that visits each measured value but once, and that covers the shortest possible distance between all positions. Agterberg’s 1970 geologic prediction problem popped up as “a typical kriging problem” in his 1974 Geomathematics.

A geologic prediction problem in 1970! A typical kriging problem in 1974! The eulogy for Matheron in 2000! The silence of NRCan’s Emeritus Scientist in 2010! What’s the matter with NRCan’s brass? When will Dr Frederik P Agterberg be asked to explain why his distance-weighted average doesn’t have a variance?

Back when NRCan was Canmet I knew several of its scientists. Most of all I remember Dr Jan Visman, a Dutch mining engineer with a keen interest in coal processing. He was an accidental sampling expert of sorts because of his need to understand coal processing. We were members of ASTM D05 on sampling and analysis of coal. He headed the Western Regional Laboratories of the Department of Mines and Technical Survey until his retirement in 1976. I owe him a debt of gratitude. His 1947 PhD thesis made it clear that the variance of the primary sample selection stage is the sum of composition and distribution components. That's why I’m keeping his memory alive on Wikipedia.

Dr Robert Sutarno was a true expert on applied statistics in general and interlaboratory test programs in particular. He taught me a lot about statistical analysis of interlaboratory test programs, and about preparation and certification of reference samples. I met him when we were members of the Canadian Advisory Committee to ISO Technical Committee 102 on iron ore. We traveled to Japan in 1974. The fact that Bob spoke Dutch was a bonus for me. I was more conversant with German and French than with English when we came to Canada in 1969. But that's not the end of my case against geostatistics!

The Age of Statistics is upon us

How I wish it were true! But that’s what W J Reichmann thought in 1961. It’s the very title of Chapter 1 in his delightful Use and abuse of statistics. The first line of its Preface points out: “Very few people nowadays can progress very far without at some point coming in contact with statistics”. Now that’s what I have been trying to tell the geostatistocracy since the early 1990s. So I pointed to Reichmann’s work in Abuse of Statistics. I have a wide range of books on sampling and statistics. Applied statistics underpins sampling practice just as much as probability theory does sampling theory. As such, degrees of freedom play a key role in sampling practice but none at all in sampling theory. No ifs or buts! Except in CIM Bulletin. And in Matheron’s tour de force of course!

The Editor of CIM Bulletin wrote on September 21, 1992 that “articles of a controversial nature” may be published under the heading Forum. It’s hard to believe that applied statistics was controversial in those days. Of course, The Geological Society of CIM has got to be vigilant when deciding what GEOSOC members get to read. So I was pleased that Abuse of Statistics was approved for publication with the proviso that the source of my most favorite quote be disclosed. That’s what Dr Frits P Agterberg wanted. In those days, Agterberg was Associate Editor with CIM Bulletin. In due course, he would wish not only geologists but geoscientists at large to work with geostatistics. Here’s what H G Wells (1866-1946) once said, “Statistical thinking will one day be as necessary for efficient citizenship and the ability to read and write.” How about that? I do not know when or even if Wells said it and to whom. But Darrell Huff thought Wells had said so. Surely, the author of How to lie with statistics would not tell a lie. So I put Huff’s book on my short list of references. And I put Wells’ thought on statistical thinking on top of my brief but somehow controversial article.


Peer review at The Geological Society of CIM in those days was a sham. Not only was it blatantly biased but it was also shamelessly self-serving. Why did it see fit to reject Precision Estimates for Ore Reserves? It did so because we didn’t refer to twenty years of geostatistical literature! And why didn’t we do that? We didn’t because Professor Dr Michel David’s 1977 Geostatistical Ore Reserve Estimation didn’t show how to derive unbiased confidence limits for metal grades and contents of in-situ ores. As a matter of fact, geostatisticians still do not know how to derive unbiased confidence limits for metal contents and grades of ore reserves. All the same, The Geological Society of CIM rejected Dependencies and degrees of freedom and The properties of variances just as brazenly as did the Journal for Mathematical Geology.

CIM’s GEOSOC failed to grasp in 1992 why the properties of variances and the concept of degrees of freedom cannot be ignored with impunity. That’s when Bre-X Minerals was getting ready to explore for gold in Kalimantan, Indonesia. I’m sick and tired thinking of Bre-X’s salting scam. The more so because Bre-X’s massive phantom gold resource was cooked up by assuming the same gold grades between step-out lines of boreholes as within lines of boreholes. It's the doctrine of Stanford’s Journel. Agterberg argued in 2009 that Bre-X’s test results for gold are “no real data”. The problem was that the Ontario Securities Commission thought they were. And many Bre-X investors thought likewise!

I’m still tickled orange that applied statistics played a key role in unraveling the Bre-X fraud. I have large sets of bogus gold assays determined by cyanide leaching 250 g test portions taken from crushed and salted drill core sections. I’ll show why it makes sense to test for rather than assume spatial dependence between measured values in ordered sets.

The first step was to insert three (3) lines of kriged boreholes between two lines of (2) salted boreholes in the South-East Zone. The second step was to derive statistics for two (2) lines of salted boreholes and three (3) lines of kriged boreholes. A little applied statistics was enough to show that geostatistics creates spatial dependence where it does not exist.

Neither line of salted boreholes displays a significant degree of spatial dependence. In contrast, all lines of kriged boreholes display a significant degree of spatial dependence. What a shame it’s all bogus gold. What’s real is that even test results for gold in salted boreholes do give degrees of freedom. In contrast, functionally dependent values such as kriged estimates are never blessed with degrees of freedom. All of that is old hat in applied statistics! The age of applied statistics may still not be upon us but the age of the internet is. The worldwide net has made it easy to find out who messed up what, when, where and why. Geostatistics is about to look a lot worse.

Stanford's Journel shed light on geostatistics

Journel got down to shedding some light on October 15, 1992. He did so in a six-page letter to the Editor-in-Chief of the Journal for Mathematical Geology. At that time, JMG’s Editor-in-Chief was Dr Robert Ehrlich, a professor at the Department of Geological Sciences with the University of South Carolina. Journel had “…a bit reluctantly...” agreed to go through my various notes. He left it up JMG’s Editor-in-Chief to decide whether his carefully crafted response should be sent to me. He did point out “…however, I strongly feel that Math Geology has had more than its share of detracting invectives”. Good grief! What could possibly be wrong with geostatistics?

My son and I knew exactly what was wrong in David’s 1977 Geostatistical Ore Reserve Estimation. But it would be the next Millennium before Matheron’s work was posted on the internet. So, I wrote ten letters to JMG’s Editor-in-Chief between November 14, 1990 and August 4, 1992. I had mailed drafts of several papers on different topics, and called him several times. I had also mailed him a copy of my book on Sampling and Weighing of Bulk Solids. I was quite pleased to have received a copy of Journel’s response to my various notes. The more so since JMG’s Editor-in-Chief wrote, “Your feeling that geostatistics is invalid might be correct”. Such a pity that he left. What I still don’t know is whether he jumped or was pushed.

Journel had written “It seems to me that Merks’ anger arises from a misreading of geostatistical theory, or a reading too encumbered by classical “Fischerian” [sic!] statistics”. He really got into the nitty-gritty of geostatistics when he pointed out “The very reason for geostatistics or spatial statistics in general is the acceptance (a decision rather) that spatially distributed data should be considered a priori as dependent one to another, unless proven otherwise.” Journel couldn’t have said it any worse. It’s a textbook case of circular logic. Accept spatial dependence between ordered sets of measured values in sample spaces and sampling units. Do not prove otherwise by applying Fisher’s F-test. What a bunch of blatant blarney! Who wouldn’t get angry?


A peeved student of geostatistics and a friend of mine had gifted me his pre-read copy of Journel & Huijbregts 1978 Mining Geostatistics. On page 1 the authors define what the book is all about. Here’s literally what they wrote: Etymologically, the term geostatistics designates the statistical study of natural phenomena. G Matheron (1962) was the first to use this term extensively, and his definition will be retained: “Geostatistics is the application of the formalism of random functions to the reconnaissance and estimation of natural phenomena.”

In a footnote on the very same page Journel & Huijbregts mentioned J Serra and A Maréchal. David’s 1977 Geostatistical Ore Reserve Estimation, too, mentioned J Serra and A Maréchal. In fact, David took this example from Serra and Maréchal’s 1970 Random Kriging. Matheron’s 1970 Random Functions and their application in Geology, too, was presented at the 1970 Geostatistics colloquium on campus at The University of Kansas in June 1970. And so was Agterberg’s 1970 Autocorrelation Functions in Geology.

That’s when I made up my mind to study the lives and times of the greatest geostatistical minds on our little planet. Journel had gone to Stanford University in 1978 as a Visiting Associate Professor of Applied Earth Sciences. He liked it so much that he never left. As a matter of fact, he has spent more time teaching geostatistics at Stanford than he studied it under Matheron’s humdrum guidance. Journel was Mining Project Engineer at Matheron’s Centre de Morphologie Mathematique from 1969 to 1973. Journel was Matheron’s most gifted disciple by far. His list of honors and awards is striking to say the least. From 1973 to 1978 he did time as Maitre de Recherches at Matheron’s Centre de Geostatistique. Professor Dr A G Journel has been teaching Stanford’s students from 1978 to the present. He has taught them all of his own silly nitty-gritty on Matheron’s novel science of geostatistics.

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!