Topological methods for the representation and analysis of exploration data in oil industry [Elektronische Ressource] / by Oleg Artamonov
129 Pages
English

Topological methods for the representation and analysis of exploration data in oil industry [Elektronische Ressource] / by Oleg Artamonov

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Published 01 January 2010
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Language English
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University of Kaiserslautern
Department of Mathematics
Research Group Algebra, Geometry and Computeralgebra
Dissertation
Topological Methods for the
Representation and Analysis of
Exploration Data in Oil Industry
by
Oleg Artamonov
Supervisor: Prof. Dr. Gerhard P ster
Kaiserslautern, 20101. Reviewer: Prof. Dr. Gerhard P ster
2. Reviewer: Prof. Dr. Bernd Martin
Day of the defense: 13 August 2010To my dear Grandfather.Abstract
The purpose of Exploration in Oil Industry is to \discover" an oil-containing ge-
ological formation from exploration data. In the context of this PhD project this
oil-containing geological formation plays the role of a geometrical object, which
may have any shape. The exploration data may be viewed as a \cloud of points",
that is a nite set of points, related to the geological formation surveyed in the ex-
ploration experiment. Extensions of topological methodologies, such as homology,
to point clouds are helpful in studying them qualitatively and capable of resolving
the underlying structure of a data set. Estimation of topological invariants of the
data space is a good basis for asserting the global features of the simplicial model
of the data. For instance the basic statistical idea, clustering, are correspond to
dimension of the zero homology group of the data. A statistics of Betti numbers
can provide us with another connectivity information. In this work represented
a method for topological feature analysis of exploration data on the base of so
called persistent homology. Loosely, this is the homology of a growing space that
captures the lifetimes of topological attributes in a multiset of intervals called a
barcode. Constructions from algebraic topology empowers to transform the data,
to distillate it into some persistent features, and to understand then how it is or-
ganized on a large scale or at least to obtain a low-dimensional information which
can point to areas of interest. The algorithm for computing of the persistent Betti
numbers via barcode is realized in the computer algebra system \Singular" in the
scope of the work.
vContents
Contents vii
1 Introduction 1
1.1 Motivations of the research. . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Input exploration data. . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Structures for Point Sets 7
2.1 Homological approximation of real objects. . . . . . . . . . . . . . 7
2.2 Preliminary constructions. . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Nerves of coverings, a geometric realization of the point cloud data
and the similarity theorem. . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Cech and Rips complexes. . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Witness complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Voronoi diagrams and Delaunay triangulations. . . . . . . . . . . . 18
2.6.1 The dual complex. . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.2 The -shape complex. . . . . . . . . . . . . . . . . . . . . . 23
3 Multiresolution and Persistence 25
3.1 Levels of resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Persistence homology. . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Persistence Structures 33
4.1 Persistence Betti numbers of di erent dimensions and barcode. . . 33
4.2 The persistence module. . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 The Artin-Rees correspondence. . . . . . . . . . . . . . . . 40
4.2.2 A structure theorem for graded modules over a graded PID. 41
5 The Realized \Singular" Software 47
5.1 The program structure. . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 The persistence algorithm. . . . . . . . . . . . . . . . . . . . . . . . 49
5.2.1 Matrix representations. . . . . . . . . . . . . . . . . . . . . 49
viiviii CONTENTS
5.2.2 A pseudo-code of the revised version
of the persistence algorithm. . . . . . . . . . . . . . . . . . 53
5.3 Examples of data processing. . . . . . . . . . . . . . . . . . . . . . 56
5.4 Summary and concluding remarks. . . . . . . . . . . . . . . . . . . 59
A Basic Notions and Concepts 63
B The \Singular" Code 71
B.1 Computation of persistence Betti numbers of noisy point cloud data. 71
B.2 of the Gr obner basis by a realization of the Buchberger-
M oller algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
B.3 Computation of the Gr obner basis by a realization of the approx-
imative version of the Buchberger-M oller algorithm. . . . . . . . . 83
C A Reference Mapping Way and a Representative Graph 91
C.1 Filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
C.2 Clustering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
C.3 The cluster complex. . . . . . . . . . . . . . . . . . . . . . . . . . . 95
C.4 The Mayer-Vietoris blowup. . . . . . . . . . . . . . . . . . . . . . . 97
C.5 The similarity graph. . . . . . . . . . . . . . . . . . . . . . . . . . . 99
D Brief Description of the Project 103
References 105
List of Symbols and Abbreviations 109
List of Figures 111
List of Tables 113
Index 115Acknowledgements
This work was dome with the nancial support of Deutscher Akademischer Aus-
tausch Dienst. The nancial support of the University of Kaiserslautern is also
gratefully acknowledged.
First of all, I would like to thank my supervisor Prof. Dr. Gerhard P ster for
his encouragement, support, and valuable help with a Sngular programming. He
is one who gave me a great opportunity to do my Ph.D. research in a supporting
and comforting atmosphere.
Next, I would like to say \Thank you so much!" to Dr. Hennie Poulisse, Prin-
cipal Research Mathematician in Shell. It may be said with absolute certainty
that the results of this thesis would not appear without my cooperation with
Hennie. He become the thesis advisor during my one-year internship at the De-
partment of Exploratory Research of Shell Research, Rijswijk, The Netherlands.
My special thanks to Prof. Dr. Dirk Siersma from Utrecht University. It
was nice of him to help with a solution of unexpectedly arisen problems in The
Netherlands.
I want to express my sincere appreciation of the fruitful concomitant discus-
sions to Matthew Heller.
Finally, I want to acknowledge large majority of people from collective of the
Research Group Algebra, Geometry und Computer-algebra of the Department of
Mathematics of the University of Kaiserslautern, between whom elapsed my life
all these years.
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