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.

CIM Bulletin has been my core source of fickle stats since the 1990s. A few of my papers were published in CIM Bulletin and may be downloaded from CIM’s website. Life Members, too, do pay for downloads. In contrast, Matheron’s work may be downloaded from CdG’s website free of charge. A paper for which no download fee ought to be charged is Armstrong and Champigny’s A Study on Kriging Small Blocks. David must have been pleased that mine planners were to blame for over-smoothed estimates. But he was not at all pleased that Precision Estimates for Ore Reserves was short of references to geostatistics. So, I submitted the same paper on November 14, 1990 to the Journal for Mathematical Geology. JMG’s reviews turned out to be a toss-up. That’s why JMG’s Editor had asked his Associate Editor to review our paper. JMG’s Associate Editor was the same Margaret Armstrong who studied kriging of small blocks. Of course, she saw fit to reject a paper that was praised by and published in Erzmetall.

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.

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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!

NRCan stuck with geostatistics

Dr Frederik P Agterberg went to work with geostatistics long before NRCan stood short for Natural Resources Canada. He is still Emeritus Scientist with NRCan`s Geological Survey of Canada. He is one of the most gifted geostatisticians in the world. As such, he has a soft spot for Professor Dr Georges Matheron and a penchant for his magnum opus. So much so that he called him the Founder of Spatial Statistics. He did so after Matheron had passed away in 2000. Matheron’s disciples didn’t agree with Agterberg’s view. Matheron taught them how to assume, krige and smooth with infinite confidence. So, they thought of him as the mastermind behind the Centre de Géostatistique and the Centre de Morphology Mathematique. What Matheron taught his disciples was inspired by one or other innovative theme that would call on his most creative thinking. That`s why they thought of him as the Creator of Geostatistics.

Professor Dr Georges Matheron (1930-2000)
Creator of Geostatistics
Founder of Spatial Statistics
Self-made Wizard of Odd Statistics

My son and I checked out what sort of new science Matheron had created. We found out in 1989 that his new science of geostatistics is an invalid variant of applied statistics. As a matter of fact, the author of the very first textbook on geostatistics couldn’t possibly have scored a passing grade on Statistics 101. He praised the “famous Central Limit Theorem” but failed to work with it when he should have. He never tested for spatial dependence between measured values in ordered sets. That’s why it didn’t take us long to get to the bottom of what was wrong. The author had seen fit to shelve degrees of freedom. But it took a long time to unravel what else was shelved, who did it, when, where and why. The power of the internet did make it possible to trace Matheron’s new science to its roots in the early 1950s. In those bygone days young Matheron was a budding geologist who went to work with what he then thought was applied statistics. As such he proved a tenuous grasp of applied statistics early on during his calling.

What struck me as a bad omen for Matheron’s new science of geostatistics was the fact that his grasp of the properties of variances can be traced to the French school of sampling in-situ ores and mined ores. What gets me hopping mad is sloppy sampling and statistics. That’s why I want to put in plain words what Matheron did in the early 1950s. In those days he was a novice geologist with the French Geological Survey (BRGM) in Algeria. His very first paper was called Formule des Minerais Connexes. It was dated November 25, 1954 and marked Note Statistique No 1 straight above its title. It would seem that Matheron saw himself as a statistician of sorts. All the same, CdG’s webmaster early in this century saw fit to mark his very first paper as Note Géostatistique No 1. Perhaps a dash of deception but too little and too late to fool anyone but the odd hardcore kriger.

Matheron’s Note Statistique No 1 was not reviewed by his peers. It took me quite a while to find out that Matheron was without peers. That was just as well since he didn’t quote literature on statistics. What was worse is that he didn’t report any primary data for l’Oued-Kebir. In fact, reporting primary data ranked just as low on his list of things to do as did counting degrees of freedom. Matheron derived population means of μ1=0.45% for lead and μ2=100 g/t for silver, and population variances of σ1=1.82%2 for lead and σ2=1.46 (g/t)² for silver. How he could have done so much with so little is a mystery. A finite set cannot possibly give population means and variances. But that’s what Matheron thought he got! He tested for associative dependence between lead and silver grades and got a correlation coefficient of ρ=0.85. He didn’t point out whether this correlation coefficient was statistically significant at 95%, 99% or 99.9% probability. What put a monkey wrench in Matheron’s first crack at statistics is that he didn’t test for spatial dependence. As luck would have it, Stanford’s Journel and Matheron’s most talented disciple put forward in 1992 that spatial dependence between measured values in ordered sets may be assumed. Testing for spatial dependence was just as trendy in 1992 as it was in 1954.

Matheron may have had a bit of an epiphany when it hit him that l’Oued-Kebir core samples did vary in length. That’s why on January 13, 1955 he tagged on a Rectificatif to his Note Statistique No 1. What he didn’t tag on were the lengths of his core samples. How he derived weighing factors was as clear as drill mud. The same weighting factors should have been applied when he tested for associative dependence between lead and silver. Weighting factors should also have been applied to test for spatial dependence between metal grades determined in ordered core samples of variable length from a single borehole. He didn’t know that degrees of freedom are positive irrationals when core samples vary in length. In other words, he didn’t even know how to fingerprint boreholes. What a shame that peerless Matheron kept on writing more of the same.

No matter what odd statistics Matheron did cook up it would smoothly pass his own peer review. It did make a mockery of statistics that his new science of geostatistics made it all the way to the USA in 1970. Tagged along on the trip were A Marechal and J Serra. They would assist Matheron in making a strong case for his novel science. The stage was set on campus at the University of Kansas for a geostatistics colloquium on 7-9 June 1970. In those days, Matheron, Marechal and Serra were geostatistical scholars at the Centre de Morphology Mathematique at Fontainebleau, France. Matheron himself had thought up Brownian motion along a straight line. It would set the stage for random functions to be continuous between measured values. He did so in his rambling Random Functions and their Application in Geology. What was still beyond Matheron’s grasp is how to verify 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. It did so since counting degrees of freedom had not yet made Matheron’s brief list of significant things to do.

Marechal and Serra in Figure 10 of Random Kriging show how to derive a set of sixteen distance-weighted averages from a set of nine measured values. What M&S didn’t derive was the variance of each and every distance-weighted average. David in 1977 took a long look at M&S’s data but didn’t derive the variance of each and every distance-weighted average either. What David did run into were infinite sets of distance-weighted averages. Journal in 1978 was so taken with David’s infinite set of distance-weighted averages that he took the zero variance with hook, line and sinker. Matheron in 1960 found out about D G Krige’s work at the Witwatersrand gold complex in South Africa and came up all sorts of krige-inspired adjectives and verbs. That’s in a nutshell why Matheron’s new science of geostatistics took on a life of its own for no reason whatsoever.

Agterberg’s 1970 Autocorrelation Functions in Geology was about what he then called “a geologic prediction problem”. He defined a set of measured values at unevenly distributed positions in a sample space. His problem was not so much how to derive the value of the stochastic variable at the selected position. His real problem was that he didn’t derive the variance of his distance-weighted value at the selected position. On a positive note, he didn’t make a point of the fact that his set of measured values didn’t define an infinite set of variance-deprived distance-weighted average values. He didn’t test for spatial dependence by taking a systematic walk that visits each measured value but once, and that covers the shortest possible distance between all positions. Agterberg’s 1970 geologic prediction problem popped up as “a typical kriging problem” in his 1974 Geomathematics.

A geologic prediction problem in 1970! A typical kriging problem in 1974! The eulogy for Matheron in 2000! The silence of NRCan’s Emeritus Scientist in 2010! What’s the matter with NRCan’s brass? When will Dr Frederik P Agterberg be asked to explain why his distance-weighted average doesn’t have a variance?

Back when NRCan was Canmet I knew several of its scientists. Most of all I remember Dr Jan Visman, a Dutch mining engineer with a keen interest in coal processing. He was an accidental sampling expert of sorts because of his need to understand coal processing. We were members of ASTM D05 on sampling and analysis of coal. He headed the Western Regional Laboratories of the Department of Mines and Technical Survey until his retirement in 1976. I owe him a debt of gratitude. His 1947 PhD thesis made it clear that the variance of the primary sample selection stage is the sum of composition and distribution components. That’s why I’m keeping his memory alive on Wikipedia.

Dr Robert Sutarno was a true expert on applied statistics in general and interlaboratory test programs in particular. He taught me a lot about statistical analysis of interlaboratory test programs, and about preparation and certification of reference samples. I met him when we were members of the Canadian Advisory Committee to ISO Technical Committee 102 on iron ore. We traveled to Japan in 1974. The fact that Bob spoke Dutch was a bonus for me. I was more conversant with German and French than with English when we came to Canada in 1969. But that’s not the end of my case against geostatistics!

The Age of Statistics is upon us

How I wish it were true! But that`s what W J Reichmann thought in 1961. It’s the very title of Chapter 1 in his delightful Use and abuse of statistics. The first line of its Preface points out: `Very few people nowadays can progress very far without at some point coming in contact with statistics`. Now that’s what I have been trying to tell the geostatistocracy since the early 1990s. So I pointed to Reichmann’s work in Abuse of Statistics. I own several scores of books on sampling and statistics. Applied statistics underpins sampling practice just as much as probability theory does sampling theory. As such, degrees of freedom play a key role in sampling practice but none at all in sampling theory. No ifs or buts! Except in CIM Bulletin. In Matheron’s tour de force of course. And in Cressie’s Statistics with upper-case S.

The Editor of CIM Bulletin wrote on September 21, 1992 that “articles of a controversial nature” may be published under the heading Forum. It’s hard to believe that applied statistics was controversial in those days. Of course, The Geological Society of CIM has got to be vigilant when deciding what GEOSOC members get to read. So I was pleased that Abuse of Statistics was approved for publication with the proviso that the source of my most favorite quote be disclosed. That’s what Dr Frits P Agterberg wanted. In those days, Agterberg was Associate Editor with CIM Bulletin. In due course, he would wish not only geologists but geoscientists at large to work with geostatistics. Here’s what H G Wells (1866-1946) once said, “Statistical thinking will one day be as necessary for efficient citizenship and the ability to read and write.” How about that? I do not know when or even if Wells said it and to whom. But Darrell Huff thought Wells had said so. Surely, the author of How to lie with statistics would not tell a lie. So I put Huff’s book on my short list of references. And I put Wells’ thought on statistical thinking on top of my brief but somehow controversial article.

Peer review at The Geological Society of CIM in those days was a sham. Not only was it blatantly biased but it was also shamelessly self-serving. Why did it see fit to reject Precision Estimates for Ore Reserves? It did so because we didn’t refer to twenty years of geostatistical literature! And why didn’t we do that? We didn’t because Professor Dr Michel David’s 1977 Geostatistical Ore Reserve Estimation didn’t show how to derive unbiased confidence limits for metal grades and contents of in-situ ores. As a matter of fact, geostatisticians still do not know how to derive unbiased confidence limits for metal contents and grades of ore reserves. All the same, The Geological Society of CIM rejected Dependencies and degrees of freedom and The properties of variances just as brazenly as did the Journal for Mathematical Geology.

CIM’s GEOSOC failed to grasp in 1992 why the properties of variances and the concept of degrees of freedom cannot be ignored with impunity. That’s when Bre-X Minerals was getting ready to explore for gold in Kalimantan, Indonesia. I’m sick and tired thinking of Bre-X’s salting scam. The more so because Bre-X’s massive phantom gold resource was cooked up by assuming the same gold grades between step-out lines of boreholes as within lines of boreholes. It’s the doctrine of Stanford’s Journel. Agterberg argued in 2009 that Bre-X’s test results for gold are “no real data”. The problem was that the Ontario Securities Commission thought they were. And many Bre-X investors thought likewise!

I’m still tickled orange that applied statistics played a key role in unraveling the Bre-X fraud. I have large sets of bogus gold assays determined by cyanide leaching 250 g test portions taken from crushed and salted drill core sections. I’ll show why it makes sense to test for rather than assume spatial dependence between measured values in ordered sets.

The first step was to insert three (3) lines of kriged boreholes between two lines of (2) salted boreholes in Busang’s South-East Zone. The second step was to derive statistics for two (2) lines of salted boreholes and three (3) lines of kriged boreholes. A little applied statistics was enough to show that geostatistics creates spatial dependence where it does not exist.

Neither line of salted boreholes displays a significant degree of spatial dependence. In contrast, all lines of kriged boreholes display a significant degree of spatial dependence. What a shame it’s all bogus gold. What’s real is that even test results for gold in salted boreholes do give degrees of freedom. In contrast, functionally dependent values such as kriged estimates are never blessed with degrees of freedom. All of that is old hat in applied statistics! The age of applied statistics may still not be upon us but the age of the internet is. The worldwide net has made it easy to find out who messed up what, when, where and why. Geostatistics is about to look a lot worse.

Stanford’s Journel shed light on geostatistics

Journel got down to shedding some light on October 15, 1992. He did so in a six page letter to the Editor-in-Chief of the Journal for Mathematical Geology. At that time, JMG’s Editor-in-Chief was Dr Robert Ehrlich, a professor at the Department of Geological Sciences with the University of South Carolina. Journel had “…a bit reluctantly…” agreed to go through my various notes. He left it up JMG’s Editor-in-Chief to decide whether his carefully crafted response should be sent to me. He did point out “…however, I strongly feel that Math Geology has had more than its share of detracting invectives.” Good grief! What could possibly be wrong with geostatistics?

My son and I knew exactly what was wrong in David’s 1977 Geostatistical Ore Reserve Estimation. But it would be the next Millennium before Matheron’s work was posted on the internet. So, I wrote ten letters to JMG’s Editor-in-Chief between November 14, 1990 and August 4, 1992. I had mailed drafts of several papers on different topics, and called him several times. I had also mailed him a copy of my book on Sampling and Weighing of Bulk Solids. I was quite pleased to have received a copy of Journel’s response to my various notes. The more so since JMG’s Editor-in-Chief wrote, “Your feeling that geostatistics is invalid might be correct”. What a pity that he left. What I still don’t know is whether he jumped or was pushed.

Journel had written “It seems to me that Merks’ anger arises from a misreading of geostatistical theory, or a reading too encumbered by classical “Fischerian” [sic!] statistics”. He really got into the nitty-gritty of geostatistics when he pointed out “The very reason for geostatistics or spatial statistics in general is the acceptance (a decision rather) that spatially distributed data should be considered a priori as dependent one to another, unless proven otherwise. Journel couldn’t have said it any worse. It’s a textbook case of circular logic. Accept spatial dependence between ordered sets of measured values in sample spaces and sampling units. Do not prove otherwise by applying Fisher’s F-test. What a bunch of blatant blarney! Who wouldn’t get angry?

A peeved student of geostatistics and a friend of mine had gifted me his pre-read copy of Journel & Huijbregts 1978 Mining Geostatistics. On page 1 the authors define what the book is all about. Here’s literally what they wrote: Etymologically, the term geostatistics designates the statistical study of natural phenomena. G Matheron (1962) was the first to use this term extensively, and his definition will be retained: “Geostatistics is the application of the formalism of random functions to the reconnaissance and estimation of natural phenomena.”

In a footnote on the very same page Journel & Huijbregts 1978 Mining Geostatistics mentioned J Serra and A Maréchal. David’s 1977 Geostatistical Ore Reserve Estimation, too, mentioned J Serra and A Maréchal. In fact, David took this example from Serra and Maréchal’s 1970 Random Kriging. Matheron’s 1970 Random Functions and their application in Geology, too, was presented at the 1970 Geostatistics colloquium on campus at The University of Kansas in June 1970. And so was Agterberg’s 1970 Autocorrelation Functions in Geology.

That’s when I made up my mind to study the lives and times of the greatest geostatistical minds on our little planet. Journel had gone to Stanford University in 1978 as a Visiting Associate Professor of Applied Earth Sciences. He liked it so much that he never left. As a matter of fact, he has spent more time teaching geostatistics at Stanford than he studied it under Matheron’s humdrum guidance. Journel was Mining Project Engineer at Matheron’s Centre de Morphologie Mathematique from 1969 to 1973. Journel was Matheron’s most gifted disciple by far. His list of honors and awards is striking to say the least. From 1973 to 1978 he did time as Maitre de Recherches at Matheron’s Centre de Geostatistique. Professor Dr A G Journel has been teaching Stanford’s students from 1978 to the present. He has taught them all of his own silly nitty-gritty on Matheron’s novel science of geostatistics.

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