Going GIGO with CRIRSCO

Snappy acronyms add spice to the way we blog and talk. GIGO has been tagging along with computing science without losing its punch. CRIRSCO is but one tong twisting tour de force for Combined Reserves International Reporting Standards Committee. Its Chairman is Niall Weatherstone of Rio Tinto. Larry Smith of Vale Inco asked Weatherstone about Setting International Standards. Weatherstone said CRIRSCO was set up in 1993 but its website says it was 1994. CRIRSCO's website makes a tough read because of its dreadfully long lines. So what have Weatherstone and his Crirsconians been doing during all those years?

Smith should have but didn’t ask what CRIRSCO has accomplished. It would seem some sort of semi-international reporting template has been set up. The problem is the Russian Federation has a code of its own, and China’s is sort of similar. As it stands, Crirsconians have yet to develop valuation codes for mineral properties. At the present pace, valuation codes that give unbiased confidence limits for contents and grades of reserves and resources might be ready in 2020, the year of perfect vision. It had better be based on classical statistics!

Here’s what was happening in my life when CRIRSCO came about either in 1993 or in 1994. I talked to CIM Members in Vancouver, BC, about the use and abuse of statistics in ore reserve estimation. Bre-X Minerals raised money to acquire the Busang property. Clark wanted me to go from Zero to Kriging in 30 Hours at the Mackay School of Mines. I didn’t go because her semi-variograms are rubbish. The international forum on Geostatistics for the Next Century at McGill University didn’t want to hear about The Properties of Variances. David S Robertson, PhD, PEng, CIM President, failed to, “… find support for your desire to debate.” What irked me was Jean-Michel Rendu’s 1994 Jackling Lecture on Mining geostatistics - Forty years passed. What lies ahead? He rambled on about, “…an endless list of other ‘kriging’ methods…” and prophesied geostatistics, “… is here to stay with all its strengths and weaknesses.” At that time, Rendu knew about infinite sets of kriged estimates and zero kriging variances.

Rendu’s lecture stood in sharp contrast to A Geostatistical Monograph of The Mining and Metallurgical Society of America. Robert Shurtz, a mining engineer and a friend of mine, wrote The Geostatistics Machine and the Drill Core Paradox. Harry Parker, a Stanford-bred geostat sage, was to find fault in Shurtz’s work. This great debate got nowhere because neither grasped the properties of variances. Otherwise, both of them could have put in plain words why kriging variances drop off. A few of Parker’s geostat pals had already found out why in 1989.

Figure 2 is rather odd in the sense that, “The kriging variance rises up to a maximum and then drops off.” That’s precisely what Armstrong and Champigny wrote in A Study of Kriging Small Blocks published in CIM Bulletin of March 1989. What I saw kriging variances do is what real variances never do. Armstrong and Champigny alleged kriging variances drop off because mine planners over-smooth small blocks. More research brought to light that kriged block estimates and actual grades were “uncorrelated.” That would make a random number generator of sorts for kriged block grades. It was David himself who approved that blatant nonsense for publication in CIM Bulletin.

Figure 2 gives kriging variances as a function of variogram ranges. As such, it was more telling than Parker’s. Neither Shurtz nor Parker scrutinized Armstrong and Champigny’s 1989 A Study of Kriging Small Blocks. Otherwise, Shurtz might have pointed out Parker’s kriging variances looked a touch over-smoothed. Neither did Parker confess he does over-smooth the odd time.

Corrected and uncorrected sampling variograms for Bre-X’s bonanza grade borehole BSSE198 show where spatial dependence between bogus gold grades of crushed, salted and ordered core samples from this borehole dissipates into randomness. The adjective “corrected” implies that the variance of selecting a test portion of a crushed and salted core sample, and the variance of analyzing such a test portion, are extraneous to the in situ variance of gold in Bre-X’s Busang resource. Subtracting the sum of extraneous variances gives an unbiased estimate for the intrinsic variance of bogus gold in Busang’s phantom gold resource. Fisher’s F-test proved this intrinsic variance to be statistically identical to zero.

Harry Parker and Jean-Michel Rendu appear to speak for the Society for Mining, Metallurgy and Exploration (SME) in the USA. What it takes to cook up ballpark reserves and resources are soothsayers who know how to failingly infer mineralization between boreholes, hardcore krigers and cocksure smoothers. What CRIRSCO ought to have done after the Bre-X fraud is set up an ISO Technical Committee on reserve and resource estimation. It’s never too late to do it! GIGO may be a bit dated but Garbage In does stand the test of time. Nowadays, Good Graphics Bad Statistics Out is a much more likely outcome. What a pity that GIGGBSO lacks GIGO’s punch!

Hooked on junk statistics

Our parents told us not to put all our eggs in one basket. This lesson has passed the test of time ever since the Easter Bunny got to working with real eggs. The world’s mining industry has put its basket full of junk statistics and got egg on its façade. Junk statistics does not give unbiased confidence limits for grades and contents of mineral reserves and resources. Annual reports, unlike opinion polls, do not sport 95% confidence intervals and ranges as a measure for the risks mining investors encounter. Many years ago I put classical statistics in my own basket. I thought I couldn’t go wrong because Sir Ronald A Fisher was knighted in 1953. But was I wrong? Matheron, who is called The creator of geostatistics, knew very little about variances, and even less about the properties of variances.

Matheron deserved some credit because he didn’t put all core samples of a single borehole in one baskett. He would have lost all his degrees of freedom but wouldn't have missed them anyway. He did derive the length-weighted average grade of a set of grades determined in core samples of variable lengths. What he didn’t derive was the variance of this length-weighted average. Matheron wrote a Synopsis for Gy’s 1967 Minerals sampling. Gy, in turn, referred to Visman’s 1947 thesis on the sampling of coal, and to his 1962 Towards a common basis for the sampling of materials. Visman bridged the gap between sampling theory with its homogeneous populations and sampling practice with its heterogeneous sampling units and sample spaces. Matheron never knew there was a gap.

Should a set of primary increments be put in one basket? Or should it be partitioned into a pair of subsets? Gy proposed in his 1977 Sampling of Particulate Matter a set of primary increments be treated as a single primary sample. He claimed the variance of a primary sample mass derives from the average mass and number of primary increments in a set, the properties of the binomial distribution, and some kind of sampling constant. I explained in Sampling in Mineral Processing why Gy’s sampling theory and his sampling constant should be consumed with a few grains of salt.


When I met G G Gould for the first time at the Port of Rotterdam in the mid 1960s, he told me how Visman’s sampling theory impacted his work on ASTM D2234-Collection of a Gross Sample of Coal. Visman’s sampling experiment is described in this ASTM Standard Method. Visman’s 1947 thesis taught me that the sampling variance is the sum of the composition variance and the distribution variance. I got to know Jan Visman in person here in Canada. I treasure my copy of his thesis. I enjoyed his sense of humor when we were griping about those who try to play games with the rules of classical statistics.

On-stream data for slurries and solids taught me all I needed to grasp about spatial dependence in sampling units and sample spaces. Fisher’s F-test is applied to test for spatial dependence, to chart a sampling variogram, and to optimize a sampling protocol.


Selecting interleaved primary samples by partitioning the set of primary increments into odd- and even-numbered subsets is described in several ISO standards. A pair of A- and B-primary samples gives a single degree of freedom but putting all primary increments in one basket gives none. Shipments of bulk solids are often divided in sets of lots so that lower t-values than t0.05; 1=12.706 apply. Those who do not respect degrees of freedom as much as statisticians do may cling to the notion that the cost for preparing and testing a second test sample is too high a price for some invisible degree of freedom. They just don't grasp why confidence limits and degrees of freedom belong together as much as do ducks and eggs.

The interleaved sampling protocol gives a reliable estimate for the total variance at the lowest possible cost. It takes into account var2(x), the second variance term of the ordered set. It makes sense to take interleaved bulk samples in mineral exploration because they give realistic estimates for intrinsic variances in sample spaces. Both Visman and Volk, the author of Applied Statistics for Engineers, were conversant with classical statistics. The geostatistical fraternity made up some new rules and fumbled a few others. They got hooked on a basket of junk statistics and are doomed to end up with egg on their faces.

Sorting out Matheron's junk statistics

Matheron claimed in the Rectificative to his Note Statistique No 1 that he had derived the length-weighted average lead and silver grades of core samples with variable lengths. I couldn’t verify whether he did or not because primary data and weighted average grades were missing. Matheron didn’t derive unbiased confidence limits for weighted average grades. Here’s what he should have done but didn’t do. He should have derived the variances of length-weighted average lead and silver grades. He should have tested for spatial dependence between metal grades of ordered core samples by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. He should have used the lowest variance to derive unbiased confidence limits for weighted average grades. He should have taken into account the variable lengths of core samples. It was beyond his grasp to count degrees of freedom either for the set of core samples, or for the ordered set. It is safe to assume Matheron did know how to count core samples.

Matheron’s Note Statistique No 1 proved he was well on his way to become a self-made wizard of odd statistics. Matheron worked by himself and made but few references to other authors when he was stacking the odds against classical statistics. He didn’t have what it took to grasp “la statistique classique.”

Just the same, he wrote 85 papers between 1954 and 1965. Rapport N-96 was a 1965 paper by Matheron and Formery. I took an instant liking to its rich title! It might shed light on Matheron’s work between 1954 and 1965. Did he add a touch of Visman’s sampling theory or a dash of Volk’s applied statistics to his search for structure and randomness in that new science of geostatistics? Not so fast!

Matheron and Formery brought up that De Wijs, Krige and Sichel worked with geometric concepts unknown in “la statistique classique” and to its practitioners. Yet, those authors did refer to classical statistics in their own work. Matheron and his coauthor did agree statistics had a role to play in quality control of manufactured products. Just the same, they prattled a lot about all that’s wrong with classical statistics. Here’s but one line I’ve struggled to convert into English prose, “The properties of classical statistics are often transposed in a rather rough manner.” I’ll say! And here’s more drivel, “(Classical statistics) resulted sometimes in naivety or even silliness.” Don’t take my word for it but do read that rather rough and silly paper.

Matheron's structure and randomness

Matheron and his coauthor set out to study structure and randomness at regular intervals. They did so with the aid of ordered and randomly distributed integers. Readers were told to put a pragmatic spin on structure and randomness, and to infer integers are in fact grades. My son and I worked with genuine gold grades of ordered rounds in a drift. We derived Riemann sums and proved a significant degree of spatial dependence between ordered grades by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. It was that simple! Yet, geostatistical minds are taught to infer grades between coordinates

Riemann's method is precisely what Matheron and his coauthor should have applied in 1965. Riemann sums would have given the j^th variance term of an ordered set (Matheron’s structured set) as follows: var_j(x)=∑(x_i−x_i+j)²÷[2(n−j)]. The first variance term of the ordered set is var₁(x)=0.50, and the variance of the set is var(x)=2.82. The observed value of F=2.82/0.50= 5.64 exceeds the tabulated value of F0.05;10;20=2.35 at 95% probability and with applicable numbers of degrees of freedom. Hence, the ordered set displays a statistically significant degree of spatial dependent. And dont' take my stats on face value! Set up a spreadsheet template and figure out what I did!

Riemann sums also underpin sampling variograms. A sampling variogram is a graph that shows where orderliness in a sample space or a sampling unit dissipates into randomness. Matheron and Formery mentioned variograms but didn’t explain how to derive lags that underscore where orderliness disperses into randomness. Matheron’s search for structure and randomness made him march in place to the beat of kriging drums. Matheron knew he ought to do something but never knew what Visman had done already. He babbled gibberish when contemplating what to do next. Matheron’s problem was he didn’t have the foggiest notion what Sir Ronald A Fisher had been doing across the Channel ever since the storm with Pearson about degrees of freedom. Matheron and his disciples didn’t have a clue how they got into junk statistics.

Born to infer and rig rules of real statistics

The mad world of geostatistics is still as bad as it was when Bre-X’s phantom gold resource was cooked up. All it took was a little placer gold, a lot of barren rock, and a load of junk statistics. When Bre-X’s shareholders were counting their losses, geostatistical ore reserve practitioners were keeping low profiles. The Toronto Stock Exchange and the Ontario Securities Commission tried in vain to sort out the mess. Geostatisticians failed to foil the Bre-X scam early in the salting game. Nowadays, they talk with confidence about mineral reserves and resources. What they won’t talk about is how to derive unbiased confidence intervals and ranges for metal contents and grades of mineral reserves and resources. They would like to bury Bre-X but I don’t want them to forget the role of geostatistics in the Bre-X hoax. Here’s why! To infer ore between boreholes is as much alive today as it was during Bre-X’s hay days. And that was some hoax! Yet, the world’s mining industry is still working with the very same junk statistics. What I want to do is show why geostatistics makes junk statistics of the worst kind. That’s why it ought not to be taught at any university on this planet.

After I found out about the Centre de Geosciences/Geostatistique’s Online Library, I took to a liking to looking at Matheron’s seminal work. I want to grasp what Matheron was thinking when he thought he was working with statistics. When I looked at Matheron’s Note Statistique No 1, I was surprised to find out it had been reborn as Note Géostatistique No 1. When and why did that happen? The scanned copy of Matheron’s Formule des minerais connexes is still marked Note Statistique No 1. Matheron signed it in Algiers on November 25, 1954. Why would anyone want to predate the birth of Matheron’s new science of geostatistics?

What I want to know most of all is why and when Matheron strayed from real statistics into his self-made new science of geostatistics. It was easy to find out when he lost touch with real statistics. When I was a consultant to Cominco long before it got keeviled, I met a geologist who got a headache reading Mining Geostatistics and gave me his copy. That’s when I finally found out why Matheron created geostatistics. It’s all in his Foreword to that 1978 textbook the lead author of which is Professor Dr A G Journel. Journel was Matheron’s protégé and most gifted disciple but also a Stanford professor.

Matheron brooded over structure and randomness, and hypothesized, “Since geologists stress the first of these aspects, and statisticians stress the second, I proposed, over 15 years ago, the name geostatistics to designate the field which synthesizes these two features and opens the way to the solution of problem of evaluation of mining deposits.” I think it’s more intuitive to contrast orderliness and randomness. I prefer text and syntax such as “order in a sample space or in a sampling unit” to clarify what I mean. I took my petite TI calculator and found that Matheron’s new science of geostatistics had been either created or synthesized some time before 1962. Sadly, Matheron in those days didn’t have an HP or TI calculator to derive his odd statistics.

I took several looks at Matheron’s very first paper entitled Formule des minerais connexes and signed in Algiers on November 25, 1954. What Matheron explored in his first paper was the degree of associative dependence between lead and silver grades of core samples of one or more boreholes drilled in a lead deposit. He provided no primary data and but a few questionable statistics. What stands out in his work is obsession with probabilistic symbols and the lognormal distribution. Matheron didn’t show how to check and compare his observed “coefficient de correlation” with the proper value of r0.05;df at 95% probability and with applicable degrees of freedom. Matheron’s predicament in 1954 was that counting degrees of freedom didn’t rank anywhere on his list of things to teach. As a matter of fact, Matheron never in his life got into counting degrees of freedom.

Matheron appended on January 13, 1955 a Rectificative á la Note Statistique No 1 to his Formule des minerais connexes. Neither read like real statistics at all. It did look like geostatistics because degrees of freedom were gone. Matheron talked about µ and σ but those symbols derive from probability theory. The concept of degrees of freedom makes the difference between probability theory with its population parameters and applied statistics with its finite sample statistics. Just the same, Matheron never counted degrees of freedom.

Matheron’s Note Statistique No 2 left me cold but his Note Statistique No 3 did pique my interest. Why did Matheron talk about the standard deviation of a deposit? Why didn’t he derive the 95% confidence interval for the lead and silver contents and grades of his deposit in Algiers? His one-page paper was about the Poisson distribution as it applies to particles of pure minerals and of barren rock, and about the upper limit for the standard deviation of some hypothetical deposit.

Matheron was a mining engineer but also a probabilist of sorts. Six symbol-packed pages with sparse text proved he was also a self-made wizard of odd statistics. How he found a following is bound to go down as a deep mystery in the history of science. His disciples seem to have been born to infer, krige, smooth, rig the rules of real statistics, and beat immeasurable odds. They do all that stuff despite fumbled variances and missing degrees of freedom! Behind Bre-X was not just a salting scam but a unique scientific fraud. And the world’s mining industry just can’t get enough of it!

Teaching junk statistics at UBC

The stage was set in 1964 to teach junk statistics at the University of British Columbia. It was the year Professor Dr Alastair J Sinclair took on his duty to teach earth sciences to UBC’s students. It was but a few years after Matheron dabbled at his own kind of unreal statistics and fumbled real variances. Just the same, Matheron’s junk statistics was hailed as new science on campus at the University of Kansas in June 1970. His tour de force at that time was to call on Brownian motion to infer the continuity of his famed stationary random function. UK’s campus was a fitting venue because that’s where Agterberg failed for the first time to derive the real variance of his distance-weighted average point grade. Here’s why it gives too rich an abundance of data in mineral exploration. As few as a pair of measured values, when determined in samples taken at positions with different coordinates in a finite sample space, gives an infinite set of Agterberg’s point grades, a zero voodoo variance, and not a single degree of freedom. How about that? Some kind of perpetual motion in mineral exploration!

Sinclair details 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 had “the good fortune to work with Journel, Huijbregts and Deraisme.” Those were Matheron’s earliest students who took his musings for dogma, and who didn’t have a clue which variances were lost on Matheron’s watch. Sinclair’s list of folks he was “fortunate to have worked with at various times” reads like a Who’s Who in the geostatistical fraternity. He credits all of them to have contributed to his education. I’m all in favor of giving credit where credit is due. But to give credit to everybody who taught him junk statistics is over the top. Some geostatistocrats on Sinclair’s list now know each weighted average has its own variance. And the odd one might even know why! One cannot help wonder how the cream of Matheron’s crop saw fit to make junk statistics look so good to Sinclair starting in 1972. So much so that Sinclair felt compelled to write his own textbook. Of course, all of that spelled bad news for UBC’s students.

When I met Sinclair at his UBC office in August 1992, I talked about real statistics. I showed how to count degrees of freedom for the set of nine holes in Figure 203 of David’s 1977 Geostatistical Ore Reserve Estimation. In Sinclair’s world, the concept of degrees of freedom breaks down in matters of spatial dependence. But it’s alive and well in my world. Sinclair did not see much of a difference between Matheron’s surreal geostatistics and Fisher’s real statistics. In fact, he knew as much about real statistics in August 1992 as he did in September 1989. That’s when CIM Bulletin entrusted Sinclair and David with the review of Precision Estimates for Ore Reserves. David blew a fuse because our paper was “without a single reference to 20 years of work in geostatistical ore reserve estimation.” And we didn’t even know we had written a geostatistical paper! So, we were baffled when Dr L R Fyffe, Editor, CIM’s Geology Division wrote on November 23, 1989, “Both reviewers recommend publication with major revisions.”

But big troubles were looming in the esoteric universe of those who infer, krige, smooth, and rig the rules of real statistics with reckless abandon. When I was working on Sampling and Weighing of Bulk Solids in the early 1980s, I studied David’s 1977 Geostatistical Ore Reserve Estimation. I found way too many symbols and far too few measured values. Brownian motion, too, played some kind of cameo role in this work of geostatistical fiction. The author confessed his work is "not for professional statisticians." In fact, he even predicted, “…statisticians will find many unqualified statements…” What David didn’t predict was he would deny anything was wrong in surreal geostatistics.

So what were we to do? Spice our paper with symbols? Scrap measured values? Delete Fisher’s F-test for spatial dependence? Call David to the task? Ask him to put in plain words his “good test to find out whether one really understands geostatistics” on page 286 of his 1977 textbook? Or try to pacify CIM Bulletin’s keepers of Matheron’s tablets with a few tidbits of token stuff? So we huffed and puffed a lot and added but a few references to works of geostatistical scholars such as Dagbert, David, Journel and Huijbregts. Our marginally revised paper was rejected on February 7, 1990. My son completed his PhD in computing science. I resolved to raise a stink. I did it then. And I still do now! Sinclair is but one reason. Bre-X’s phantom gold resource is another!

On November 23, 1989, CIM Bulletin’s editor wrote “Both reviewers recommend publication with major revisions.” Sinclair started some charade of sorts on November 22, 1989, at 08:30AM. He welcomed those who attended my short course on Sampling Precious Metal Deposits: Metrology-A New Look. The venue was Room 330A at UBC’s Department of Geological Sciences. The course was sponsored by its Mineral Deposits Research Unit. Sinclair didn’t have time to listen and moved about a lot. In fact, he popped in and out of Room 330A like a Jack-in-the-Box. Sinclair didn’t ask any questions. Was it because the paper he rejected was part of my notes? Did he worry others might ask questions? Did he worry I would talk too much about real statistics and too little about Matheronian geostatistics?

Dr J A McDonald, Interim Director, Mineral Deposits Research Unit, on February 21, 1990, wrote, “We certainly were pleased with the response to your course and have elected to maintain the theme with a 5-day course to be held April 23-27, 1990, entitled Geostatistics for the Mining Industry, New Concepts, New Tools.” How about that for cruel and unusual punishment? Sinclair was in damage control mode. So much more has happened in our stand-off on real statistics ever since I met Sinclair in his UBC Office in August 1992. Much of it will stay untold for some time to come.

Dr Alastair J Sinclair, PEng, PGeo, has striking credentials. He is a former Member of the Discipline Committee of the Association of Professional Engineers and Geoscientists of British Columbia with its Code of Ethics to protect the public at large. He was CIM’s Distinguished Lecturer for 2000-2001. He taught a short course at the UBC Robson Square Campus, Vancouver, BC, on May 15-16, 2008. What he didn’t teach was that each distance-weighted average has its own variance. He didn’t teach how to verify spatial dependence by applying analysis of variance and how to count degrees of freedom. Neither did he teach how to derive unbiased confidence interval and ranges for metal grades and contents of mineral inventories. Sadly, Sinclair is still teaching junk statistics!

Teaching junk science by consensus

The Centre de Géosciences/Géostatistique deserves praise for posting to its Online Library a treasure trove of writings. A great deal came from the seminal work of Professor Dr Georges Matheron (1930-2000). Most of it merits long overdue scrutiny and review. The problem is not so much that Matheron put a few spurious findings on paper but that his students took it for doctrine. The Online Library has made it easy to pinpoint what Matheron did wrong and when he did so.

Matheron derived the length-weighted average grade of a set of metal grades determined in core samples with variable lengths. He did so in his Rectificatif of January 13, 1955, to Formule des Minerais Connexes of November 25, 1954 (see Note Statistique No 1). What he didn’t derive was the variance of this length-weighted average grade. Neither did he show how to test for spatial dependence between grades of ordered core samples by applying analysis of variance. He didn’t report primary data sets because of his penchant for working with symbols rather than with real measured values.

Matheron concocted the honorific eponym krigeage in his 1960 Krigeage d’un Panneau Rectangulaire par sa Périphérie. In this Note géostatistique No 28, Matheron derived k*, his “estimateur”, and a precursor to kriged estimate or kriged estimator. In real statistics, Matheron’s k* is in fact the length-weighted average grade of a single block. In this case, too, he didn’t derive var(k*), the variance of his “estimateur”. Sadly, kriging became a curse of sorts for Professor D G Krige.

Matheron’s Stationary Random Function seemed not to have troubled those who were at the first geostatistics colloquium in the USA in 1970. Matheron even called on Brownian motion to infer by hook or by crook the continuity of his Riemann integral. He didn’t explain what Brownian motion and mineral deposits have in common. Matheron, unlike John von Neumann in 1941 and Anders Hald in 1952, didn’t work ever in his life with Riemann sums. On the contrary, he would rather infer spatial dependence than apply Fisher’s F-test to the variance of a set and the first variance term of the ordered set.

It is to Matheron’s credit that it was not him who lost variances of all weighted averages. It was Dr Frederik P Agterberg who failed to derive the variance of his distance-weighted average. He did derive the distance-weighted average grade of a set of five (5) points at positions with different coordinates but failed to derive the variance of this central value. What he didn’t point out was that as few as two such points define an infinite set of distance-weighted averages. He fumbled the variance of his central value for the first time in his 1970 colloquium paper and once again in his 1974 Geomathematics.

Matheron’s length-weighted average grade was reborn as an honorific kriged estimate or estimator. But then Agterberg’s distance-weighted average grade was honored in the same way! And here’s the clincher! An infinite set of Agterberg's zero-dimensional point grades fits along any borehole, and within any ore block, sampling unit or sample space. That’s why distance-weighted average point grades without variances became the heart and soul of geostatistics. Matheron’s seminal work merely set the stage for Agterberg’s giant step into the abyss of mineral reserve and resource estimation with confidence but without confidence intervals and ranges.

The above figure is a facsimile of Fig. 203 on page 286 of David's 1977 Geostatistical Ore Reserve Estimation. It shows the infinite set of "estimated" values within B derived from the same set of nine (9) holes.

The more geostatistocrats tinkered with real statistics, the more flawed geostatistics grew. It’s a scientific fraud to derive confidence limits from pseudo kriging variances. It’s as silly to talk about confidence without limits as it is to infer spatial dependence within or between boreholes. To discount degrees of freedom would make no sense at all in real statistics. To count degrees of freedom makes no sense in geostatistics. That’s the very reason why geostatistics does not give unbiased confidence limits for metal contents and grades of mineral reserves or mineral resources.

Professor Dr Roussos Dimitrakopoulos is a catch of sorts for the Department of Mining, Metals and Materials Engineering at McGill University. I don’t know why! I told him in 1993 that weighted averages have variances because one-to-one correspondence between functions and variances is sine qua non in statistics. This basic rule is still beyond his grasp in 2008. All the same, he is Editor-in-Chief, Journal of Mathematical Geosciences. Agterberg, President, International Association for Mathematical Geosciences, left his fingerprints when he failed to derive the variance of his distance-weighted average point grade. Dimitrakopoulos talks about “gazillion types” of probabilistic models. What he doesn’t talk about is that the odds to select the least biased subset of some infinite set of kriged estimates are immeasurable. The problem is not so much he himself believes it but the world’s mining industry believes it. The more so because he does all of that with voodoo variances.

Mining engineers, mine geologists, resource analysts, and project managers were invited to a course on Applied Risk Assessment for Ore Reserves and Mine Planning at McGill University. The same course deals with Strategic Risk Quantification and Management for Ore Reserves and Mine Planning and with Conditional Simulation for the Mining Industry. That’s a lot of buzz for a bundle of bucks! Too bad that voodoo variances underpin all that risk assessment and quantification stuff! That’s why one should come with a buddy. For it’s more difficult to baffle a few birds of a feather than a single sitting duck. Dimitrakopoulos should explain why Agterberg’s distance-weighted average point grade aborted its variance during its rebirth as an honorific kriged estimate on Matheron’s watch.

NRC shelled out real dough for bogus statistics

Professor Dr Michel David was awarded Grant NRC7035 to advance geostatistics. In those days NRC stood for National Research Council. So David plugged away and his work was printed in 1977. But why did David come up with such a curious caution? He cautioned that professional statisticians would find unqualified statements. How about that? He touched upon it on page VII of what he had come to call Geostatistical Ore Reserve Estimation. He didn’t point out any unqualified statement. But I found out he was right after I had bought my own copy of his book.

David was as smitten with geostatistics in the 1970s as young Matheron was with applied statistics in the 1950s. Alas, Matheron’s pursuit of applied statistics was not to last. On the contrary, Professor Dr Georges Matheron in the 1970s praised geostatistics as a new science. He did so because he had failed to grasp that all functions do have variances. It is a fact that each distance-weighted average has its own variance in applied statistics. All the same, Matheron got stuck with the variance-deprived distance-weighted average. So much so that he got into calling it a kriged estimate. He did it to honor D Krige who put distance-weighted averages to work at Witwatersrand gold mines in South Africa. The problem is that the variance of the kriged estimate vanished on Matheron’s watch.

A strong case can be made that eulogies be written long before one’s time on this planet comes to an end. One might ask a lawyer to assist in assuring the veracity of one’s credentials. A case in point is Professor Dr George Matheron’s 2000 eulogy. Dr F P Agterberg was his eulogist. He remembered him as the founder of spatial statistics. As luck would have it, Matheron never tested 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. Given that Matheron’s magnum opus is posted on its own website, anyone could spend a long time to study his life and times.

Professor Dr Michel David passed away on May 10, 2000. Professor Dr Roussos Dimitrakopoulos and Michel Dagbert put together his obituary. The International Association for Mathematical Geology had awarded him in 1988 the W C Krumbein medal. David became a Fellow of the Royal Society of Canada in the same year. CIMMP recognized his worldwide achievements in 1989 with the award of the Selwyn G Blaylock medal.

My son and I knew precisely what had gone wrong with geostatistics when I was face-to-face with David on Saturday, March 23, 1991. It was at a seminar on Sampling and Ore Reserves at the Royal York Hotel, Toronto, Ontario. CIM Bulletin in 1990 had rejected Precision Estimates for Ore Reserves. We had shown how to test for spatial dependence and how to derive unbiased confidence limits for gold content and grade. David saw fit to nitpick that twenty years of geostatistical literature is missing in our work. He did not ask me a single question. Take a look at what is wrong in Matheron’s new science of geostatistics.

Marechal & Serra, Random Kriging, 1970, Figure 10
David, Geostatistical Ore Reserve Estimation, 1977, Figure 203

Professor Dr Michel David claimed: "Writing all the necessary covariances for that system of equations might be a good test to find out whether one really understands geostatistics"! David didn’t know that deriving the variance of each equation and counting degrees of freedom for his system of equations is a good test to find out whether one really grasps applied statistics.

The National Research Council is the Government of Canada’s institute for research and development. As such it has been active since 1916. NRC’s task is to stand on guard for ethics and integrity. It did not know in the 1970s that Matheron’s new science of geostatistics is an invalid variant of applied statistics. Neither did Dr Roger A Blais, a Professor in Economic Geology and a Fellow of the Royal Society of Canada. He made it possible for David to write so much about so little. David confessed to be indebted to Matheron. David’s writing added up to a batch of bogus statistics. NRC7035 was in place not only for his 1975 Geostatistical Ore Reserve Estimation but also for his 1988 Handbook of Applied Advanced Geostatistical Ore Reserve Estimation.

Dr Isobel Clark in her 1979 Practical Geostatistics derived the variance of the distance-weighted average AKA kriged estimate. Professor Dr Michel David wrote about the “famous Central Limit Theorem” but did not know how to apply it. The Royal Society of Canada no longer list David under Obituaries of Deceased Fellows. Sic Transit Gloria Mundi!