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Modification and automation of fractal geometry methods [Elektronische Ressource] : new tools for quantifying rock fabrics and interpreting fabric-forming processes / Axel M. Gerik

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Published 01 January 2009
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Technische Universität München
Fachgebiet Tektonik und Gefügekunde



Modification and Automation of Fractal Geometry
Methods: New Tools for Quantifying Rock Fabrics
and Interpreting Fabric-Forming Processes


Axel M. Gerik


Vollständiger Abdruck der von der Fakultät für Bauingenieur- und Vermessungswesen
der Technischen Universität München zur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften (Dr. rer. nat.)

genehmigten Dissertation.



Vorsitzender: Univ.-Prof. Dr. phil. nat. Urs Hugentobler


Prüfer der Dissertation:
1. Univ.-Prof. Dr. rer. nat., Dr. rer. nat. habil. Jörn H. Kruhl
2. Assoc. Prof. Dr. Cristian A. Suteanu, Saint Mary’s University, Halifax/Kanada



Die Dissertation wurde am 7. April 2009 bei der Technischen Universität München
eingereicht und durch die Fakultät für Bauingenieur- und Vermessungswesen am
2. Juni 2009 angenommen.
Acknowledgments
First of all, I’d like to thank my rst supervisor, J orn Kruhl, who initiated this project. Thanks
for the patient guidance and helpful advice, for new ideas for aspects that could be looked
into, for the encouragement, for many fruitful discussions and for your motivation in di cult
times along the way. I’ll keep my ngers crossed for that his next PhD student will be a co ee
a cionado again...
Many thanks are also due to my second supervisor, A. Cristian Suteanu { not just for the good
time at Saint Mary’s University and for showing me around the area. His short course and many
interesting discussions have greatly helped improving my understanding of the subject.
Moreover, I’m much obliged to Frank Fueten for inviting me to Brock University, for introducing
me to the rotating polarizer stage (and providing me with one), as well as for the cooperation
in the framework of the RISE internship for Allen Poapst { and, of course, for the patient tech
support!
Special thanks go to the International Graduate School THESIS, through which this work was
generously funded. The students and sta of THESIS have contributed in many dierent ways
{ from the warm welcome on the day of my application to providing motivational support until
the very end. Special thanks go to the THESIS science coordinator, Helen Pfuhl, for taking
great care of her students and ensuring that we were always problem-free!
Many great colleagues have made working at the Munich GeoCenter even more enjoyable. I’m
especially grateful to Sabine Volland, for her continous encouragement, lunchtime company,
shared woes and always being there to talk to; to Beno^ t Cordonnier, for sparking new ideas,
interesting discussions and his ways to cheer others up, even when he is not in a good mood
himself; to Heather McCreadie for writing support, proof-reading, fun conversations as well as
shared rants, raves and Tresznjewski burgers; to Yan Lavallee for the pleasant cooperation, his
radiating optimism and a lot of fun; and to my former colleague Mark Peternell for his tough
and constructive reviews, enlightening discussions, great cocktails and a fun stay at his new
place down under. The Tuesday night MPVG seminar series and especially the post-colloquium
across the street has helped a lot with bringing everyone together { thanks to the organizers,
Kai-Uwe Hess and Werner Ertel-Ingrisch!As my work is very computer-oriented, I owe a lot of gratitude to the always helpful IT-team at
LMU Geophysics - namely Jens Oeser, Alex Hornung and Alexandra Fischl. They have provided
the best infrastructure that I had ever the pleasure to work with! And while I’m there: this
Awhole thesis would not be typeset as nicely without the resourceful and patient LT X-supportE
provided by Jens Oeser and Marcus Mohr.
Nothing could work in a research facility without the people in the background who keep the
whole thing running: the kind angels in the main o ces { Yvonne Ne ler, Marion Bachh aubl,
Sandra Bauer, Margot Lieske, Dagmar Hossfeld and Ana Salamano; the sample preparators
{ Vladimir Ruttner and Cathleen Helbig; the allrounders { Kurt Doppler and Klaus Haas...
Thank you all!
Furthermore, I’m deeply indebted to my friends and family, for whom I did not always have as
much time as I would have liked, but who never lost their patience, even though I failed to keep
in touch. Finally, I owe many thanks to my parents, who always supported me in every way
they could.
Last { but most { I would like to express my deepest gratitude to my wife, Fernanda Maria da
Silva Mendes. For your love, your kind nature, your encouragement and your ability to let the
sun shine on a rainy day { with a single smile. Without you, this would not have been possible.
Obrigad ssimo!
To you I dedicate this thesis.Abstract
Practically all processes in Geology leave their signatures in the form of fabrics on di erent
scales from single crystal to continental crust. In addition to the qualitative description of such
fabrics, their quanti cation is an essential step towards understanding of geological processes.
However, many geological fabrics have a complex structure that cannot be analyzed with com-
mon statistical approaches. In the scope of this work, methods that are speci cally suitable for
the analysis of pattern anisotropy and inhomogeneity were automated and their applicability on
di erent geological fabrics was tested.
This thesis is organized in two parts. The rst part focuses on the methodological aspects of
anisotropy and inhomogeneity in geological fabrics and the quanti cation of such. It de nes the
term \fabrics" in a geological context and gives an overview of the current state-of-the-art with
respect to quanti cation of anisotropy and inhomogeneity in geological fabrics. Since geological
fabrics often show complex geometries, fractal geometry provides important tools for their quan-
ti cation. Principles of Fractal Geometry are introduced, classic techniques for quanti cation
of fractals are presented and strategies for their automation are pointed out. Modi ed methods
for quanti cation and visualization are examined and software implementations are presented
along with exemplary applications.
The second part presents practical applications of the developed tools on mathematically derived,
experimentally produced and natural patterns. The automated analyses’ ability to extract
information from the patterns is demonstrated and the implications of this information with
respect to the pattern-forming processes are inferred.
The developed software tools represent a new and promising step towards a time-e cient analysis
of complex geological fabrics. They allow for analyses of large patterns and data sets, enable
the quantitative comparison of patterns from nature, experiment and simulation and, therefore,
give access to more profound investigations of geological processes.Zusammenfassung
Praktisch alle geologischen Prozesse hinterlassen charakteristische Spuren in Form von Gefugen
auf verschiedenen Skalen von einzelnen Kristallen bis hin zur kontinentalen Kruste. Als Erg an-
zung der ublic hen qualitativen Beschreibung solcher Gefuge erm oglicht deren Quanti zierung
ein besseres Verst andnis der geologischen Prozesse, die zu ihrer Entstehung beigetragen haben.
Da aber viele geologische Gefuge komplexe Strukturen aufweisen, die nicht ohne weiteres mit
g angigen statistischen Methoden analysiert werden k onnen, wurden im Rahmen dieser Arbeit
Methoden automatisiert, die eine Quanti zierung von Anisotropie und Inhomogenit at speziell
in solchen Mustern erm oglichen. Darub er hinaus wurde deren Eignung zur Untersuchung un-
terschiedlicher geologischer Gefuge unter Beweis gestellt.
Diese Arbeit gliedert sich in zwei Teile. Der erste, methodische Teil handelt von Anisotropie
und Inhomogenit at in geologischen Gefugen und deren Quanti zierung. Zun achst wird der Be-
gri des \Gefuges" imhen Sinne de niert und ein Uberblick des aktuellen Standes der
Forschung auf dem Gebiet der Quanti zierung von Anisotropie und Inhomogenit at in geolo-
gischen Gefugen gegeben. Da diese Gefuge oft komplexe geometrische Strukturen aufweisen,
bietet sich die Verwendung von Methoden aus dem Bereich der Fraktalen Geometrie an. Die
Grundlagen der Fraktalen Geometrie werden dargelegt, klassische Methoden zur quantitativen
Beschreibung von Fraktalen erl autert und Strategien zu deren Automatisierung skizziert. Modi-
zierte Methoden zur Quanti zierung und Visualisierung werden diskutiert und deren software-
technische Umsetzung wird anhand von Anwendungsbeispielen vorgestellt.
Im zweiten Teil der Arbeit wird der praktische Einsatz der neuen Werkzeuge an mathematisch
hergeleiteten, experimentell erzeugten und naturlic hen Mustern veranschaulicht. Die sich durch
die automatisierten Methoden neu ergebenden M oglichkeiten, Informationen aus Mustern zu
gewinnen, werden aufgezeigt und die sich daraus ergebenden Ruc kschlusse auf die gefugebil-
denden Prozesse gezogen.
Die neu entwickelten Softwarel osungen sind ein weiterer, vielversprechender Schritt hin zu
schnelleren und e zienteren Analysem oglichkeiten fur komplexe geologische Gefuge. Sie er-
lauben die Untersuchung gro er Muster und Datens atze, erm oglichen den Vergleich von Mustern
aus Natur, Experiment und Simulation auf quantititativer Basis und erschlie en dadurch neue
Wege fur tiefer gehende Untersuchungen geologischer Prozesse.Contents
1 Introduction 1
1.1 Geological fabrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Quantitative analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Fractal Geometry-based quanti cation approaches . . . . . . . . . . . . . 13
2 Fractal Geometry 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Occurrence of fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Fractal geometry in 2D and 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Classic methods of Fractal Geometry 28
3.1 Cantor-dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 Occurence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.2 Automated analysis of Cantor-dust . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Divider method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Occurence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Automated determination of the ’right’ ruler dimension . . . . . . . . . . 34
3.3 Box-counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Occurence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.2 Automated determination of the box-counting dimension . . . . . . . . . 37
4 New Fractal Geometry-based quanti cation methods 40
4.1 Modi ed Cantor-dust method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Software implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2 Sample application on a magma mingling pattern . . . . . . . . . . . . . . 42
4.2 Map-counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Automation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 MORFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Automation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Modi ed divider method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.1 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.2 Specifying the analyzed area . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.3 Determination of shortest path . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.4 Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Anisotropy assessment with AMOCADO 62
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 Data extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.3 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 Analyses of natural patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.1 Grain-scale recrystallization pattern . . . . . . . . . . . . . . . . . . . . . 69
5.3.2 Micro-scale fracture pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Fabric quanti cation in sheared tonalite 79
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2 Geological framework and history of the Squillace Tonalite . . . . . . . . . . . . . 80
6.3 Material and preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.2 Image acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3.3 Phase pattern binarization . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.4 Pattern quanti cation { methods and results . . . . . . . . . . . . . . . . . . . . 88
6.4.1 Results of box-counting analyses . . . . . . . . . . . . . . . . . . . . . . . 90
6.4.2 of modi ed Cantor-dust method . . . . . . . . . . . . . . . . . . . 92
6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7 Anisotropy in experimentally deformed rocks 100
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.2 Performed analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8 Conclusion 106List of Figures
1.1 Grain boundaries within a dynamically recrystallized quartz layer . . . . . . . . . 2
1.2 Recrystallized quartz lens in a quartz-porphyr . . . . . . . . . . . . . . . . . . . . 2
1.3 Mineral distribution pattern of K-feldspar phenocrysts in a granitoid rock . . . . 3
1.4 Mingled ma c and felsic magma . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Network of late-Hercynian pegmatoid dikes in a tonalite . . . . . . . . . . . . . . 4
1.6 Large scale ruptures zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 Utah’s Green River entering Desolation Canyon . . . . . . . . . . . . . . . . . . . 5
1.8 Application of the inverse SURFOR wheel . . . . . . . . . . . . . . . . . . . . . . 7
1.9 The inertia tensor method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.10 Di use magmatic ow fabric in a mesozoic syntectonic granite . . . . . . . . . . 9
1.11 Application of the intercept method . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.12 Brecciation density maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.13 D-mapping of fault rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1 The Chaos Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Visualization of an exemplary orbit trace . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Fractal dimensions of Euclidean objects . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Romanesco broccoli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Dendrites on a sandstone bedding plane . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Peaks and ridges of the eastern Himalayas Mountains . . . . . . . . . . . . . . . 23
2.7 Application of cube-counting in osteoporosis research . . . . . . . . . . . . . . . . 24
2.8 Sensitivity comparison of X-ray-based tomography and neutron tomography . . . 26
2.9 3D-reconstruction of biotite aggregates in a tonalite . . . . . . . . . . . . . . . . 27
3.1 Geometrical construction of the Cantor set . . . . . . . . . . . . . . . . . . . . . 29
3.2 First 5 iterations of the Smith-Volterra-Cantor set . . . . . . . . . . . . . . . . . 30
3.3 Measuring the fractal dimension D of a Cantor set . . . . . . . . . . . . . . . . . 31
3.4 the coastline of Britain with the divider method . . . . . . . . . . . . 32
3.5 Richardson plot of the divider measurements in gure 3.4 . . . . . . . . . . . . . 33
3.6 The box-counting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1 Outcrop photographs of mingled ma c and felsic phases . . . . . . . . . . . . . . 43
4.2 Line sketches of the areas indicated in gure 4.1 and the corresponding results . 43
4.3 The Map-counting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Exemplary results of a Map-counting analysis . . . . . . . . . . . . . . . . . . . . 47
4.5 Application of MORFA (Mapping of rock fabric anisotropy) . . . . . . . . . . . . 50
4.6 Exemplary results of a MORFA analysis . . . . . . . . . . . . . . . . . . . . . . . 51
4.7 The modi ed divider method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.8 Basic routing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.9 Determination of shortest path in gure 4.8 using a depth- rst search (DFS) . . 57
4.10 of path in gure 4.8 using Dijkstra’s algorithm . . . . . . 58
4.11 of shortest path in gure 4.8 using a A . . . . . . . . . . . . . . . 59
5.1 Application of the modi ed Cantor-dust method on holes of a Sierpinski carpet . 65
5.2 Determination of representative slope value from cumulative segment length plot 68
5.3 Binary phase distribution of magnetite and ilmenite minerals . . . . . . . . . . . 70
5.4 Results of analyses of the phase-distribution pattern in gure 5.3a . . . . . . . . 71
5.5 Thin section photograph of part of a fractured garnet porphyroblast . . . . . . . 74
5.6 Line drawing of the fracture pattern in gure 5.5 . . . . . . . . . . . . . . . . . . 75
5.7 Results of analyses performed on fractures and fragments depicted in gure 5.6 . 76
6.1 Geology of the Squillace tonalite body and its surroundings . . . . . . . . . . . . 82
6.2 Photomicrograph of tonalite sample KR4819B . . . . . . . . . . . . . . . . . . . . 83
6.3 Scanned sample KR4979 (YZ section) and corresponding thresholded image . . . 84
6.4 Threshold determination for image binarization . . . . . . . . . . . . . . . . . . . 87
6.5 Slope estimation for cumulative segment length distribution . . . . . . . . . . . . 89
6.6 In uence of scanline spacing on AMOCADO results’ quality . . . . . . . . . . . . . . 89
6.7 Distinguished intervals in box-counting results for sample KR4994 (YZ section) . 91
6.8 Analyzed areas of XZ and YZ sections of the studied tonalite samples . . . . . . 92
6.9 Binary patterns of the analyzed regions shown in gure 6.8 . . . . . . . . . . . . 94
6.10 Results of pattern-anisotropy quanti cation for the regions shown in gure 6.9 . 96
7.1 High-load, high-temperature uniaxial press . . . . . . . . . . . . . . . . . . . . . 101
7.2 Thin section scans with di erent polarizer con gurations . . . . . . . . . . . . . . 103
7.3 Deformation induced changes of fabric anisotropy . . . . . . . . . . . . . . . . . . 104