Georges Matheron
Encyclopedia
Georges François Paul Marie Matheron (December 2, 1930 – August 7, 2000) was a French mathematician
and geologist, known as the founder of geostatistics
and a co-founder (together with Jean Serra
) of mathematical morphology
. In 1968 he created the Centre de Géostatistique et de Morphologie Mathématique at the Paris School of Mines in Fontainebleau
. He is known for his contributions on Kriging
and mathematical morphology. His seminal work is posted for study and review to the Online Library of the Centre de Géostatistique, Fontainebleau, France.
and later Ecole des Mines de Paris, where he studied mathematics
, physics
and probability
theory (as a student of Paul Lévy
).
From 1954 to 1963, he worked with the French Geological Survey
in Algeria and France, and was influenced by the works of Krige
, Sichel
, and de Wijs, from the South African school, on the gold deposits of the Witwatersrand
. This influence led him to develop the major concepts of the theory for estimating resources he named Geostatistics.
s. In his Rectificatif of January 13, 1955, he revised the arithmetic mean lead and silver grades because his core samples varied in length. He did derive the length-weighted average lead and silver grades but failed to derive the variances of his weighted averages. Neither did he derive the degree of associative dependence between metal grades of ordered core samples as a measure for spatial dependence
between ordered core samples. He did not disclose his primary data set and worked mostly with symbols rather than real measured values such test results for lead and silver in Matheron's core samples. Matheron's Interprétations des corrélations entre variables aléatoires lognormales of November 29, 1954, was marked Note statistisque No 2. In this paper, Matheron explored lognormal variables and set the stage for statistics by symbols. Primary data would have allowed him to assess whether or not lead and silver grades departed from the lognormal distribution, or displayed spatial dependence
along core samples in his borehole.
Matheron coined the eponym krigeage (Kriging
) for the first time in his 1960 Krigeage d’un Panneau Rectangulaire par sa Périphérie. In this Note géostatistique No 28, Matheron derived k*, his estimateur and a precursor to the kriged estimate or kriged estimator. In mathematical statistics, Matheron’s k* is the length-weighted average grade of a single panneau in his set. What Matheron failed to derive in this paper was var(k*), the variance of his estimateur. Matheron presented his Stationary Random Function at the first colloquium on geostatistics
in the USA. He called on Brownian motion
to conjecture the continuity of his Riemann integral
but did not explain what Brownian motion and ore
deposits have in common. Matheron, unlike John von Neumann
in 1941 and Anders Hald
in 1952, never worked with Riemann sums. It was not Professor Dr Georges Matheron but Dr Frederik P Agterberg who derived the distance-weighted average of a set of measured values determined in samples selected at positions with different coordinates in a sample space. What Agterberg did not do was derive the variance of this function.
Matheron did indeed derive length-weighted average grades of core samples and ore blocks but did not derive the variance of these functions. In time, the length-weighted average grade for Matheron's three-dimensional block grade was replaced with the distance-weighted average grade for Agterberg's zero-dimensional point. Both central values turned into honorific kriged estimates or kriged estimators. An infinite set of Agterberg's zero dimensional points fits within any ore block, along any borehole, or inside any sampling unit or sample space. Matheron's block grades and Agterberg's point grades are unique because both are functions without variances.
of Jean Serra, dedicated to quantifying the ore
properties of the iron deposit of Lorraine
. Serra came up with the idea of using structuring element
s for the analysis, which lead to the concept of hit-or-miss transform
. The theoretical analysis of this transform led Matheron to derive and investigate the concepts of erosion
, dilation
, opening
and closing
, which became known later as the basic morphological operators. He also developed a tool for granulometry
, i.e., the computation of a "size distribution", where he mathematically characterizes the concept of size. In December 1964, Matheron and Serra, together with Philippe Formery, named this approach mathematical morphology. It has since evolved into a theory and method that is applied in a variety of image processing
problems and tasks, and is researched worldwide (main article: Mathematical morphology
). Matheron continued to contribute to mathematical morphology during the years, his best-known contribution being the morphological filtering theory, which he developed with Serra in the 1980s.
, located in Fontainebleau
, France, and named Matheron its first director. In 1979, the center was renamed Centre de Géostatistique et de Morphologie Mathématique, and, in 1986, the latter was split into two separate centers: Centre de Géostatistique, directed by Matheron, and Centre de Morphologie Mathématique, directed by Serra.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and geologist, known as the founder of geostatistics
Geostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology,...
and a co-founder (together with Jean Serra
Jean Serra
Jean Paul Frédéric Serra is a French mathematician and engineer, and known as one of the co-founders of mathematical morphology.-Education:...
) of mathematical morphology
Mathematical morphology
Mathematical morphology is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions...
. In 1968 he created the Centre de Géostatistique et de Morphologie Mathématique at the Paris School of Mines in Fontainebleau
Fontainebleau
Fontainebleau is a commune in the metropolitan area of Paris, France. It is located south-southeast of the centre of Paris. Fontainebleau is a sub-prefecture of the Seine-et-Marne department, and it is the seat of the arrondissement of Fontainebleau...
. He is known for his contributions on Kriging
Kriging
Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
and mathematical morphology. His seminal work is posted for study and review to the Online Library of the Centre de Géostatistique, Fontainebleau, France.
Early career
Matheron graduated from Ecole PolytechniqueÉcole Polytechnique
The École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...
and later Ecole des Mines de Paris, where he studied mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
theory (as a student of Paul Lévy
Paul Pierre Lévy
Paul Pierre Lévy was a Jewish French mathematician who was active especially in probability theory, introducing martingales and Lévy flights...
).
From 1954 to 1963, he worked with the French Geological Survey
Geological survey
The term geological survey can be used to describe both the conduct of a survey for geological purposes and an institution holding geological information....
in Algeria and France, and was influenced by the works of Krige
Danie G. Krige
Danie G. Krige is a South African Mining Engineer who pioneered the field of geostatistics and, until recently, was professor at the University of the Witwatersrand, Republic of South Africa. The technique of kriging is named after him...
, Sichel
Herbert Sichel
Dr Herbert Sichel was a statistician who made great advances in the areas of both theoretical and applied statistics.He developed the Sichel-t Estimator for the Log-normal distribution's t-statistic...
, and de Wijs, from the South African school, on the gold deposits of the Witwatersrand
Witwatersrand
The Witwatersrand is a low, sedimentary range of hills, at an elevation of 1700–1800 metres above sea-level, which runs in an east-west direction through Gauteng in South Africa. The word in Afrikaans means "the ridge of white waters". Geologically it is complex, but the principal formations...
. This influence led him to develop the major concepts of the theory for estimating resources he named Geostatistics.
Geostatistics
Matheron’s [Formule des Minerais Connexes] became his Note Statistique No 1. In this paper of November 25, 1954, Matheron derived the degree of associative dependence between lead and silver grades of core sampleCore sample
A core sample is a cylindrical section of a naturally occurring substance. Most core samples are obtained by drilling with special drills into the substance, for example sediment or rock, with a hollow steel tube called a core drill. The hole made for the core sample is called the "core hole". A...
s. In his Rectificatif of January 13, 1955, he revised the arithmetic mean lead and silver grades because his core samples varied in length. He did derive the length-weighted average lead and silver grades but failed to derive the variances of his weighted averages. Neither did he derive the degree of associative dependence between metal grades of ordered core samples as a measure for spatial dependence
Spatial dependence
In applications of statistics, spatial dependence is the existence of statistical dependence in a collection of random variables or a collection time series of random variables, each of which is associated with a different geographical location...
between ordered core samples. He did not disclose his primary data set and worked mostly with symbols rather than real measured values such test results for lead and silver in Matheron's core samples. Matheron's Interprétations des corrélations entre variables aléatoires lognormales of November 29, 1954, was marked Note statistisque No 2. In this paper, Matheron explored lognormal variables and set the stage for statistics by symbols. Primary data would have allowed him to assess whether or not lead and silver grades departed from the lognormal distribution, or displayed spatial dependence
Spatial dependence
In applications of statistics, spatial dependence is the existence of statistical dependence in a collection of random variables or a collection time series of random variables, each of which is associated with a different geographical location...
along core samples in his borehole.
Matheron coined the eponym krigeage (Kriging
Kriging
Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
) for the first time in his 1960 Krigeage d’un Panneau Rectangulaire par sa Périphérie. In this Note géostatistique No 28, Matheron derived k*, his estimateur and a precursor to the kriged estimate or kriged estimator. In mathematical statistics, Matheron’s k* is the length-weighted average grade of a single panneau in his set. What Matheron failed to derive in this paper was var(k*), the variance of his estimateur. Matheron presented his Stationary Random Function at the first colloquium on geostatistics
Geostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology,...
in the USA. He called on Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
to conjecture the continuity of his Riemann integral
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The Riemann integral is unsuitable for many theoretical purposes...
but did not explain what Brownian motion and ore
Ore
An ore is a type of rock that contains minerals with important elements including metals. The ores are extracted through mining; these are then refined to extract the valuable element....
deposits have in common. Matheron, unlike John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
in 1941 and Anders Hald
Anders Hald
Anders Hald was a Danish statistician who made contributions to the history of statistics.He was a professor at the University of Copenhagen from 1960 to 1982.- Bibliography :...
in 1952, never worked with Riemann sums. It was not Professor Dr Georges Matheron but Dr Frederik P Agterberg who derived the distance-weighted average of a set of measured values determined in samples selected at positions with different coordinates in a sample space. What Agterberg did not do was derive the variance of this function.
Matheron did indeed derive length-weighted average grades of core samples and ore blocks but did not derive the variance of these functions. In time, the length-weighted average grade for Matheron's three-dimensional block grade was replaced with the distance-weighted average grade for Agterberg's zero-dimensional point. Both central values turned into honorific kriged estimates or kriged estimators. An infinite set of Agterberg's zero dimensional points fits within any ore block, along any borehole, or inside any sampling unit or sample space. Matheron's block grades and Agterberg's point grades are unique because both are functions without variances.
Mathematical morphology
In 1964, Matheron was supervising the PhD thesisThesis
A dissertation or thesis is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings...
of Jean Serra, dedicated to quantifying the ore
Ore
An ore is a type of rock that contains minerals with important elements including metals. The ores are extracted through mining; these are then refined to extract the valuable element....
properties of the iron deposit of Lorraine
Lorraine (région)
Lorraine is one of the 27 régions of France. The administrative region has two cities of equal importance, Metz and Nancy. Metz is considered to be the official capital since that is where the regional parliament is situated...
. Serra came up with the idea of using structuring element
Structuring element
In mathematical morphology, a structuring element is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image...
s for the analysis, which lead to the concept of hit-or-miss transform
Hit-or-miss transform
In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements...
. The theoretical analysis of this transform led Matheron to derive and investigate the concepts of erosion
Erosion (morphology)
For use of "Erosion" in dermatopathology, see Erosion Erosion is one of two fundamental operations in Morphological image processing from which all other morphological operations are based...
, dilation
Dilation (morphology)
Dilation is one of the basic operations in mathematical morphology. Originally developed for binary images, it has been expanded first to grayscale images, and then to complete lattices...
, opening
Opening (morphology)
In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:A\circ B = \oplus B, \, where \ominus and \oplus denote erosion and dilation, respectively....
and closing
Closing (morphology)
In mathematical morphology, the closing of a set A by a structuring element B is the erosion of the dilation of that set,A\bullet B = \ominus B, \, where \oplus and \ominus denote the dilation and erosion, respectively....
, which became known later as the basic morphological operators. He also developed a tool for granulometry
Granulometry (morphology)
In mathematical morphology, granulometry is an approach to compute a size distribution of grains in binary images, using a series of morphological opening operations...
, i.e., the computation of a "size distribution", where he mathematically characterizes the concept of size. In December 1964, Matheron and Serra, together with Philippe Formery, named this approach mathematical morphology. It has since evolved into a theory and method that is applied in a variety of image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
problems and tasks, and is researched worldwide (main article: Mathematical morphology
Mathematical morphology
Mathematical morphology is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions...
). Matheron continued to contribute to mathematical morphology during the years, his best-known contribution being the morphological filtering theory, which he developed with Serra in the 1980s.
The Centre de Géostatistique et de Morphologie Mathématique
In 1968, the Paris School of Mines created the Centre de Morphologie MathématiqueCentre de Morphologie Mathématique
Centre de Morphologie Mathématique is a research center of the École des Mines de Paris, France, devoted to the research and promotion of mathematical morphology...
, located in Fontainebleau
Fontainebleau
Fontainebleau is a commune in the metropolitan area of Paris, France. It is located south-southeast of the centre of Paris. Fontainebleau is a sub-prefecture of the Seine-et-Marne department, and it is the seat of the arrondissement of Fontainebleau...
, France, and named Matheron its first director. In 1979, the center was renamed Centre de Géostatistique et de Morphologie Mathématique, and, in 1986, the latter was split into two separate centers: Centre de Géostatistique, directed by Matheron, and Centre de Morphologie Mathématique, directed by Serra.
Books by Matheron
- Traité de géostatistique appliquée, Editions Technip, France, 1962–63, where Matheron lays the fundamental tools of linear geostatistics: variography, variances of estimation and dispersion, and kriging.
- His doctoralDoctorateA doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...
thesisThesisA dissertation or thesis is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings...
: Les variables régionalisées et leur estimation: une application de la théorie des fonctions aléatoires aux sciences de la nature, published in 1965 by Masson, Paris. - Elements pour une théorie des milieux poreux, Masson, Paris, 1967, which includes Matheron's work on hydrodynamics.
- The theory of regionalised variables and its applications, 1971, a reference book on geostatistics for students and researchers.
- Random sets and integral geometry, John Wiley & Sons, 1975, ISBN 978-0471576211, conveying his contribution to the theory of random sets.
- Estimating and Choosing: An Essay on Probability in Practice, Springer, 1989, ISBN 978-0387500874, a newer reference book on geostatistics.
External links
- Georges Matheron at the Centre de Géostatistique
- Obituary by Dominique Jeulin (Centre de Morphologie Mathématique Ecole des Mines de Paris, October 2000) probably from Image Analysis & Stereology.
- Georges Matheron – Founder of Spatial Statistics by Frederik P. Agterberg (Proceedings of the International Association for Mathematical Geology, 2003)
- A chronology of Matheron's seminal work
- History of Mathematical Morphology, by Georges Matheron and Jean Serra
- A website that criticizes Matheron's geostatistics