Lecture Notes in Statistics 183Edited by P. Bickel, P. Diggle, S. Fienberg, U. Gather,I. Olkin, S. Zeger

Michel BilodeauFernand MeyerMichel Schmitt (Editors)

Space, Structure and Randomness

Contributions in Honor of Georges Matheronin the Field of Geostatistics, Random Setsand Mathematical Morphology

With 121 Figures

Michel Bilodeau Fernand Meyer Michel SchmittEcole des Mines de Paris Ecole des Mines de Paris Ecole des Mines de ParisCentre de Morphologie Centre de Morphologie Direction des recherches

Mathématique Mathématique 60 Boulevard Saint Michel35 rue Saint Honoré 35 rue Saint Honoré 75272 Paris Cedex 0677305 Fontainebleau 77305 Fontainebleau FranceFrance France

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Personal Reminiscences of Georges Matheron

Dietrich Stoyan

Institut fur Stochastik, TU Bergakademie Freiberg

I am glad that I had the chance to meet Georges Matheron personally oncein my life, in October 1996. Like many other statisticians, I had learned alot in the years before from his books, papers, and research reports producedin Fontainebleau. From the very beginning, I had felt a particular solidaritywith him because he worked at the Ecole des Mines de Paris, together withBergakademie Freiberg one of the oldest European mining schools.

The first contact to his work occurred in talks with Hans Bandemer, whoin the late 1960s had some correspondence with Georges Matheron. Proudlyhe showed me letters and reprints from Georges Matheron. He also had acopy of Georges Matheron’s thesis of 1965, 300 pages narrowly printed. Itsfirst part used Schwartz’ theory of distributions (Laurent Schwartz was thesupervisor) in the theory of regionalized variables, while the second part de-scribed the theory of kriging. Hans Bandemer was able to read the Frenchtext, translated it into German language and tried, without success, to finda German publisher. Unfortunately, the political conditions in East Germanyprevented a meeting between Hans Bandemer and Georges Matheron.

A bit later, in 1969, I saw George Matheron’s book “Traite de Geostatis-tique Appliquee”. It is typical of the situation in East Germany at the timethat this was a Russian translation published in 1968, based on the Frenchedition of 1967. (We could not buy Western books and most of us could notread French. At the time the Soviet Union ignored the copyright laws.) I wasvery impressed by the cover illustration showing a gold-digger washing gold.

Georges Matheron’s Traite is a fascinating book, and for me and manyothers it was the key reference in geostatistics at the time. The progressachieved in this work is perhaps best expressed by the following quote fromthe postscript (written by A.M. Margolin) of the Russian translation:

“We find in the monograph the fundamental ideas and notions of amathematical theory of exploration, called by Matheron ‘geostatistics’.Geostatistics is characterized by a concept which corresponds to the

VI Dietrich Stoyan

mathematical problems of exploration, by fundamental basic notionsand by a large class of solvable problems.

In comparison to classical methods of variation statistics [the sim-ple methods using mean and variance and Gaussian distribution],which were until the 1950s nearly the only way of application of math-ematics in geological exploration, geostatistics is directed to the truemathematical nature of the objects and tools of exploration. . .

According to geostatistics, the uncertainty which is characteristicfor results of exploration is a consequence of incompleteness of ex-ploration of the object but not of its randomness. In this probably theincreasing possibilities of the geostatistical theory consist and the prin-cipal difference to variation statistics, which is based on the analysisof geological variables irrespectively to the locations of observation. . . ”

Not at the time, but later on when I had learnt more about public relationsin science, I admired Matheron’s cleverness in coining terms: It was verysmart to call statistics for random fields ‘geostatistics’ (to use a very generalword, which suggests ‘statistics for the geosciences’, for a more limited class ofproblems) and least squares linear interpolation ‘kriging’ (originally ‘krigeage’;in Freiberg it was for a long time not clear whether it should be pronounced‘krig-ing’ or ‘kraig-ing’ – as suggested by the Russian translation – or ‘kreedge-ing’.)

Georges Matheron’s “Random Sets and Integral Geometry” of 1975 was alandmark in my own scientific development. The mid 1970s were a great timefor stochastic geometry and spatial statistics, which then became more thanjust geostatistics. In that time mathematical morphology was also shaped,and Jean Serra’s work in image analysis became known outside Fontainebleau,both in theory and in applications. The first image analyser, the famous LeitzTAS, was produced, based on ideas of Georges Matheron and Jean Serra.

We, the scientists in East Germany, were unable to order the book in abook store, but I got a copy from Dieter Konig in exchange for a book onqueueing theory. Up to this day, for me (and many others, I believe) GeorgesMatheron’s book is the key reference in random set theory; the cover of mycopy is now in pieces, and many pages are marked with notes. Unfortunately,the book has not been reprinted since.

This book gives an excellent exposition of the topics named in the title, it isvery clearly written, and of just the right theoretical level. Georges Matheron’swork also led me to Hadwiger’s, whose monograph on set geometry is nowone of my most beloved mathematical books in German language. So GeorgesMatheron had an excellent base for the integral-geometric part of his work. Itis a great combination of many mathematical fields, such as integral geometry,set geometry, Choquet’s theory of capacities (in fact Matheron developedit independently), and ideas of Poisson process based stochastic geometry(created in particular by Roger Miles) to obtain a new, rich and fruitful theory.Georges Matheron’s book contains many gems. I name here only the theory

Personal Reminiscences of Georges Matheron VII

of the Boolean model (he writes ‘boolean model’, with small ‘b’), the Poissonpolyhedron, and the granulometries.

It was Georges Matheron who made notions of measurability propertiesrigorous in the context of random sets. This can be explained in terms ofRobbins’ formula for the mean volume of a compact random set:

Eν(X) =∫Rd

p(x)dx

where p(x) = P (x ∈ X). Its proof is based on Fubini’s theorem and wasalready given – in part – in Kolmogorov’s famous book of 1933. It was GeorgesMatheron who showed that the mapping set → volume is measurable. Theσ-algebra with respect to which measurability is considered (based on Fell’stopology) is today called Matheron’s σ-algebra. Some of his results for modelsrelated to Poisson processes were the starting point for further scientific work,when it became clear that methods from the theory of marked point processescan be used to generalize them. Through his book, Georges Matheron has beena teacher and inspirer for a large number of mathematicians in the last threedecades.

In 1989 Georges Matheron published the book “Estimating and Choosing”.Grown older, I was offered the honour to serve as one of its reviewers. This is aphilosophical book, discussing the fundamental question of spatial statistics:“Why does it make sense to perform statistical inference for spatial data,when only a sample of size n = 1 is given?” Indeed, very often a statisticianhas only data from one mineral deposit, or from one forest stand; a secondsample taken close to it can typically not be considered as a sample fromthe same population, because of different geological or ecological conditions.He developed the idea of ‘local ergodicity’, which is plausible and justifiesthe statistical approach. Each spatial statistician should read the book. I alsoenjoyed its sarcastic humor.

In June 1983 Dominique Jeulin came to Freiberg as an invited speakerat a conference. For me and colleagues like Joachim Ohser and Karl-HeinzHanisch this was the first chance to meet a scientist from Georges Matheron’sschool. Dominique Jeulin spoke about multi-phase materials, rough surfaces,and image transformations. He is likely the first French person I ever met, andI learned that French English differs greatly from German English. Dominiquehad to repeat three times his ‘Stojang’ until I understood that he asked forme. The contact to him, which is still lively, finally led to the meeting withGeorges Matheron.

After the big changes of 1989/90 we had many West German PhD studentsat Freiberg. One of them, one of the best, was Martin Schlather. Martinhad studied one year at Fontainebleau, had written a diploma thesis there,and had been given oral exams by Georges Matheron personally. Thus, hecould tell me first hand about Georges Matheron’s personality and aboutthe situation at Fontainebleau. Like anybody who ever had personal contact

VIII Dietrich Stoyan

with Georges Matheron Martin admired him, both as a mathematician andas a person. Martin told me that Georges Matheron’s work is much morecomprehensive than his books and journal publications suggest. So we are allextremely grateful to Jean Serra for publishing a CD with the reports andpapers by Georges Matheron. I agree with Martin and Jean that one willfind wonderful ideas and solutions to hard problems as one goes through thismaterial.

It was Martin Schlather who encouraged me to travel to the Fontainebleauconference in 1996. This conference was organized by Dominique Jeulin in hon-our of Georges Matheron, and the lectures are published in the volume Jeulin(1997). There we met Georges Matheron, who did not give a lecture, but wassitting in the front row in the lecture room, obviously listening with attention.One afternoon there was a reception in the municipal hall of Fontainebleau.The maire decorated Georges Matheron (and others) with a medal, for reasonsthat I did not completely understand, either scientific or political. GeorgesMatheron was polite enough to bear the ceremony, but quite obviously he didnot take it very seriously. He told me anecdotes, and his body language wasclear enough. I had no difficulty in communicating with him, he accepted mypoor English, and I understood him very well.

In the evening I was honoured by an invitation to the family of Jean Serra,where I also met Mesdames Matheron and Serra. It was an enjoyable, not verylong evening, with friendly non-mathematical talk, and a bit of French wine.

In the end, I never orally discussed mathematical problems with GeorgesMatheron, but I appreciated the contact over many years, through studyinghis work, and through a few comments that he made on my work. I believe thatthere is only a small number of international mathematical conferences whichGeorges Matheron attended. His example shows that a mathematician whodoes not visit conferences can be nevertheless be influential and widely known.Perhaps, Georges Matheron could have had even more influence. However, didhe really want this?

Back to Freiberg, I had the idea of honouring him there. At the time I wasthe president of the little Technische Universitat Bergakademie Freiberg andsaw the chance of honouring him with a Dr.h.c. degree. I asked DominiqueJeulin. He liked the idea but warned me that Georges Matheron would proba-bly never come to Freiberg, and if he did, I could not expect his collaborationin a public relations event to the benefit of my university, as I was hoping for.To my regret, I did not pursue the idea any further.

In October 2000, the sad news of Georges Matheron’s death spread. I re-gretted that I had not seen him again after 1996. As a keepsake to GeorgesMatheron, I asked Jean Serra for Georges Matheron’s personal copy of Had-wiger’s book – expecting a book with many pencil notes. To my surprise Ilearned that Georges Matheron had used a library copy.

I am very happy that this volume is now ready, which will honour GeorgesMatheron, one of the great mathematicians of the 20th century.

A few words about Georges Matheron(1930-2000)

Jean Serra

Centre de Morphologie Mathematique, Ecole des Mines de Paris

That day, Paris was steaming hot, as it may happen in July when there isnot the slightest breeze of wind to cool the air. Despite the vigour of hisnineteen years, Georges Matheron, while waiting for his turn, was suffocatingand finally lost his composure before the board of examiners of the EcoleNormale Superieure. Fortunately, he came second at the Ecole Polytechnique,which he entered a few weeks later. That was in 1949. He probably had missedthe opportunity to work in the best possible environment for him, and at thetime he did not know that the path he was heading for would involve him inearth sciences for the rest of his professional life.

Those who have practised geostatistics all know how fascinating it is todiscover the mineral world underground, to figure the structures from drillholevariograms, like a blind person fingering an object to guess its shape. HoweverG. Matheron was the first one to know that excitement, all the more as hewas creating the mathematical tool while using it to comprehend the earthsubstratum. Moreover he designed it so that the description of the mineralspace and the estimation of mining resources be indissolubly linked, like thetwo sides of the same coin.

After two years spent at the Ecole Polytechnique, two more years at theEcole des Mines and another one in the military service, it was in Algeria thatthe Corps des Mines sent him for his first appointment. He had married oneyear before, and landed in Algeria with wife and child. He quickly took overthe scientific management (1956), then the general management (1958) of theAlgerian Mining Survey.

It takes imagination to realize how important it was for a young Frenchengineer. The huge Algerian territory stretches as far as the Saharan Southand abounds in orebodies of all kinds. It is one of the reasons why the Inter-national Geological Congress has been held in Algiers in 1952, just two yearsbefore he arrived. It was also in the early 50’s that papers written by threeSouth-African authors, Krige, Sichel and de Wijs, laid the statistical founda-tions from which G. Matheron would base his theory of geostatistics, that wasrevolutionary at the time.

X Jean Serra

What does not kill you makes you stronger. In the early 60’s, FrenchAlgeria collapsed, France recalled its executives to the home country andreorganized its mining research by creating the BRGM (Geological and MiningSurvey) in Paris. G. Matheron was assigned a “Geostatistical department”,practically reduced to himself. The BRGM did not believe in geostatistics, andthe only mining partners came from the CEA (∼ Atomic Energy Commission),namely A. Carlier and Ph. Formery. This solitude was in fact a blessing whichenabled him to devote himself to the final development of what will be calledlater linear geostatistics.

The genesis of the following works is instructive, and tells a lot aboutthe personality of G. Matheron. In Algeria, he invented the random func-tions with stationary increments, while being sceptical about the probabilisticframework. Every deposit is a unique phenomenon, that occurred once onlyin geological times; besides, when one estimates its reserves, one does notcompare its drillings with those of more or less similar deposits. This uniquephenomenon, studied in itself, does not offer any more hold to probabilities,than if one wanted to know the proportion of hearts in a deck of cards bydrawing only one card once.

It was within that “semi-random” framework that G. Matheron wrote thefirst volume of his “Traite de Geostatistique Appliquee” (treatise of appliedgeostatistics) in 1962, which was based upon the Algerian experience, then thesecond volume (1963) on kriging, which solved the problems of local estimationraised by the uranium deposits of the CEA. Even today, the reader is amazedat such a skilful mastery of first the mathematics, then of the physics ofthe topic, with the right simplifying approximations. The rule of one-to-onecorrespondence for instance, in volume I, is still a masterpiece, where eachterm of the limited expansion of the theoretical variogram (whose estimationis empirically accessible) corresponds, with a known invariable weight, to aterm of the limited expansion of the deposit estimation variance (which issought for).

However this rule could be formulated in a deterministic framework aswell as in probabilistic terms. Hence a third book, entitled “La theorie desvariables regionalisees et leur estimation”, which became his PhD thesis in1963. The deterministic and random parts were developed successively, withmuch rigour, and all theoretical conclusions drawn. But G. Matheron waitedtwenty years before expressing himself, in “estimating and chosing”, upon thechoice of either approach according to the context – the mathematician wasahead of the physicist.

As the BRGM continued to ignore the practical interest of his sextupleintegrals, G. Matheron looked elsewhere to collect followers, through teach-ing. A “geostatistical option” was created at the Ecole des Mines de Nancy,which provided him with his first PhD student. The latter, Jean Serra, rapidlybranched off and oriented methods and applications towards the new field ofmathematical morphology (random event or deterministic fate?). The qualityof the iron ore of Lorraine was defined as much by its grade as by its suit-

A few words about Georges Matheron (1930-2000) XI

ability to grinding, which could only be quantified from the mosaic of thepetrographic phases, as they appear under the microscope. Hence the idea ofmeasuring petrographic variograms, and extending the concept of variogramto that of the hit-or-miss transformation, then to that of opening, etc. Formore than one year, the master and the disciple, though separated by threehundred kilometers, met each month, sharing their enthusiasm, their notes,and the advancement state of the “texture analyser”, which the student wasdeveloping with Jean-Claude Klein. Many years later, in 1998, when describ-ing this intense period at the research committee of the Ecole des Mines deParis, G. Matheron would say : “these were the most beautiful years of mylife”.

In april 1968, the Ecole des Mines de Paris gave G. Matheron the opportu-nity to create the “Centre de Morphologie Mathematique” in Fontainebleau,with J. Serra as the other permanent researcher. The events of may 1968 werefavourable to them, thanks to all the public founds they released, so that intwo years the team grew from two to twelve persons. It spread on both fronts ofgeostatistics and mathematical morphology. From that time, the first gainedinternational recognition, and proposals for mining estimations coming fromthe five continents arrived at the Centre. Moreover, from the early 70’s, theCMM had been asked to map the sea bed, atmospheric pressures, etc. Theapplication fields broadened and with them the variety of the problems to besolved ; for example, that of submarine hydrography led G. Matheron to in-vent the universal kriging (1969), then the random functions with generalizedcovariances (FAI-k, 1973), which both released the constraint of the stationar-ity hypothesis. Another example : the integration of local mining estimationsinto operating management programs led G. Matheron to conditional simula-tions, less accurate than kriging, but which did not smooth the data. Finally,during the 70’s, G. Matheron formalised and proposed a definitive answer tothe major issue of mining estimation, namely the change of support. In thiscase, the problem is no longer to estimate the variance of panels with respectto their size, as in linear statistics, but to be able to predict their whole distri-bution function, in order to fit mining exploitations to the economic conditions(1976).

In parallel with these developments, the mathematical morphology groupbecame independent and evolved to an autonomous centre in the early 80’s.Indeed, from the beginning, its applications covered the whole field of opticalmicroscopy, and while metallography and porous media still pertained to earthsciences, medical histology and cytology addressed quite another audience.

G. Matheron did not show more than a polite interest in such applicationsof mathematical morphology. He did not penetrate them like he had donewith mining technology. Morphological applications were too varied, and whatexcited him was to extract from them some general approaches, which couldbe conceptualised. This is why were produced the theories of granulometries,of increasing operators, of Poisson hyperplanes and of Boolean sets, whichhe gathered into the book Random sets and Integral Geometry, in 1975. The

XII Jean Serra

topic of porous media was the sole exception. This application already had awhole specific physical framework, varying according to the considered scale(Navier-Stokes equation at the microscopic level, Darcy equation at largerscales). How can such changes be linked ? To what extent can random setsprovide tractable models ? Throughout his career at the CMM, G. Matheronkept involved with these questions and indicated fruitful directions.

At the beginning of the 80’s, his professional life took a more “morpho-logical” turn. On the theoretical level, the geostatistical vein seemed to beexhausted, whereas morphologists had just designed new operators, productsof openings and closings, which had the property of being both increasingand idempotent (here the word “morphologists” refers to the members of theCMM team as well as to the American S.R. Sternberg, the German D. Stoyanor the Australian G.S. Watson, among others). As these operators used to ap-ply to both frameworks of sets and numerical functions, G. Matheron situatedhis approach at the broader level of the complete lattices and constructed ageneral theory of the increasing and idempotent operators that he called mor-phological filtering (1982-1988). In spite of appearances, these new levels caneasily be integrated into the structure of his overall work. Morphological fil-tering gave a simplified and denoised vision of the numerical functions, as didkriging for mapping. Increasingness and idempotence had replaced linearityand the master observed the consequences of that genetic change.

Since the 90’s, multimedia have been the dominant theme in Mathemati-cal Morphology, bringing into focus the three topics of motion, segmentationand colour. Today, these topics still concentrate most of the CMM activities.However, G. Matheron did not take interest in them. Since he inserted mathe-matical morphology into the lattice framework, he pursued the idea to extendalso his random set theory. In order to do so, complete lattices must be firstequipped with adequate topologies. G. Matheron’s efforts were devoted to thistask until his retirement, which was punctuated by an unpublished and lastbook on compact lattices (1996).

Jean SerraSeptember 2004

Contents

Personal Reminiscences of Georges MatheronDietrich Stoyan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

A few words about Georges Matheron (1930-2000)Jean Serra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

IntroductionFrom the editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part I Geostatistics

The genesis of geostatistics in gold and diamond industriesDanie Krige, Wynand Kleingeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Concepts and Methods of GeostatisticsJacques Rivoirard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Prediction by conditional simulation: models and algorithmsJean-Paul Chiles, Christian Lantuejoul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Flow in porous media: An attempt to outline GeorgesMatheron’s contributionsJ.P Delhomme, G. de Marsily . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Over Thirty Years of Petroleum GeostatisticsPierre Delfiner, Andre Haas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

The expansion of environmental geostatisticsRoberto Bruno, Chantal de Fouquet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Part II Random Sets

XIV Contents

Random Closed SetsI. Molchanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

The Boolean Model: from Matheron till TodayDietrich Stoyan, Klaus Mecke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Random Structures in PhysicsDominique Jeulin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Part III Mathematical Morphology

Mophological Operators for the Segmentation of ColourImagesJean Serra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Automatic design of morphological operatorsJunior Barrera, Gerald J. F. Banon, Edward R. Dougherty . . . . . . . . . . . . 257

Morphological Decomposition Systems with PerfectReconstruction: From Pyramids to WaveletsHenk J.A.M. Heijmans, John Goutsias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Morphological segmentation revisitedFernand Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Ubiquity of the Distance Function in MathematicalMorphologyMichel Schmitt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Partial Differential Equations for Morphological OperatorsFrederic Guichard, Petros Maragos, Jean-Michel Morel . . . . . . . . . . . . . . . 369

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

List of Contributors

Gerald J. F. BanonDivisao de Processamento deImagensInstituto Nacional de PesquisasEspaciaisSao Jose dos Campos, SP, Brazilbanon@dpi.inpe.br

Junior BarreraDepartamento de Ciencia daComputacaoInstituto de Matematica e EstatısticaUniversidade de Sao Paulo. SaoPaulo, SPBraziljb@ime.usp.br

Michel BilodeauEcole Nationale Superieure desMines de ParisCentre de MorphologieMathematique35 rue Saint Honore77300 FontainebleauFranceMichel.Bilodeau@ensmp.fr

Prof. Ing. Roberto BrunoUniversita di BolognaDipartimento di Ingegneria ChimicMineraria e delle TecnologieAmbientali

Viale Risorgimento,2 - 40136 BolognaItalyroberto.bruno@unibo.it

Jean-Paul ChilesEcole Nationale Superieure desMines de ParisCentre de Geostatistique35 rue Saint Honore77300 FontainebleauFranceJean-Paul.Chiles@ensmp.fr

Chantal de FouquetEcole Nationale Superieure desMines de ParisCentre de Geostatistique35 rue Saint Honore77300 FontainebleauFranceChantal.De Fouquet@ensmp.fr

Pierre DelfinerTOTAL2 Place de la CoupoleLa Defense 692078 Parix La Defense CedexPierre.DELFINER@total.com

XVI List of Contributors

G. de MarsilyUniversite Paris VI et Academie desSciencesGDemarsily@aol.com

Edward R. DoughertyDepartment of Electrical Engeneer-ingTexas A & M UniversityCollege StationUSAedward@ee.tamu.edu

Dominique JeulinEcole Nationale Superieure desMines de ParisCentre de MorphologieMathematique35 rue Saint Honore77300 FontainebleauFranceDominique.Jeulin@ensmp.fr

John GoutsiasCenter for Imaging ScienceThe Johns Hopkins UniversityBaltimore, MD 21218USAgoutsias@jhu.edu

Frederic GuichardDO Labs, 3 rue Nationale92100 BoulogneFrance.fguichard@dolabs.com

Andre HAAS16 rue Paul Mirat64000 Pauhaas.pau@wanadoo.fr

Henk HeijmansCWI, P.O. Box 94079 GBAmsterdam, The Netherlandshenkh@cwi.nl

Wynand KleingeldDe Beers MRM;Mendip Court – Bath RoadWells, Somerset, BA53DGUnited KingdomWynand.Kleingeld@dtc.com

Danie KrigePO box 121 – Florida Hil1716 South Africaomricon@iafrica.com

Christian Lantu’ejoulEcole Nationale Superieure desMines de ParisCentre de Geostatistique35 rue Saint Honore77300 FontainebleauFranceChristian.Lantuejoul@ensmp.fr

Petros MaragosSchool of ECENational Technical University ofAthens15773 AthensGreece.maragos@cs.ntua.gr

Klaus MeckeInstitut fur Theoretische PhysikUniversitat Erlangen-NurnbergStaudtstrasse 7, D-91058 ErlangenGermanymecke@fluids.mpi-stuttgart.mpg.de

Fernand MeyerEcole Nationale Superieure desMines de ParisCentre de MorphologieMathematique35 rue Saint Honore77300 FontainebleauFranceFernand.Meyer@ensmp.fr

List of Contributors XVII

Ilya MolchanovDepartment of MathematicalStatistics and Actuarial Science,University of BernSidlerstrasse 5, CH 3012 BernSwitzerlandilya.molchanov@stat.unibe.ch

Jean-Michel MorelCMLA, Ecole Normale Superieurede Cachan94235 CachanFrance.morel@cmla.ens-cachan.fr

Jacques RivoirardEcole Nationale Superieure desMines de ParisCentre de Geostatistique35 rue Saint Honore77300 FontainebleauFranceJacques.Rivoirard@ensmp.fr

Michel SchmittEcole Nationale Superieure desMines de Paris60 Boulevard Saint-Michel75272 Paris cedex 06Michel.Schmitt@ensmp.fr

Jean SerraEcole Nationale Superieure desMines de ParisCentre de MorphologieMathematique35 rue Saint Honore77300 FontainebleauFranceJean.Serra@ensmp.fr

Dietrich StoyanInstitut fur StochastikTU Bergakademie Freiberg09596 FreibergGermanystoyan@orion.hrz.tu-freiberg.de

Geo

rges

Math

eron

Part I

Geostatistics

The genesis of geostatistics in gold anddiamond industries

Danie Krige1 and Wynand Kleingeld2

1 Florida Hill, South Africa2 De Beers MRM, UK

1 Introduction

Geostatistics has had a phenomenal half a century of development and achieve-ments, and to which George Matheron has made invaluable contributionsthroughout. The genesis of geostatistics is clearly linked to South Africa andmore specifically to gold mining in the Witwatersrand basin, which startedtowards the end of the 19thcentury. In the later stages of the developmentof geostatistics the problems inherent in the valuation of diamond depositspresented a new field for geostatistical contributions.

2 The Influence of Gold and the Origin of Geostatistics

For economic reasons this gold mining was and still is conducted on a selectiveblock basis and calls for intensive and regular sampling of underground expo-sures of the ore bodies. This resulted in the accumulation over many years ofmassive data sets conducive to statistical analysis and the study of frequencydistribution models. In the pre-geostatistics period ore reserve blocks werevalued on the arithmetic averages of samples from the block peripheries; asthese blocks were being mined the advancing stope faces inside the blocks werealso sampled regularly to yield extensive follow-up block values. Comparisonsof these follow-up grades with the original block estimates provide an obviousopportunity for statistical analyses such as frequency distribution studies andclassical correlations. However, this opportunity remained dormant until the1940’s.

At that time, extensive exploration of virgin properties in the new SouthAfrican gold fields by deep drilling was also taking place. Grade estimatesfor these new mines had to be based on limited sets of drill hole grades withno proper basis for estimating the effects of selective mining to economiccut-off grades. Fortunately, the ideal venue for access to all this data fromnumerous existing mines and from the new gold fields was provided by the

6 Danie Krige and Wynand Kleingeld

records in the Government Mining Engineer’s Department where the firstauthor was privileged to be employed in the late 1940’s. He was introducedto the statistical approach for the processing of this data by the earlier initialwork by Sichel [27], Ross [26] and De Wijs [3, 4]. This led to a first setof publications ([8, 9]) which in turn introduced, inter alia, Allais [1] andMatheron [15] to the subject.

This paper is, thus confined to ore valuations in the mining field.

2.1 Frequency Distribution Models

The skew nature of the gold grade frequency distributions was first observedby Watermeyer [33] and later studied by Truscott [32]. But these studies weredone without the knowledge of the lognormal model and were unsuccessful.

Real progress was absent until the 1940’s when Sichel [27] suggested theuse of the lognormal model. He was a classical statistician and in themining field he concentrated his efforts on frequency distribution models. Hedeveloped the T-estimator with its appropriate confidence limits [28]. This es-timator is more efficient than the arithmetic mean, but is strictly valid only fora random set of data which follows the lognormal model exactly. Departuresfrom the 2-parameter lognormal model, as observed in practice, were largelyovercome with the introduction in 1960 of the 3-parameter lognormal model[10] which requires an additive constant before taking logarithms. However,there were still cases which could not be covered properly by the 3-parameterlognormal, and led to the introduction of the more flexible Compound Lognor-mal Distribution [30], originally developed by Sichel for diamond distributions.This model is very flexible and caters specifically for a tail of high values whichis much longer than that for earlier models. This development is covered inthe second part of this paper.

2.2 The genesis and early development of geostatistics

Geostatistical concepts originated in the late 1940’s when Ross applied thelognormal model to a variety of actual gold grade data [26], de Wijs showedhow the differences between individual grades depended on their distancesapart [3, 4] and particularly when the basic concept of gold ore grades as avariable with a spatial structure was introduced in 1951/2 [8, 9]. The objectiveof this latter work was to develop more efficient grade estimates for new minesand for ore blocks on existing mines.

The first paper [8] was aimed at finding an explanation for the experienceon all the gold mines for many decades, of ore reserve estimates during subse-quent mining consistently showing a significant under-valuation in the lowergrade categories and the reverse for estimates in the higher grade categories.Classical statistical correlation and regression analyses proved this to be anunavoidable result of block estimates subject to error and to conditional biases[8]. In the proper perspective it was essential to no longer view the peripheral

Genesis of geostatistics 7

data used for individual block estimates and the ore blocks themselves in iso-lation. It was essential to see the peripheral data as part of an extensivespread of data (the data population) in stopes and development ends inthe relevant mine section; also to accept the grade of the ore block concernedas part of a collection of block grades (both intact and already mined out),i.e. as a member of a population of oreblock grades.

In this way, the spatial concept was introduced as well as the con-cept of support in moving from individual sample grades (point supports),to block grades. A mathematical model was first set up of the lognormaldistribution of actual block values in a mine section. The errors in assign-ing the limited peripheral grades to the blocks were super imposed on theactual grades to yield the corresponding distribution of block estimates. Oncorrelating these two sets of block values on a classical statistical basis, theaverages of the actual block values relative to the corresponding block esti-mates in grade categories could be observed, i.e. the curvilinear regression ofactuals on estimates. This was a theoretical follow-up exercise to simulate theresults actually observed in practice. It provided the statistical explanation ofthe natural phenomena of the unavoidable under- and over-valuation featuresas mentioned above, i.e. the inherent conditional biases. The use of thelognormal model also covered the curvilinear nature of the regressiontrend as observed in practice.

The very fact that a correlation exists between the block estimates andthe internal actual grades emphasises the presence of a spatial structure. Withthe explanation of these conditional biases, the initial application in practicewas to apply the trend, or regression, of actuals (or follow-ups) versus esti-mates –as observed from the mine records or modelled geostatistically– tothe orthodox block estimates so as to eliminate these biases. As the regressedestimates were, in effect, weighted averages of the peripheral estimates andthe global mean grade of the mine section, it was the first application ofwhat became known as kriging. It can be labelled Simple Elemen-tary Kriging, being based on the spatial correlation between the peripheralvalues and the actual grades of the ore inside the ore blocks, and giving properweight to the data outside the block periphery via the mean.

During the 1950’s several large gold mines introduced regression techniquesfor their ore reserve estimates on a routine basis. It is instructive to observethat on the gold mines the improvement in the standard of block valuationsdue to the elimination of conditional biases accounts for some 70% of the totallevel of improvement achievable today with the most sophisticated geostatis-tical techniques. It is for this reason, that so much stress is placed on theelimination of conditional bias (so called “conditional unbiasedness”).

In the second paper [9], the spatial structure of the grade data from91 drill holes in the main sector of the new Orange Free State gold field wasdefined by the log-variance/log-area relationship (see Fig. 1). This demon-strated the so-called Krige formula (point variance within a large area minusthe average point variance within ore blocks = the variance of block values

8 Danie Krige and Wynand Kleingeld

within the large area) as well as the so-called permanence of the lognor-mal model for different support sizes. On this basis the lognormal modelfor the expected distribution of ore block grades was modelled with a globalmean grade as estimated from the 91 drill hole grades. It led successfully tomeaningful tonnage and grade estimates for a range of cut-off grades, i.e. thefirst version of the now well known tonnage-grade curve. Withouta proper block distribution model the orthodox approach would have basedthe tonnage-grade estimates directly on the individual drill hole grades withseriously misleading results globally and for individual mines.

−5 0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

Common logarithm of reef area

Var

ianc

e of

log−

valu

es

Half core section

Deflection area

Ore blocks

Individual mines

Main sector

Fig. 1. Dispersion variance versus domain size: example of gold data from the Or-ange Free State. The horizontal axis represents the area of the domain in logarithmicscale, from 10−5 to 1010 ft2

These developments were published in 1951/1952 and aroused world-wideinterest in the subject now known as geostatistics, particularly in French cir-cles. Matheron and Duval translated these two papers and re-published themin 1955 together with two personal contributions by them ([15, 5]). This wasfollowed by a paper on exploration prospects in the Sahara by Allais [1]. Math-eron [15] in particular covered the more theoretical background underlying thetwo basic South African papers and the models involved in all this work. Heshowed, for example, that the permanence of the lognormal model canonly logically apply where a spatial structure is present and that the positivecorrelation between the log variances and the mean grades of lognormal dis-tributions – the so-called proportional effect – is an inherent feature of thelognormal model. This contribution by Matheron was accompanied and/orfollowed in the 1950’s and 1960’s by numerous other notes and publications

Genesis of geostatistics 9

in French and the introduction of the theory of regionalised variables.Matheron’s first geostatistical papers in English were in the 1960’s ([16, 17]),followed by an English monograph in 1971 ([18]) which covered the theory indetail.

Modelling of the spatial structure is basic to any geostatistical approach.The original approach in South Africa [9] was followed by extensive correla-tions of pairs of grades for different lags and the results were modelled bycorrelograms and covariograms [11, 12] and used in multiple regressiontechniques to arrive at the relative weights to be applied to the data avail-able for a block valuation. This was already introduced on a routine basis onsome of the large gold mines for ore reserve estimates in the early 1960’s andwas called weighted moving average estimates until at Matheron’s insistencethe term kriging prevailed. In the mean time Matheron covered a continu-ous series of further developments of geostatistical models based on the nowgenerally applied variogram for defining spatial structures.

The critical need for identifying likely changes in the characteristics ofthe spatial structure between sections within the ore body, such as gradecontinuity levels, anisotropy directions, etc. was also already stressed duringthe 1960’s. This basic tenet of geostatistics has been and is still widely metin practice via the linkage of these characteristics with changes in geologicaland/or mineralogical parameters which can more readily be modelled.

2.3 The main basic tenets of geostatistics

Virtually all the fundamental concepts and tenets of geostatistics were estab-lished in these early years and are still applicable today.

1. The use of appropriate parametric distribution models when prac-tical for confidence limits of estimates of the mean grade. Various non-parametric approaches have been developed, but face the common prob-lem that the pattern of the observed point distribution accepted as themodel can be misleading for the upper tail of the distribution unless avery large data base is available.

2. Spatial structures generally present in ore bodies and with character-istics associated with geological and mineralogical features.

3. The concepts of support sizes and types, the proportionally effectand models for estimating the SMU (Selected Mining Unit) blockdistribution parameters directly or indirectly from the point value dis-tribution.

4. Kriging for block estimates. Many types of kriging have been developedbut all allow to eliminate - or at least reduce - the conditional bias.

5. If block valuations are done before the actual selective mining stage - whenthe final data becomes available - the estimates will be smoothed andhave to be post-processed. Meaningful post-processing techniques wereset up after the early stages of the development of geostatistics, as well

10 Danie Krige and Wynand Kleingeld

as the so-called non-linear geostatistics (see the paper by Rivoirard in thepresent volume, page 17).

2.4 Conditional unbiasedness

Early developments in South Africa generally retained strong links with classi-cal statistics, particularly through the preference on the gold mines for simplekriging. This is essentially a classical multiple regression technique based onthe mean grade for the ore body or local part thereof and on the correspond-ing spatial structure to provide the necessary covariance’s for the solution ofthe matrix equations involved. However, where block valuations are based onexploratory data to be supplemented at the final mining stage by additionaldata more closely spaced, the earlier block estimates will be smoothed com-pared to the final estimates and have to be post-processed before declaringore reserves and doing mine planning and feasibility studies.

Arising from this problem and the fact that the mean grade as used in sim-ple kriging itself changes within an ore body, a general preference developpedfor ordinary kriging, which relies only on the data as accessed for the krigingof each block. There is no objection, in principle, to this approach providedthe data accessed is adequate to effectively provide a close grade level for thelocal area encompassing the block. This will ensure conditional unbiasednessbut will not overcome the smoothing problem. Also the effort time and costinvolved in kriging each block on a relatively large data base required to meetthis objective (say 50 instead of only 5, 10 or even 20 values), led to thewidespread use of a limited search routine, and lately also to simulation asan alternative. These practices can reduce or eliminate smoothing, but unfor-tunately, re-introduce conditional biases as prevailed in the pre-geostatisticalperiod and cannot be post-processed unless the conditional biases are firstremoved. Although such estimates could provide acceptable global tonnage-grade figures for the whole ore body, they could still be seriously conditionallybiased for sections of the ore body as will be mined sequentially over shorttime periods [13] and thus be unacceptable.

3 The influence of the specificity of the diamond miningindustry on the development of geostatistics

3.1 Introduction

Diamonds are a unique commodity in the realms of mining and its associateddisciplines. The particulate nature of the diamond affects the processes of ex-ploration, evaluation, mining and metallurgy and especially the way in whichdiamonds are valued and sold.

The evaluation of alluvial diamond deposits specifically drew the attentionof some of the greatest minds in the field of geostatistics and this would

Genesis of geostatistics 11

eventually give rise to novel ways of dealing with sampling and estimationtechniques in the case of discrete particle deposits.

The diagram shown in Fig. 2 illustrates the complex nature of diamonddeposits compared with other mineral commodities. The average grade fordiamond deposits is generally very low and a high degree of geological discon-tinuity exists, particularly in the case of marine placer deposits where selectivemining is absolutely essential.

Fig. 2. Plotting the average grade versus the mineralization continuity of variousminerals

Groundbreaking work by Sichel during the early seventies [29] gave riseto a statistical approach on how to evaluate these deposits, and models suchas the compound Poisson for diamond density distributions [20, 21] and theCompound Lognormal for diamond size distributions were developed. Theadaptation of these models had significant impact later on the estimation ofother minerals [31].

In the 10 years from 1980 to 1990 a substantial research effort [24] wasdirected towards understanding the following issues;

1. The complex nature of the geology that gave rise to the discrete particlemineral deposit.

2. The problems associated with the sampling of deposits of this nature andthe fact that the sampling could produce non representative results sincethe sample size is smaller than the trap sites in which the particles occur.

12 Danie Krige and Wynand Kleingeld

3. The statistical models required to cater for the extremely skewed samplingdata, with the emphasis on smoothing the shape of the curve, and toincrease sample representavity.

4. Methods to obtain local reserve estimates, including confidence limits,which require a local distribution density function.

5. The need to produce bivariate representation of the probability mass func-tions for the non-linear kriging procedures used.

The outcome of the research culminated in a geostatistical approach tothe development of the ideas of cluster sampling, discrete isofactorial modelsthat could be used in the Disjunctive Kriging estimation process to developestimates and the Cox Process for discrete particle simulation.

In essence it could thus be stated that the research undertaken on theevaluation of alluvial diamond deposits gave rise to discrete geostatistics.

The research would not have been possible without the substantial inputfrom people such as Matheron, Sichel and others such as Lantuejoul andLajaunie.

3.2 Geology

In the research, ancient beach deposits were considered where mineralizationis largely confined to basal gravel horizons. Those are located in one or moremarine abrasion platforms, usually cut into schists and phyllites.

The shist bedrock is extensively gullied by wave action assuming a char-acteristic pattern well developped. The gullies are controlled by the slope andthe structure of the bedrock and by the presence of boulders and gravel. Thepot-holes and gullies act as particle trap sites and can contain high concen-trations of the mineral.

Though some trap sites can contain high concentrations they could besurrounded by sites that have low concentration or even be barren. This highdegree of variability is related to the complex interaction of geological controlsduring deposition. The chance of finding a particle in this type of deposit isrelated to the chance of sampling a trap site and the distribution of particlesin the trap site.

The distribution of particles is different for each beach, corresponding tothe different marine transgressions and is related to the length of stillstand ofthe sea which influenced the degree of abrasion of the marine platform andthe degree of reworking that took place. It is also influenced by the amountof mineral bearing gravel that was available during the transgression period.

Thus the distribution of particles is directly related to the presence ofparticles in the gravel, the quantity and quality of the trap sites and to thedegree of reworking of the gravel.

In a certain area a characteristic gully pattern is normally formed withparallel gullies at relatively constant distances apart. The trap sites in thesegullies also show characteristic patterns with the typical size of a trap site 5malong and 3m across a gully, but variable from area to area.

Genesis of geostatistics 13

Fig. 3. Mining an alluvial deposit. The diamonds are trapped in the pot-holes andthe gullies of the bedrock

3.3 Sampling

From the outset the sampling of these types of deposits posed significant prob-lems, even at elevated sample support sizes (5 square meters) up to 90% of thesamples did not contain any particle. In contrast, there are rare occurrenceswhere several hundred particles were recovered in one sample unit locatedover a natural trap site such as a deep pot-hole.

Based on the geological model it became obvious that normal samplingtheory as applied to homogeneous mineralization was not applicable in thecase of marine placers where mineralization was concentrated in trap sites[29].

Other examples occur in vein and alluvial deposits of diamonds and gold.In such deposits, two different factors account for the grade variability at twodifferent scales, firstly the spatial distribution of the trap sites, and secondlythe dispersion of the mineralization within the trap sites.

The existence of several scales of variability makes sampling a very complexoperation. As a matter of fact, a set of samples of a given size may not account

14 Danie Krige and Wynand Kleingeld

for all the scales of variability. Using many small samples, traps are welldetected, but their mineralization contents are poorly assessed. Using a limitednumber of larger samples, the quality of the traps is better known, but itbecomes more difficult to assess the distribution of the traps sites.

The methodology involved in sampling such highly dispersed type ore bod-ies was addressed in [7]. The paper presents several results on the samplingof highly dispersed type ore bodies and highlights the two major problemsencountered when sampling under such conditions.

It also addresses the problem of defining a representative sample supportsize and making it operatory by resorting to a cluster sampling approach.

During the research it became evident that limited experience exists in thefield of in situ sampling, especially when stratification is present.

3.4 Estimation

In his book ”Estimer et choisir” [19], Matheron discusses certain fundamentaldifferences between the approaches adopted in statistics and geostatistics.Fundamentally statistics is involved in the estimation of parameters for achosen probability model, whereas geostatistics is involved with the estimationof a spatial average for a natural phenomenon.

However, in the case of discrete particle deposits where the bases for es-timation (variogram and histogram) are not well defined due to the non rep-resentative nature of sampling, the necessity of using statistical models inestimation was evident [6]. Matheron highlights that the model is not the de-posit and notes that substantial research is needed to explain the variation ofmodel parameters with the geology of a deposit. Such work was carried outby Oosterveld et al. [25].

The need for introducing a statistical model for local reserve estimationwas clearly indicated and research was done to provide a method to estimatethe number of particles expected in an in situ reserve block of specific supportsize.

The introduction of geostatistics contributed to the understanding andquantification of the risk associated with grade estimation. Uncertainty wasdefined in terms of confidence limits derived from a modelled probability dis-tribution for the grades of the mining blocks.

The skewness of the sample grades gave rise to skew distributions for theblock estimates and their error distributions. This led to research in the fieldof non-linear kriging, more specifically disjunctive kriging under appropriatediscrete isofactorial models [22]. A suitable statistical model which representsa discrete type of particle density distribution had to take into account thedistributional characteristics of the trap sites as well as the particles containedwithin the trap sites. The most important problem in mining geostatistics, i.e.that of change of support, was also addressed. The inference of the parame-ters is a challenging problem and practical aspects of implementing discreteisofactorial methodology are presented in [14].