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!