Stochastic mine planning at McGill

Mrs Heather Monroe-Blum, Principal and Vice-Chancellor at McGill University, has not yet responded to my email of November 9, 2010. Neither did Dr Jacques Hurtubise, Chair, Department of Mathematics and Statistics, or Dr Christophe Pierre, Chair, Department of Engineering. I explained why geostatistics is an invalid variant of applied statistics. Here’s what I wrote ad verbatim:

On November 5, 2010 I perused Canada’s University Innovation Leaders. Something struck me as odd at McGill University. Stochastic mine planning is taught by Professor Dr Roussos Dimitrakopoulos at the Department of Mining and Materials Engineering. Stochastic mine planning with kriging variances is at odds with genuine variances in applied statistics as McGill students are taught at Mathematics and Statistics. Here’s what McGill students should know about applied statistics and geostatistics. Applied statistics morphed into geostatistics under the leadership of Professor Dr Georges Matheron, a French geologist-cum-probabilist and a self-made wizard of odd statistics. My 20-year campaign against the geostatocracy and its army of degrees of freedom fighters is chronicled on my blogs and my website.

Dr Frederik P Agterberg is Emeritus Scientist with Natural Resources Canada and Past President of the International Association for Mathematical Geosciences. He called Professor Dr Georges Matheron (1930-2000) in his eulogy the Founder of Spatial Statistics and ranked him on a par with giants of mathematical statistics such as Sir Ronald A Fisher (1890-1962) and Professor Dr J W Tukey (1915-2000). Agterberg was wrong! Young Matheron in 1954 derived the degree of associative dependence between lead and silver grades determined in core samples of variable lengths. On second thought he even derived length-weighted average lead and silver grades. What he did not derive were weighted variances of sets of metal grades, and the variances of length-weighted average metal grades. The Founder of Spatial Statistics never tested for spatial dependence between measured values in ordered sets. He never counted degrees of freedom. He did not know that 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 himself didn’t derive the variance of the distance-weighted average in his 1970 Autocorrelation Functions in Geology or in his 1974 Geomathematics. Agterberg’s problem is that as few as a pair of measured values, determined in samples selected at positions with different coordinates in a finite sample space or sampling unit, gives an infinite set of zero-dimensional, variance-deprived distance-weighted averages. Distance-weighted averages morphed into kriged estimates. Agterberg’s work ignores that his distance-weighted average point grade converges on David’s …famous central limit theorem… when all measured values do have identical weights.

Infinite sets of kriged estimates and zero kriging variances are the very reason why the world’s mining industry embraced geostatistical mineral resource/ore reserve estimation with reckless abandon. Geostatistics converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource. I applied Fisher’s F-test to prove that the intrinsic variance of Bre-X’s gold at its Busang property was statistically identical to zero. I did so for Barrick Gold several months before Bre-X’s boss salter passed away.

Lord Kelvin (William Thomson 1824-1907) once said, “…when you can measure what you are speaking about, and express it in numbers, you know something about it, but when you cannot express it in numbers your knowledge is of the meagre and unsatisfactory kind…” Lord Kelvin knew more about degrees Kelvin and degrees Celsius than about degrees of freedom and the study of climate change. Lord Kelvin and Sir Ronald A Fisher (1890-1960) were marginal contemporaries. Lord Kelvin would have wondered about the wisdom behind assuming spatial dependence between measured values in ordered sets. Sir Ronald A Fisher could have verified spatial dependence by applying his F-test to the variance of a set of measured values and the first variance term of the ordered set.

Not all scientists need to know as much about Fisher’s F-test as do geoscientists. Far too few know how to test for spatial dependence by applying Fisher’s F-test, and how to derive sampling variograms that show where orderliness in our own sample space of time dissipates into randomness. So much concern about climate change! So little concern about sound sampling practices and proven statistical methods! I make a clear and concise case against geostatistics on my blogs and on my website. What I have been teaching most of my life ought to be taught at all universities on our little planet and be implemented in all international standards. I’m working hard to make it happen. Applied statistics should be given a chance! McGill students should not be taught stochastic simulation with kriging variances. Please peruse my case against geostatistics. Please do respond to my message before the end of this year.

Yours truly,
Jan W Merks

http://cg.ensmp.fr/

Praise for a scientific fraud

The First Coming of Dr Roussos Dimitrakopoulos from Down Under all the way to McGill University came to pass in 1993. He had come to chair an International Forum on Geostatistics for the Next Century. The stage for this early forum was set at McGill’s Conference Office on June 3-5, 1993. Geostatistical scholars from far and wide had come to praise Professor Dr Michel David. For it was he who had crafted the very first textbook. His 1977 Geostatistical Ore Reserve Estimation does refer to “the famous central limit theorem”. Even degrees of freedom make a token appearance in Table 1.IV. His work did qualify for support from the National Research Council of Canada (Grant NRC7035). I do have my own copy of his book and have worried a lot about it. That’s why I mailed an abstract on “The Properties of Variances”. In fact, I did send it twice by registered mail. The first was lost and the second rejected. All I wanted to ask was why each and every distance-weighted average AKA kriged estimate did not have its own variance in his 1977 textbook. The more so since one-to-one correspondence between functions and variances is so sine qua non in my work!

David in November 1989 had rejected Merks and Merks Precision Estimates for Ore Reserves. He had done so because our paper was short on references to 20 years of geostatistical literature. In contrast, Erzmetall praised its “splendid preparation” and published it in October 1991. David didn’t know how to test for spatial dependence, how to count degrees of freedom, or how to derive unbiased confidence limits for gold grades and contents of in-situ ore. We did what David never got around to doing. We derived precision estimates for the mass of contained gold based on assays determined in a set of ordered rounds in a drift. The set of primary increments from each mined round had been put in the same basket. That’s why we couldn’t estimate the intrinsic variance of gold.

My son and I had taken at different times the same stats courses at Simon Fraser University. I had done so shortly after we came to Canada in October 1969. My problem in those days was that I spoke German and French better than English. I left SGS to work with Cominco. I wrote a lot on sampling and statistics and lectured all over the world. These days I still write about sampling and statistics but travel little. My son Ed has a PhD in Computing Science and was twice awarded the Dean’s Silver Medal. He worked with IBM in Toronto. Nowadays he leads the top-level Eclipse Modeling Project and the Eclipse Modeling Framework subproject. Ed set up Macro Modeling, a small but independent company. My wife and I are pleased that he and his partner have settled in Vancouver, BC. Applied statistics is still fun and games for both us. Here’s what keeps us thinking about McGill University.

Variance of a general function

This formula finds its origin in calculus and probability theory. It shows that the population variance of a general function is the sum of n variance terms, each of which is the squared partial derivative toward an independent variable multiplied by its variance. It bridges the gap between probability theory with its infinite set of possible outcomes and sampling practice with its finite sets of measured values. The formula underpins the variance of any central value. The arithmetic mean is the central value of a set of measured values with constant weights. Area-, count-, density-, distance-, length-, mass-, and volume-weighted averages are central values of sets of measured values with variable weights. The transition from sampling theory to sampling practice with finite sets of measured values demands that degrees of freedom be counted. No ifs or buts! Functions without variances have gone where dodos fly!

In 1970 Professor Dr Georges Matheron brought his new science of geostatistics to North America. In his Random Functions and their Application in Geology Matheron invoked Brownian motion along a straight line. It was just as richly embellished with symbols and as short on primary data as is his magnum opus. Maréchal and Serra in Random Kriging applied the same symbols that Matheron had taught to all of his disciples. Figure 10 puts in plain view how to do more with less.

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

David may have thought that what Maréchal and Serra were doing was kind of cool. So, he explains it on page 286 of his 1977 textbook in Chapter 10 The Practice of Kriging. He dressed up M&S‘s Figure 10 with a slightly different caption.

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

David added a dash of subterfuge when he called his points within B “estimated”. Each point within B derives from the same set of measured values for nine (9) holes. As such, each and every one of them is a function of the same set of nine (9) holes. Of course, each distance-weighted average does have its own variance in applied statistics. Thus it came about that variance-deprived distance-weighted average point grades morphed into kriged estimates.

In Section 12.2.1 Using a simulated model of Chapter 12 Orebody Modelling (see page 324) David prevaricates, “The criticism to this model is obvious. The simulation is not reality. There is only one answer: The proof of the pudding is…! So far the few simulations made which it has been possible to check have a posteriori proved to be adequate”. Good grief! Why didn’t they ask Merks and Merks?

McGill University had set the stage in 1993 to praise Professor Dr Michel David. Those who had come to its Conference Center to praise him did not have the faintest clue what was wrong with geostatistics. But rigs were drilling at Bre-X’s Busang property! The Bre-X fraud came about because the geostatocracy had failed to grasp the properties of variances.

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Unscrambling the French sampling school

My grandma taught me not to put all my eggs in one basket. She was a caring matriarch who told inspiring stories. She played card games but odds were beyond her grasp. She played for pennies but not with other people’s pennies. She didn’t have a PhD in anything. But I took her word and never put all my eggs in one basket.

Dr Pierre Gy (1924-…) and Professor Dr Georges Matheron (1930-2000) put the French sampling school on the world map. Matheron never put core samples from a single hole in one basket so to speak. But Gy did put a set of primary increments taken from a sampling unit in one basket. So he didn’t even get a single degree of freedom. The interleaved sampling protocol is described in several ISO Standard Methods. It is also described in Chapter 6 Spatial Dependence in Material Sampling of a textbook on Approaches in Material Sampling. Dr Bastiaan Geelhoed edited the text. IOS Press published the book in 2010.

Matheron marched to a new low when he sampled in situ ores. So he didn’t put in one basket a set of core samples from a single borehole. But he failed to derive measures for precision, to test for spatial dependence between grades of ordered core sections, and to count degrees of freedom. Quelle dommage! Matheron thought that Gy knew a lot about sampling theory and sampling practice. Gy’s L’Échantillonage des Minerais en Vrac was printed in two parts and on 656 pages. Tome 1 is dated January 15, 1967, and Tome 2 hit the shelves on September 15, 1971.

Gy’s sampling slide rule

Gy pioneered a slide rule of sorts to simplify the sampling of mined ores. His sampling constant C is a function of c, the mineralogical composition factor, of l, the liberation factor, of f, the particle shape factor, and of g, the size range factor. Hence, Gy’s sampling “constant” is a function of a set of four (4) stochastic variables. As such, Gy’s constant C does have its own variance.

Some sampling constant!

Matheron wrote a three-page Synopsis to Gy’s Tome 1 Theory Generale. He praised Gy’s work for defining, “… accuracy and precision, bias and random error, etc…” Gy, in turn, praised Matheron’s 1965 PhD thesis. Gy did refer to Visman’s 1947 PhD Thesis and to his 1962 Towards a common basis for the sampling of materials. Gy didn’t mention Sir R A Fisher, Anders Hald, Carl Pearson, and William Volk. Why did Gy deserve Matheron’s praise?

Dr Pierre M Gy is a chemical engineer with a deterministic take on sampling. He is the most prolific author of works on sampling. He sent me a copy of his 1979 Sampling of Particulate Materials, Theory and Practice. It was marked Christmas 1979 and signed underneath. Gy pointed to degrees of freedom in Chapter 14. His Index does not list degrees of freedom between “degenerate splitting processes” and “degree of representativeness”. Another odd entry in this Index is “SF = Student-Fisher”. Student’s t-test proves or disproves bias between paired data. Fisher’s F-test proves or disproves whether two variances are statistically identical or differ significantly. Both statistical tests demand that degrees of freedom be counted!

Matheron praised Gy’s work in 1967 and Gy, in turn, praised Matheron’s work in 1979. Here’s what Gy wrote literally:

“The sampling of compact solids and more specifically mineral deposits
is covered by the science known as ”Geostatistics”. The fundamentals
of this science, established by Krige, Sichel, deWijs were developed by
Matheron and his team (references in appendix). Worked out in France,
Matheron’s theories are slowly but steadily gaining acceptance in
English speaking countries around the world thanks to
an increasing teaching and to technical textbooks such as
Michel David’s “Geostatistical Ore Reserve Estimation” (1977)”.

Now that’s a nice little tit-for-tat between scholars who created the French sampling school! Matheron and his disciples cooked up quite a variant of applied statistics! Thank goodness, his magnum opus is posted on CdG’s website. I have scanned his 1965 PhD Thesis for degrés de fidelité but didn’t find any at all in 301 pages of dense probability theory. But I did find two sets of numerical data. Matheron’s Set A looks a lot less variable than Set B but both sets have the same central value. So, I applied Fisher’s F-test to the variances of the sets and the first variance terms of the ordered sets.

Data sets in Matheron’s PhD thesis

I have pasted Matheron’s A- and B-sets on a truncated title page of his 1965 PhD Thesis. This title page and Fisher’s F-tests for his A- and B-sets are posted on my website. Matheron and Gy didn’t know how to test for spatial dependence in sampling units and sample spaces. The root of the problem is these scholars didn’t grasp the properties of variances. But then, neither did my grandma!

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Accuracy of pneumatic conveying calculations

The reliability of a pneumatic conveying calculation is depending on:

– The accuracy of the input parameters and the degree in which they reflect the reality.

– The accuracy of the performance data of the pneumatic installation components.

– The completeness of the theory on which the calculation algorithm is built.

– The degree of approximations in the calculation algorithm.

Ambient conditions:

– Intake conditions:

For a pressure pneumatic conveying system, the intake conditions are important as they determine the mass flow of gas.

• The intake conditions are:
• Temperature

The intake temperature determines the intake density of the compressor. A positive displacement compressor will compress a less dense volume and  therefore delivers less mass flow.

A diffuser controlled turbo compressor delivers a constant mass flow.

The intake temperature varies with the weather conditions.

• pressure

The intake pressure determines the intake density of the compressor. A positive displacement compressor will compress a less dense volume and  therefore delivers less mass flow.

A diffuser controlled turbo compressor delivers a constant mass flow.

The intake pressure varies with the weather conditions.

• Relative Humidity (RH)

The intake relative humidity determines the intake density of the compressor. A positive displacement compressor will compress a less dense volume and  therefore delivers less mass flow.

A diffuser controlled turbo compressor delivers a constant mass flow.

In addition, the RH determines the amount of water vapor, which influences the amount of condensation in the pneumatic pressure conveying line.

In a vacuum system, there is normally no condensation, as the absolute pressure of the gas is decreasing.

When the conveyed material is very cold compared to the intake temperature of the conveying gas, then condensation can occur.

The intake RH varies with the weather conditions.

The mass flow of gas determines the Solid Loading Ratio (SLR) and this ratio determines the pressure drop for material collision and friction losses.

In addition the pressure drop for gas resistance is influenced by the mass flow of gas.

–          Ambient conditions.

Ambient conditions are not necessarily the same as the intake conditions. The intake conditions can be indoor and the pipe routing can be outdoor with completely different ambient conditions.

• The ambient conditions are:
• Temperature

The temperature(s) of the surroundings (especially around the pipe) determine, in combination with the pipe material and pipe insulation, the cooling (or heating up) of the material/gas mixture.

The temperature of the material/gas mixture determines the gas density in the pipeline and thereby the gas velocity and through impulse transfer, the material velocity.

This, in the end influences the partial pressure drops.

• Pressure

The ambient pressure(s) of the surroundings are not that influential, apart from a slight effect on convection cooling.

• Relative Humidity (RH)

The RH of the surrounding has no or a slight influence on the heat transfer from the material/gas mixture to the environment (or vice versa)

Material properties:

The pneumatic conveying properties of the material can have a great influence on the created pressure drop in combination with the Solid Loading Ratio (SLR).

In the calculation, the basic pneumatic conveying parameters are the suspension velocity and the Solid Loss Factor (SLF).

• Material properties are:
• Particle size distribution
• Particle shape
• Particle density

These 3 properties determine the suspension velocity of the material particle. The suspension velocity is related to the minimum gas velocity at the considered location in the pneumatic conveying system. Moreover, the suspension velocity, in relation to the gas velocity, determines the acceleration of the particles and the, quasi stationary, slip velocity (gas velocity – particle velocity).

The suspension velocity is also influenced by the cohesion of a material.

The cohesion causes particles to agglomerate, thereby forming a virtually higher suspension velocity.

The required pressure drop for keeping the particles in suspension increases with higher suspension velocities.

• Bulk density

The influence of the bulk density is very small when all particles are conveyed in suspension. When there is sediment formed (saltation), the blocked area, calculated with the bulk density, is deducted from the flowing area, increasing the gas velocity. This happens also in a bend, where the material is decelerated by friction in the outer bend curve.

• Internal friction coefficient.

The internal friction coefficient determines the flow properties of the material. The flow properties around a suction nozzle determine the column pressure drop outside the intake nozzle and thereby determine the available pressure drop for conveying. This effect is also important for the material flow in a pressure tank, determining the pressure drop for feeding the conveying pipe line.

• External friction coefficient

The external friction coefficient is influencing the velocity drop in bends and the end flow in pressure tanks.

• Flow function

Apart from the internal friction coefficient, the cohesion of a material contributes to the flow function of the material. This is important for inflow in nozzles and pipe intakes.

• Fluidizability

Good fluidizability enhances the flow properties towards suction nozzles and pipe intakes, thereby reducing the required pressure drop for material feeding.

• Solid Loss Factor

The solid loss factor, which is part of a formula comprising the SLR and Reynolds’s number, determines the energy losses, due to particle collisions and particle friction.

The SLR and the used formula, is unique for each material composition. F.i. a different particle size should be reflected in the SLR.

The SLR can be derived from a set of conveying test (scale or preferably at scale=1), ranging over various conveying lengths, diameters, rates and airflows.

• Temperature

The material temperature is an influential factor at the material/gas mixture, because of the higher specific heat capacity compared to the conveying gas.

The resulting gas temperature determines the gas density (or volume) and thereby the gas velocity. The gas velocity influences all partial conveying pressure drops.

• Heat capacity

Determines the resulting material/gas mixture temperature and thereby influences ultimately the partial conveying pressure drops.

Installation component data:

• Installation components are:

• Compressors

The compressor type (positive displacement, turbo, centrifugal fan, oil free, adiabatic compression, isochoric compression, isothermal compression) determine the airflow and gas temperature at different pressures.

Especially oil free positive displacement compressors (blowers and screw compressors with internal compression) have a performance accuracy of +/- 5%, due to the fact that the rotor clearance after production cannot be guaranteed 100%. Only a slight difference in the rotor clearance causes a significant influence on the rotor leakage gas volume.

• Feeders (pressure), dischargers (vacuum)
• Screw feeders

Screw feeders are considered leak free and do not influence the gas mass flow through the conveying system.

The material temperature is increased by the material wall- and flight friction.

• Rotary locks

Rotary locks have 2 types of leakage:

• Pocket leakage.

The empty pockets transport gas from the “high” pressure zone to the “low” pressure zone. The amount of leakage is depending on the pressure drop over the rotary lock.

The leaked air is lost for the pneumatic conveying purposes.

• Gap leakage.

The rotor/housing gaps (2 or 3 in series) also leak gas from the “high” pressure zone to the “low” pressure zone. This leakage is pressure drop dependent and is strongly related to the actual clearance and thereby strongly related to the wear condition of the rotary lock.

The leaked air is lost for the pneumatic conveying purposes.

• Pressure tanks

Pressure tanks are considered leak free and do not influence the gas mass flow through the conveying system.

• Filter assemblies

The filter pressure drop depends on the degree of filter contamination.

The filter contamination is represented by a filter resistance factor.

The filter resistance factor changes in time as the filter becomes more and more clogged, despite of regular filter pulse cleaning.

Thereby, the filter pressure drop increases, leaving less pressure drop available for pneumatic conveying.

While the compressor gas flows are pressure related, the filter pressure drop is also related to the actual conveying pressure.

In a vacuum system, the filter pressure drop is maximum, when the vacuum is minimum.

• Gas only piping

The gas only piping causes a relatively small pressure drop (when properly designed) and is not accounted for in the computer calculation.

• Silencers

The silencers cause a relatively small pressure drop (when properly designed) and is not accounted for in the computer calculation.

• Valves

The energy losses in the valves are considered relatively small and are not accounted for in the computer calculation.

• Diverters

The pressure drop caused by diverters is accounted for as a bend.

• Influence of wear and tear of the installation components

The wear and tear of installation components in a pneumatic conveying system, which influence the pneumatic conveying process are related to gas flow. The relevant components are compressors and feeders.

A worn rotor of a screw compressor acting as a vacuum pump can be damaged so badly by continuously leaking filters that the rotor clearance increases to a level that the gas flow becomes zero at maximum vacuum.

Mathematical description of the theory of pneumatic conveying:

Although the basic principle of pneumatic conveying is simply the transfer of impulse from a moving gas to a moving solid, whereby the occurring losses are covered by the internal energy from the gas at the cost of pressure drop of that gas, the involved physical processes are quite complex.

The calculation model is based on the law of conservation of energy, Newton’s laws and thermodynamic laws for gases and heat exchanges.

• Involved physical processes are
• gas compressing
• gas expansion
• condensation of water vapor
• heat exchange with material
• heat exchange with surroundings
• acceleration by impulse transfer between gas and material particle.
• deceleration by inter particle- and wall collisions
• Gas density changes by condensation of water vapor in the conveying gas.
• pressure drop for keeping the particle in suspension
• extra pressure drop for sedimentation (vertical, slope and horizontal)
• pressure drop for gas resistance
• pressure drop for acceleration
• pressure drop for elevation
• pressure drop for solid collision- and friction losses.
• tank pressurizing
• pipe line purging

Although many variables are considered as input, the programming syntax is a simplification of the real interactions between the gas, material and the environment. Especially in the heat exchanges, the time factor is ignored and assumed to happen instantaneously.

Released heat from water vapor condensation is not accounted for.

The errors as a result of the simplifications and omissions are hidden in the Solid Loss Factor.

The SLF is thereby serving as a (partially) “fudge” factor.

The more effects are accounted for in the calculation algorithm, the less the SLF is acting as a

“fudge” factor and the more reliable the outcome is, but never 100%.

After having executed a pneumatic conveying installation for the specific used input variable values, the calculated result is seldom observed in practice, due to the above mentioned considerations.

Not only the assumed environmental conditions will not always be met, the degree of modeling and the approximations in the calculation algorithm cause the calculation result to be an approximation of the reality.

Calculating a pneumatic conveying installation for different inputs for intake temperature, ambient temperature, ambient pressures and gas flows +/- 5%, the variation in calculation results was approx. +/- 4%.

A computer program presents the calculation outcome as exact numbers, suggesting a non existing 100% accuracy.

Computer programs or algorithms that calculate a pneumatic conveying installation with a minimum of input variables, neglecting complex heat exchanges, condensation, the occurring of sedimentation, the dependency of product losses of the SLR and the turbulence (Re-number) and the calculation of the slip velocity are much less accurate than a computer program that accounts for all these effects.

Caution in interpreting the calculation outputs is therefore strongly advised and a comparison of the calculation results with the observed performances of existing installations or cases is a worthwhile check.

Different times but same fools

Travelling back in time was what H G Wells made nightmares off. I have read the latest biography about his life and time. Michael Sherborne called it Another Kind of Life. Here’s what I learned. Wells did not praise statistical thinking because Ronald A Fisher won the case for degrees of freedom. Sherborne pointed out that the quotation which Darryl Huff did attribute to H G Wells came from Samuel S Wilks. When Wilks gave his 1954 presidential address to the members of the American Statistical Association, he said: ”The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood for complete initiation as an efficient citizen of one of the great new complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima, as it now to be able to read and write.” Sherborn tracked that rather rambling sentence down to Chapter 6 of Wells’s Mankind in the Making. It was Wells himself who brought it down to, “Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” Wells had a way with words and women. But who would not want Wells’s way with words?

Samuel S Wilks in 1954 praised statistical thinking in the USA. Meanwhile in Algeria Georges Matheron thought he was working with applied statistics. He called his first paper Formule de Minerais Connexes and marked it Note Statistique No1. CdG’s webmaster has posted it as Note géostatistique No 1. Matheron tested for associative dependence between lead and silver grades determined in drill core samples of variable length. He didn’t show his set of primary data nor did he give references. He deemed himself without peers. And that’s just as well. Matheron in a Rectificatif to his paper derived length-weighted average lead and silver grades for core samples of variable length. Here’s where he failed. His degree of associative dependence between lead and silver didn’t take into account variable lengths of core samples. He didn’t derive the variances of lead and silver grades. Neither did he derive variances of length-weighted average lead and silver grades. Matheron, too, lost David‘s “famous Central Limit Theorem”.

Lost and never found

He didn’t even try to test for spatial dependence between ordered sets of metal grades. Young Matheron had a long way to go. Yet, he kept marching in place to the din of kriging drums.

the founder of Spatial Statistics

So it was that he never did what he had failed to do in 1954. It was Dr Frederik P Agterberg in his eulogy who called Professor Dr Georges Matheron (1930-2000) the founder of Spatial Statistics. Here’s what may baffle many a mind! Matheron and his minions would rather assume spatial dependence between measured values than apply Fisher’s F-test to the variance of the set and the first variance term of the ordered set. To assume spatial dependence where it doesn’t exist is the very reason why the study of climate dynamics is such a mess.

Professor Dr Margaret Armstrong

Armstrong’s up-to-date list of publications does not refer to the study on over-smoothed small blocks. She doesn’t even mention her stirring Freedom of Speech? She wrote it as Editor of De Geostatisticis. It was a little leaflet with a few pages of text about the blessings of geostatistics. From time to time one more leaflet would be published to praise Matheron’s gift to the mining industry. Mark Twain knew a bit about mining. He may never have claimed that a mine is a hole in the ground with a liar on top. What I would do if I were a mining investor is question the validity of mineral inventories in annual reports. I would do so until the mining industry sets up an ISO Technical Committee on reserve and resource estimation. That would have made the 31st President of the United States proud!

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BHP Billiton fell for stochastic sham

Assume that I have invested in BHP Billiton. Surely, I would want to know where my mining giant is going. And most of all I would want to know whether or not Chairman Marcus Kloppers knows what he is doing. Does he have a buddy on his Board who knows a bit about applied statistics? Would she or he know that functions do have variances, and that only measured values do give degrees of freedom? Take a long look at my short story. Assume, krige, smooth, and rig the rules of applied statistics. The roots of geostatistics rest in the archives of CIM Bulletin. What I have done for more than twenty years is keep my story alive. It’s about an invalid variant of applied statistics. It was geostatistics that converted Bre-X’s bogus grades and Busang’s barren rock so smoothly and effortlessly into a massive inferred gold resource. Just the same, Fisher’s F-test proved that the intrinsic variance of gold at Busang was statistically identical to zero. I do not know where Kloppers was when Bre-X Minerals blew up. What I do know is that BHP Billiton was not the only one who put up plenty of play dough to do more with fewer boreholes. So let’s go forward to the play!

McGill Professor Dr Roussos Dimitrakopoulos is Canada Research Chair and BHP Billiton Chair in Mine Planning Optimization at the Department of Mining, Metals and Materials Engineering. He came all the way from Down under in June 1993 to celebrate at McGill University a forum called Geostatistics for the Next Century. When I read about this forum I thought it somewhat premature. That’s why I put together The Properties of Variances and submitted the abstract by registered mail on January 4, 1992. My son and I had studied David’s 1977 Geostatistical Ore Reserve Estimation. The author did not know how to derive unbiased confidence limits for metal contents and grades of volumes of in-situ ore. That’s why we derived confidence limits in Precision Estimates for Ore Reserves. But Dr RD was not at all interested in properties of variances in 1993. And he’s still not interested in 2010!

Professor Dr Roussos Dimitrakopoulos is a Member of IAMG. Once upon a time, IAMG stood for International Association for Mathematical Geology. Nowadays, it stands for International Association for Mathematical Geosciences. The novel tag does make sense. More geoscientists than geologists study this little planet. Dr RD is Editor-in-Chief, the Journal for Mathematical Geosciences.

Editor-in-Chief

As such, he approves what meets his requirements and rejects what is at variance with stochastic mine planning with kriging variances. Much of Behind Bre-X, The Whistleblower’s Story, reviews the works of Matheron, Agterberg, David, Dimitrakopoulos, Journel, and scores of likeminded geostatistical thinkers.

Here’s what I wrote on March 4, 2010 to Dr Marius Kloppers, Chief Executive Officer,
BHP Billiton Plc, London, United Kingdom.

A great deal of my experience was put together in Sampling and Weighing of Bulk Solids. I did so after I had been Assistant to the Chairman of Cominco Ltd. For example, I derived unbiased confidence limits for the mass of metal contained in a mass of mineral concentrate or mined ore. ISO/TC 183 checked and approved this method. ISO published it as ISO 13543:1996 Determination of mass of contained metal in a lot.
Attached is a copy of my letter of November 30, 1994, to the Chairman, CIM Ad Hoc Reserve Definitions Committee. I pointed out that the mass of contained metal in a volume of in-situ ore is a function of volume, in-situ density, and a grade factor. This function, too, does have its own variance. In fact, one-to-one correspondence between functions and variances is a condition sine qua non in mathematical statistics. In geostatistics, however, not all functions do have variances. The variance of the distance-weighted average aka kriged estimate went missing. Drill core sections were crushed and salted at Bre-X’s Busang site when I wrote the enclosed letter in November 1994. Geostatistics converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource. Mathematical statistics proved the intrinsic variance of gold to be statistically identical to zero.
The purpose of my letter is to suggest that the mining industry should set up and support an ISO Technical Committee on reserve and resource estimation. It is simple to derive confidence limits for masses of metals in reserves, and to derive confidence limits for proved masses of metals in resources. My son and I did set the stage in 1992.

Dr Marius Kloppers, Chairman, BHP Billiton, has not yet replied to my letter.

Geostatistics in a nutshell

Assume, krige, smooth, and rig the rules of applied statistics. That’s all! It’s simple to assume spatial dependence between measured values in ordered sets. And to krige or not to krige is never a question. Not since Matheron himself in the 1960s cooked up the krigeage eponym. It took on a life of its own when the kriged estimate and the kriging variance became the heart and soul of Matheron’s new science of geostatistics. Smoothing sounds so soothing! Yet, the smoothing stage should not be taken lightly. In fact, the kriging variance of the least biased subset of some infinite set of kriged estimates should not be “over-smoothed”. Of course, one would not expect those who are taught to assume, krige and smooth to do too much of it with too few data. Matheron thought in 1954 that he worked with applied statistics. He was wrong! Journel in 1992 taught that spatial dependence between measured values in ordered sets may be assumed. Matheron’s most gifted disciple was wrong! It’s a piece of cake to assume spatial dependence between measured values in sampling units and sample spaces. To apply Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set is much more intuitive. It was Journel who pointed out that one’s reading ought not to be “… too encumbered by classical Fischerian [sic!] statistics”.

To krige or not to krige has never been a dilemma in my work. Kriging between measured values in sampling units or sample spaces either enhances spatial dependence or gives a false positive. It was a cinch to prove that how the geostatocracy rigged the rules of applied statistics. Scores of textbooks show what went wrong, when and why. Geostatistics is all the rage with the world’s mining industry. It is true that the practice of assuming, kriging and smoothing does a lot with small sets of boreholes. But geostatistics, unlike applied statistics, does not give unbiased confidence limits for metal contents and metal grades of mineral inventories in annual reports. And that’s a fact!

Professor Dr Michel David wrote a few textbooks on geostatistics. My son and I studied his 1977 Geostatistical Ore Reserve Estimation. What we learned is that David did not derive confidence limits for metal contents and grades of ore reserves. So, we did it in our own paper on Precision Estimates for Ore Reserves. My son and I took at different times the same stats courses at Simon Fraser University. Ed earned his PhD in Computing Science and went to work at Big Blue in Toronto. Our paper was sent on September 28, 1989 to CIM Bulletin. David did reject it because we had shown “our own method”. Now whose method had he expected? He foretold in 1977 that “statisticians will find many unqualified statements”. Why then was he surprised when we did? Precision Estimates for Ore Reserves praised by and published in Erzmetall 44, (1991).

The National Research Council of Canada was so taken with Matheron’s new science that it sponsored David’s 1977 Geostatistical Ore Reserve Estimation. Patronage played a role in Grant NRC7035. The very first page of his 1977 book shows an epiphany that had come to David. That’s where he wrote, “To our statistician readers, we apologize”. And so he should! In Section 2.1.1 The Standard Error of the Mean, the author pointed to what he then praised as “the famous Central Limit Theorem”. David was inspired by Figure 10 in Maréchal and Serra’s 1970 Random Kriging. M&S brought a bag of geostat stuff to the USA in 1970. It was made up at the Centre de Morphologie Mathématique, Fontainebleau, France. M&S had crafted it under Matheron’s guidance. So it came about that Random Kriging saw the light at the first krige-and-smooth shindig at the University of Kansas, Lawrence, USA on 7-9 June 1970. Dr Frederik P Agterberg, Dr Daniel F Merriam and a gathering of geostatistocrats were tickled pink. The central limit theorem had morphed into the kriged estimate. The few statisticians didn’t question Brownian motion along a straight line.

Professor Dr Michel David in his 1977 textbook explained how M&S had derived sixteen (16) “famous Central Limit Theorems” from a set of nine (9) holes.

Chapter 10   Figure 203   page 286

The author clarified, “Pattern of all the points within B, which are estimated from the same nine holes”. Each and every one of David’s “estimated points” is a function of the same nine (9) holes. As such, each estimated point (otherwise known as Central Limit Theorem) does have its own variance. In fact, one-to-one correspondence between functions and variances is sine qua non in applied statistics. In short order, David’s estimated points morphed into kriged estimates. In Section 10.2.3.3 Combination of Point and Random Kriging on the same page he suggested, “Writing all the necessary covariances for that system of equations might be a good test to find out whether one really understands geostatistics”. Here’s what I have already pointed out some twenty years ago, “Counting degrees of freedom for that system of equations is a good test to find out whether really understands applied statistics”.
Somehow it seems to make sense that geoscientists apply geostatistics. Some geoscientists call it “mathematical statistics” but did get rid of degrees of freedom. I work with Volk’s Applied Statistics for Engineers and with Visman’s sampling theory and practice. Applied statistics defines the sampling variogram, which, in turn, defines spatial dependence in sampling units and sample spaces alike.

Dr Frederik Pieter Agterberg

He seems as ambitious and ambiguous today as he was in 1970. He ignores fundamental rules of applied statistics. He has yet to explain why he stripped the variance off his distance-weighted average. He did review and approve Abuse of Statistics. I asked Natural Resources Canada on September 21, 2010 for permission to interview Dr Frederik P Agterberg in writing. I have yet to receive a response to my request. It made me wonder whether or not he is still Emeritus Scientist with Natural Resources Canada.

//

NRCan scientists told to shut up

That’s what I read in the Vancouver Sun of September 14, 2010. In fact, the heading read: “Muzzling scientists offends principle of public service.” Why would Natural Resources Canada muzzle its scientists? And why so shortly after Labor Day? A few NRCan scientists would not mind to be muzzled. But Dr Frederik P Agterberg, Emeritus Scientist with the Geological Survey of Canada, ought to speak up. The trouble is he doesn’t want to. He is but one of a few scientists who could shed light on an inconvenient truth. Why did his distance-weighted average not have a variance in 1970? And why did all of Maréchal and Sierra’s distance-weighted averages not have variances in 1970? In time, the distance-weighted average morphed into a kriged estimate. Geostatistics is all about kriging variances of sets of kriged estimates. It has made such a mess of the study of climate change.

Dr Graeme Bonham-Carter and Dr Eric Grunsky are Agterberg’s soul mates at the Geological Survey of Canada. This three-some has been thinking alike for quite a while. They were members of the International Association of Mathematical Geology (IAMG) long before it morphed into the International Association of Mathematical Geosciences (IAMG). It’s all part of the tangled tale behind one and the same acronym. Bonham-Carter chairs the Publications Committee, Grunsky represents Computers & Geosciences, and Agterberg is Member ex officio.

Stanford Associate Professor DrJef Caers
Conditional Simulation with Patterns
2007 Best Paper Award

By far the most dedicated geostatistician on IAMG’s Publication Committee is Professor Dr Roussos Dimitrakopoulos. He is Editor-in-Chief, Mathematical Geosciences. He is the Canada Research Chair and BHP Billiton Research Chair in Mine Planning Optimization at the Department of Mining, Metals and Materials Engineering at McGill University. He teaches McGill students all he knows about stochastic modeling with kriging variances. What he does not show is how to test for spatial dependence by applying Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set. Neither does he teach McGill students how to count degrees of freedom.

McGill Professor Dr Roussos Dimitrakopoulos

Dr RD came all the way from Down Under to McGill in June 1993. He came to chair a Forum to honor Professor Dr Michel David’s contribution to geostatistics. My abstract for The Properties of Variances failed to arouse his interest. Geostatistical software was about to convert Bre-X’s bogus grades and Busang’s barren rock into a gold resource. Analysis of variance proved that the intrinsic variance of Busang’s gold was statistically identical to zero.

A long while ago I asked Dr Nathan J Divinsky, a UBC Professor Emeritus of Mathematics, whether or not degrees of freedom may be ignored. Here’s literally what he said: “But without degrees of freedom statistical inferences are impossible.” One of Canada’s Prime Ministers paid close attention to Dr Divinsky. Not surprising since the Right Honourable Kim Campbell, Canada’s first female Prime Minister, and Dr Divinsky were once married. I did not dare ask Dr Divinsky whether infinite sets of kriged estimates and zero kriging variances do make any statistical sense.

Of tangled tales and geostatistics

Here are but a pair of my lingering questions. Why did Dr Frederik P Agterberg strip the variance off his distance-weighted average? Why did Professor Dr Noel A C Cressie dismiss degrees of freedom? I brought my concern to the attention of Minister Lisa Raitt on March 9, 2009, and of Minister Christian Paradis on March 6, 2010. NRCan’s technocrats were instructed to deal with my concern. Mark Corey, Assistant Deputy Minister, pointed out on June 4, 2009, “Geostatistics continues to evolve as a discipline, and we appreciate your contribution in this field.” Dr David Boerner, Acting Assistant Deputy Minister, declared on May 10, 2010, “Natural Resources Canada is a science-based organization and values the scientific rigor of the peer review process”. Now that’s a whole load of chutzpah! NRCan’s library is a treasure trove of works on geostatistics. Each and every one of them proves that geostatistics is but bogus geosciences. Look at Agterberg’s 1970 and 1974 works. He derived the distance-weighted average of a set of five (5) measured values determined at positions with variable coordinates in a sample space. He didn’t derive the variance of this distance-weighted average. He didn’t count the number of degrees of freedom for the set and for the ordered set. So much for Agterberg’s grasp of applied statistics. Of course, geostatistics couldn’t possibly get any worse, could it? But it did in 1970 when Matheron and his disciples came all the way to the USA!

Maréchal & Serra – Figure 10

Maréchal and Serra in 1970 derived a set of sixteen (16) distance-weighted averages on a 4 by 4 matrix. They did so from a set of nine (9) measured values determined at positions with variable coordinates. What M&S didn’t derive was the variance of any of those sixteen (16) distance-weighted averages. What Professor Dr Michel David didn’t do in 1977 was what M&S didn’t do in 1970, what Agterberg didn’t do in 1970 and in 1974, and what young Matheron didn’t do in 1954. David predicted that professional statisticians would find “unqualified statements” in his 1977 work. Now there’s one fact he got right. But once push came to shove he saw no wrong. Degrees of freedom did pop up on in his work but did so in a table others had put together. Little else made sense at all let alone statistical sense. M&S’s Figure 10 morphed into David’s Figure 203. He got into the nitty-gritty of geostatistics in Chapter 12 Orebody Modelling. That’s where he cooked up the tangled tale of conditional simulations and ran into infinite sets of simulated values. Finally, distance-weighted averages first morphed into kriged estimates and then into simulated values. Stanford’s Professor Dr Andre Journel was Matheron’s most gifted disciple. It is fitting then that he was the one who defined in 1978 the zero variance of the infinite set of distance-weighted averages-cum-kriged estimates. Voila, Matheron’s new science of geostatistics!

Why stop kriging when there are infinite sets of kriged estimates to play with? The problem is one-to-one correspondence between functions and variances. Have a function? Stuck with a variance! No ifs or buts! It is a fact that one-to-one correspondence between functions and variances is sine qua non in mathematical statistics. And don’t take my word for it. Read Volk’s Applied Statistics for Engineers. NRCan’s technocrats are blessed for they may borrow Volk’s 1980 book at NRCan’s library. Volk explains in exhaustive detail what geostatisticians love to hate. I already owned a 1958 copy when I was toiling in the Port of Rotterdam. It has fallen apart but my faith in Volk’s work never wavered. My second copy went missing when I traveled the world and taught sampling and statistics in all sort of settings. And I do have a 1980 copy. Chapter Seven is called Analysis of Variance and Section 7.1.4 is called Variance of a General Function.

Scientific rigor of the peer review process was shamelessly self-serving and blatantly biased already when NRCan was CANMET and Agterberg was Associate Editor with CIM Bulletin. He reviewed Abuse of Statistics but all he worried about was whether and when H G Wells praised statistical thinking. That’s how serious NRCan’s Emeritus Scientists took geostatistical peer review at CIM Bulletin in the 1990s.

Geostatistics recast Statistics with upper-case S

Young Matheron in 1954 took a shine to what he then thought was statistics. Professor Dr Georges Matheron in the 1970s saw it at that time as his new science of geostatistics. Professor Dr Noel A C Cressie in 1993 saw it more as what he came to call Statistics with upper-case S. He is the brains behind that sort of stats stuff at the Ohio State University. He teaches Statistics with upper-case S at OSU’s Department of Statistics. But why does he teach Statistics with upper-case S? Here’s in plain prose how Cressie put it in his Preface: “Notice that Statistics is capitalized to distinguish it from its other meaning: a collection of numbers that summarize a complex phenomenon – such as baseball or cricket”. Good grief! Could that really be the reason why he brought Statistics with a capitalized S to those who interpret statistics for spatial data? Has he paid any attention to the study of climate change? Turned out to be a bit of a mess, didn’t it? He cautioned elsewhere in his Preface, “We should not forget our roots”. But why then did Cressie forget his roots in mathematical statistics?

Dr Frederik P Agterberg, NRCan’s Emeritus Scientist, brought to my attention in November 2009 that Noel Cressie is a mathematical statistician. He suggested that I consult Cressie’s 1993 Revised Edition of Statistics for Spatial Data. He pointed out that it deals with “kriging variance and equivalent numbers of independent observations”. I was stunned to say the least. I do respect the properties of variances and the concept of degrees of freedom much more than do most scientists. Yet, I studied Cressie’s Statistics for Spatial Data with a capital S. In retrospect I should have studied his original work to find out how far he had already wandered away from the straight and narrow of mathematical statistics.

Why did Agterberg want me to study Cressie’s Statistics for spatial data? I do not have the faintest idea! I have asked Agterberg why he stripped the variance off his distance-weighted average in 1970 and in 1974. He has never told me why. French scholars such as A Maréchal and J Serra worked with distance-weighted averages whose variances, too, had vanished in thin air. Matheron had taught Maréchal and Serra everything he knew before they came to the USA in 1970 for the very first krige and smooth fest. Matheron and his disciples failed to grasp what Professor Dr Michel David called in 1977 the “famous” Central Limit Theorem. But then neither did David.

Geologic prediction problem in 1970
Typical kriging problem in 1974
Statistical fraud since 1954

Agterberg writes about the Central Limit Theorem in Chapter 6 Probability and Statistics of his 1974 Geomathematics. Yet, in Chapter 10 Stationary Random Variables and Kriging he paid no heed to the theorem that underpins sampling theory and sampling practice. That’s why NRCan should ask him to explain why his distance-weighted average does not have a variance.

At this stage I’ll go back in time some twenty years. That’s when I got to know Professor Dr Robert Ehrlich. He taught at the Department of Geological Science at the University of South Carolina. He was Editor-in-Chief of what was then known as the Journal for Mathematical Geology. I had mailed him on November 14, 1990 a paper called Precision Estimates for Ore Reserves. It was the same paper that our peers at CIM Bulletin saw fit to thrash. They had done so simply because we had ignored twenty years of geostatistical literature. We wrote it because geostatistics was then and is still today a scientific fraud. The Bre-X’s salting scam has not yet been used in a court of law to prove that geostatistics is a scientific fraud.

The point I want to make is that Professor Dr Robert Ehrlich is a scholar and an independent thinker. He was a perfect fit for the position of JMG’s Editor-in-Chief. Much to my delight he saw through much of the fluff that underpins Matheronian geostatistics. I had mailed him drafts of a few wicked papers and a copy of my 1984 Sampling and Weighing of Bulk Solids. And we did talk a few times about counting degrees of freedom and testing for spatial dependence. But that wasn’t to last alas!

All efforts to have our take on how to derive unbiased confidence limits for ore reserves reviewed by and published in the Journal for Mathematical Geology came to naught. What’s more, a paper titled The Properties of Variances went missing. What also went missing late in 1994 was Professor Dr Robert Ehrlich, JMG’s Editor-in-Chief. So, I asked Dr Daniel F Merriam, JMG’s new Editor-in-Chief, on January 13, 1995 why that paper had not yet been reviewed. He wrote a brief note on April 14, 1995 that it was rejected. So it was that JMG’s peer review process had thrashed the same paper just as thoroughly as did CIM Bulletin’s in November 1989. A few problems surfaced. Agterberg was CIM Bulletin’s Associate Editor and JMG’s Book Review Editor. Agterberg reviewed and approved  Abuse of Statistics. A copy was attached to The Properties of Variances. One of JMG’s reviewers pointed out that it “should never have appeared in print”. One cannot help but wonder who said so!

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