Setting new standards?

The Bre-X fraud inspired the Toronto Stock Exchange (TSE) and the Ontario Securities Commission (OSC) to set up a task force. Its objective was to take a close look at National Instrument 43-101. The Members of the task force are given in this Interim Report. Mr Morley P Carscallen, OSC’s Vice Chair, and Mr John W Carson, TSE’s Senior Vice-President, took on this task in April 1997. It is a fact that Bre-X’s bogus gold grades and Busang’s barren rock were made to look by hook and by crook like a gold resource. But who were the crooks? And who set the hook for Bre-X’s shareholders? OSC’s own qualified persons have yet to grasp the fact that geostatistics is a scientific fraud! Perhaps ironically, it was geostatistical software that made Bre-X’s bogus grades and Busang’s barren rock to look like massive gold resource!

I have put on paper why geostatistics is a scientific fraud. A few simple steps were all it took to cook it up! The first step was to strip the variance off the distance-weighted average. The second step was to call what was left a kriged estimate to honor D G Krige and his work. Matheron taught his disciples how to work with infinite sets of kriged estimates and zero kriging variances. What a shame that such a simple scientific fraud underpinned what was called a new science. Matheron himself never got into counting degrees of freedom. Neither did Stanford’s Journel, UBC’s Sinclair, and similarly gifted scholars.  

Young Dr A J Sinclair took to geostatistics in the 1970s. He may well have thought that Matheron had a fresh take on applied statistics. In those days Sinclair was entrusted with teaching UBC’s students all about Earth Sciences. CIM Bulletin asked Sinclair in 1990 to review Precision Estimates for Ore Reserves. My son and I had shown how to test for spatial dependence between a set of gold grades determined in ordered rounds in a drift. Given that interleaved bulk samples had not been selected, it was impossible to estimate the intrinsic variance of gold. Professor Dr A J Sinclair, PEng, PGeo rejected our article. We were pleased that it was praised by and published in Erzmetall, October 1991.

What a surprise that David’s peers wanted to praise his 1977 Geostatistical Ore Reserve Estimation! Why would his peers want to praise infinite sets of simulated values? The stage for an international forum was set at McGill University on June 3-5, 1993. It was called Geostatistics for the Next Century. What is so striking in retrospect is the fact that Bre-X Minerals was already drilling in Borneo when David was praised by his peers! Nobody was interested in the properties of variances in 1993! Yet, the additive property of variances in a measurement chain played a key role in unscrambling the Bre-X fraud.

 Measurement variance included

 Measurement variance subtracted

The Mining Standard Task Force released its Final Report in January 1999. Why had MSTF not pointed out that geostatistical software had convert Bre-X’s bogus grades and Busang’s barren rock so slickly into a massive phantom gold resource? MSTF’s Final Report was made public in January 1999. On a positive note, Dr A J Sinclair no longer graces National Instrument 43-101. On a negative note, Setting New Standards still didn’t explain at all how the Bre-X fraud could have been nipped in the bud. So it was that the Mining Standard Task Force ended up as a farce. The properties of variances were nowhere to be found. Sinclair still teaches students at UBC's Department of Earth and Ocean Sciences how to assume spatial dependence, krige, smooth, and rig the rules of applied statistics with impunity. So much for scientific integrity!

I have set up several sources of information on my website. Under Correspondence are listed all sorts of letters in a context of source and time. Academic freedom to teach a scientific fraud makes no sense at all. The fact that "geostatistics has flourished in the scientific literature for more than four decades" does not imply that spatial dependence between measured values in ordered sets may be assumed. Neither does it imply that degrees of freedom need not be counted.

To krige or not to krige?

Not only is it a verb with a touch of a noun but it is also a true eponym. Matheron had written in 1960 what he himself had called Krigeage d’un panneau rectangulaire par sa périphérie. Nowadays it is posted as Note géostatistique No 28. An anthology of Matheron’s life and time, and of his creation of geostatistics, is posted on a massive website. Danie G Krige had put together a Preface to David’s 1977 Geostatistical Ore Reserve Estimation. References to Krige pop up on many pages. Journel’s 1978 Mining Geostatistics, too, refers not only to D G Krige but also to the zero kriging variance. 

Geostatistical software made Bre-X’s bogus grades and Busang’s barren rock look like a massive gold resource. So why had geostatistics been hailed as a new science in the 1970s. The Bre-X scam was well on its way when geostatisticians got together to praise David’s 1977 Geostatistical Ore Reserve Estimation. He was praised at a celebration called Geostatistics for the Next Century at Montreal on June 3-5, 1993. My take on The Properties of Variances clashed with the celebrations at McGill University. What applied statistics did do is prove that the intrinsic variability of Bre-X’s gold was statistically identical to zero. How about that? The geostatocracy is still poised in 2012 to assume, krige, smooth, and rig the rules of applied statistics with impunity.  

David’s 1977 textbook displayed his tenuous grasp of applied statistics. The author points on page 33 of Chapter 2 to what he calls “the famous central limit theorem”. On page 286 in Figure 203 he shows how to derive a set of sixteen (16) “famous central limit theorems” from the same set of nine (9) holes. Next, he points out on this page, “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics”. Counting degrees of freedom would have  shown that the author of the first textbook on geostatistics did grasp applied statistics.

It is simple to verify spatial dependence between measured values in an ordered set by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. The F-test requires that degrees of freedom be counted. Stanford’s Journel claims that spatial dependence between measured values may be assumed. For crying out loud! He did so in his letter to JMG’s Editor. Now how’s that for a nouveau science! Surely, spatial dependence in sample spaces should be proved beyond reasonable doubt. It took but two steps to go from goofy geostatistics to a genuine fraud. The first step was to strip the variance off the distance-weighted average. The second step was to call a kriged estimate what had once been a distance-weighted average with a variance. Now that’s simple comme bonjour, n’est ce pas? Kriging is a stacked game of chance. Thou shall not krige when scientific integrity matters!

Mineral Inventory Studies of Precious Metal Deposits in British Columbia is one work of geostatistical fiction that I have kept on file. The study that peeked my interest most of all was Ordinary Block Kriging with Geological Control, A Practical Approach to Estimating Mineral Inventory, Nickel Plate Mine, Hedley, British Columbia. I did so simply because primary data are given. The authors of this study were A J Sinclair et al. It hit the spotlight on June 3-5, 1993 when “Geostatistics for the Next Century” was hailed for no reason whatsoever!   

Dr A J Sinclair, Professor Emeritus (Geological Engineering), was 2000-2001 recipient of a distinguished lecturer award. Sinclair talked about “Geology and data analysis: essential components of high quality resource/reserve estimation”.  He talked across the country in both official languages. His paper on Ordinary Block Kriging with Geological Control, A Practical Approach to Estimating Mineral Inventory, Nickel Plate Mine, Hedley, BC was presented when David was praised at McGill in June 1993. I applied Fisher F-test to test for spatial dependence.

 

Fisher's F-test for spatial dependence

The set of production data didn’t display a significant degree of spatial dependence. Neither did the set of ordinary block kriging data. Bartlett’s chi-squared test would have shown significant discrepancies not only between variances of sets but also between first variance terms of ordered sets. 

    

95% Confidence limits for arithmetic means

The central values in this table are arithmetic means. Confidence intervals and ranges are derived in Excel spreadsheet files. Shortly, a link to both files will be be posted.

  

The Society for Mining, Metallurgy, and Exploration published in Volume 308 Transactions 2000 a reviewed paper entitled Borehole statistics with spreadsheet software. The paper shows how to fingerprint boreholes. Its reviewer expected it would “stir up a hornets’ nest” but it never did! This paper underpins a report in which confidence limits for a large gold reserve had been derived. It was submitted to Barrick Gold early in 1998. 

Metrology in mining and metallurgy

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

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

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

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

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

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

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

 

Borehole statistics with spreadsheet software
SME Volume 308, Transactions 2000

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

From human error to scientific fraud

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

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

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

Figure 203 – Pattern showing all points within B,

which are estimated from the same nine holes

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

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

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

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

UBC still stuck with geostatistics

 Professor Dr Alastair J Sinclair has been teaching earth sciences at the University of British Columbia since 1964. It was but a dozen years after Matheron tried his hand at applied statistics. Young Georges Matheron in 1952 was an up-and-coming geologist in Algiers. He had a penchant for applied statistics in those days. For example, he knew how to test for associative dependence between lead and silver grades in drill core samples of variable lengths. What he did not know was how to derive variances of length-weighted average lead and silver grades. Perhaps ironically, young Matheron at that time thought he was working with applied statistics. Yet he didn’t know how to test for spatial dependence in sample spaces and sampling units by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set.  Neither did he derive length-weighted average lead and silver grades for his data set.

Professor Dr Georges Matheron
Abuser of applied statistics
Creator of geostatistics
Founder of spatial statistics

Professor Dr Georges Matheron was not at all into sharing primary data with his students. Even for his 1965 PhD Thesis he saw fit to cook up a funny pair of minuscule primary data sets.

 Matheron's primary data sets for 1965 PhD Thesis

So I have decided to show how to test for spatial dependence between Matheron’s make-believe primary data. All I did was apply Fisher's F-test to the variance of the set and the first variance terms of the ordered set and compare the observed F-values with tabulated F-values.

Stats for Matheron's 1965 PhD Thesis

Matheron’s magnum opus is posted on a massive website. Its webmaster made a few minor changes to suggest that Matheron had applied geostatistics somewhat sooner than he had done in real time.

It is ironic to the extreme that geostatistics was hailed as a new science when Matheron and his disciples brought it to campus at the University of Kansas in June 1970. Matheron’s own tour-de-force at this colloquium was to invoke Brownian motion along a straight line. He did so to infer that his random functions are continuous between measured values. The study on Random kriging by A Marechal and J Serra at the Centre de Morphology Mathematique seemed successful under Matheron’s supervision. Figure 10 in this 1970 study metamorphosed in Figure 203 on page 286 in Chapter 10 The Practice of Kriging in Professor Dr Michel David’s 1977 Geostatistical Ore Reserve Estimation.

David’s 1977 textbook and Gy’s 1979 Sampling of Particulate Materials, Theory and Practice, stand side-by-side on a shelf in my office. One time soon I’ll use them to prove how the French sampling school has messed up statistical thinking. And all it really took was to ignore one-to-one correspondence between functions and variances, to assume spatial dependence between measured values in ordered sets, and to pay no attention to counting degrees of freedom.

Dr Alastair J Sinclair, PEng, PGeo

UBC Emeritus Professor

Professor Dr Alastair J Sinclair described in Applied Mineral Inventory Estimation how his “exciting and invigorating career” took off when he was exposed to Matheron’s ideas, and how he had “the good fortune to work with Journel, Huijbregts and Deraisme”. Good grief! Those were Matheron’s earliest students who took his musings for dogma, and who didn’t have a clue that the variance of the distance-weighted average cum kriged estimate had vanished into thin air on Matheron’s watch. Sinclair’s list of those who he was “fortunate to have worked with at various times” reads like a Who’s who in the world's  geostatistical fraternity. Sinclair credits all of them to have contributed to his education. For once I do agree! I’m all in favor of giving credit where credit is due. But to give credit to everybody who has taught Professor Dr Alastair j Sinclair, PEng, PGeo how to apply a flawed variant of applied statistics is a bit over the top. Some geostatistocrats on Sinclair’s list know that each and every distance-weighted average cum kriged estimate does have its own variance. No ifs or buts! Whether Al likes it or not!

I wrote one more letter to Dr Martha C Piper, President, The University of British Columbia. I pointed out that H G Wells (1866-1946) had predicted, “Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write”. I mentioned that statistical thinking served me well indeed as a consultant, a lecturer, an author and a publisher, and as a global citizen of sorts on IMO and ISO Technical Committees such as TC69-The application of statistical methods.

Professor Dr Nathan Divinsky was charged in 1949 with the teaching of mathematics to UBC students. He retired as a professor in the mathematics department in 1991. I met a few of his former students who enjoyed his teaching and appreciated the power of applied statistics. Once upon a time I called him to ask whether statistical inferences are possible without degrees of freedom. I’ll always remember that he said, “But without degrees of freedom statistical inferences are impossible”. Dr Nathan Divinsky passed away at 86. He was married for eleven years to former Prime Minister Kim Campbell. Who would dare doubt such a short, crisp and to the point response by a Professor of Mathematics? May he rest in peace!

McGill toils with Markov chains

McGill University claims to be at the cutting edge of defining ore deposits with Markov chains. The National Post on August 15, 2005 published an article with the caption “It’s mining by the numbers”. McGill’s Professor Dr Roussos Dimitrakopoulos pointed out that, “Uncertainty means probabilistic models, and there are a gazillion types of them”. He has yet to show how to select the least biased model. He has a $3.5 million budget to put Markov chains to work. I’ll call him Dr RD for short. I do respect his blatant chutzpah! Dr RD cited a study by the World Bank that alleged 73% of North American mines had failed. What he didn’t point out is that geostatistical software is to blame!

It is a bit of a mystery when, where and why Dr RD made up his mind to travel all the way back to Markov chains. Stringing Markov chains overnight on a fast computer seems to somehow pin down ore deposits. But Dr RD didn't know that Markov chains cannot possibly give unbiased confidence limits for metal contents and grades of ore deposits! Markov and his chains may have made some sense before Fisher and Pearson feuded about degrees of freedom for the chi-square distribution. Why is it that counting degrees of freedom is still baffling the most gifted geostatistical gurus?

What’s more, Dr RD’s grasp of the properties of variances was already flawed in June 1993. At that time he was in a rush to get Geostatistics for the Next Century going. I had submitted by registered mail on March 10, 1993 an abstract for The Properties of Variances. I received an unsigned letter dated March 31, 1993. As luck would have it “a number of potential participants and their very interesting abstracts couldn’t be accommodated”. It so happened that I was one of those! All I wanted to do was show  how to derive unbiased confidence limits for metal contents and grades of ore reserves. We had shown how to do it in 1990. Professor Dr Michel David blew a fuse because we had applied “our own method”. Whose method had he expected? Given geostatistical peer review at CIM Bulletin in the 1990s I had asked JASA’s Editor for a courtesy review of The Properties of Variances. It passed JASA’s litmus test! A copy of The Properties of Variances is posted on my website. Peruse the properties of variances, count degrees of freedom, and derive confidence limits for mineral inventories. It is simple comme bon jour! I did it for Barrick Gold in 1998.

Professor Dr Michel David and his 1977 Geostatistical Ore Reserve Estimation were honored at Montreal, Quebec on June 3-7, 1993. Geostatistical scholars had come to praise the author of the first textbook. It deals with Matheron’s new science in mind numbing detail. David brought up “the famous central limit theorem “ on page 33 in Chapter 2 Contribution of Distributions to Mineral Reserves Problems. Chapter 10 The Practice of Kriging shows how to derive sixteen (16) famous central limit theorems from the same nine (9) holes. David pointed out on page 286, “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics!” Good grief! Counting degrees of freedom for his system of equations would have been a good test to find out whether David did grasp applied statistics. David's Index does not refer to Markov chains. But who would want to bring up Markov chains at David’s bash?

Lost: variance of kriged estimate

Found: zero kriging variance 

It was none other than Stanford’s Professor Dr Andre G Journel who did! He had put forward a paper to shore up his own vision. It was called “Modeling Uncertainty, Some Conceptual Thoughts”. He had embellished his thoughts with prettified statements such as stochastic simulation, random models, Bayes’ updating, likelihood functions, sequential simulation and non-Gaussian models. That’s what preoccupied the mind of Matheron’s most gifted disciple in June 1993. It may well have turned off some of those who had come to praise David’s 1977 Geostatistical Ore Reserve Estimation! 

Every Spring quarter Emeritus Professor Dr A G Journel teaches an advanced PhD level seminar. What he does not teach is that each and every distance-weighted average AKA kriged estimate does have its own variance in applied statistics. What he ought to study is Dr Isobel Clark’s 1979 Practical Geostatistics. She derived the variance of a distance-weighted average AKA kriged estimate. Alas, Dr Clark didn’t test for spatial dependence between hypothetical uranium concentrations in her ordered set. Neither did she know that degrees of freedom for her set are positive irrationals rather than positive integers.

SADG with Markov chains?

SAGD stands for Steam Assisted Gravity Drainage. It makes oil easier to recover. What has SAGD to do with Markov chains? That’s what I want to discuss in this blog! I was into consulting at Fort McMurray long before Markov chains were strung together. I have worked with applied statistics since the 1960s. It would seem that geostatistocrats have forgotten that geostatistics converted Bre-X 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 gold was statistically identical to zero. No if or buts! The Ontario Securities Commission and the Toronto Stock Exchange set up a Mining Standards Task Force to protect mining investors. Canada’s most gifted geostatisticians got this task force to work without Fisher’s F-test. What boggles the mind is that the mining industry took to Stochastic Mine Planning with Markov Chains! It’s but one more flavor of geostatistics.  It was bred at Stanford University and put to work at McGill University. Geostatistocrats need not assume spatial dependence between measured values in ordered sets. CPUs crunch numbers overnight and stochastic mining plans pop up in the morning. It’s Markov’s gift for those who are not into counting degrees of freedom!

Here are a few notes on SAGD Reservoir Characterization Using Geostat: Application of the Athabasca Oil Sands, Alberta Canada. Its authors are Jason A McLennan and Clayton V Deutsch. The latter may well remember that once upon a time at some event we shook hands. What he does not remember is one-to-one correspondence between functions and variances. It is impossible to score a passing grade on Statistics 101 by stripping the variance off the distance-weighted averages AKA kriged estimate! So I decided to look up what Clayton V Deutsch had been taught where, when, why and by whom. He earned a BSc in Mining Engineering at the University of Alberta in April 1985. Next, he got a Mac in Applied Earth Sciences (Geostatistics) at Stanford University in April 1987. Finally, he was granted his PhD in Applied Earth Sciences (Geostatistics) at Stanford University in June 1992. Now how’s that for kriging out loud!

I had mailed on November 14, 1990 a copy of Sampling and Weighing of Bulk Solids to Professor Dr R Ehrlich, Editor, Mathematical Geology. Here’s what he wrote on October 26, 1992: “Your feeling that geostatistics is invalid might be correct”. Attached to his letter was Professor Dr A G Journel’s response. The Editor’s letter and Journel’s response are posted on my website. Journel pointed out,”I’ll leave it to you to decide whether this letter should be sent to J W Merks; however, I strongly feel that Math Geology has had more than its share of detracting invectives”. Journel’s circular logic was a brazen tour de force.

I want to show what McLennan and Deutsch didn’t do in this SAGD study before putting in plain words  who set the stage for Markov chains, when, where and why.

Top Surface and Bottom Surface: Realization 50

These figures show Northing and Easting coordinates and sets of measured values for top and bottom surfaces. What comes to mind when I look at such plots are door-to-door peddlers of days gone by. They would walk such that the shortest distance is covered when each and every door is called on but once. Today’s door-to-door peddlers are into saving souls. And I’m into peddling on-line.  My eBook on Sampling and Weighing of Bulk Solids has been posted. Foremost on my mind is Metrology in Mining and Metallurgy. But I tend to slow down a bit when voodoo science drives me up a hanging wall!

 SAGD Reservoir Characterization with Applied Statistics

A spreadsheet template with SAGD statistics will be posted on geostatscam.com. In due course I’ll show how to derive the mass of oil in each block and the variance of that mass. The same method can be applied not only to in-situ ores and oils but also to mined ores and oils. All it takes is to put the additive property of variances to work. Neither Markovian chains nor Matheronian geostatistics have a role to play in mineral exploration and mining.

David’s 1977 Geostatistical Ore Reserve Estimation shows in Figure 203 on page 286 a set of sixteen (16) points. Each point is a function of the same set of nine (9) holes. One-to-one correspondence between functions and variances dictates that each point does have its own variance. David on page 323 points to the infinite set of simulated values and ponders how to make it smaller. Journel and Huibregts 1978 Mining Geostatistics on page 308 points to a zero kriging variance. None of these geostatistocrats got into counting degrees of freedom!

Here’s what Dr Isobel Clark acknowledged in the Preface to her 1979 Practical Geostatistics, “And finally to André Journel and others at Fontainebleau who taught me I know about the theory of the Theory of Regionalized Variables". It was Dr Clark who taught that each distance-weighted average AKA kriged estimate does indeed have its own variance. Stanford’s Journel didn’t know simply because Matheron didn't know. It was Matheronian thinking that has messed up ore and oil reserve estimation all over the world. A few mining giants are sold on Markov chains. Canadian regulators do not know which end of a Markov chain is up!

When to work with Markov chains

Once upon a time a keen geologist measured the degree of associative dependence between lead and silver in lead ore. Next, he put on paper Formule des minerais connexes and called it Note statistique No 1. He didn’t report his primary data but did correct an error. In time, he became famous. So much so that he set up the Centre de Géosciences/Géostatistique at Fontainebleau, France. Professor Dr G Matheron will be remembered either as the creator of geostatistics or as the founder of spatial statistics. As fate would have it; never in his life did he test for spatial dependence between measured values in ordered sets by applying Fisher’s F-test.

Professor Dr George Matheron (1930-2000)
Creator of geostatistics
Founder of spatial statistics

Abuser of applied statistics

Matheron’s most gifted disciple was Dr A G Journel. He put forward on October 15, 1992, “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”. It was a prima facie case of circular logic. He did respond to a request from Professor Dr Robert Ehrlich, Editor, Mathematical Geology. Stanford’s Journel also deemed my reading too encumbered with classical “Fischerian” statistics. So the coauthor of Mining Geostatistics put forward, “In presence of dependence the classical notion of degrees of freedom vanishes: n spatially dependent data do not provide n degrees of freedom”.

Now that’s where I didn't see eye to eye with Professor Dr A G Journel. A set of n measured values always gives df=n-1 degrees of freedom whereas an ordered set of n measured values gives dfo=2(n-1) degrees of freedom for the first variance term. Degrees of freedom are positive integers for sets of measured values with the same weight but positive irrationals for sets of measured values with variable weights. The concept of degrees of freedom has left little space for ifs and buts!

Index A. Geostatistical Concepts in my copy of 1978 Mining Geostatistics does not refer to Degrees of freedom between Deconvolution and Discontinuity at the origin of a sampling variogram. What went missing on Matheron’s watch was the variance of the distance-weighted average AKA kriged estimate. Incredibly, Matheron’s students never told him that too much was lost. On the contrary, the zero kriging variance of an infinite set of kriged estimates took on a silly life of its own. Neither does it list Markov chains above Massive deposits. Stanford’s Professor Dr A G Journel may not have been as hot on Markov chains as McGill’s Professor Dr R Dimitrakopoulos is on its role in stochastic mine planning. Unbiased confidence limits for metal contents and grades of mineral deposits can only be derived with applied statistics.

Andrey Markov (1856-1922)

A Markov chain is a mathematical system that transitions from one state to another between countable (finite) numbers of possible states. One ought to peruse the properties of variances before toiling with Markov chains. Study what McGill’s Dr RD didn’t want to know about the properties of variances when Geostatistics for the Next Century came about at the McGill Conference Office on June 3-5, 1993. What a pity that deriving unbiased confidence limits for metal grades and contents of ore reserves is still beyond Dr RD’s grasp.

Count Leo Tolstoy (1828-1910)

“I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives”.

Sir Ronald A Fisher (1890-1962) and Karl Pearson (1857-1936)

For quite a while these statisticians feuded about the chi-square distribution. Pearson worked with large data sets whereas Fisher worked with small data sets. Fisher was right! That's why the chi-square distribution takes degrees of freedom for small data sets into account. Take a long look at David’s 1977 Geostatistical Ore Reserve Estimation, Table 1.IV, Copper grade Prince Lyell. How about that? So why not reunite the distance-weighted average and its lost variance? Mining investors are bound to like it! In fact, Barrick Gold liked it before Bre-X's boss salter passed away.

Geostatistocrats such as Professor Dr Michel David (1945-2000) and UBC Emeritus Professor Dr Alastair J Sinclair, PEng, PGeo never got into counting degrees of freedom. Why is it that one-to-one correspondence between functions and variances is sine qua non in applied statistics but irrelevant in geostatistics. Dr Michel David was once listed as a Deceased Fellow with the Royal Society of Canada. He is no longer listed but I still do not know why!

Some institutions of higher learning such as COSMO McGill Mining and Stanford University work with Markov chains to derive stochastic mining plans. What they cannot possibly derive are unbiased confidence limits for metal contents and grades of ore reserves. Geostatisticians stripped the variance off the distance-weighted average AKA kriged estimate. That’s how real functions got surreal variances!

A study on kriging small blocks

Dr Margaret Armstrong and Mr Normand Champigny had put this study on paper when they were toiling at the Centre de Géostatistique at Fontainebleau, France. Professor Dr Georges Matheron himself may have inspired them to compile their study in a paper. Be that as it may, this simple study was never added to Matheron’s magnum opus. It was kriging small blocks that inspired Armstrong and Champigny to elaborate on what they had detected. “Mine planners tended to define ore/waste limits as finely as possible”.

How about that? Perfect people are hard to find. So, the average mine planner was often tempted to over-smooth small blocks. The central tenet of this study was that over-smoothed estimates should not be used to derive recoverable reserves. That sort of research may well be the reason why Normand Champigny was awarded a Diploma in Geostatistics.

And why was this study published in CIM Bulletin, March 1989? Here’s why! Professor Dr Michel David and Professor Dr Alastair J Sinclair, PEng, PGeo, reviewed each and every paper in which kriging popped up in those days. Dr Frederik F Agterberg, Associate Editor with CIM Bulletin would not have hesitated to approve Armstrong and Champigny’s study. Here are a few facts and figures that neither the authors nor the reviewers knew about.

Dr Isobel Clark, in Chapter 4 Estimation of her 1979 Practical Geostatistics, derives not only the distance-weighted average AKA kriged estimate but also its variance. What she didn’t do was test for spatial dependence in the sample space defined by her hypothetical uranium concentrations. She sets the stage on page 3 of Chapter 1 Introduction under Figure 1.1. Hypothetical sampling and estimation situation. On page 5 she puts forward “the convenient assumption that there is no trend within the scale in which we are interested…” On the same page she fiddled with the factor 2 for “mathematical convenience” and fumbled her fickle “semi-variogram”. What went missing in her Index on page 127 above Disjunctive Kriging is Degrees of freedom. Dr Isobel Clark credits Professor Dr Andre Jounel and others at Fontainebleau who taught her all she knows “about the theory of the Theory of Regionalised Variables”. What a pity that Dr Clark did not know how to test for spatial dependence within the sample space defined by her set of hypothetical uranium concentrations. But then neither did any geostatistical reviewer for CIM Bulletin know how to test for spatial dependence between measured values in ordered sets!

Dr Isobel Clark, author of Practical Geostatistics
BSc, MSc, DIC, PhD, FSS, FSAIMM, FIMMM, CEng

Bringing Matheron’s new science of geostatistics to the world was quite a tour de force. Scores of geologists thought it odd that so much could be done with so few boreholes. But too few knew applied statistics well enough to figure out what was wrong with geostatistics. What Matheron and his disciples had failed to grasp was not only that all functions do have variances but also that sets of measured values do give degrees of freedom. That’s about all it took to do so much with so few boreholes! CIMMP’s archive has what Matheron’s magnum opus does not have. And that’s an authentic copy of Armstrong and Champigny’s study for a mere C$20.00.


What irked was that CIM Bulletin rejected Precision Estimates for Ore Reserves. I had mailed on September 28, 1989 four (4) copies to The Editor of CIM Publications. Not surprisingly, peer review of a paper that is at variance with the central tenets of geostatistical thinking turned out to be a blatantly biased and shamelessly self-serving sham. Our peers at CIM Bulletin were Professor Dr Michel David (1945-2000) and UBC Emeritus Professor Dr Alastair J Sinclair, PEng, PGeo. What a shame that unbiased confidence limits for metal contents and grades of ore reserves remain as rare as hen’s teeth.