Space, Structure and Randomness: Contributions in Honor of Georges Matheron in the Field of Geostatistics, Random Sets and Mathematical Morphology

402 Pages·2005·9.274 MB·English

by  Danie Krige, Wynand Kleingeld (auth.), Michel Bilodeau, Fernand Meyer, Michel Schmitt (eds.)

Lecture Notes in Statistics 183
Edited by P.Bickel,P.Diggle,S.Fienberg,U.Gather,
I.Olkin,S.Zeger
Michel Bilodeau
Fernand Meyer
Michel Schmitt (Editors)
Space, Structure and Randomness
Contributions in Honor of Georges Matheron
in the Field of Geostatistics, Random Sets
and Mathematical Morphology
With 121 Figures
Michel Bilodeau Fernand Meyer Michel Schmitt
Ecole des Mines de Paris Ecole des Mines de Paris Ecole des Mines de Paris
Centre de Morphologie Centre de Morphologie Direction des recherches
Mathématique Mathématique 60 Boulevard Saint Michel
35 rue Saint Honoré 35 rue Saint Honoré 75272 Paris Cedex 06
77305 Fontainebleau 77305 Fontainebleau France
France France
[email protected]
[email protected] [email protected]
Library ofCongress Control Number:2005927580
ISBN 10:0-387-20331-1 Printed on acid-free paper
ISBN 13:978-0387-20331-7
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Personal Reminiscences of Georges Matheron
Dietrich Stoyan
Institut fu¨r Stochastik, TU Bergakademie Freiberg
I am glad that I had the chance to meet Georges Matheron personally once
in my life, in October 1996. Like many other statisticians, I had learned a
lot in the yearsbefore from his books,papers, and researchreports produced
in Fontainebleau. From the very beginning, I had felt a particular solidarity
with him because he worked at the Ecole des Mines de Paris, together with
Bergakademie Freiberg one of the oldest European mining schools.
The first contact to his work occurred in talks with Hans Bandemer, who
in the late 1960s had some correspondence with Georges Matheron. Proudly
he showed me letters and reprints from Georges Matheron. He also had a
copy of Georges Matheron’s thesis of 1965, 300 pages narrowly printed. Its
first part used Schwartz’ theory of distributions (Laurent Schwartz was the
supervisor) in the theory of regionalized variables, while the second part de-
scribed the theory of kriging. Hans Bandemer was able to read the French
text, translated it into German language and tried, without success, to find
a Germanpublisher. Unfortunately, the political conditions in EastGermany
prevented a meeting between Hans Bandemer and Georges Matheron.
A bit later, in 1969,I saw George Matheron’s book “Trait´e de G´eostatis-
tique Appliqu´ee”. It is typical of the situation in East Germany at the time
that this was a Russian translation published in 1968, based on the French
edition of 1967. (We could not buy Western books and most of us could not
read French. At the time the Soviet Union ignoredthe copyrightlaws.) I was
very impressed by the cover illustration showing a gold-digger washing gold.
Georges Matheron’s Trait´e is a fascinating book, and for me and many
others it was the key reference in geostatistics at the time. The progress
achieved in this work is perhaps best expressed by the following quote from
the postscript (written by A.M. Margolin) of the Russian translation:
“Wefindin themonograph thefundamentalideas andnotions of a
mathematical 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 notions
and 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 untilthe 1950s nearly the only way of application of math-
ematics in geological exploration, geostatistics is directed to the true
mathematical nature of the objects and tools of exploration...
According to geostatistics, the uncertainty which is characteristic
for results of exploration is a consequence of incompleteness of ex-
ploration of the object but not of its randomness. In this probably the
increasingpossibilities ofthegeostatisticaltheoryconsist andtheprin-
cipal difference to variation statistics, which is based on the analysis
of geological variables irrespectively tothelocations of observation...”
Notatthetime,butlateronwhenIhadlearntmoreaboutpublicrelations
in science, I admired Matheron’s cleverness in coining terms: It was very
smart to call statistics for random fields ‘geostatistics’ (to use a very general
word,whichsuggests‘statisticsforthegeosciences’,foramorelimitedclassof
problems)andleastsquareslinearinterpolation‘kriging’(originally‘krigeage’;
in Freiberg it was for a long time not clear whether it should be pronounced
‘krig-ing’or‘kraig-ing’–assuggestedbytheRussiantranslation–or‘kreedge-
ing’.)
GeorgesMatheron’s“RandomSets andIntegralGeometry”of1975wasa
landmarkinmy ownscientific development.The mid1970swereagreattime
for stochastic geometry and spatial statistics, which then became more than
just geostatistics. In that time mathematical morphology was also shaped,
andJeanSerra’sworkinimageanalysisbecameknownoutsideFontainebleau,
bothintheoryandinapplications.Thefirstimageanalyser,thefamousLeitz
TAS, was produced, based on ideas of Georges Matheron and Jean Serra.
We, the scientists in East Germany, were unable to order the book in a
book store, but I got a copy from Dieter Ko¨nig in exchange for a book on
queueing theory. Up to this day, for me (and many others, I believe) Georges
Matheron’s book is the key reference in random set theory; the cover of my
copy is now in pieces,and many pagesare markedwith notes. Unfortunately,
the book has not been reprinted since.
Thisbookgivesanexcellentexpositionofthetopicsnamedinthetitle,itis
veryclearlywritten,andofjusttherighttheoreticallevel.GeorgesMatheron’s
work also led me to Hadwiger’s, whose monograph on set geometry is now
oneofmymostbelovedmathematicalbooksinGermanlanguage.SoGeorges
Matheronhadanexcellentbaseforthe integral-geometricpartofhiswork.It
isagreatcombinationofmanymathematicalfields,suchasintegralgeometry,
set geometry, Choquet’s theory of capacities (in fact Matheron developed
it independently), and ideas of Poisson process based stochastic geometry
(createdinparticularbyRogerMiles)toobtainanew,richandfruitfultheory.
Georges Matheron’s book contains many gems. I name here only the theory
Personal Reminiscences of Georges Matheron VII
of the Booleanmodel (he writes ‘booleanmodel’, with small‘b’), the Poisson
polyhedron, and the granulometries.
It was Georges Matheron who made notions of measurability properties
rigorous in the context of random sets. This can be explained in terms of
Robbins’ formula for the mean volume of a compact random set:
(cid:2)
Eν(X)= p(x)dx
Rd
where p(x) = P(x ∈ X). Its proof is based on Fubini’s theorem and was
alreadygiven–inpart–inKolmogorov’sfamousbookof1933.ItwasGeorges
Matheron who showed that the mapping set → volume is measurable. The
σ-algebra with respect to which measurability is considered (based on Fell’s
topology)istodaycalledMatheron’sσ-algebra.Someofhisresultsformodels
relatedtoPoissonprocesseswerethestartingpointforfurtherscientificwork,
whenitbecameclearthatmethodsfromthetheoryofmarkedpointprocesses
canbeusedtogeneralizethem.Throughhisbook,GeorgesMatheronhasbeen
a teacher and inspirer for a large number of mathematicians in the last three
decades.
In1989GeorgesMatheronpublishedthebook“EstimatingandChoosing”.
Grownolder,Iwasofferedthehonourtoserveasoneofitsreviewers.Thisisa
philosophical 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 statistician
has only data from one mineral deposit, or from one forest stand; a second
sample taken close to it can typically not be considered as a sample from
the same population, because of different geological or ecological conditions.
He developed the idea of ‘local ergodicity’, which is plausible and justifies
the statisticalapproach.Eachspatialstatisticianshouldreadthe book.I also
enjoyed its sarcastic humor.
In June 1983 Dominique Jeulin came to Freiberg as an invited speaker
at a conference. For me and colleagues like Joachim Ohser and Karl-Heinz
HanischthiswasthefirstchancetomeetascientistfromGeorgesMatheron’s
school. Dominique Jeulin spoke about multi-phase materials, rough surfaces,
andimagetransformations.HeislikelythefirstFrenchpersonIevermet,and
IlearnedthatFrenchEnglishdiffersgreatlyfromGermanEnglish.Dominique
had to repeat three times his ‘Stoj´ang’ until I understood that he asked for
me. The contact to him, which is still lively, finally led to the meeting with
Georges Matheron.
Afterthebigchangesof1989/90wehadmanyWestGermanPhDstudents
at Freiberg. One of them, one of the best, was Martin Schlather. Martin
had studied one year at Fontainebleau, had written a diploma thesis there,
and had been given oral exams by Georges Matheron personally. Thus, he
could tell me first hand about Georges Matheron’s personality and about
the situation at Fontainebleau. Like anybody who ever had personal contact
VIII Dietrich Stoyan
with Georges Matheron Martin admired him, both as a mathematician and
as a person. Martin told me that Georges Matheron’s work is much more
comprehensive than his books and journalpublications suggest.So we are all
extremely grateful to Jean Serra for publishing a CD with the reports and
papers by Georges Matheron. I agree with Martin and Jean that one will
find wonderful ideas and solutions to hard problems as one goes through this
material.
ItwasMartinSchlatherwhoencouragedmetotraveltotheFontainebleau
conferencein1996.ThisconferencewasorganizedbyDominiqueJeulininhon-
ourofGeorgesMatheron,andthe lecturesarepublishedinthe volumeJeulin
(1997).There we met GeorgesMatheron,who did not givea lecture,but was
sittinginthefrontrowinthelectureroom,obviouslylisteningwithattention.
One afternoon there was a reception in the municipal hall of Fontainebleau.
ThemairedecoratedGeorgesMatheron(andothers)withamedal,forreasons
that I did not completely understand, either scientific or political. Georges
Matheronwaspolite enoughtobeartheceremony,butquiteobviouslyhedid
not take it very seriously. He told me anecdotes, and his body language was
clear enough. I had no difficulty in communicating with him, he accepted my
poor English, and I understood him very well.
IntheeveningIwashonouredbyaninvitationtothefamilyofJeanSerra,
whereIalsometMesdamesMatheronandSerra.Itwasanenjoyable,notvery
long evening, with friendly non-mathematical talk, and a bit of French wine.
In the end, I never orally discussed mathematical problems with Georges
Matheron, but I appreciated the contact over many years, through studying
hiswork,andthroughafewcommentsthathemadeonmywork.Ibelievethat
thereis onlyasmallnumberofinternationalmathematicalconferenceswhich
Georges Matheron attended. His example shows that a mathematician who
doesnotvisitconferencescanbeneverthelessbeinfluentialandwidelyknown.
Perhaps,GeorgesMatheroncouldhavehadevenmoreinfluence.However,did
he really want this?
BacktoFreiberg,Ihadtheideaofhonouringhimthere.AtthetimeIwas
the president of the little Technische Universita¨t Bergakademie Freiberg and
saw the chance of honouring him with a Dr.h.c. degree. I asked Dominique
Jeulin.HelikedtheideabutwarnedmethatGeorgesMatheronwouldproba-
bly nevercometo Freiberg,andifhe did,Icouldnotexpecthis collaboration
inapublic relationseventtothe benefitofmyuniversity,asI washopingfor.
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 Georges
Matheron, 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 I
learned that Georges Matheron had used a library copy.
Iamveryhappythatthisvolumeisnowready,whichwillhonourGeorges
Matheron, one of the great mathematicians of the 20th century.