Pros and cons of Practical Geostatistics

Dr Isobel Clark is the author of Practical Geostatistics. What she did in this 1979 textbook took me by surprise. It would have baffled many a thinker who has never bothered to read works of others. She deserved praise because she had derived the variance of the distance-weighted average. Agterberg in 1974, David in 1977, and Journel in 1978 never took the trouble to derive this variance. It’s a bad omen that the distance-weighted average morphed into a kriged estimate on Matheron’s watch. It was Matheron who had failed to derive the variance of this kriged estimate long before geostatistics was hailed far and wide as a new science.

Dr IC pointed out on the jacket of her textbook, “Geostatistics is the popular name for the application of statistical methods to problems in mining and geology”. She confessed in her Preface that Journel and others at Fontainebleau, France had taught her all she knows about the Theory of Regionalized Variables. Now that’s what got me worried. Matheron’s way of drawing strings of symbols on a blackboard may well have driven his most gifted disciple to routinely assume spatial dependence. All it took is to assume, krige, smooth, a leap of faith, and a lot of luck!

Author 1974 Geomathematics
ex NRCan Emeritus Scientist

Dr Frederik P Agterberg, too, took a leap from applied statistics to geostatistics. He went from serial correlations in 1958 to functions without variances in 1974. In contrast, Dr Isobel Clark derived not only the distance-weighted average grade of a set of measured values but also the variance of the set and the variance of its central value. Much of it is detailed in Figure 4.1 and Table 4.1 of her textbook and a few pages about her data set. She fretted a bit on page 72 whether or not the Central Limit Theorem would hold. But it always does! Surely, David himself in 1977 would not have written about if it were false.

Dr IC’s set of hypothetical uranium concentrations seem to have been measured in hypothetical samples selected at positions with real coordinates in a two-dimensional sample space. I wish she had given the mass of each of her hypothetical samples. She reports that her 95% confidence range for a distance-weighted average hypothetical uranium grade of 400 ppm has a lower limit of 95% CRL=350 ppm and an upper limit of 95% CRU=450 ppm. It doesn’t look too bad, does it? Of course, t0.95;4=2.776 from the t-distribution rather than z0.95=1.96 from the normal distribution should have been used to derive the 95% confidence interval for xbar=400 ppm. But who would split hairs at this stage? Sir R A Fisher was already counting degrees of freedom for small data sets when Dr IC was a cute tot.

Assumes with poise!

The question is not so much whether this distance-weighted average hypothetical grade of 400 ppm is an unbiased estimate for the unknown true grade at 1,244 m Easting and 713 m Northing. Of course, I wish I knew her sampling protocol. The question is whether it is an unbiased estimate for the sample space defined by this set of five (5) hypothetical uranium concentrations. The most effective method to test for spatial dependence is to do what door-to-door sales people have always done. Travel from point to point such that each point is called on but once and the shortest distance is traveled. The next step is to derive the variance of the set and the first variance term of the ordered set. Finally, Fisher’s F-test is applied to the variance of the set and this first variance term. The observed F-value of 2.07 between var(x)=4,480 for the set and var1(x)=2,161 for the ordered set is below the tabulated F-value of 4.53 at 95% probability. Hence, her set of measured values does not display a significant degree of spatial dependence. By implication, the distance-weighted average grade of 400 ppm is not an unbiased estimate for this sample space. Dr IC did what Journel had taught her to do. All she did was assume spatial dependence between measured values in the ordered set. Here’s what happens when spatial dependence is assumed and functionally dependent kriged estimates are added to measured values.

Assume, krige, smooth, and be happy!

Dr Isobel Clark is due to teach a 3-day course at Global InfoMine in Vancouver, BC on May 3-5, 2011. She will take students from "no knowledge of statistics or geostatistics to understanding the mysteries of ordinary kriging and its variants in 30 hours (or less)". I wish I were there to shed light on that shameless practice of converting bogus grades and barren rock into a phantom gold resource. But then, why rush the good stuff!

ISO to tackle trueness

ISO has set the stage to tackle trueness. It did so by issuing an NWIP ( New Work In Progress). I am pleased that ISO/TC69/SC6/WG1 has been entrusted with the task. Accuracy (trueness and precision) is way wide of the mark. Trueness (accuracy and precision) reads but a bit better. I would rather work with either Precision and accuracy or Precision and bias. Trueness makes more sense in a court of law. I have worked on a number of ISO Committees since 1974. I would find it a senseless task to keep track of trueness. Testing for bias always makes sense. Student’s t-test not only shows what’s biased and what’s not but also gives intuitive measures for statistical risks. I am pleased that ISO never took to kriging and smoothing. Unlike ASTM which went along with geostatistical thinking in 1994. ASTM was set up in 1898 by chemists and engineers. A precursor of CIMMP was set up in 1898. CIMMP's geostatistical peer review has been biased since the 1990s.

ISO/TC183 has set up a standard method to derive metal contents and grades of mineral concentrates and ores. It would have been just as simple to set up an ISO Technical Committee to derive metal contents and grades of mineral reserves and resources. But the COSMO brains behind the mining industry were too keen to assume, krige, smooth, and rig the rules of applied statistics. That’s why Bre-X’s bogus grades and Busang’s barren rock morphed so smoothly into a massive phantom gold resource. And that’s when Barrick Gold asked me to assist in testing for bias between paired gold assays determined by cyanide leaching and by fire assaying.

Student's t-test for bias

The observed t-value of 11.258 is statistically significant at 99.9% probability. It’s impossible to salt drill core with placer gold. In contrast, salting crushed core with placer gold is a cinch. Early in 1997 I derived confidence limits for the mass of gold for one of Barrick’s many deposits. My report was deemed worth its weight in gold. Here’s what Richard Rohmer wrote in Golden Phoenix, “For Munk and others affected by Bre-X and Busang, the strange news about de Guzman was the first hint that the find might well not be what Walsh and Felderhof were claiming”. He also pointed out, “Peter Munk was appalled when he read the Strathcona report. The damage inflicted on Canada’s national and international mining industry was beyond what anyone would have thought possible a few short weeks before”. Now that’s what crying out late is all about! I took a look in my crystal ball. It seems as if a Munk Debates for and against geostatistics has been put on ice!

Here's what Peter Munk himself pontificated on the jacket of his Golden Phoenix:
"You have to be courageous; you have to learn to take advantages of change. Be non-conventional; don't fritter your energies - be focused; remember to share. Most important, use the biggest weapon of all weapons, the least appreciated yet the most important tool for success, and this is moral integrity; and don't be afraid to dream and don't be afraid to dream big."

Never mind Munk's moral integrity! I dream about scientific integrity in mineral exploration and mining. It’s a piece of cake to derive borehole statistics with spreadsheet software. I like to call such stats the fingerprint of a borehole. Those who have tried like it a lot! Following is a synopsis derived of the fingerprint of one mind-boggling borehole.

Fingerprint of monster borehole

It shows some 5.5 million ounces of gold in 78 million metric tons of ore. SME published in Transactions 2000 a paper on Borehole Statistics with Spreadsheet Software. One reviewer called it “an excellent paper” and the other thought “it would stir up a hornet’s nest”. But the hornets stayed put!

It makes no sense to assume spatial dependence in sample spaces. Fisher’s F-test ought to be applied to the variance of the set of measured values and the first variance term of the ordered set. Interpolation between measured values gives a false positive for spatial dependence. And that’s where geostatistocrats are taking not only mineral reserve and resource estimation but also the study of climate change. Or climate dynamics as I like to call it! I found Loehle’s 2000-year global temperature reconstruction by accident. Loehle and McCulloch pointed out in 2008 that data sets were smoothed with a 30-year running mean. It does create a pretty smooth picture. But here’s what happens with variances when measured values are enriched with variance-deprived kriged estimates.

Test for homogeneity of variances

I partitioned Loehle’s 2000-year set into twenty (20) sets of no more than 100-year each, derived the first variance term of each set, and applied the chi-square test to this set of twenty (20) variances. These variances constitute too homogeneous a set. The probability that this inference is true exceeds 99.9%. The probability that it is false is less than 0.1%.

I trust that ISO/TC69/SC6/WG1 will never assume spatial dependence between measured values in ordered sets. ISO/TC17 on Steel and ISO/TC34 on Food Products are interested in this New Work in Progress. I shall continue to write whatever needs to be put in writing to ensure that applied statistics prevails not only in ISO standards but also in mineral reserve and resource estimation.

Teaching real statistics at TU Delft

A smart student of sorts taught at TU Delft in 1958 a seminar on the skew frequency distribution of ore assay values. It was none other than young Agterberg. And did he teach real statistics in those days! Scores of students at the University of Utrecht grew up with real statistics! Agterberg was no exception. I only found out when I read his 2000 eulogy for Professor Dr Georges Matheron. He brought up that Professor H J de Wijs thought the ratio of element concentration values to be constant regardless of the volume of the block. Here’s ad verbatim Agterberg’s criticism of what Professor H J de Wijs taught at TU Delft in 1958, “…it would be better to apply the conventional method of serial correlation to series of assay values.” Now that was in 1958 Agterberg’s point of view on spatial dependence between measured values in ordered sets. Why then has he swallowed Matheron’s spatial statistics with hook, line, and sinker?

Neither Agterberg nor de Wijs knew in 1958 that Dr Jan Visman’s 1947 PhD thesis on coal sampling was on file at TU Delft. I, too, was unaware of Visman’s work when I was a teaching assistant and a mature student in the early 1960s. I had been chief chemist with Dr Verwey but wanted to know more about sampling and statistics. The exchange of test samples and test results between trading partners was a thankless task to say the least. I knew all about the analytical variance but didn’t know how to estimate the variance of the primary sampling stage. I thought TU Delft would teach me what I wanted to know about sampling and statistics. One professor was H J de Wijs and the other a coal mining engineer. Neither knew of Visman’s work or of the properties of variances. In fact, H J de Wijs was Rector Magnificus when a student of his defended in 1965 a thesis in which transformation matrices played a key role. So, I left TU Delft, went to work for SGS in the Port of Rotterdam, and found out in 1967 about Visman’s 1947 sampling theory. So, Agterberg and I knew that mathematical statistics was shortchanged at TU Delft in those days. I don’t know why matrix and vector analysis were taught but sampling and statistics were ignored.

Dr Frederik P Agterberg had all but forgotten in this century what he had taught in 1958 at TU Delft. Here’s ad verbatim the very first paragraph of his eulogy, “Professor Georges Matheron (1930-2000) made fundamental contributions to science by establishing new theoretical frameworks in spatial statistics, random sets, mathematical morphology and the physics of random media”. Matheron was a French geologist who was called creator of geostatistics and founder of spatial statistics. I would never have praised Matheron’s surreal geostatistics let alone Journel's assumed spatial dependence. Whatever small minute contribution Matheron has made to science fell far short of real statistics. Surely, it did add up to surreal geostatistics. How ironic that he never got around to testing for spatial dependence between measured values in ordered sets. Why then did Dr Frederick P Agterberg see fit to praise Matheron’s work? He is Emeritus Scientist with the Geological Survey of Canada. Most of his 1974 textbook on Geomathematics has passed the test of time. And most of it will last much longer than Matheron’s magnum opus. What a shame that real statistics behind his 1970 and 1974 figures crumbles under scrutiny.

Figure 1 - Geologic prediction problem in 1970
Figure 64 - Typical kriging problem in 1974

Agterberg solved his geologic prediction problem by linear prediction in time series and assuming a two-dimensional autocorrelation function between his set of five (5) points. He has to explain how it came to pass that a geologic prediction problem in 1970 turned into a typical kriging problem by 1974. That was the very year that Elsevier published Agterberg’s Geomathematics. Cramped between its covers are some 600 pages of mostly real statistics, a lot of sound mathematics and a dash of Matheron’s surreal geostatistics. But why did Agterberg add Stationary random variables and kriging to an otherwise readable textbook?

What I see in each figure is a set of five (5) measured values in a two-dimensional sample space. If each of Agterberg’s points were equidistant to Po, then the central value of his set would be its arithmetic mean. David's famous Central Limit Theorem defines the functional relationship between a set of measured values with identical weights and its central value.

Agterberg refers to the Central Limit Theorem on pages 166, 206, 207 and 231. The number of degrees of freedom is a positive integer for a set of measured values with identical weights but a positive irrational for a set of measured values with variable weights. Agterberg refers to degrees of freedom on pages 174, 190 and 254. The Central Limit Theorem and the concept of degrees of freedom do not play a role in Chapter 10 - Stationary random variables and kriging. Dr Frederik P Agterberg, the author of Geomathematics and Emeritus Scientist with Natural Resources Canada, ought to explain why his Central Limit Theorem is not blessed with a variance and why his set of measured values is not blessed with degrees of freedom.

ISO erred on trueness

When ISO was set up in April 1947 at Paris, France, it was all about nuts and bolts. As a matter of fact, ISO/TC1 Screw heads came first and ISO/TC2 Fasteners was second. Ever since has ISO been setting up a broad range of standards while the world is putting its standards to the test. But I wonder why ISO did err on trueness. Here’s what ISO announced in its Technical Corrigendum 1 on 2005-08-15.

Accuracy (trueness and precision) of
measurement methods and results
Part 5: Alternative methods for the determination of
the precision of a standard measurement method

ISO/TC69, Applications of Statistical Methods, Subcommittee SC 6, Measurement methods and results published the above Technical Corrigendum 1. So what error was SC6 to correct? Of course, trueness and precision should never have been between brackets! What ought to be between brackets are precision and accuracy! A true test for bias would need first of all an unbiased variance estimate. Those who have kriged and smoothed cannot possibly test for bias or estimate precision. Neither can they test for spatial dependence by applying Fisher’s F-test to the variance of bogus data and the first variance term of the bogus data set. So much for kriging and smoothing when we study climate change on our planet!

What I would want between brackets is precision and bias. Derive the variance and then test for bias if enough degrees of freedom are available. Bias detection limits (BDLs) and Probable bias ranges (PBRs) for Type I risks and Type I&II risks are intuitive and powerful measures for the observed bias. Ignorance of precision and bias has irked me as long as have central values without variances. Surely, Matheron and his disciples have brought a big catch of bad science to our world.

I have juxtaposed precision and bias since 1974. That’s when I became a member of CAC/ISO/TC102-Iron ore. I am also a Member of ISO/TC27-Solid mineral fuels, of ISO/TC69-Applications of statistical methods, and of ISO/TC183-Copper, lead, zinc and nickel ores and concentrates. Much of what I have written on sampling and weighing of bulk solids became part of ISO/TC183. My son and I have written a software module on Precision and Bias for Mass Measurement Techniques. ISO has published it as an ISO Standard. I was told Canadian Copyright was not violated. Merks and Merks found it easy to work with precision and bias. What’s more, we are pleased to be encumbered with Fischerian (sic!) statistics.

The International Organization for Standardization was much on my mind when I posted false and true tests for bias. ISO comes from the Greek word isos which means “equal”. Scores of countries have set up national institutions to interface with ISO. The Standards Council of Canada Act received Royal Ascent in 1970. That’s when the CAC prefix was placed before ISO. I have nothing but praise for Standards Council of Canada. CAC/ISO/TC69 Applications of statistical methods has played a key role in my work. I have always juxtaposed Precision and Bias. But it’s a long a story. And it’s bound to get longer while I’m trying to keep it short. I do want to kill two nutty practices with the same stone. The first is to assume spatial dependence between measured values in an ordered set. The second is to not apply Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set. Geostatistocrats assume, krige, smooth and select the least biased subset of any infinite set of kriged estimates. It may well have dazzled those who have never scored a passing grade on Statistics 101. I still find it funny how so few could write so much about so little.

Biased but high degree of precision

Here’s where ISO has created the error to be corrected. Are true and false antonyms or not? Wouldn’t the antonym of trueness be falseness, or perhaps falsehood? Of course, I would call this ISO document Trueness (precision and bias) of measurement methods and results. Surely, a significant degree of spatial dependence between measured values in an ordered set does impact precision. But I upped the odds of finding a false positive. I did so by inserting David’s “famous Central Limit Theorem” between each pair of measured values. Pop in more kriged estimates between measured values and bogus spatial dependence may make the odd mind spin. Is it a minor miracle or Matheronian madness?

Spatial dependence between measured values in an ordered set ought to be verified by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. When applied to sets of test results for single boreholes I came to call it fingerprinting boreholes. SME’s reviewers liked it a lot. And so will members of ISO/TC69/SC6 once the upshot of spatial dependence on confidence limits for central values is clear. Assuming spatial dependence between measured values and interpolation by inserting functionally dependent values between measured values has made a mess of the study of climate change. Surely, CAC/ISO/TC69/SC6 has a role to play in selecting the most fitting statistical methods.

True test for bias

Every scientist and engineer ought to grasp the properties of variances. Those who don’t should not even try to apply a true t-test for bias. And nobody could do it without counting degrees of freedom. It is an irrefutable fact that a true t-test for bias cannot be applied without counting degrees of freedom. That’s why geologists ought to question the validity of geostatistics. The more so since the author of the very first textbook predicted that "…statisticians will find many unqualified statements…” What David didn’t predict was that he would blow a fuse if somebody did. By the way, keep your copy in a safe place. It may well become a collector’s treasure before this millennium is history. Counting degrees of freedom comes to mind as a topic that does not get the respect it so richly deserves. But I’m getting off my train of thought! Here’s a simple but true test for bias applied to an ancient set of paired data.

Observed t-value significant at 99.9% probability

Scientists and engineers ought to apply Student’s t-test in the same way as W S Gosset himself did. All textbooks on applied statistics teach the t-test. It was Volk’s Applied Statistics for Engineers that taught me all about the t-test. Those who want to apply Student’s t-test for bias the proper way should study Chapter Six The t Test. Study not only Section 6.1.4 The Null Hypothesis but even more so Section 6.1.3 Degrees of Freedom. Here’s what Volk wrote, “…we accept a risk of a 5 per cent chance of being in error”. Next, he pointed out, “This error, of falsely rejecting the null hypothesis, is called an error of Type 1”. What I have done in my work is avoid the term “error” without some appropriate adjective. Risk analysis and loss control have played a key role in my work. That’s why I report Type I risk and combined Type I&II risks as intuitive measures for the power of the t-test.

Bias Detection Limits and Probable Bias Ranges

A simple analogy exists between those types of risks and the role of a fire alarm. The Type I statistical risk refers to the event that a fire occurs but the alarm does not sound. The Type II statistical risk refers to the event that the alarm sounds but no fire occurs. The combined Type I and Type II statistical risks refer to the event that a fire occurs and the alarm sounds. Simple comme bonjour! The next step was to define Probable Bias Ranges. It is true that PBRs may sound counterintuitive to those who are not used to working with applied statistics. But surely, PBRs fit the observed bias bon d’un t!

Dr Pierre Gy’s view on accuracy is spelled out on page 17 of his 1979 Sampling of Particulate Materials. Here’s his take, “Accurate: when the absolute value of the bias is not larger than a certain standard of accuracy”. He could have but didn’t mention Standard Reference Materials. SRMs play an important role in calibrating analytical methods and systems. Analytical chemists need SRM’s with confidence intervals for one or more constituents. Gy’s so-called sampling constant is, in fact, a function of a set of four (4) variables. As such, it does have its own variance. What’s more, Gy’s own Student t-Fisher test has raised more eyebrows than interest.

Posted on my website are scores of statistical tests and techniques. I used to attached it as an Appendix to my reports long before Professor Dr Michel David took such a dim view of applied statistics. My son and I in 1992 had to put together Precision and Bias for Mass Measurement Techniques. It was Part 1 of a series on Metrology in Mining and Metallurgy. I hold Canadian copyrights for Metrology in Mining and Metallurgy and for Behind Bre-X, the Whistleblower’s Story. I have scores of blogs to write before I make up my mind what to complete first.

False test for bias

Testing for bias plays a key role in science and engineering. Student’s t-test is the par excellence test for bias. The t-test for paired data has always played a role in my work. A bias between test results determined at loading and discharge is a cause of conflict between trading partners. The question is always whose test results are biased. A matter of concern in 1967 was dry ash contents of anthracite shipments from the mines in Pennsylvania to the port of Rotterdam. I went to the USA and determined that loss of dust during sample preparation was the most probable cause of bias between dry ash contents. I had done time at TUDelft. So, I knew that carbonaceous shale is softer than anthracite, and that hammer mills tend to crush and grind autogenously. That’s why I thought that loss of fine dust during preparation of primary samples at loading would cause test samples to show lower dry ash contents at discharge in the Port of Rotterdam.

Holmes hammer mill

SGS’s coal testing laboratory in Rotterdam, too, had a Holmes hammer mill. It was similar to the one at loading but ours worked with its spring-loaded container closed. Settlements between buyer and seller were based on test results determined at discharge. So, we couldn’t afford to mess up primary samples by running our hammer mill ajar. What we did do was prepare test samples for analysis in the usual manner. We would then pass the reject of each primary sample through the hammer mill with its container left slightly ajar. We collected dust on sheets of paper placed at 0.5 m and 1.5 m from the hammer mill. Dust that had settled at 0.5 m weighed 26.2 grams and contained 14.6% dry ash. Dust that had settled at 1.5 m weighed 12.1 g and contained 16.5% dry ash. The settlement sample showed 10.40% ash on dry basis whereas our messed-up sample showed 10.26% ash on dry basis. With but one degree of freedom our experiment was not much of a true test for bias. But it proved that the integrity of our settlement samples passed scrutiny. We didn’t determine dry ash in dust collected on our coveralls and face masks. I had a fine team to work with. But I wanted more than a team! I wanted SGS to build a new laboratory as far away as possible from where we were. But SGS was not ready yet. What SGS did do was ask me to set up a laboratory in Vancouver. Now guess what?

When I met Greg Gould for the first time at Rotterdam in 1967 he did already chair ASTM Committee D05 on coal and coke. Greg praised Volk’s Applied Statistics for Engineers so I bought my first copy. He told me about Dr Jan Visman, his work at the Dutch State Mines during the war, his 1947 PhD thesis, and his input in ASTM. I was pleased to meet him in person after we had moved to Canada in October 1969. Dr Jan Visman was an independent thinker. He was as a true a scientist as Greg Gould was a professional engineer. And he was a true PEng! I shall always treasure my copy of Visman’s PhD thesis and our correspondence. I remember our talks with fondness. We talked about the composition and segregation components of his sampling variance. I pointed out the term “segregation” suggests a sampling unit may have been more homogeneous in the past. My friend agreed. That's the reason why the distribution variance and the composition variance add up to the sampling variance. It happened when two Dutchmen talked about sampling in a foreign language. But unlike French sampling experts we did grasp the properties of variances.

The odd reader of my blogs may think I’m a pack rat. I do plead guilty! I want to get back to testing for bias with Student’s t-test. But I need to tell one more tale before talking about false bias testing. Once upon a time Matheron’s new science of geostatistics somehow slipped into bias testing. It came about after ASTM awarded me in 1996 a plaque for 25 years of services. Greg Gould had asked ASTM’s Board of Directors to recognize Dr Jan Visman and his work. ASTM did so but misspelled Jan's first name as Jane! It was the same year that Barrick Gold signed me on to figure out what kind of gold resource Bre-X Minerals had cooked up in the Kalimantan jungle. And it was the time when Greg Gould sent me bias test data that Charles Rose had enhanced by kriging. He had taken to liking to geostatistics. So much so that of one of his papers was approved for David’s 1993 bash at McGill University. Rose talked about A Fractal Correlation Function for Sampling Problems. But one of his many problems was that Mohan Srivastava had lent him a helping hand. I met Rose for the first time in Columbia many years ago. He joined SGS some time after I had left in 1979. ASTM awarded Rose in 2004 the R A Glen Award. He represents the USA on ISO/TC27 on coal. He talked about his take on bias testing during the meeting at Vancouver in 2009. What a waste of my time! SGS announced on April 24, 2008 the strategic acquisition of Geostat Systems International, Montreal, Canada. For crying out loud!

False test for bias

That’s why I decided to show how to apply a false bias test. Firstly, I got the set of paired dry ash contents determined in eleven (11) shipments of Pennsylvanian anthracite at loading and at discharge. Next, I played the kriging game by inserting a kriged estimate between each pair of measured values. Take a look at what I cooked up! The variance of differences between paired data dropped from var(Δx)=0.1078 for a set of eleven (11) measured values to var(Δx)=0.0396 for the embellished set. That's what happens when a set of measured values is enriched with a set of kriged estimates. So much for kriging when testing for bias. Stay tuned for a true test for bias.

How to fingerprint boreholes

I know how to fingerprint boreholes. What’s more, Merks and Merks not only know how to fingerprint boreholes but how to derive unbiased confidence limits for metal contents and grades. We have known all of that since Professor Dr Michel David in February 1990 rejected Precision Estimates for Ore Reserves. He was but one of many similarly gifted geostatistical reviewers with CIM Bulletin. The same paper was praised by and published in Erzmetall, October 1991. What’s more, Borehole Statistics with Spreadsheet Software was approved by and published in SME Transactions 2000. I applied this technique to a large set of test results for a gold deposit in Kazakhstan. When I asked my contact at Barrick Gold what he thought of my report, his response was: “It’s worth its weight in gold”. I didn’t charge quite that much on February 9, 1998.

Here’s what Merks and Merks have decided to propose. We give a short course on how to derive unbiased confidence limits for grades and contents of mineral inventories. Professor Dr Roussos Dimitrakopoulos and all of his staff and students should participate. Of course, Dr Frederik P Agterberg, Emeritus Scientist with Natural Resources Canada, should be invited. Those who are Members of the International Industry Advisory Board ought to attend. McGill staff and students should be given free access. Our fee for this crash course in statistical thinking is C$100,000.
Here's why! It took a long while to unscramble the French sampling school. Here are but a few of the most salient facts. Dr Pierre Gy would put a set of primary increments in a single basket in a manner of speaking. That’s why he didn’t get a single degree of freedom. It may well be why Gy engineered his sampling constant. What he didn’t know was that his sampling constant does have its own variance. Professor Dr Georges Matheron never put a set of core samples from the same borehole in but one basket. So, he was blessed with almost as many degrees of freedom as there were core sections in a borehole. Not quite as many because a set of n boreholes gives df=n-1 degrees of freedom. But Matheron got stuck in a rut. He may have taught his disciples to assume spatial dependence between measured values in ordered sets, and to strip the variance off the distance-weighted average AKA kriged estimate. It would explain why Stanford's Professor Dr Andre G Journel is so unfamiliar with Fisher’s F-test. He never knew that ordered sets of n measured values give dfo=2(n-1) degrees of freedom. Neither did he know that the number of degrees of freedom for a set of measured values with variable weights is a positive irrational. It took a lot of independent statistical thinking to unscramble what French sampling school had cooked up between 1954 and 1974.

Doing more with less at McGill

The First Coming of Professor Dr Roussos Dimitrakopoulos to McGill University came to pass in June 1993. He had come all the way from Down Under to chair Geostatistics for the Next Century. It was a bash where geostatistical thinkers of the world had flocked together. They had come to praise Professor Dr Michel David and his 1977 Geostatistical Ore Reserve Estimation. David wrote in his Introduction on page VII that “…statisticians will find many unqualified statements here”. Not a single thinker asked why David had done so. My son and I found some odd gaffes when we studied David’s stuff. That’s the very reason why we wrote Precision Estimates for Ore Reserves. My abstract for The Properties of Variances raised more eyebrows than interest. I have posted on my website both the paper and its review by CIM Bulletin.

When all praise was said and done and duly recorded Dr RD went back to the University of Queensland, Brisbane, Australia. That’s where he honed his skill to do more with less. So what did McGill University think it got when Dr RD came to stay? Here’s how McGill’s University Relations Office on July 13, 2005 saw his Second Coming: “…It’s no surprise, then, that a world leader in the field of mining engineering has chosen Canada as a place to pursue his scholarship.” What world leader would be driven to merge stochastic mine planning and holistic mining? And Dr RD did all that while the properties of variances stayed as far beyond his grasp as they were in June 1993. Those were the very properties I had wanted to talk about at David’s shindig.

Professor Dr Roussos Dimitrakopoulos
Spanning disciplines, spanning the globe

Doing more with less sounds so forward looking and snobbish. But Stanford had already bested McGill at doing more with less. It did so after Matheron and a few of his disciples took the new science of geostatistics all the way to North America in 1974. Some novel science it turned out to be! The distance-weighted average lost its variance and morphed into a kriged estimate. Spatial dependence between measured values in ordered sets was to be assumed. Degrees of freedom went the way of the dodo. That’s how Professor Dr Georges Matheron and his mates have managed to make a mockery of applied statistics!

Professor Dr Roussos Dimitrakopoulos had come to chair Geostatistics for the Next Century. But he was also keen to speak about Spatiotemporal Modelling: Covariances and Ordinary Kriging Systems. It was a thoughtful touch to talk about David’s work at his bash. The more so since David had written much about covariances and kriged estimates in his 1977 Geostatistical Ore Reserve Estimation. We knew in the late 1990s that David was wrong on page 286 of Chapter 10 The Practice of Kriging. Here’s where the author of the very first textbook on geostatistics had strayed away from applied statistics. Figure 10 in Maréchal and Serra’s 1974 Random Kriging made a comeback in 1977 as David’s Figure 203.

Fig. 203. Pattern showing all the points within B,
which are estimated from the same nine holes.

Here’s what the author wrote in Section 10.2.3.3 Combination of Point and Random Kriging on page 286, “Writing all the necessary covariances for that system of equations might be a good test to find out whether one really understands geostatistics!” David’s exclamation mark may well have been inserted to imply the veracity of the whole quote. The problem is not so much with this statement but with the caption under Figure 203. All the points within B are not estimated but derived from the same nine holes. As such, each and every point is a function of the same set of nine (9) holes. And each and every one of them does have its own variance in applied statistics. No ifs or buts! Counting degrees of freedom would have been a good test to find out whether one really understands applied statistics. If all holes in a set of nine (9) were equidistant, then the number of degrees of freedom would be df=n-1 for the set, and dfo=2(n-1) for the first term of the ordered set. If distances between holes are variable, then the numbers of degrees of freedom are no longer positive integers but positive irrationals. Degrees of freedom do not disappear because Professor Dr Roussos Dimitrakopoulos says so.

I have told my story on the power of applied statistics and the flaws of geostatistics to the Members of CIM’s Vancouver Branch on Friday, January 29, 1993. CIM Bulletin and its peer review process have played a key role in the proliferation of geostatistics. It’s about time to tell my story about sound sampling practices and proven statistical tests.