Fault-Related Deformation Over

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FAULT-RELATED DEFORMATION OVER GEOLOGIC TIME: INTEGRATING FIELD OBSERVATIONS, HIGH RESOLUTION GEOSPATIAL DATA AND NUMERICAL MODELING TO INVESTIGATE 3D GEOMETRY AND NON-LINEAR MATERIAL BEHAVIOR

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF GEOLOGICAL AND ENVIRONMENTAL SCIENCES AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Peter James Lovely January 2011

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/yb440sg1391

Includes supplemental files: 1. High resolution copy of Figure 1.4a: ALSM hillshade image and outcrop map of Sheep Mountain anticline, WY (Figure_1.4_HiRes.pdf)

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

David Pollard, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

George Hilley

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Mark Zoback

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

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Abstract

A thorough understanding of the kinematic and mechanical evolution of fault- related structures is of great value, both academic (e.g. How do mountains form?) and practical (e.g. How are valuable hydrocarbons trapped in fault-related folds?). Precise knowledge of the present-day geometry is necessary to know where to drill for hydrocarbons. Understanding the evolution of a structure, including displacement fields, strain and stress history, may offer powerful insights to how and if hydrocarbons might have migrated, and the most efficient way to extract them. Small structures, including faults, fractures, pressure solution seams, and localized compaction, which may strongly influence subsurface fluid flow, may be predictable with a detailed mechanical understanding of a structure's evolution. The primary focus of this thesis is the integration of field observations, geospatial data including airborne LiDAR, and numerical modeling to investigate three dimensional deformational patterns associated with fault slip accumulated over geologic time scales. The work investigates contractional tectonics at Sheep Mountain anticline, Greybull, WY, and extensional tectonics at the Volcanic Tableland, Bishop, CA. A detailed geometric model is a necessary prerequisite for complete kinematic or mechanical analysis of any structure. High quality 3D seismic imaging data provides the means to characterize fold geometry for many subsurface industrial applications; however, such data is expensive, availability is limited, and data quality is often poor in regions of high topography where outcrop exposures are best. A new method for using high resolution topographic data, geologic field mapping and numerical interpolation is applied to model the 3D geometry of a reservoir-scale fold at Sheep Mountain anticline. The Volcanic Tableland is a classic field site for studies of fault slip scaling relationships and conceptual models for evolution of normal faults. Three dimensional elastic models are used to constrain subsurface fault geometry from detailed maps of fault scarps and topography, and to reconcile two potentially competing conceptual models for fault growth: by coalescence and by subsidiary faulting. The Tableland fault array likely initiated as a broad array of small faults, and

v as some have grown and coalesced, their strain shadows have inhibited the growth and initiation of nearby faults. The Volcanic Tableland also is used as a geologic example in a study of the capabilities and limitations of mechanics-based restoration, a relatively new approach to modeling in structural geology that provides distinct advantages over traditional kinematic methods, but that is significantly hampered by unphysical boundary conditions. The models do not accurately represent geological strain and stress distributions, as many have hoped. A new mechanics-based retrodeformational technique that is not subject to the same unphysical boundary conditions is suggested. However, the method, which is based on reversal of tectonic loads that may be optimized by paleostress analysis, restores only that topography which may be explained by an idealized elastic model. Elastic models are appealing for mechanical analysis of fault-related deformation because the linear nature of such models lends itself to retrodeformation and provides computationally efficient and stable numerical implementation for simulating slip distributions and associated deformation in complicated 3D fault systems. However, cumulative rock deformation is not elastic. Synthetic models are applied to investigate the implications of assuming elastic deformation and frictionless fault slip, as opposed to a more realistic elasto-plastic deformation with frictional fault slip. Results confirm that elastic models are limited in their ability to simulate geologic stress distributions, but that they may provide a reasonable, first-order approximation of strain tensor orientation and the distribution of relative strain perturbations, particularly distal from fault tips. The kinematics of elastic and elasto- plastic models diverge in the vicinity of fault tips. Results emphasize the importance of accurately and completely representing subsurface fault geometry in linear or nonlinear models.

Preface

Faults and folds are inherently three dimensional structures that form according to the laws of physics in response to mechanical forces within the earth. However, structural analysis is often based upon two dimensional models and ad hoc kinematic methods. Recent advances in computing power and numerical methods provide the means to consider structural evolution in three dimensions with sophisticated mechanical models. Great strides have been made; however, routine structural analysis remains primarily a two dimensional and kinematic science. This thesis aims to advance our ability to model geologic structures in three dimensions and using mechanical approaches. In Chapter 1 of this thesis, I present a new method for creating a three dimensional geometric model of a folded geologic surface at Sheep Mountain anticline, WY, using airborne LiDAR data, outcrop mapping, and numerical interpolation. Accurate geometric characterization of the present day structure is a necessary first step for any 3D analysis of fault-related folding. Chapter 2 focuses on faulting processes in the Volcanic Tableland, Bishop, CA. Mechanical models are used to infer three dimensional subsurface fault geometry from maps of fault scarps and deformed topography, represented by airborne LiDAR data. Strain perturbations resulting from slip on the largest faults are then applied to investigate the relative importance of several different conceptual models for fault evolution. Chapter 3 considers mechanics-based restoration, a relatively new modeling approach that realizes the retrodeformational benefits of kinematic restoration in a fully mechanical framework. However, the new method is not without flaws. I demonstrate detrimental kinematic implications that stem from unphysical boundary conditions used in published restoration studies, and consider alternative methods that produce more physically appropriate results. In Chapter 4, I consider the limitations of simulating geological processes using elastic models without friction. Cumulative fault-related rock deformation is not elastic in nature, but elastic models are appealing for their numerical stability and computational efficiency. Mechanics-based restoration methods assume elastic deformation because it is reversible. In the

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Appendix, I use mechanical models to study fault slip, but the models are applied to earthquake processes rather than cumulative deformation. Models indicate that regions of reduced static stress drop in the vicinity of fault tips in large earthquakes rupturing multiple fault segments may help explain slip distributions inferred from geophysical inversion. While these research goals may appear diverse at first glance, all focus on characterization of the geometry and mechanics of faults and fault-related folds, which are common elements of hydrocarbon traps. In the first chapter of this thesis, I present a new three dimensional model of fold geometry at Sheep Mountain anticline, WY. David Pollard suggested using airborne LiDAR data and outcrop mapping to model fold geometry at Sheep Mountain anticline before I arrived at Stanford, and this project evolved under my direction over the following four years. I spearheaded the field mapping, data processing, and interpolation efforts, and wrote the initial manuscript. Co-author Christopher Zahasky, an undergraduate intern from the University of Minnesota in the summer of 2009, did much of the field mapping and helped develop codes to extract outcrop points from the DEM and to project these points to the position of a single stratigraphic unit. David Pollard, beside inspiring this work, provided valuable feedback and suggestions throughout the process, and was instrumental in revising the manuscript. The manuscript has benefited from thoughtful and thorough reviews from Richard Lisle and Ken McCaffrey, and has been published in the Journal of Geophysical Research: Solid Earth. [Reprinted from Journal of Geophysical Research, Lovely, P., C. Zahasky, and D. D. Pollard, Fold geometry at Sheep Mountain anticline, WY, constructed using ALSM data, outcrop scale geologic mapping, and numerical interpolation, B12414, doi:10.1029/2010JB007650 Copyright 2010, with permission from the American Geophysical Union.] Chapters 2 and 3 both originated from an internship at Chevron during the summer of 2008. Eric Flodin and Chris Guzofski were my technical mentors during this internship. In Chapter 2 I apply three-dimensional elastic models to investigate the relative importance of subsidiary faulting versus the previously documented process of fault coalescence in the evolution of the Volcanic Tableland extensional

viii fault system. This chapter evolved after co-author Eric Flodin and I observed interesting relations between the stress shadows of large faults in forward models of the Tableland and the local density of smaller faults. I refined the models, analyzed results, and wrote the manuscript. Eric, Chris and David Pollard are co-authors; each has provided insights and guidance that have been invaluable to shape the manuscript. This manuscript is nearly ready for submission, most likely to Tectonophysics, Journal of Geophysical Research, or the Journal of Structural Geology. Chapter 3 describes significant kinematic implications of unphysical boundary conditions applied to the earth's surface in mechanics-based restoration models. I suggest a more physically appropriate, alternative approach to retrodeformational modeling based on the reversal of tectonic loading. Optimal reverse tectonic loads may be determined by mechanics-based paleostress analysis. I am the principle author of this manuscript, with co-authors Eric Flodin, Chris Guzofski, Frantz Maerten and David D. Pollard. In late 2007, prior to my qualifying exam, Chevron offered me an intriguing internship. I would work with Chris Guzofski and Eric Flodin to evaluate the application of mechanics-based restoration methods for predicting geologic stress and strain. The initial approach was to construct a three dimensional restoration model of the Volcanic Tableland, and to observe if there was a correlation between restoration strain distributions and the locations of smaller structures (faults and fractures) not explicitly included in the model. However, this approach was abandoned quickly when I discovered that forward models failed to explain the location or density of smaller structures. I changed course, and have evaluated mechanics-based restoration methods by comparison of forward and retrodeformational models instead. The fundamental ideas associated with limitations of and alternatives to restoration models are my own. I completed all of the numerical analysis and wrote the original manuscript. David Pollard has been of great assistance as I have refined my thoughts concerning the theoretical limitations of restoration boundary conditions and revised the manuscript. Beyond inspiration, Chris and Eric provided instrumental guidance in the early stages of this project and have been a great help crafting a flowing manuscript with clear direction. Frantz Maerten

ix provided the paleostress code, performed some of the early paleostress analysis, and also assisted with Dynel3D restoration models. This manuscript is nearly ready for submission to the Journal of Structural Geology. Chapter 4 provides an evaluation of elastic models and their limitations. Elasticity is appealing to structural geologists for its numerical stability and computational efficiency, and restoration models have assumed elasticity because it is reversible. However, elasticity is a dramatically simplified representation of cumulative rock deformation. The initial idea for this study came from a discussion with David Pollard over two years ago. I sat on the idea for some time, considering the best approach. I performed the numerical models, analyzed the results, and wrote the manuscript. David has provided valuable feedback along the way. This manuscript is in preparation for submission to a yet to be determined journal. Finally, the Appendix is a published manuscript that grew out of a term project for Mark Zoback's tectonophysics course. It is not a chapter because it focuses on individual earthquakes, rather than cumulative deformation. I demonstrate that regions of reduced stress drop near fault segment tips in large earthquakes that rupture multiple fault segments may explain slip distributions inferred from geophysical inversion. I wrote the manuscript with co-authors David Pollard and Ovunc (Uno) Mutlu. The idea spawned from a conversation with Uno, and I used code he had developed, which applies principles of complementarity to simulate frictional slip on linear elements of displacement discontinuity in a homogeneous, linear elastic plane. I completed the modeling, analyzed the results, and wrote the manuscript. David Pollard provided great insights and suggestions to improve the study throughout the process. The manuscript also benefited from a comprehensive and constructive review from Ruth Harris. It has been published in the Bulletin of the Seismological Society of America. [Reprinted from Bulletin of the Seismological Society of America, v. 99, no. 3, Lovely, P. J., D. D. Pollard and O. Mutlu, Regions of Reduced Static Stress Drop near Fault Tips for Large Strike-Slip Earthquakes, p. 1691-1704, Copyright 2009, with permission of the Seismological Society of America.]

Acknowledgements

My experience at Stanford has been intellectually stimulating and academically enriching beyond expectations. This thesis would not have been possible, were it not for the personal support and intellectual contributions of numerous individuals, both at Stanford and along my path to Stanford. First and foremost, I thank my advisor, David Pollard, who has been a tremendous mentor. Dave has played an instrumental role in teaching me how to approach scientific problems, and in developing focused objectives for each chapter of this thesis. Many times over the past four years I have entered his office feeling frustrated, confused, or lost, and left with renewed confidence and direction; Dave's intellectual enthusiasm is contagious. His "students first" attitude and personal qualities as a mentor and role model are admirable. I would like to thank my defense committee, including readers Mark Zoback and George Hilley, committee member Paul Segall, and committee chair Ronnie Borja. Each of them has provided valuable insight that has improved the intellectual quality of this thesis. While at Stanford, I have had the pleasure of collaborating with many individuals, both within and beyond the Stanford community. Eric Flodin and Chris Guzofski, of Chevron, introduced me to the Volcanic Tableland field site and provided the opportunity to make mechanics-based restoration a significant component of this thesis. Each has been a great collaborator and mentor, and I look forward to joining them as colleagues. Frantz Maerten, of IGEOSS, provided valuable technical assistance and intellectual advice on restoration modeling and paleostress analysis. Uno Mutlu inspired and collaborated on the appendix to this thesis, as a post-doc at Stanford. Later, he, along with Rod Meyers and Pablo Sanz, was an excellent mentor during a stimulating summer internship with ExxonMobil. Chris Zahasky, as an undergraduate visiting from the University of Minnesota, provided tremendous assistance with field mapping and data analysis at Sheep Mountain anticline. I would be amiss not to recognize the academic contributions of two individuals who played a pivotal role in my development as a scientist prior to my arrival at Stanford: John Shaw, my undergraduate advisor at Harvard and Frank

Bilotti, my mentor during my first internship at Chevron. John introduced me to structural geology and inspired me to pursue a PhD and Frank introduced me to the broad applications of structural geology in the petroleum industry. Day-to-day interaction with other students has been among the most enriching parts of my experience at Stanford. Academic discussions relating to coursework, research, and unrelated topics have been intellectually stimulating, and occasional diversions from studying have kept life in the office fun. I have no doubt that many of the friendships and working relationships I have developed at Stanford will last a lifetime. Ole Kaven, Fil Nenna, Betsy Madden, Libby Ritz, Solomon Seyum, Josie Nevitt, Daniel Zhou, Pablo Sanz, Ashley Griffith, Ian Mynatt, Uno Mutlu and Tricia Allwardt have been great friends and colleagues. I have received generous financial support from a number of sources while at Stanford. I am grateful for the unrestricted funding provided by the Robert and Marvel Kirby Stanford Graduate Fellowship, which allowed me the opportunity to develop my own research projects, a valuable part of my experience at Stanford. I have received additional support for tuition, stipend, field work, and meetings from the National Science Foundation's Collaborations in Mathematical Geosciences grant no. CMG-0417521, the Stanford Rock Fracture Project, and the McGee and Levorsen grants. ALSM data, which facilitated much of the research in my thesis, was provided courtesy of Chevron Corporation (Bishop Tableland) and through a collaboration between the National Science Foundation's Collaborations in Mathematical Geosciences program and the National Center for Airborne Laser Mapping (Sheep Mountain anticline). My parents, Andy and Laurie, have encouraged my curiosity and academic pursuits since I was young, nurturing my love of nature and introducing me to geology on our family road trips through the western US when I was in high school. They have been a source of endless support and love. My sisters, Ann and Karen, have provided encouragement from afar and welcome respites from study on their occasional visits to California. Finally, the greatest thing that has happened in my time at Stanford, was meeting my lovely wife, Christina. We started dating as I prepared for my qualifying exams and were married as I prepared to defend this thesis. She has stuck with me, a constant source of love and support, through both joyful and

xii difficult times. Her encouragement, particularly through the last few months, has been invaluable.

Peter J. Lovely January 2011

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Table of Contents

Chapter 1: Fold geometry at Sheep Mountain anticline, WY, constructed using ALSM data, outcrop scale geologic mapping, and numerical interpolation.....1 Abstract...... 1 Introduction...... 1 Geologic setting...... 8 Airborne laser swath mapping (ALSM) data...... 9 Field methods...... 11 Geologic mapping...... 11 Stratigraphy...... 12 Stratigraphic thickness...... 15 Numerical methods...... 20 Projection...... 20 Numerical interpolation and regularization...... 24 Data considerations...... 27 Results and discussion...... 30 Conclusions...... 43 Appendix: ALSM technical specifications...... 44 Acknowledgements...... 45 References...... 46

Chapter 2: Integrating geomechanics and field observations to unravel the evolution of an extensional fault system, Volcanic Tableland, Bishop, CA.....55 Abstract...... 55 Introduction...... 56 Geologic Setting...... 60 Slip profiles as evidence for coalescence...... 61 Optimizing subsurface 3D fault geometry...... 64 Evaluation of elastic models...... 72 Role of subsidiary faulting...... 79

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Paleostress analysis...... 80 Forward model results...... 85 Discussion...... 89 Conclusions...... 93 Acknowledgements...... 95 References...... 95

Chapter 3: Pitfalls among the promises of mechanics-based restoration: addressing implications of unphysical boundary conditions...... 107 Abstract...... 107 Introduction...... 108 Theory...... 111 Synthetic example...... 114 Analysis of field data...... 117 Geologic setting...... 118 Geologic model...... 121 Paleostress analysis using mechanics...... 122 Numerical models...... 125 Results...... 129 Discussion...... 136 Conclusions...... 144 Acknowledgements...... 146 References...... 146

Chapter 4: Implications of assuming a linear, frictionless, elastic constitutive model for simulation of fault-related deformation...... 155 Abstract...... 155 Introduction...... 155 Model description...... 158 Single fault models...... 162 Extensional model results...... 164

Contractional model results...... 175 Implications of a linearized constitutive law...... 185 Elastic, frictionless models...... 185 Partially linearized constitutive models...... 190 Strain localization...... 192 Multi-fault models...... 195 Evaluating the implications of elastic models on a site-specific basis...... 200 Conclusions...... 201 Acknowledgements...... 202 References...... 202

Appendix: Regions of reduced static stress drop near fault tips for large strike slip earthquakes...... 213 Abstract...... 213 Introduction...... 213 Methods...... 218 Results...... 227 Discussion...... 233 Conclusions...... 237 Acknowledgements...... 238 References...... 238

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List of Tables

Table 2.1: RMS residuals between model and observed topography for trial fault models without Fish Slough Fault...... 69

Table 2.2: RMS residuals between model and observed topography for trial fault models with Fish Slough Fault ...... 71

Table A.1: RMS error of slip distributions for various models representing the Landers and Hector Mine earthquakes...... 228

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List of Illustrations

Figure 1.1: Geologic map of Sheep Mountain anticline...... 5

Figure 1.2: Bedding surface outcrop photographs at Sheep Mountain anticline...... 6

Figure 1.3: Stratigraphic column for Sheep Mountain anticline area...... 7

Figure 1.4: ALSM hillshade image and mapped bedding surface outcrops...... 10

Figure 1.5: Schematic illustration of bed thickness calculation...... 16

Figure 1.6: Selected bed thickness measurements in the Paleozoic section...... 18

Figure 1.7: Selected bed thickness measurements in the Mesozoic section...... 18

Figure 1.8: Down-plunge view of Sheep Mountain anticline...... 21

Figure 1.9: Schematic illustration of bed orientation calculated from ALSM data...... 21

Figure 1.10: Difference between strike and dip measured from ALSM and outcrop...23

Figure 1.11: Structure contour maps of Sheep Mountain anticline surface models.....32

Figure 1.12: Difference between new and old models of the base Sundance fm...... 33

Figure 1.13: Cross-sections comparing Sheep Mountain anticline fold models...... 35

Figure 1.14: Results of cross-validation analysis of new fold model...... 41

Figure 2.1: ALSM hillshade image of the Volcanic Tableland and fault map...... 58

Figure 2.2: High resolution fault map from DGPS field mapping...... 62

Figure 2.3: Throw profile along a long, continuous fault...... 63

Figure 2.4: Profiles comparing model and observed topography on two transects...... 67

Figure 2.5: Comparison of linear and nonlinear 2D models...... 73

Figure 2.6: Cross section strain distribution...... 81

Figure 2.7: Strain distribution map and smaller-scale fault density...... 87

Figure 3.1: Strain fields for synthetic model with a single fault, calculated from forward, reverse & restoration boundary conditions...... 115

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Figure 3.2: ALSM hillshade image and fault map of the Tableland study area...... 120

Figure 3.3: FEM mesh for Dynel3D models of the Volcanic Tableland...... 128

Figure 3.4: Maximum principal strain from forward and retrodeformational models of the Volcanic Tableland...... 130

Figure 3.5: Vertical strain from forward and retrodeformational models of the Volcanic Tableland...... 131

Figure 3.6: Displacement fields from forward and retrodeformational models of the Volcanic Tableland...... 132

Figure 3.7: Tableland topography before and after retrodeformation by application of reverse tectonic loads...... 135

Figure 3.8: Maximum principal strain map calculated by restoration boundary conditions and by reverse tectonic boundary conditions...... 141

Figure 4.1: Mesh and model configuration for single fault models...... 163

Figure 4.2: Slip distributions from single fault extensional models...... 165

Figure 4.3: Maximum principal strain from single fault extensional models...... 166

Figure 4.4: Maximum principal stress from single fault extensional models...... 167

Figure 4.5: Maximum principal strain evolution in inelastic extensional model...... 172

Figure 4.6: Maximum principal stress evolution in inelastic extensional model...... 174

Figure 4.7: Slip distributions from single fault contractional models...... 176

Figure 4.8: Minimum principal strain from single fault contractional models...... 179

Figure 4.9: Minimum principal stress from single fault contractional models...... 181

Figure 4.10: Minimum principal strain evolution in inelastic contractional model....184

Figure 4.11: Maximum principal strain from two models subject to mesh-dependent strain localization...... 193

Figure 4.12: Mesh and model configuration for multi-fault models...... 196

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Figure 4.13: Maximum principal strain from multi-fault extensional models...... 198

Figure 4.14: Maximum principal stress from multi-fault extensional models...... 199

Figure A.1: Maps of the Landers and Hector Mine rupture traces...... 216

Figure A.2: Comparison of slip distributions and stress drop for multi-segment models representing the Landers and Hector Mine earthquakes...... 223

Figure A.3: Comparison of slip distributions and stress drop for single segment models representing the Landers and Hector Mine earthquakes...... 229

Figure A.4: 3D model results representing the Bullion Fault segment of the Hector Mine earthquake...... 231

Figure A.5: Slip distributions from curvilinear models representing the Landers and Hector Mine earthquakes...... 231

Chapter 1

Fold geometry at Sheep Mountain anticline, WY, constructed using ALSM data, outcrop scale geologic mapping, and numerical interpolation

Abstract Constraints available at the Earth’s surface on the geometry of geologic folds often are discontinuous, consisting of scattered bedding surface outcrops representing the position and orientation of different stratigraphic surfaces. For this reason, characterization of fold geometry from surface data, by methods such as sequential balanced cross sections, remains a subjective and interpretive process. We present a new method for modeling the geometry of kilometer-scale folds using dense, precise topographic data available from airborne laser swath mapping (ALSM), outcrop scale geologic mapping, and a reproducible numerical interpolation method that is free of subjectivity. Geologic mapping at Sheep Mountain anticline, Wyoming, enables the association of hundreds of thousands of ALSM data points with fourteen weathering- resistant bedding surfaces from seven different stratigraphic units ranging in age from Mississippian to Cretaceous. These data, supplemented by several hundred strike and dip measurements constraining local orientation of folded strata, are projected to the stratigraphic position of a single surface, and used as constraint for a smoothed minimum curvature spline interpolation to generate a continuous representation of fold geometry. Both strengths and weaknesses of the new method are discussed in the context of comparison between the new surface model and an existing model constructed by traditional methods.

Introduction Three dimensional surface models of folded geologic strata, often portrayed as structure contour maps, are a fundamental tool for geologists and geophysicists. In the hydrocarbon industry surface models have been used for nearly a century to define trap geometry, identify spill points, and estimate trap potential [e.g. DeGolyer, 1928; Howell, 1934]. Models of folded surfaces are necessary input for curvature analysis,

1 which has been used to describe folded surfaces geometrically [Bengtson, 1981; Bergbauer and Pollard, 2003; Bergbauer et al., 2003; Lisle, 1992; Lisle and Robinson, 1995; Lisle and Martinez, 2005; Mynatt et al., 2007a; Pearce et al., 2006; Roberts, 2001], and as a proxy for strain [Bevis, 1986; Ekman, 1988; Lisle, 1994; 1999; Nothard et al., 1996; Samson and Mallet, 1997; Wynn and Stewart, 2005] and for fracture density and orientation [Allwardt et al., 2007; Bergbauer, 2002; Fischer and Wilkerson, 2000; Hennings et al., 2000; Lisle, 1994; Murray, 1968; A Thomas et al., 1974]. Structure contour maps may be compared to synthetic structure contour maps constructed using displacement fields from numerical models to infer fault geometry [e.g. Hilley et al., 2010; Maerten et al., 2000] and to confirm the subsurface interpretation of fault geometry made using seismic and well data [e.g. Fiore et al., 2007]. Subsurface imaging from seismic reflection data enables accurate constraint of fold geometry where such data are available [e.g. Brown, 2004; Shaw et al., 2005]. Other important sources of data to constrain fold geometry are geological mapping [e.g. Groshong, 2006; Woodward et al., 1989] and well logging [e.g. Serra, 1984]. Here we focus on the contribution of geologic mapping augmented by high resolution geospatial data. At Sheep Mountain anticline (SMA), the field site for this study, a few 2D seismic lines are available [Gries, 1983]; however, image quality is poor and the extent of the 2D lines insufficient to constrain the full 3D geometry of the fold. In the absence of good subsurface imaging and well control, discontinuous surface outcrops must be used to constrain continuous models of surfaces, the vast extent of which are eroded or buried. Such fold surface models often are created from sequential balanced cross sections as has been done at SMA [Forster et al., 1996; Stanton and Erslev, 2004] and elsewhere [e.g. Hennings et al., 2000]. A series of parallel or subparallel cross sections are created, and a smooth surface is interpreted, which honors the location of the selected stratigraphic contact on each cross section. Cross sections are constrained by scattered strike and dip measurements and the locations of exposed stratigraphic contacts. Subtle features of the fold between cross sections are not captured by these methods. Fold characteristics along cross sections

2 may be omitted or misinterpreted if data points are not sufficiently dense or if the underlying assumptions of fold kinematics on which the cross section is based are incorrect [Woodward et al., 1989]. While reproducibility is a key component of rigorous scientific investigation [e.g. Laine et al., 2007; Peng et al., 2006], most methods of cross section construction lead to non-unique and often non-reproducible results. Early methods for constructing cross-sections of folded strata, such as the Busk method [Busk, 1929], in which folded surfaces are interpreted as concentric circular arcs, rely upon simple geometric relations to interpolate fold surfaces between sparse data points. In many subsequent methods for cross section construction, the interpolation of fold geometry between data points is dependent upon fault geometry [e.g. Dahlstrom, 1969; Suppe, 1983], which may be poorly constrained. This is particularly problematic at SMA, where there are no well data and the seismic data provide poor resolution of the underlying faults. As a consequence, multiple viable but imperfect fault interpretations exist [Forster et al., 1996; Hennier and Spang, 1983; Stanton and Erslev, 2004] (see Geologic Setting). Additionally, balanced cross sections assume plane-strain deformation. While this may be an appropriate assumption for long, nearly cylindrical folds, SMA is a more complex, non-cylindrical fold [Bellahsen et al., 2006a; Forster et al., 1996; Hennier and Spang, 1983; Savage and Cooke, 2004; Stanton and Erslev, 2004]. The generation of 3D surface models from subparallel 2D cross sections results in a directional bias, so surfaces constructed in this manner may fail to capture the full 3D character of fold geometry. 3D geometric techniques such as the equivalent dip domain method [Carrera et al., 2009] have been suggested to overcome some of the limitations of sequential balanced cross- sections; however, such methods are adequate only for relatively simple folds. Precise, high-resolution geospatial data provide a new means to constrain 3D fold geometry. For example, Pearce et al. [2006] and Allwardt et al. [2007] use differential GPS to constrain the geometry of small folds and small patches of larger folds. Mynatt et al. [2007b] and Kaven et al. [2009] use airborne laser swath mapping (ALSM), also known as airborne LiDAR, and numerical interpolation to

3 constrain fold geometry at Raplee Ridge, Utah. In folds such as Raplee Ridge or SMA erosion results in the exposure of bedding surfaces at the top of weathering-resistant strata. These outcrops provide scattered data on the location and orientation of the folded surfaces. By mapping such outcrops on the digital elevation model produced from the ALSM survey, locally dense and precise geospatial coordinates defining the location and orientation of hundreds of bedding surface outcrops across a fold may be obtained [Mynatt et al., 2007b]. Challenges arise, however, because not all outcrop exposures correspond to the same stratigraphic surface. Kaven et al. [2009] addresses this issue at Raplee Ridge by simultaneously interpolating multiple surfaces, while applying inequality constraints to intervening layer thicknesses to maintain a physically realistic model without surface interpenetration or excessive thinning or thickening. This method fails at SMA because outcrops of any given surface generally occur only in relatively narrow bands along each limb of the fold. No one unit is exposed with adequate spatial continuity across the fold. Surface models of SMA interpolated by the method of Kaven et al. [2009] exhibit large error at constraint points and do not accurately reflect fold geometry, even to first order. To overcome this limitation, we present a new method for constructing a fold surface model from ALSM data, in conjunction with geologic mapping and numerical interpolation, based on projection of data from 14 different outcrop forming surfaces, to generate a single surface model. The method for surface model construction presented here eliminates dependency on assumptions of the underlying fault structure, attempts to minimize model dependency upon fold kinematics, and takes advantage of the most precise and spatially extensive constraint on fold geometry available from surface outcrops. Although some data manipulation, in the form of omitting selected problematic data points in a few locations, introduces a degree of subjectivity to our models, the numerical interpolation method for generating a continuous surface from raw data offers the distinct benefit of being entirely reproducible.

Figure 1.1: Geologic and tectonic maps of Sheep Mountain anticline after Bellahsen et al. [2006a]. a) Regional map. The hexagonal polygon indicates the location of SMA. b) Geologic map of SMA, after Rioux [1994]. The bold dashed line represents the extent of the observation grid over which the interpolated surface is calculated and thin dashed line represents the location of the down-plunge view (Figure 1.8).

Figure 1.2: a) Aerial photograph of SMA study area, taken from the northwest. Major lithologic formations are labeled in white. Black letters show the locations of outcrop photographs b-e. Image adapted from Fiore [2006]. b) Aerial photograph of a large Tensleep Sandstone bedding surface outcrop (orange, Figure 1.3), in the backlimb. The outcrop is the large surface immediately below the crest. c) Gypsum Springs bedding surface outcrops such as this are found along the ridges in this formation. This is a cyan outcrop located in the backlimb. d) Typical Cloverly (pink) bedding surface outcrop found along the ridge in this formation. Due to colluvial cover, only a small swath (1-3m wide) is mappable on many outcrops in this and other Mesozoic units. e) Large hogback-forming Phosphoria (red) bedding surface outcrop in the steeply dipping forelimb. f) Phosphoria (green) bedding surface outcrop in the backlimb. The lighter colored triangle in the foreground is the mapped surface.

Figure 1.3: Stratigraphic column representing the units exposed in the study area. The colored lines represent mapped bedding surfaces (Figure 1.4). The bold black line at the base of the Sundance formation represents the surface modeled by Forster et al. [1996] and in this study. Distances on the right side represent the distance each mapped unit is projected to the base Sundance surface, assuming constant thickness everywhere on the fold.

Geologic setting Abundant outcrops at SMA have motivated numerous structural studies focusing on fault-related folding [Forster et al., 1996; Hennier and Spang, 1983; Sanz et al., 2007; Savage and Cooke, 2004; Stanton and Erslev, 2004] and fold-related fracturing [Allwardt et al., 2007; Bellahsen et al., 2006a; Bellahsen et al., 2006b; Harris et al., 1960; Johnson et al., 1965; Sanz et al., 2008; Savage et al., In review]. SMA is a northwest-southeast trending, doubly plunging, asymmetric, basement-cored fold (Figures 1.1 & 1.2). The steeply dipping forelimb faces northeast and dips generally exceed 60 degrees and locally approach 90 degrees. The more gently dipping backlimb faces southwest and exhibits dips generally ranging from 10-40 degrees, but locally exceeding 60 degrees, particularly in the vicinity of a large parasitic fold (called the “Thumb”) that branches to the south from the main fold (Figure 1.1). The northwestern extent of the fold is characterized by a very narrow hinge that becomes broader to the southeast. Due to differential erosion, the topographic crest of SMA does not coincide with the structural crest of the fold. A cross-section of the fold in the Madison Limestone is exposed in Sheep Canyon, where the Bighorn River cuts through the anticline, nearly perpendicular to the axial surface trace. This study focuses only on that part of SMA northwest of Sheep Canyon (Figure 1.1) because outcrop quality is poor to the southeast. Sheep Mountain anticline is located on the eastern flank of the Bighorn Basin (Figure 1.1a), north-central Wyoming, north of the town of Greybull. The Bighorn Basin is a structural low, with the Absaroka Range to the west, the Bighorn Mountains to the east, and the Owl Creek Mountains to the south. During Paleozoic and Mesozoic times, approximately 3km of sediments were deposited in the Bighorn Basin and surrounding region [Ladd, 1979; L E Thomas, 1965]. Rocks exposed in the study area range from Mississippian through Cretaceous in age, and span more than 1km of the stratigraphic section (Figure 1.3). During the late Cretaceous and Paleocene, the Laramide orogeny resulted in northeast-southwest compression throughout the Rocky Mountain region [Bird, 1998; 2002; Dickinson and Snyder, 1978; Engebretson et al., 1985]. SMA and neighboring

8 folds formed due to slip on underlying thrust faults during Laramide shortening, but the fault associated with SMA is not exposed. While these faults may be interpreted from seismic data [Gries, 1983; Stone, 1993], poor image quality prevents resolution of detailed fault structure [Fiore, 2006; Stanton and Erslev, 2004]. Published interpretations agree that the fault primarily responsible for the development of SMA dips to the southwest and that the fault system involves basement rock, but the size of this fault and structural context in relation to nearby faults are debated. Earlier work [Forster et al., 1996; Hennier and Spang, 1983] suggests that the Sheep Mountain thrust is a third order structure, a back-thrust off the northeast-dipping Rio thrust, which itself is a back-thrust of the southwest-dipping, crustal scale Bighorn Mountains Eastern thrust. However, geomechanical models by Sanz et al. [2007] suggest that a small, third-order fault could not produce a structure as large as SMA. Stanton and Erslev [2004] suggest that the Sheep Mountain thrust is a deeply rooted structure that pre-dates and has been cross-cut by the younger Rio thrust; however, their model, like previous models, cannot be fully restored, and the geometry of the fault system underlying SMA remains obscure.

Airborne laser swath mapping (ALSM) data ALSM, also known as airborne LiDAR (Light Detection And Ranging), utilizes a laser rangefinder mounted on a fixed-wing aircraft or helicopter, which emits pulses and records reflection returns at a high temporal frequency while scanning laterally and compiling a dense swath of data points representing the topography of Earth’s surface beneath the flight path. When the laser rangefinder data are post- processed in conjunction with recordings from an onboard differential global positioning system (DGPS) and inertial measurement unit (IMU), reflector (ground, foliage, etc.) locations may be determined with decimeter precision. The principals of and standard processing procedures for ALSM data are described by Shrestha et al. [1999]. An ALSM survey at SMA, covering approximately 86 km2, was flown by the National Center for Airborne Laser Mapping (NCALM) in July 2007. From the raw point cloud of processed laser returns, two digital elevation models with 1m resolution

Figure 1.4: a) Hillshade image from ALSM data covering entire study area. Color shaded polygons represent the extent of mapped bedding surface outcrops and red dots represent locations of strike and dip measurements. A high-resolution version of this map is available as an electronic supplement. The green rectangle represents the extent of the observation grid on which the new interpolated surface is calculated (Figure 1.11). b) A zoomed in view of the ALSM and field data. The extent is indicated by the red box in (a).

10 were assembled. One is filtered to remove vegetation and buildings, and the other is unfiltered. The unfiltered digital elevation model (DEM, hillshade image presented in Figure 1.4) was used for this study because vegetation is minimal and there are very few buildings in the study area. Technical specifications for data acquisition and processing at SMA are summarized in the Appendix. However, it is important to emphasize that the uncertainty of ALSM data is relatively inconsequential to our models. Projection and interpolation, rather than the ALSM data itself, are the primary sources of uncertainty in the surface models presented.

Field methods

Geologic mapping Bedding surface outcrops corresponding to 14 different weathering resistant stratigraphic surfaces were mapped on the ALSM data in order to obtain their precise 3D geospatial locations (Figures 1.3 & 1.4). The density of ALSM data provides more than three non-collinear points on many bedding surfaces, which theoretically provides a means for calculating local surface orientation. Most bedding surfaces were identified for mapping because they are continuous and provide high quality outcrops in most areas. Several surfaces are less continuous, but were mapped because they are in a similar stratigraphic position to other mapped surfaces, and are exposed in areas where the adjacent surface does not provide good outcrops. Photographs of a selection of mapped bedding surface outcrops are presented in Figure 1.2. Mapping was performed in the manner of Mynatt et al. [2007b], by printing hillshade images generated from the 1m DEM on a 42 inch plotter at a scale of roughly 4000:1, on which individual outcrops more than 2-3m across may be identified. Outcrops of each bedding contact surface were mapped in the field and later digitized as polygons (Figure 1.4) (high-resolution image available as an online electronic supplement). Digitized polygons representing outcrop exposure enabled extraction of 514,342 DEM points, representing outcrops of the fourteen exposed bedding surfaces. Because each of these surfaces is not unique to a stratigraphic unit

(e.g. 4 surfaces were mapped in the Phosphoria Formation, 3 in the Gypsum Springs, and 3 in the Cloverly Formation), each surface is referenced by color, as in figures 1.3 & 1.4. Approximately half of the points extracted from the DEM (258,769) represent the top of the Tensleep Sandstone (mapped orange), while the fewest points (167) represent the surface in the Gypsum Springs Formation mapped in gold. Strike and dip measurements taken at 282 sites across the study area (Figure 1.4) prove critical for projection and as slope constraints for interpolation because, while surface orientation may theoretically be derived from the DEM, errors are large when surface orientation is calculated from small outcrops. Although distinct measurements were not taken at every outcrop due to time constraints, difficult access, and inability to identify an adequate surface on some outcrops, they were taken at sufficiently regular intervals that a surface orientation might be accurately prescribed by interpolation and extrapolation to nearly every mapped outcrop. It is important to note that the mapping of outcrops and measurement of bed orientation were performed at different scales. Mapped surfaces are selected because they provide extensive outcrops at a scale of meters or more. While these surfaces accurately constrain fold geometry, many are weathered such that it is difficult to obtain accurate strike and dip measurements with a Brunton compass. For this reason, many strike and dip measurements are obtained from surfaces that are near to the mapped surfaces, and therefore have similar orientation, but they do not necessarily represent exactly the same surface.

Stratigraphy Stratigraphy at SMA has been described in detail by Ladd [1979] and Rioux [1994], and much of the following description is adapted from these detailed studies. The oldest sedimentary units exposed at SMA are found in the core of the fold, near the hinge, and form the topographic high that is Sheep Mountain. These Paleozoic units comprise, chronologically, the Madison, Amsden, Tensleep, and Phosphoria formations. The purple surface (surface colors of mapped units are identified in Figure 1.3) is the oldest bed mapped, and corresponds to the top of the massive Madison Limestone. Outcrops are sparse, but generally good quality, occurring along

12 the hinge of the fold. However, mapping was difficult because most outcrops are not accessible, making it difficult to identify whether exposed surfaces are indeed the top of the Madison Limestone, or if upper layers of the continuous and uniform limestone have been eroded. Also, because outcrops are inaccessible, strike and dip measurements were not obtained on most of these surfaces. Above the Madison Limestone is the Amsden Formation, followed by the Tensleep Sandstone. Due to a lack of continuous and resistant units forming quality bedding surface outcrops, no units were mapped in the Amsden Formation. The orange mapped unit corresponds to the top of the Tensleep Sandstone. The bed mapped is 2-3m thick and heavily cross- bedded. Cross bedding makes strike and dip measurements difficult to obtain on many outcrops; however, this unit forms the most extensive outcrops in the study area (e.g. Figure 1.2b). Outcrops are numerous in the backlimb, generally near the topographic crest of the fold, and are distributed more sparsely in the forelimb. Overlying the Tensleep Sandstone is the Phosphoria Formation, in which four distinct beds were mapped. The lower Phosphoria Formation consists largely of shales and siltstones. The yellow and blue mapped surfaces represent the tops of thin, carbonate- rich, resistant units within the lower Phosphoria Formation. The yellow mapped unit is underlain by a distinct red shale. Most yellow and blue mapped outcrops are relatively small, but of high quality. Good strike and dip measurements were taken on these beds. Outcrops are widespread along the forelimb and backlimb of the fold. The upper Phosphoria Formation consists of a massive, carbonate-rich quartz arenite [Bellahsen et al., 2006a], the top of which is mapped red. This unit is overlain by shale several meters thick, and capped by a thin limestone bed, mapped green. The red mapped unit forms massive flatirons in the forelimb (e.g. Figure 1.2e), and is exposed extensively along the backlimb. Irregular weathering patterns make measurement of strike and dip difficult. The green mapped unit forms smooth flatirons in the backlimb (e.g. Figure 1.2f), on which good strike and dip measurements were taken, but it forms few outcrops in the forelimb. The Phosphoria Formation is the youngest unit in the Paleozoic section, and the youngest unit on the prominent topographic expression of the fold. Although the

13 fold has less topographic relief in the Mesozoic section, these units were folded with the underlying strata and resistant beds crop out in distinctive ridges. Bedding surfaces that provide additional constraint on fold geometry often are exposed on the down-dip side of these ridges. The Phosphoria Formation is overlain by the massive Chugwater Shale, which forms a distinct red band around the perimeter of Sheep Mountain’s topographic high (Figure 1.2a). The Chugwater Shale is easily weathered, and contains no resistant, outcrop-forming beds. It is overlain by the Gypsum Springs Formation, in which several resistant units are mapped. The surface mapped in cyan is the oldest ridge-forming unit in the Gypsum Springs Formation. It consists of a relatively thin limestone (Figure 1.2c), overlying several meters of shale and the distinct, thick layer of gypsum at the base of the Gypsum Springs Formation [Rioux, 1994]. This unit forms a consistent, identifiable ridge in the backlimb and most of the forelimb of the fold, and is exposed again beyond the syncline northeast of the forelimb. The beds mapped as gold and peach are discontinuous limestone beds, between intervals of shale and gypsum. The gold surface is exposed only in a small area at the northern extent of the forelimb. It was mapped, despite its limited extent, because the cyan surface is poorly exposed in this area. The peach surface is continuous on much of the fold, but ends rather abruptly north of 4946600m northing in the backlimb and 4947100m northing in the forelimb (Figure 1.4a). It too is exposed northeast of the forelimb syncline. The Gypsum Springs Formation is the youngest resistant, outcrop-forming unit exposed in the forelimb study area. Units mapped higher in the stratigraphic column are exposed only in the backlimb. The Gypsum Springs Formation is overlain by the Sundance Formation; however, a precise boundary between these units is difficult to identify. The Sundance Formation consists mostly of shale, with a cross-bedded sandstone at the top. This sandstone forms a discontinuous ridge, which is mapped as brown in parts of the backlimb. Mapping is limited because cross-bedding and inconsistent weathering make it difficult to identify a unique surface within the sandstone. Cross bedding makes accurate strike and dip measurement nearly impossible. Above the Sundance are the Morrison and Cloverly formations. A precise boundary between these units is

14 difficult to distinguish in the vicinity of SMA, as the Morrison and lower Cloverly formations consist predominantly of soft siltstones and shales with a few inter-bedded sandstones. No surfaces were mapped in the Morrison Formation, because outcrops of a unique, continuous bedding surface are difficult to identify; however, several units are mapped in the upper Cloverly Formation. These ridge-forming units, mapped as pink, raspberry, and magenta, consist of resistant sandstone layers inter-bedded with shale (e.g. Figure 1.2d). The sandstone surfaces provide good strike and dip measurements of bedding. Pink and raspberry beds are mapped consistently south of 4946400m northing, but, similar to the peach unit in the Gypsum Springs Formation, abruptly end to the north (Figure 1.4a). Where the pink and raspberry surfaces are difficult to identify, the younger magenta bed is the dominant ridge-forming layer. Above the Cloverly Formation are the Thermopolis and Mowry shales, in which few resistant units are found, followed by the Frontier Formation. The unit, mapped lime green, is a resistant, ridge-forming sandstone at the base of the Frontier Formation. It is found immediately below a thick, distinct bentonite bed, which is mined locally. In areas of active mining, this surface provides excellent sites for strike and dip measurement; however, good, unweathered exposures elsewhere are limited. Younger sandstone units within the Frontier Formation provide better orientation measurements, but are not mapped because of discontinuous exposure and difficulty identifying a unique surface. The sandstone at the base of the Frontier Formation is the youngest ridge-forming unit exposed in the extent of the ALSM survey.

Stratigraphic thickness Projection of data points from disparate stratigraphic positions to the position of a single folded surface requires accurate knowledge of stratigraphic thickness between each of the mapped units. Several such measurements were obtained in the field using a Jacob’s staff; however, these were used only for verification of a larger number of measurements obtained directly from the ALSM data. Bed-perpendicular stratigraphic thickness between surfaces a and b (surface a is stratigraphically lower than surface b) may be estimated (Figure 1.5):

Figure 1.5: Schematic illustration of the stratigraphic thickness calculation (eq. 1). a) Map view. Polygons indicate two outcrops. Bedding orientation is indicated to the upper left of each outcrop. Points Xa and Xb are located on the older and younger outcrops, respectively, along a line parallel to the average dip direction. b) Cross- section view illustrating the various parameters of eq. 1. The bold line represents topography along the dashed line in (a).

 zab  Tab =  xab − sinϕ (1)  tanϕ  from two points, Xa and Xb that are located on outcrops of mapped horizons a and b and fall on a line parallel to the local dip direction. Tab is the thickness, xab is the horizontal component of the distance between points Xa and Xb, zab = za-zb is the difference in elevation between the two points, and φ is the dip, which is taken as the mean of φa and φb. By assuming a constant (i.e. mean) dip, equation 1 technically applies only to unfolded rocks. Thus, thickness measurements were made in locations where folding between points Xa and Xb is minimal and error calculations account for potential folding. Thickness measurements obtained from the ALSM data are in agreement with field measurements using a Jacob’s staff. However, thickness measurements show significant variability (standard deviation is typically about 10% of the mean thickness for a given stratigraphic package) and error bars are relatively large. Plots of thickness for several stratigraphic intervals taken along the fold trend suggest that this variability is not particularly systematic (Figures 1.6 & 1.7). Although significant heterogeneity among measurements is observed, none of the measured intervals suggest a strong correlation between thickness and local dip (Figures 1.6b & 1.7b). It is particularly interesting that there is no trend toward thinning in regions of greatest dip because such an observation might suggest non- parallel folding. However, this conclusion is not well constrained in softer Mesozoic units where we would most likely expect a non-parallel fold profile because none of the sampled intervals provide exposures covering a full range of dip values. Thickness measurements between the cyan and red surfaces obtained in the forelimb have a mean value of 246m, and thickness measurements obtained in the backlimb between the cyan and green surfaces have a mean value of 255m. The green surface is uniformly about 8m above the red surface everywhere on the fold, suggesting that in fact, the Chugwater Shale, which comprises most of the interval between the green and cyan surfaces, may be thinned in the steeply dipping forelimb by approximately 17m (7%).

Figure 1.6: Thickness measurements obtained from the ALSM data for several intervals in the Paleozoic section. Open markers represent measurements taken in the backlimb and solid markers represent forelimb measurements. Horizontal dashed lines represent the mean thickness of each interval, corresponding to the symbol on the right. a) Thickness vs. UTM northing coordinate of the center point of the line along which thickness is measured. b) Thickness vs. local dip.

Figure 1.7: Thickness measurements obtained from the ALSM data for several stratigraphic intervals in the Mesozoic section. Open markers represent measurements taken in the backlimb and solid markers represent forelimb measurements. Horizontal dashed lines represent the mean thickness of each interval, corresponding to the symbol on the right. a) Thickness vs. UTM northing coordinate of the center point of the line along which thickness is measured. b) Thickness vs. local dip.

This contradicts the parallel folding hypothesis, and is accounted for as described in the Projection section. Ramsay [1967] classifies folded strata into five categories based on relative curvature of the inner and outer arcs and the relative orientation of dip isogons along the folded layer between the inner and outer arcs. Parallel (class 1b) and similar (class 2) folds are two ideal fold types in this scheme, which allow for simple implementation of a projection algorithm. In parallel folds, dip isogons are perpendicular to bedding and bed-perpendicular thickness of folded units is uniform. In similar folds, dip isogons are everywhere parallel to the axial surface and bed thickness parallel to the dip isogons is uniform. Intermediate (class 1c) and extreme (classes 1a and 3) fold types are characterized by more complicated relationships between bed thickness and dip isogon orientation, and therefore do not lend themselves to straightforward projection between folded surfaces. The fold at SMA is most nearly idealized as a parallel fold. Given an ideal similar fold, bed perpendicular thickness scales with undeformed thickness by cos(φ) [Ramsay, 1967], in which case thinning of 50% would be expected where the forelimb dip is 60⁰ and thinning in excess of 74% should be observed where dip locally exceeds 75⁰. Expected thinning in the more gently dipping backlimb would be generally less than 25%. Thus, we would expect to observe forelimb thinning of at least 30%, and locally greater than 60%, relative to thicknesses observed in the backlimb. Because units above the Gypsum Springs are not exposed in the forelimb portion of the study area, it is unknown if other shale units such as the Morrison, Cloverly, Thermopolis, or Mowry, may have been thinned in this steeply dipping region. UTM northing coordinates are used as a proxy for position along the fold to investigate potential stratigraphic thickness gradients independent of folding and local dip. Thickness measurements were taken along exposures that roughly parallel the fold trend. Therefore, increasing northing measurements represent a progression in a northwestward direction. No significant trend is observed in the Paleozoic section (Figure 1.6a), but in the Mesozoic section (Figure 1.7a) subtle thinning to the southeast is suggested by measurements in the backlimb. However, thickness

19 heterogeneity is of a comparable magnitude to the error bars, so it is not possible to ascertain if the trend is real. Mean thickness values for each stratigraphic interval fall within the error range of most measurements. Therefore, we treat bed thickness as everywhere uniform, with the exception of accounting for subtle thinning of the Chugwater Shale in the forelimb. Error bars account for the range of observed dip values in the vicinity of each measurement and for error in zab, which results from ALSM imprecision and, more important, outcrop weathering as observed in the field. Because the component of thickness calculation error resulting from dip measurement error scales with bed thickness, and the contribution from zab measurement error does not (eq. 4), dip provides a relatively greater contribution to error in the thicker units of the Mesozoic section and zab is the dominant contributor in the Paleozoic section. The largest errors are observed in the Mesozoic section where dips at up-section and down-section outcrops differ significantly.

Numerical methods

Projection All data points extracted from the ALSM data that represent bedding surface outcrops at different stratigraphic positions, as well as strike and dip measurement locations taken at all stratigraphic levels, are projected to the stratigraphic position of a single bedding surface for interpolation. We project points normal to bedding, based on the assumption that SMA is a flexural-slip style, parallel fold [e.g. Ramsay, 1967]. Previous studies [Forster et al., 1996; Hennier and Spang, 1983] suggest that SMA is a parallel fold, citing evidence such as relatively uniform bed thickness, even on the steeply dipping limbs. The lack of correlation between bed thickness variation and local bed dip (figures 1.6b & 1.7b) is consistent with this conclusion. A down-plunge view of the fold to the northwest also illustrates the parallel form (Figure 1.8). However, it is impossible to base the down-plunge view on data from a swath spanning more than 1km along the fold trend, due to the non-cylindrical nature of the fold, emphasizing the need to consider the fold shape in a fully 3D manner.

Figure 1.8: Down-plunge view (6⁰ plunge at N34⁰W) of the fold on cross-section A- A’ (Figure 1.1). Black curves represent the surfaces mapped in orange (top Tensleep), red (upper Phosphoria) and cyan (Gypsum Springs). Black points represent mapped outcrops of these three surfaces within 500m to either side of A-A’ and grey points represent mapped outcrops of other surfaces within the same swath.

Figure 1.9: Triangulated surfaces are generated from ALSM points representing each outcrop to calculate local surface orientation. Light gray triangles have high aspect ratio and are omitted from calculation. White vectors represent the normal to each triangle, the mean of which is taken to represent local bedding orientation. Inset: For each triangle, gradient magnitude (∇z) and orientation (θdip, equivalent to dip direction) are calculated. Dip (φ) is the arctangent of the gradient. Because the normal vector (n) and the dip vector (d) are orthogonal and in the same vertical plane, the map orientation of n is equivalent to θdip and its angle above horizontal is complementary to φ.

Subtle but apparent thinning of the Chugwater Shale in the forelimb calls into question the parallel fold assumption, but is accounted for by reducing the projection distances through the Chugwater in the forelimb by 17m. Projection of ALSM points in the Cloverly formation at the northwest tip of the fold present an additional challenge to the parallel fold assumption. Down-section projection paths of points in the forelimb and backlimb cross, which is geometrically inadmissible. For this reason, the observation grid area on which the interpolated surface is calculated (figures 1.1b & 1.4a) is restricted in this area. Projection distances are calculated from mean stratigraphic thickness measurements presented in the previous section, and supplemented by measurements of Rioux [1994]. Constant projection distances to the base Sundance Formation are presented on the right side of Figure 1.3. Projection orientation, parallel to the local bedding-normal vector, may be determined from strike and dip measurements or directly from the ALSM data (Figure 1.9). Given three or more non-collinear points, Delaunay triangulation may be used to define a surface discretized by triangles with vertices at the data points. We calculate the orientation of the surface from ALSM data as the mean gradient (magnitude and orientation) of all triangles for a given outcrop. Because the DEM consists of a 1m x 1m grid, triangles on the interior of any outcrop are, in map view, isosceles right triangles. In calculating the mean gradient, we disregard triangles with high aspect ratios, which are found only on the periphery of outcrops and are subject to error because peripheral points may easily be misidentified and therefore not represent the bedding surface. The difference between bed orientation as measured with a Brunton compass in the field, and as calculated from the ALSM data, is plotted versus outcrop size in Figure 1.10. We see that for both strike and dip, there are large differences between calculated and measured orientations for small outcrops. For larger outcrops, differences are generally less, but still significant. Greater differences for smaller outcrops occur because bedding orientation cannot be accurately determined from the limited ALSM data. In the case of large outcrops, very accurate measurements,

Figure 1.10: Absolute value of the difference between (a) dip and (b) strike measured in the field with a compass and calculated from ALSM data points extracted from outcrops versus outcrop area.

23 representative of the whole surface should be obtained from the ALSM data. However, field measurements may be influenced by cross-bedding, weathering effects and measurement errors that do not reflect folding. Also, short wavelength parasitic folding, which is relatively uncommon, though not absent, at SMA, may result in local perturbations to strike and dip measurements which are real but inappropriate for projection purposes. We conclude that bedding orientations obtained from the ALSM data are superior for large outcrops, and those obtained using a Brunton compass are superior for smaller outcrops. Outcrops smaller than 1000m2 are projected according to field measurements and outcrops with greater area are projected according to the ALSM data. A few small outcrops lacking strike and dip measurements were ignored.

Numerical interpolation and regularization We implement a two-dimensional minimum curvature (biharmonic) spline (MCS) interpolation method [Briggs, 1974; Sandwell, 1987; Wessel and Bercovici, 1998] constrained by ALSM data points and strike and dip measurements. The MCS surface in effect minimizes the strain energy of a thin elastic plate, deflected to honor constraints on either vertical position or local slope, assuming gradients of the surface are small [Timoshenko and Woinowsky-Krieger, 1959]. We acknowledge that multilayer folding to nearly vertical limb dips is a more complex physical process than the small deflection of a single elastic layer. However, we use this approach because it provides relatively smooth interpolation with modest curvatures, and we do not pursue any physical interpretations. One apparent limitation of the MCS approach is for chevron folds, in which deformation tends to concentrate within narrow zones of high curvature, which separate relatively planar dip domains [e.g. Carrera et al., 2009; Suppe, 1983]. In regions of dense data coverage, an MCS surface will capture such zones of greater and lesser curvature in order to honor constraints; however, where data are limited, the MCS method, by definition, creates a broad hinge region in which curvature is distributed to the greatest extent possible. Where such gaps exist in data coverage, it is generally difficult to determine the location or breadth of such zones of localized curvature, or even to determine if such zones exist; therefore, we suggest the MCS

24 method is appropriate, particularly given the dense ALSM data coverage available across much of the fold. Interpolation by splines in tension [Mitasova and Hofierka, 1993; Smith and Wessel, 1990; Wessel and Bercovici, 1998] is an extension of biharmonic splines, and has been suggested as a method to reduce the high frequency oscillations that may plague biharmonic splines. Splines in tension add mathematical and computational complexity, and for the SMA model, made a significant but not necessarily positive difference in interpolated surfaces only in regions of poor data constraint. Therefore, our approach uses simpler biharmonic splines. The MCS interpolation is implemented using a Green’s function approach, in which the continuous interpolated surface is defined by linear combination of two- dimensional Green’s functions for the biharmonic equation, centered at each constraint point, and weighted such that the interpolated surface exactly fits the data [Sandwell, 1987; Wessel and Bercovici, 1998]. Given n data points, n corresponding weights are determined from the matrix equation G ⋅ c = w (2) where G is the n x n Green’s matrix, c is an n x 1 vector of weights, and w is an n x 1 vector of data constraints. The interpolated surface is calculated by

z = G1 ⋅ c (3) where G1 is an m x n Green’s matrix relating m observation (grid) points to Green’s functions centered at n data points, c is the vector of weights calculated from eq. 2, and z are the interpolated values at discrete observation points. Although finite difference implementations of MCS interpolation provide improved stability and increased computational efficiency relative to this method (G is a dense matrix and often close to singular), the Green’s function implementation offers the ability to constrain surface gradients, in addition to standard spatial constraint from ALSM data points [Sandwell, 1987; Wessel and Bercovici, 1998]. This is useful for geologic surface interpolation because it admits the use of strike and dip measurements to provide explicit constraint on surface geometry. Additionally, the Green’s function approach enables the implementation of a variety of regularization

(smoothing) techniques, by which it is possible to relax the requirement that the interpolated surface exactly honor all data constraints. While ALSM data are extremely precise, a substantial amount of noise is introduced to MCS constraints by the projection process. Heterogeneity of bedding thickness, non-parallel folding, and projection orientation error result in projection vector error, which increases roughly linearly with projection distance. Regularization is critical to generate a smooth interpolated surface from noisy data. Several regularization methods were investigated to appropriately smooth the SMA surface model and avoid unnatural surface oscillations in such a manner that the detail and precision provided by ALSM data are maintained. Candidate methods included the truncated singular value decomposition (SVD) [e.g. Wessel and Bercovici, 1998], least squares approximations in which the linearly combined Green’s functions are fewer in number than the data constraints [e.g. Billings et al., 2002; Sandwell, 1987], least squares approximations with weighted data [e.g. Aster et al., 2005], and generalized cross validation [e.g. Billings et al., 2002]. Greatest success was attained using the truncated SVD. By this method, G, from eq. 2, may be represented as G = U ⋅ S ⋅V t (4) where U and V are n x n orthonormal matrices, and S is a diagonal matrix of n singular values, ordered from greatest to least and each of which is positive. Eq. 2 may be solved by c = G −1 w ≈ c* = G *w (5) where

* −1 t G = V p S p U p (6)

Up and Vp are n x m matrices consisting of the first m columns of U and V, respectively, and Sp is an m x m diagonal matrix populated by the first (largest) m singular values of S. G* is a singular Moore-Penrose pseudoinverse [Moore, 1920; Penrose, 1955]. In G*, the basis vectors corresponding to the smallest singular values of G are neglected. These basis vectors, which contribute minimally to G, otherwise would become the dominant components of G-1. By discarding the smallest singular

26 values, components of the interpolated surface characterized simultaneously by large amplitude and high spatial frequency are smoothed at the expense of exactly honoring data constraints. Discrete values for the vertical coordinate of the interpolated surface may thus be calculated at any point in the horizontal plane by

* z = G1 ⋅ c (7) We calculate surface coordinates on a rectangular grid with 50m spacing, spanning the study area and oriented sub-parallel to the fold axis.

Data considerations The study area, defined by the observation grid on which the interpolated surface is calculated, is presented in figures 1.1b and 1.4a. Its extent is limited to the area in which ALSM data representing bedding surface outcrops are widely available to constrain fold geometry. The breadth of the area includes most of the fold, including backlimb, hinge, forelimb, and the synclinal hinge northeast of the forelimb. The length is restricted to the southeast by the Bighorn River, where it cuts a canyon through SMA. Southeast of this point, only scattered outcrops, insufficiently distributed to accurately constrain fold geometry, are available. To the northwest, the study area is terminated because problems arise apparently due to non-cylindrical fold geometry. Additionally, outcrops to the northwest are poorly distributed, limited almost entirely to the ridge forming Frontier formation. This single arcuate band of data points is insufficient to accurately constrain fold geometry. Although the study area is limited to the northwest and southeast, data points beyond the limited area of the observation grid are used for interpolation constraint. While these points are not sufficient to accurately constrain fold geometry in their immediate vicinity and their influence on surface geometry within the grid area is relatively small, we suggest that including additional data points outside the primary study area is more likely to improve than to detract from the final surface model. For example, such points can help to prevent the surface from plunging infinitely toward the periphery of the study area as minimum curvature surfaces may do in the absence of data constraint beyond the observation grid.

Limitations of the 2D MCS method and instabilities of the numerical implementation, as well as limitations of computational power and numerical precision in MATLAB, prevent executing the interpolation with all 514,342 ALSM points representing bedding surface outcrops and 282 strike and dip measurements. As discussed in the following paragraphs, it was necessary to discard selected, problematic data points from the constraint set and to significantly decimate the dense ALSM data in order to interpolate a realistic model of fold geometry at SMA. We acknowledge that data manipulation is not ideal because it introduces human subjectivity and limits model reproducibility. Therefore, data points are omitted only when a specific, identifiable point or subset of points may be attributed to model behavior that is clearly and significantly unrealistic based on the known geology. Because the biharmonic spline interpolation algorithm is inherently 2D, calculations are performed on a fixed, horizontal grid, and elevation is treated as the only unknown. Therefore, in steeply dipping regions, such as the forelimb of SMA, small horizontal error of constraint points correlates to large vertical error. Small horizontal errors are prevalent among the projected data in this region, particularly for data points projected more than 200m across the Chugwater shale, because a large component of the projection vector for steeply dipping surfaces is in the horizontal plane. While the interpolated surface may in fact pass very close to the given data point in 3D space, because the interpolation scheme minimizes error in the vertical direction, rather than normal to the nearest surface point, excessive smoothing was required to prevent unrealistic oscillations. These oscillations are characterized by a dominant fold-parallel waviness, in the forelimb of the interpolated surface. This fold parallel waviness was unsubstantiated in raw data, including the ALSM survey, and therefore deemed an artifact of projection and numerical methods. The regularization necessary to overcome these artifacts caused degradation of other aspects of the surface model that was deemed unacceptable. This artificial, fold-parallel waviness is prevented in the final surface models by omitting forelimb data points mapped in the Gypsum Springs, above the Chugwater Shale. In the top Tensleep Sandstone model, these data points contribute the greatest error among forelimb data points because they

28 have been projected the greatest distance, across the Chugwater shale. In the Base Sundance model, it is the points representing Paleozoic surface outcrops, projected upward, through the Chugwater, that contribute error. Gypsum Springs data points are discarded despite this, because doing so addresses the primary issue of unrealistic waviness and because of their paucity relative to data points from older Paleozoic surfaces. As discussed previously, additional data points from the Cloverly formation in the nose of the fold beyond the study area were discarded due to issues with the parallel folding hypothesis. In regions of poor ALSM data constraint, slope constraints were liable to cause large, broad, geologically unrealistic vertical perturbations in the surface model, apparently due to limitations of the MCS interpolation technique and numerical instabilities inherent to the Green’s function implementation. About thirty such points were identified and discarded from the constraint set. Many of these points were located to the northwest, beyond the extent of the observation grid, where the fold appears to become less cylindrical and where ALSM data constraint was limited to sparse points from the Frontier Formation. About six other discarded points were scattered around the periphery of the study area. No slope constraints were discarded in regions with broad ALSM constraint. These few slope constraints and the Gypsum Springs Formation ALSM data points in the forelimb were the only constraint points omitted within the extent of the study area. Data decimation is necessary to achieve a numerically feasible and stable interpolation. Decimation is performed after projection because constraint points too near each other may result in numerical instabilities in the MCS interpolation [Sandwell, 1987; Wessel and Bercovici, 1998]. The decimation algorithm examines each data point sequentially in the order input, calculates its distance from all subsequent points input, and deletes subsequent points whose map (XY) coordinates are less than the threshold distance from the current observation point. Data points are decimated such that strike and dip measurements (slope constraints) are input first. Therefore, only those slope constraints that are within the threshold distance of another slope constraint are discarded. Following this, ALSM constraints within the

29 threshold distance of the remaining slope constraints are discarded and finally the decimation algorithm is applied to all remaining ALSM points. In one sense, this approach gives preference to the slope constraints; however, because slope constraints are much fewer than spatial constraints (282 vs. 514,342), it provides a more balanced distribution of each type of constraint. If slope constraints were distributed randomly throughout the data or at the end of the decimation input, the interpolation would be dominated by spatial constraint with very few, if any, slope constraint points surviving the decimation process. We contend that including slope constraint points is important, because, as demonstrated in the Projection section, ALSM data alone cannot accurately constrain surface orientation given small outcrops. Models were investigated in which data was decimated to 5m, 10m, and 25m resolution. The resulting data sets consist, respectively, of 22,798 spatial and 251 slope constraints, 7,361 spatial and 251 slope constraints, and 1,940 spatial and 248 slope constraints. To calculate the weights of the Green’s functions for each of these data sets requires calculating the SVD for matrices of 23,049x23,049, 7,612x7,612, and 2,188x2,188, respectively. MATLAB fails to calculate the SVD of G for 5m data resolution. Data decimated to 10m and 25m produce comparable interpolated surfaces; however, the 10m data results in unrealistic edge effects in regions outside the periphery of data constraint. These issues likely occur because of close neighboring noisy data points. Further decimation, for example 50m, eliminates too many data points, and important constraint on surface geometry is lost. 25m decimation is selected for the optimal model in preference to greater decimation because it maintains sufficient resolution to capture fold details resolvable above noise, and in preference to lesser decimation because it produces a model that is less sensitive to noisy data and because the increased decimation reduces computational time from hours to minutes.

Results and discussion Two new surface models have been generated by MCS interpolation honoring projected ALSM data points that represent bedding surface outcrops and strike and dip measurements scattered across SMA. Models of the top Tensleep sandstone and of

30 the base Sundance Formation are presented in figures 1.11a and 1.11b, respectively. Because data constraining these models are projected by assuming constant bed thickness, one can theoretically model a surface representing any stratigraphic position simply by adding an appropriate constant to all projection distances. Models representing the top Tensleep Sandstone and the base Sundance Formation each have strengths and weaknesses. Errors in the projection vector increase with projection distance. Uncertainty of the raw ALSM coordinates due to weathering and measurement imprecision is negligible relative to projection error. Thus, each interpolated surface is most precise nearest where physical outcrops of the surface are found. The Base Sundance Formation is a good choice to minimize overall projection error, as it is near the middle of the stratigraphic column and therefore minimizes the maximum projection distance. This model is most precise near where Gypsum Springs Formation outcrops are exposed. Additionally, we elect to model the base Sundance Formation because it is the surface modeled by Forster et al. [1996] (Figure 1.11c) and therefore enables a direct comparison to the model generated by sequential cross sections. Contours of the difference between new and old models of the base Sundance Formation are presented in Figure 1.12. The base Sundance Formation stratigraphic contact, however, is somewhat ambiguous in the field. For this reason, it would have been preferable to model one of the nearby ridge-forming units of the Gypsum Springs Formation (e.g. cyan). The top Tensleep Sandstone model should be very precise near the fold hinge, where the Tensleep Sandstone is exposed, but is subject to greater error around the periphery where outcrops high in the stratigraphic column (i.e. lime green) are projected nearly 1km down section. Points representing the extensive outcrops of the top of the Tensleep Sandstone (orange) are not projected, and therefore maintain the decimeter precision of the ALSM data, and points from other surfaces in the Paleozoic section are projected relatively small distances (<100m). Because most data points are projected down and in, toward the core of the fold, a surface model at this stratigraphic level does not leave much distance between forelimb and backlimb data points open to interpolation. As a result, there is only a small area around the hinge without data

Figure 1.11: Structure contour maps generated by the new method representing a) the top of the Tensleep Sandstone and b) the base of the Sundance Formation. c) Structure contour map of the Base Sundance formation from Forster et al. [1996]. Contour interval is 40m. Bold contour interval is 200m. Open circles represent the locations of projected slope constraints from field measurements of strike and dip, and black dots represent the locations of projected and decimated ALSM data points.

Figure 1.12: Contours of the difference (zForster-zALSM) between the base Sundance Formation model of Forster et al. [1996] and the new model representing the same surface. Contour interval is 40m. 200m contour intervals are bold and the zero contour, representing the curve along which the two surfaces intersect, is extra bold. Black dots represent the locations of projected and decimated ALSM data constraint used to generate the new surface model.

33 constraint, and the MCS method has no alternative but to generate an appropriately narrow zone of high curvature in order to honor data. It is worth noting, however, that even when data are projected to the position of the base Sundance Formation and there is greater than 500m space between backlimb and forelimb constraint points, the MCS surface does not suggest a significantly broader hinge than is suggested by the Forster model (Figure 1.13). Before continuing the comparison between existing and new surface models, it is important to consider more thoroughly the implications of assuming a parallel fold in order to project data points to the position of a single folded surface. Abundant thickness measurements from both the backlimb and forelimb, spanning a relatively wide range of dips in the thin, competent beds of the Paleozoic Amsden, Tensleep, and Phosphoria formations (Figure 1.6b) demonstrate that the parallel fold model is quite appropriate there. However, in the Data considerations section, above, two specific concerns pertaining to overlying Mesozoic units were discussed, apparent thinning of the Chugwater shale in the forelimb and crossing projection paths when points in the Cloverly Formation near the fold nose are projected down-section to the top Tensleep Sandstone position. This occurs because the parallel fold hypothesis is necessarily violated if the down-section projection distance exceeds the local radius of curvature in an anticline (or up-dip projection distance in a syncline). The model tends to break down in thick layers where bedding surfaces are not present to accommodate necessary flexural slip, and field evidence demonstrates that different types of folding may occur in adjacent layers[e.g. Donath and Parker, 1964; Farzipour-Saein et al., 2009], particularly in the presence of mechanically heterogeneous stratigraphy. The massive, shale-rich Mesozoic units at SMA fit the above description, and the idealized parallel fold model may not be that appropriate. Because units above the Gypsum Springs Formation have been eroded throughout the steeply dipping forelimb and most of the hinge (except the plunging extremes), folding in these units cannot be characterized as parallel or otherwise. Nonetheless, having accounted for apparent thinning of the Chugwater Shale in the forelimb, and because local, appropriate thickness measurements are available for down-section projection of data

Figure 1.13: Cross sections of SMA along transects identified in Figure 1.11a. The Solid black line and dashed black line represent MCS interpolated surfaces of the top Tensleep Sandstone and Base Sundance Formation, respectively. Black circles indicate ALSM data constraint points within 25m of the cross-section. The dotted black line represents the Base Sundance Formation as interpreted from the MCS surface representing the top Tensleep Sandstone, assuming an ideal parallel fold and 366m between the two surfaces (Figure 1.3). This parallel fold projection for cross sections B-B' and C-C' accounts for a 10° plunge of the fold to the northwest. The thin grey line represents the Forster et al. [1996] model of the Base Sundance Formation.

35 from younger Mesozoic surfaces in the backlimb, assuming constant bed- perpendicular projection distances throughout this study seems appropriate. The parallel folding assumption might not be appropriate, however, for modeling younger surfaces than the base Sundance Formation because it would require up-section projection of points in the hinge and forelimb through younger units that, prior to being eroded, may not have been folded in parallel form. Careful comparison of the difference in elevation of the two new surface models at the fold's crest presents another challenge to the parallel fold hypothesis. Thickness there should be equal to the stratigraphic thickness (366m, Figure 1.3), but comparison of contours in Figure 1.11 suggests thickness in excess of 400m. Isopach contours representing the thickness between modeled surfaces are difficult to comprehend due to steeply dipping strata across much of the fold, so we use cross- sections of the fold (Figure 1.13) to evaluate whether we successfully interpolate two parallel surfaces. On each cross-section, the interpolated surfaces representing the top Tensleep Sandstone and base Sundance Formation are plotted, as well as a representation of the base Sundance formation calculated by up-section projection of the Top Tensleep Sandstone surface, assuming an ideal parallel fold. Comparison of these two representations of the Base Sundance Formation enables a qualitative evaluation of how parallel the fold profile is between the two modeled surfaces; if the fold were perfectly parallel, the two surface representations should be identical. Fold profiles are parallel within reason (though not perfectly) everywhere that data are abundant; however, as might be expected, they tend to deviate from parallel where data are lacking. On cross sections C-C', D-D', and G-G', the fold is remarkably parallel. Indeed, the interpolated representation of the Base Sundance is somewhat higher (~40m) than the representation by projecting the top Tensleep Sandstone surface up-section in the vicinity of the fold crest on sections C-C' and D-D', but considering the lack of data in the fold hinge and the uncertainty of any surface model at SMA, this difference is relatively small. On these same sections, similar magnitude deviation from perfectly parallel may be observed elsewhere on the fold. Cross- sections B-B', E-E', and F-F' deviate from parallel significantly more. On sections E-

E' and F-F', the non-parallel geometry is limited to the forelimb and forelimb syncline. In this vicinity, because data are lacking, we acknowledge that one or both of the new surface models is likely inaccurate. Where ALSM constraint points are not available to enforce the parallel fold model, MCS interpolation need not generate parallel surfaces. Cross section B-B' displays the greatest deviation from parallel folding, which persists from the hinge, northwest through the forelimb and forelimb syncline. Throughout this region, few outcrops are available to provide ALSM constraint. At the northwest end of the section where ALSM constraint from the Gypsum Springs Formation is available, the fold model returns to a more parallel form. Many differences between the new surface model and that of Forster et al. [1996] are apparent, some less subtle than others, and these are highlighted by the contours in Figure 1.12 and evident on cross sections in Figure 1.13. The greatest differences between new and old models, in some cases more than 400m, occur where ALSM data representing bedding surface outcrops are unavailable. In regions of good data coverage, the difference is smaller, generally less than the 40m contour interval. However, finer contours, which are omitted for the sake of figure clarity, reveal that even where ALSM data coverage is good, differences between the two models commonly range from 20-30m. Despite error introduced by projection, which we acknowledge is difficult to quantify, the new model, in these regions of dense data constraint, should capture subtle features of 3D fold geometry that might be omitted from older models created from sequential balanced cross sections. Much of the difference between models in areas of dense data coverage may be attributed to increased resolution and accuracy of the ALSM constrained MCS method. A good example of the improvement offered by the new model, although subtle, may be observed in the change of bedding strike along the fold in the backlimb northwest of the thumb structure (around point A in Figure 1.11b). In the Forster et al. [1996] model the strike changes relatively abruptly; however, in the new model this change is gradual. The new surface exhibits a much broader curvature along strike. This difference is substantiated by nearly continuous ALSM data coverage, and therefore is not an artifact of the MCS tendency to generate broad, smooth curves.

The two models clearly differ in the syncline between the thumb and the main fold (point B in Figure 1.11b, cross-section F-F' in Figure 1.13), a feature that is more accentuated in the Forster et al. [1996] model. The balanced cross section approach is limited in this region because the two folds with non-parallel axial surfaces exhibit deformation that is not plane-strain. Also, the cross-sections rely on interpretation of the fault structure underlying the Thumb fold, which is poorly understood. Despite consensus that the fault underlying the thumb does not involve basement rocks, Hennier and Spang [1983] suggest the fault involves a broad, upward widening zone of cataclasis, Forster et al. [1996] suggest the fault tip involves a complex wedge structure, and Stanton and Erslev [2004] suggest that the thumb is a fault propagation fold associated with a single fault tip. On the other hand, bedding surface outcrops in much of the syncline are relatively sparse, so it is difficult to judge the superiority of one model over the other in this structural domain. The new ALSM derived model and the Forster et al. [1996] model differ most dramatically around the periphery of the observation area, and in other areas where data constraint is limited. The greatest differences occur beyond the forelimb syncline in areas where ALSM data are unavailable (point C & beneath the scale bar in Figure 1.11b and cross-sections B-B', E-E' & F-F' in Figure 1.13). In the region in between, where data points from Gypsum Springs outcrops are available, the difference between new and old models is relatively small; however, where data are lacking the minimum curvature criterion forces the surface to continue plunging indefinitely at a similar slope to the forelimb. Differences in excess of -400m result (Figure 1.12). While explicit data points were unavailable to constrain the surfaces in these areas, qualitative field observations make it clear that the syncline continues to the northwest and southeast, as in the Forster et al. [1996] model. This is a clear limitation of the new method that highlights the uncertainty in areas lacking ALSM constraint. Large positive differences between the models are observed in the vicinity of the forelimb syncline at the left edge of Figure 1.12 (corresponding to point D in Figure 1.11b), to the north of the plunging nose of the fold. Here, bedding surfaces are not exposed in the forelimb, and as a result no ALSM constraint is available, so the

38 modeled fold hinge is too broad. The anticline bulges unrealistically to the northeast, into what should be a steeply dipping forelimb and synclinal hinge. This shortcoming of the new model, which is most pronounced in the top tensleep surface, may be seen clearly in a comparison of MCS surfaces and the Forster model on cross section B-B' (Figure 1.13). In this region and beyond the forelimb syncline, the ALSM constrained MCS model could be improved by adding data points that estimate the surface geometry. Imprecise constraint here could be gathered by extrapolating steeply dipping forelimb points from the southeast. Beyond the forelimb syncline, imprecise constraint could be gathered from observations such as the extensive surface exposure of the Chugwater Shale in the forelimb syncline, which provides some bounds on the location of adjacent surface contacts. However, uncertainty would be very great, given the 200m thickness and homogeneous composition of the Chugwater Shale. We avoid using such imprecise, estimated points in order to minimize subjectivity in the new model and to investigate how much of the surface geometry at SMA may be inferred from precise bedding surface outcrop data alone. Large negative differences are apparent on the contours of Figure 1.12 in portions of the backlimb (e.g. points E & F in Figure 1.11b, cross-section E-E' in Figure 1.13) where dips are greater in the ALSM model than in the older model. These differences are evidenced in the field (Brunton compass measurements of strike and dip) and by the ALSM data, particularly high in the stratigraphic section (Cloverly and Frontier outcrops) up-section from the Thumb structure, suggesting that the new model may offer improvements in these vicinities. While the new method should be significantly more precise in regions of good data constraint, careful consideration of the differences between the ALSM constrained minimum curvature surface model and the sequential cross section model of Forster et al. [1996] is necessary to determine which model better represents any particular aspect of fold geometry. In some areas where the models differ significantly, the older model may be better, but in other areas, the new model may offer an improved representation of fold geometry.

The reproducible nature of the smoothed MCS interpolation enables a quantitative measure of model uncertainty in the form of cross validation. Leave-one- out cross validation [e.g. Efron and Gong, 1983] is implemented on the decimated data set on an outcrop-by-outcrop basis, and applied only to ALSM data points. All slope constraints are applied in all models. Cross-validation is implemented on an outcrop-by-outcrop, rather than point-by-point, basis in order to minimize the bias of the resulting measure of model uncertainty. Because many outcrops are represented by multiple data points (up to 51) in the decimated data set, a cross validation scheme in which only some points representing a given outcrop were omitted from the training data set would result in an optimistic bias in uncertainty estimates. Interpolation error at test points will be minimal if only 25m (decimation resolution) typically separates test points from training points. Omitting points from the training set one outcrop at a time reduces this bias; however, cross validation results likely remain optimistically biased because outcrops tend to be clustered. Thus, cross validation is unable to account for model uncertainty in more extensive regions lacking data constraint. The results of the cross-validation analysis are presented as a histogram in Figure 1.14a. RMS error is calculated between an interpolated surface (constrained by training points only) and the omitted test points on the selected outcrop for each of 560 outcrops represented in the decimated data set. 95% of outcrops are characterized by RMS error less than 25m, 92% by error less than 20m, and 83% by error less than 10m. The maximum RMS error, however, is 1327m, corresponding to omission of the most southeastern outcrop in the Gypsum Springs Formation, northeast of the forelimb syncline. This extreme error occurs because the MCS method is not adequate for extrapolating fold geometry beyond the periphery of the data constraint. It is critical to recognize that confidence intervals calculated by cross validation are valid only in regions of relatively dense data coverage because of the optimistic bias discussed previously. Model uncertainty in regions without data constraint must be evaluated qualitatively. A histogram of RMS error at each outcrop resulting from model smoothing (all decimated ALSM constraints are used but the numerical interpolation does not exactly

Figure 1.14: Histograms show the distribution of model misfit, represented by RMS error, for a) leave-one-out (outcrop-by-outcrop) cross validation of the interpolated surface and b) the smoothed (truncated SVD) interpolation using the complete decimated data set, also on an outcrop-by-outcrop basis.

41 honor all data, see Numerical interpolation and regularization) is presented in Figure 1.14b. This plot allows a comparison of the magnitude of uncertainty from cross validation with the magnitude of misfit inherent to the fully constrained MCS surface. RMS error across the model is 2.09m, and 95% of outcrops are characterized by RMS error less than 5m. Maximum RMS error for any outcrop due to smoothing of the interpolated surface is 40m. Average misfit magnitudes are several times greater where training points are omitted in cross validation than the misfit magnitudes inherent to the smoothed interpolation. The moderate uncertainty values presented by the cross validation analysis and applicable to areas of SMA where dense ALSM constraint is available should be considered an improvement over the Forster model. Forster et al. [1996] do not present the locations or density of data on which cross-sections are based, but there is little doubt that their data lacked the density or precision of ALSM. Projection of data points above or below the base Sundance Formation would have been subject to error similar to the projection error in this study. Even if the specific data points used by Forster et al. [1996] were available, the qualitative nature of cross section construction makes quantitative statistical analysis of uncertainty in the former model impossible. A clear advantage of the new method, based on geologic mapping, high resolution, precise geospatial data, and numerical interpolation, is the unique result and reproducible nature of the interpolation. Reproducibility has been a key component of the scientific method for centuries [e.g. Laine et al., 2007; Peng et al., 2006], but geologic cross-sections are subjective interpretations, particularly when data are limited. There is no doubt that cross-sections are a valuable tool among geologists; however, they are not reproducible. Despite minor subjective data manipulations necessary to overcome limitations of the parallel fold model and numerical instabilities, which resulted in significant artifacts in the new surface model, the new method is based upon a specific mathematical formulation constrained by sound data and therefore better satisfies this fundamental requirement of scientific methodology.

Conclusion The construction of continuous (or faulted) 3D surface models of folded geologic layers from scattered, discontinuous outcrop data representing diverse stratigraphic surfaces poses many challenges for the structural geologist. Traditional methods based on sequential parallel or sub-parallel balanced cross sections, particularly in the case of complex, structures such as SMA, where the subsurface structure is poorly understood, cannot capture subtle fold features perpendicular to these cross-sections. Also, balanced cross sections assume plane strain deformation, but deformation of complex folds such as SMA has not occurred under plane strain conditions. Moreover, cross section geometry between sparse data points is dependent on interpretations of subsurface structures such as fault geometry and kinematics, which are poorly understood at SMA. We attempt to overcome these limitations using the constraints on fold geometry available from a combination of high resolution topographic data acquired with ALSM technology and outcrop scale geologic mapping. Projection of data points representing bedding surface outcrops from disparate positions in the stratigraphic column to the position of a single folded surface provides relatively dense and continuous data constraint across much of the fold. MCS interpolation enables the reproducible construction of a continuous surface model from these data. Although the method has limitations, the resulting surface captures details of fold geometry that are not represented in models developed using traditional methods. In regions with good outcrops of mapped bedding surfaces, the ALSM constrained minimum curvature surface model is generally similar to the model based on sequential cross sections. Differences between models are in the range of 20-30m, and likely represent subtle features of the fold captured by the dense, precise ALSM data. Statistical confidence in the new model is quite good where ALSM constraint is available. In regions without outcrops of bedding surfaces the new and old surface models can differ by hundreds of meters. These differences must be considered critically to determine which model might be more appropriate. We acknowledge that in some such regions the sequential cross section approach to surface modeling is

43 likely to be better if the cross sections are constructed using well constrained fault geometry and the correct fold kinematics. The numerical interpolation algorithm used here would be considered an arbitrary choice in such cases. On the other hand, where outcrop data are lacking, data constraining the balanced cross-sections may be poor and cross sections become subjective interpretations, in contrast to the unique and reproducible nature of numerical interpolation. If fault geometry or fold kinematics are in doubt, surface models based on sequential cross sections, such as the Forster et al. [1996] model, may be less accurate than an ALSM constrained MCS model.

Appendix: ALSM technical specifications The ALSM survey was carried out in two flights on July 13, 2007. NCALM used an Optech Gemini airborne laser terrain mapper (ALTM) system, mounted in a twin-engine Cessna Skymaster. The ALTM system can operate at frequencies from 33-167 kHz, and resolves elevation with 5-10cm accuracy (+/- 1 standard deviation) according to the manufacturer’s specifications. NCALM researchers have verified these accuracy specifications in a study which suggests that RMS error of elevation measurements by ALSM is only 6-10 cm [Shrestha et al., 1999]. The SMA survey consisted of 14 flight lines to cover the primary area, and five additional flight lines to provide additional coverage on the steep topography of the anticlinal forelimb north of Sheep Canyon. Differential correction of the GPS trajectory of the aircraft were calculated from two local base stations operated by NCALM, located at (N44° 33.986', W108° 11.043') and (N44° 30.507', W108° 4.713'). Repeat session calculations performed using OPUS (http://www.ngs.noaa.gov/OPUS/) resulted in station coordinate differences of less than 2.2 cm in both horizontal and vertical position. Differential correction of the aircraft trajectory was performed using the KARS software [Mader, 1992] independently for each base station. Average differences between the two calculated trajectories were 2.5cm vertical and <1cm horizontal. After GPS processing, a Kalman filter algorithm, implemented in the ApplanixTM software POSProcTM, was used to integrate 1Hz differential GPS data and 200Hz IMU recordings to produce a smoothed and blended solution for aircraft position and

44 orientation at 200Hz. The raw data were calibrated in a two step process. First, a relative calibration was performed using the TerraMatch software from Terrasolid, Ltd. to correct for absolute position error between flight lines. The method uses data from a flight path perpendicular to the primary flight lines, and optimizes aircraft heading, roll, pitch, and scanner mirror scale to minimize the deviation between flight paths. After relative calibration, the average vertical difference between cross-flight and primary flight path measurements was <5cm. Second, absolute calibration was performed by nearest neighbor comparison of ALSM points with more than 800 check points acquired by vehicle-mounted GPS along highway US310, several kilometers from the study area. RMS difference was 5.9 cm. Finally, a scan cutoff angle of 4° was imposed to eliminate error-prone data points at the edge of scan lines. Extraneous low and high points (e.g. multipath and fog, respectively) were eliminated from the data set, and a filtered bare-earth model (vegetation and buildings removed) was generated using Terrasolid's software TerraScan. Filtered and unfiltered digital elevation models with 1m pixel resolution were generated from the raw point cloud with a kriging algorithm in Surfer® Version 8.04, from Golden Software, Inc. Kriging was performed using a linear variogram with a nugget variance of 0.07m, micro variance of 0.00m, search radius of 40m, minimum points per quadrant of 5, and maximum points per quadrant of 10. The complete technical report on acquisition and processing of ALSM data is available by contacting the authors or directly from NCALM (http://www.ncalm.org).

Acknowledgements The authors thank Fil Nenna and Solomon Seyum for assistance with field mapping and Michael Sartori and Ionut Iordache of NCALM for assistance with ALSM planning, acquisition, and processing. The manuscript has benefited greatly from thorough, constructive, detailed reviews by Ken McCaffrey and Richard Lisle and comments from Associate Editor W. P. Schellart. This work has been supported by the National Science Foundation’s Collaborations in Mathematical Geosciences grant no. CMG-0417521 and P. Lovely has been supported by the Robert and Marvel Kirby Stanford Graduate Fellowship Fund.

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Chapter 2

Integrating geomechanics and field observations to unravel the evolution of an extensional fault system, Volcanic Tableland, Bishop, CA

Abstract A popular conceptual model of fault evolution, which has been invoked for the Volcanic Tableland, suggests that large faults develop by coalescence of smaller faults. Another model, which has not been investigated for the Tableland fault system, suggests that smaller faults formed due to stress perturbations resulting from slip on pre-existing larger faults. Each conceptual model has distinct implications for fault pattern-development during basin evolution and the relative ages of faults. A linear elastic optimization procedure is used to constrain 3D subsurface fault geometry, and then elastic models are applied to investigate deformation fields resulting from slip on large faults on and around the Tableland, which are compared with the spatial density of smaller faults to infer the likelihood of subsidiary faulting. Geophysical data and a comparison of 2D elastic and elasto-plastic models demonstrate that the large stress perturbations suggested by elastic models are suppressed by inelastic deformation within the earth, and that cumulative strain distributions may be a better predictor of subsidiary faulting. Elastic models provide first order insights to regions of relatively enhanced and inhibited strain. Results suggest that strain perturbations are predominantly extensional strain shadows; thus, subsidiary normal faulting, in the most rigorous sense, may not be supported by the elastic models. However, the mechanical influence of the larger faults on the evolution and distribution of smaller faults is clear. We conclude that the Tableland fault system likely initiated as a broad array of small faults, whose distribution has always been influenced by stress perturbations due to slip on large faults older than the Bishop Tuff. Some of the smaller faults apparently have grown and coalesced to form larger faults, which in turn inhibited both the initiation of new faults and the growth of existing small faults in their strain shadows, but beyond these shadows smaller faults continued to grow.

Introduction Field observation [e.g. Peacock and Sanderson, 1991; Trudgill and Cartwright, 1994; Cartwright et al., 1995; Dawers and Anders, 1995; Gupta and Scholz, 2000], analog [Ackermann et al., 2001; Mansfield and Cartwright, 2001; Sims et al., 2005] and numerical models [e.g. Willemse et al., 1996; Cowie et al., 2000; Schopfer et al., 2007] support a conceptual model that suggests large normal faults develop primarily by linkage of smaller faults, as opposed to continuous propagation of single fault segments. The Volcanic Tableland, north of Bishop, CA, is a classic field laboratory for investigating the growth and evolution of normal fault systems, providing clear evidence for the mechanical interaction of independent segments and for a history of coalescence of nearby segments. Dawers et al. [1993] suggested that scaling between fault length and slip is linear and that fault growth might be a scale- independent process, in one of the first studies of slip-length scaling that includes data representing faults spanning several orders of magnitude in size and all in the same geological and tectonic setting. Dawers and Anders [1995] took their investigation a step further, demonstrating that the cumulative throw along a series of closely linked echelon faults is more characteristic of a single extended fault than the lesser slip that would be expected from a series of independent small faults, and Willemse, et al. [1996] and Willemse [1997] supported these observations with numerical models demonstrating the importance of mechanical interaction between neighboring segments. Even before linkage, echelon faults may behave more like a single large fault than independent segments. Ferrill et al. [1999] illustrate, using topography and the mapped geometry of fault scarp traces, the evolution and subsequent break- through of relay ramps as echelon segments coalesce. Linear elastic numerical models, which are commonly used to evaluate stress and strain perturbations associated with fault slip, demonstrate that tensile stress is relaxed and fault initiation and growth should be inhibited in the immediate footwall and hanging wall of planar normal faults [Willemse et al., 1997; Cowie et al., 2000]. As faults grow by coalescence, these tensile stress shadows generally become larger. However, tensile stress perturbations elsewhere, particularly in the vicinity of fault

56 tips, may be positive and encourage fault initiation and propagation. Tensile stress perturbations may occur elsewhere in the vicinity of non-planar faults [e.g. Chester and Chester, 2000]. These tensile stress perturbations are the driving force behind another conceptual model for normal fault evolution: smaller (subsidiary) faults may form in response to tensile stress perturbations due to slip on larger faults [Bourne et al., 2000; Maerten et al., 2002; Dee et al., 2007; Wilkins, 2007]. These two conceptual models need not be mutually exclusive. The two conceptual models have very different implications for basin and fault system evolution. A common theory of basin evolution postulates that basins and rift zones begin as diffuse zones of extension with many small faults, and that as these faults grow and coalesce, stress shadows grow, enveloping more small faults, and slip is eventually localized on a small number of large faults [e.g. Cowie et al., 2000]. However, the formation of subsidiary faults might suggest that deformation in fact becomes more diffuse if new faults form in response to slip on larger faults. Kinematic analysis of the Tableland suggests that the fault system may be evolving in response to flexural extension in a rollover anticline in the hanging wall of the basin bounding White Mountain fault (Figure 2.1, inset) [Pinter, 1995]. The conceptual model of fault growth by coalescence requires a system of smaller faults must exist first. While this conceptual model does not preclude subsequent infilling of smaller faults, it implies that many smaller faults are likely coeval with or older than larger faults, but have not coalesced. The growth of these small faults is likely to be inhibited by the growing stress shadows in the footwalls and hanging walls of larger faults. Conversely, the conceptual model based on subsidiary faulting implies that smaller faults are younger than their larger counterparts. An array of faults such as that observed on the Tableland, spanning three orders of magnitude in size, begs the question: Are some, or even most small faults young subsidiary structures, or, as might be suggested by coalescence, are they among the oldest structures, whose growth has been inhibited by larger faults? Understanding mechanisms of fault system evolution is of interest and practical value to structural geologists. Because faults may act as conduits or barriers

Figure 2.1: Hillshade image of Tableland topography from 2m DEM. Bold solid lines represent west-dipping faults and bold dashed lines represent east-dipping faults included in numerical models. Thin lines represent smaller fault traces. The bold gray line in the southwest corner represents the trace of an additional fault, slightly below the seismic-scale threshold, but which was necessary to include for paleostress inversion. The black dashed-line represents the region where the inversion for fault geometry was performed. Inset: regional map of Tableland vicinity. Abbreviations: WMF – White Mountain Fault; IF – Inyo Fault; OVF – Owens Valley Fault.

58 to fluid flow [e.g. Knipe et al., 1997; Aydin, 2000], fault system evolution could influence hydrocarbon migration pathways and trap potential [e.g. Clarke et al., 2006] if, for example, faults that are continuous at present were segmented at the time of migration, or if some faults had not yet formed at that time. Understanding mechanisms of fault system evolution also may provide insights concerning fault connectivity (physical or mechanical) in the subsurface, which could be valuable for estimating seismic hazard, particularly the risk of earthquakes that might rupture apparently distinct fault segments, as happened during the 1992 Landers Earthquake in California's Mojave desert [e.g. Harris and Day, 1999]. This manuscript consists of four primary components. First, we provide further evidence in the form of slip distributions that even large Tableland faults with continuous and relatively linear traces that are not indicative of coalescence may have formed by linkage of multiple, distinct fault segments. Subsequently, we use a mechanically based optimization procedure to constrain the subsurface geometry of Tableland faults. Fault growth is a 3D process [e.g. Kaven and Martel, 2007], and a 3D model is necessary to simulate complete deformation fields associated with slip on non-planar, finite fault surfaces. This is followed by a comparison of simplified 2D elastic and elasto-plastic models, in an effort to evaluate the implications of using elastic models that suggest unphysical stress fields to gain insight to geological processes. Linear elastic models with frictionless sliding faults have been applied successfully in many previous studies of fault-related deformation, although cumulative rock deformation is largely accommodated by inelastic processes. Elastic models are appealing for their computational efficiency and numerical stability, particularly for simulating the 3D effects of slip on non-planar fault surfaces. Finally, we investigate the potential role of subsidiary faulting in the evolution of the Tableland fault system. Three dimensional geomechanical models are used to infer strain perturbations, which are compared with maps of fault density to infer the likelihood of subsidiary faulting in response to slip on faults at three different scales:

1) crustal scale basin bounding faults, 2) Fish Slough fault, and 3) the larger (>20m throw) faults distributed across the Tableland.

Geologic Setting The Volcanic Tableland, a topographic plateau, north of Bishop, CA, in the Owens River Valley, boasts one of the world's premier field exposures of a young, evolving extensional fault system. In addition to being a classical field laboratory for investigations of slip-length scaling and fault system evolution [Dawers et al., 1993; Dawers and Anders, 1995; Willemse et al., 1996; Willemse et al., 1997; Ferrill et al., 1999], it has been the field site for studies investigating fault kinematics [Ferrill et al., 2000; Ferrill and Morris, 2001], micro-scale deformation mechanisms [Evans and Bradbury, 2004] and fault zone permeability [Dinwiddie et al., 2006]. Topography on the Tableland is defined by the weathering resistant surface of the Bishop Tuff, a volcanic ash flow erupted 760,000 years ago [Izett and Obradovich, 1994; Vandenbogaard and Schirnick, 1995; Sarna-Wojcicki et al., 2000] from the Long Valley Caldera. The tuff, which consists of an uppermost densely welded, weathering resistant unit 7-8m thick, underlain by units of variable but lesser degrees of welding [Wilson and Hildreth, 1997], is 70-150m thick in the study area, and is underlain by several kilometers of alluvial, lacustrine and volcanic basin sediments [Gilbert, 1938; Pakiser et al., 1964; Bateman et al., 1965]. The tuff has been deformed by several hundred normal faults (Figure 2.1) whose scarps are preserved remarkably well in the uppermost welded unit. Because erosion of the uplifted footwalls and deposition on the downthrown hanging walls is much less (1-2m [Goethals et al., 2009]) than fault topography (tens of meters), well-preserved fault scarps accurately document cumulative throw distributions. Topography on the Tableland is interpreted as vertical displacement due to fault slip since the tuff’s deposition because similar ashfall deposits are relatively planar [e.g. Griggs, 1922]. Faults generally strike north-south, throw varies from less than 1m to 150m, and length varies from less than 100m to greater than 10km. Fish Slough fault (FSF, Figure 2.1) stands out, as it is larger than any other fault on the Tableland by about a factor of three.

Located at the western margin of the Basin and Range extensional province and at the northern end of the Eastern California Shear Zone, the region is tectonically active. Deformation, characterized by east-west extension and right-lateral strike-slip, is accommodated primarily on oblique-slipping basin-bounding faults. Since the deposition of the Bishop Tuff, these faults have accumulated nearly twice as much strike-slip as dip-slip [Kirby et al., 2006]. The White Mountain fault (WMF), which outcrops 5-10 km east of the Tableland and dips west, and the Round Valley (RVF) and Hilton Creek (HCF) faults, which outcrop 10-15km to the west, along the Sierran range front, and dip east (Figure 2.1, inset), are the basin-bounding faults nearest the Tableland. Strike-slip cannot be ruled out on Tableland faults; however, they appear to accommodate primarily dip-slip, evidenced by several stream channels that likely pre-date much fault slip but that show no discernible lateral offset [Bateman et al., 1965; Dawers et al., 1993; Pinter and Keller, 1995].

Slip profiles as evidence for coalescence Three dimensional differential global positioning system (DGPS) was used to map footwall and hanging wall cutoffs of fault scarps in the southeastern corner of the Tableland. From this data, a detailed fault trace map was interpreted (Figure 2.2). Many fault traces (e.g. those labled 1, 2 & 3) are characterized by complex patterns of intersecting fault segments, splays, and faulting distributed on sub parallel segments. These features provide strong evidence for fault growth by coalescence [e.g. Trudgill and Cartwright, 1994; Ferrill et al., 1999]. However, traces of other large faults, such as fault 4, are relatively linear and continuous. Examples such as this might lead one to question whether all large Tableland faults have grown by coalescence of smaller faults, or if this mechanism applies only to some faults. Trace maps, however, are not the only evidence for fault growth by linkage. Local minima in slip along fault strike may provide further support for this conceptual model [e.g. Peacock and Sanderson, 1991]. As individual segments grow, their slip distribution must taper to zero at the tip. Thus, after linking, the newly coalesced fault is characterized by distinct local slip maxima, corresponding to the maxima on each of

Figure 2.2: Detailed map of fault traces interpreted from DGPS map of footwall and hanging wall cutoffs in the southeastern corner of the Tableland. Complex patterns of intersecting fault segments, splays, and faulting distributed on subparallel segments are apparent. Dashed curves represent west-dipping faults and solid curves represent east-dipping faults.

Figure 2.3: a) Distribution of throw along strike of fault 4 in Figure 2.2. b) The trace of the fault is plotted to scale with the horizontal axis of the throw distribution.

63 the previously distinct segments, with minima at the points of linkage. The throw distribution along fault 4 (Figure 2.3a) is characterized by three distinct local maxima that provide a strong argument that this fault too has grown by coalescence of previously individual segments. Correlation of the location of slip minima with the fault trace map (Figure 2.3b) suggests subtle jogs in the fault trace, which may indicate the points at which nearly coplanar faults linked tip-to-tip, without significant bending or breaching of relay ramps between echelon segments. Maximum slip on this fault corresponds with the southern end of the mapped fault because the Bishop Tuff has been eroded to the south. Thus, the southern termination of the fault trace, as mapped, is not actually a fault tip, but only the end of the preserved scarp. Slip distributions such as that observed on fault 4 substantiate previous conclusions that most, if not all, long faults on the Tableland have grown by coalescence of smaller segments.

Optimizing subsurface 3D fault geometry Subsurface geometry of Tableland faults is poorly constrained. Slip surfaces cannot be identified on scarps and only a few dip measurements may be made where small faults outcrop in exposed bluffs at the southern end of the Tableland. Dips range from 50-90⁰ [Bateman et al., 1965; Ferrill et al., 2000] and measurements in the near surface may not be indicative of dips at depth because faulting in the near surface is accommodated largely by pre-existing cooling joints. The depth of faulting is entirely unconstrained from surface observations. However, 3D representations of fault surfaces are necessary for numerical models. We infer the best down-dip extrapolation of fault traces from a coarse optimization procedure that minimizes the root mean square (RMS) difference between vertical displacement of the earth's surface as predicted by numerical models representing 22 faults in the southeastern corner of the Tableland and topography, as measured by airborne LiDAR, over an area of 4.3 km2 (Figure 2.1). The optimization for fault geometry considers a relatively small area, much less than the full extent of the LiDAR data, because it is important that all faults within and around the area be represented in numerical models. To do this, while

64 maintaining model resolution and avoiding numerical instabilities (not too many elements), requires a restricted area with a limited number of faults. The area identified in Figure 2.1 was selected as representative of the Tableland, with faults spanning a range of size (throw and length), and because this is the area in which the precise, field-checked fault map assembled from differential GPS data (Figure 2.2) is available. Numerical models discussed in this section are performed using the computer code Poly3D© [Thomas, 1993; Maerten et al., 2009], available from IGEOSS (http://www.igeoss.com). Faults are represented as non-planar surfaces discretized as triangular elements of uniform displacement discontinuity in a semi-infinite, homogeneous, linear-elastic half-space. Elastic Young’s modulus E = 2.5 GPa and Poisson’s ratio ν = 0.25, laboratory measurements for poorly welded tuff [Lin et al., 1993; Avar et al., 2003], are used. Slip is driven by far-field homogeneous stress or strain, and calculated simultaneously on each element such that shear traction resolved by the far-field stress tensor on each fault element is relaxed completely. While real faults are not actually frictionless, previous studies [e.g. Aydin and Schultz, 1990; Cooke and Marshall, 2006] suggest that the effects of uniform friction do not have a strong influence on mechanical interaction of fault segments, but only on the magnitude of slip. Hence, most aspects of deformation associated with frictional fault slip may be approximated by models that treat faults in a simplified manner as frictionless surfaces. In these elastic models of extensional tectonics, faults are subject to potentially unphysical tensile normal traction, and therefore friction would have no effect if it were included. Elastic continuum deformation is also a limiting assumption, as discussed in the Validating elastic models section; however, for optimization purposes, simplified frictionless, elastic models enable the evaluation of several hundred numerical models in a computationally efficient manner. Ninety-six trial fault geometries were generated by down-dip projection of surface traces. These models span the range of plausible geometric configurations with dips from 50°-80° and depths from 100-3000m. Models represent linear down- dip extrapolations in which the depth of faulting is constant, linear extrapolations in

65 which depth of faulting scales with fault trace length, and listric down-dip extrapolations with uniform depth of faulting. Each fault model is implemented in Poly3D in half-space mode and subjected to uniaxial, east-west extension [Pinter, 1995]. Alternative strain orientations more compatible with the right-lateral strike-slip observed on basin bounding faults were tried as well, but east-west strain, which minimizes strike-slip on Tableland faults, best reproduces topography. Pinter's kinematic analysis suggests a strain magnitude of 2%; however, this estimate is highly dependent upon fault dip. Poor constraint on strain magnitude is overcome by taking advantage of the linear nature of the elastic models, in which displacement, strain and stress fields, including vertical displacement observed at the earth's surface, scale linearly with far-field strain magnitude. A least square optimization procedure is implemented for each model to determine far-field strain magnitude and to account for mean elevation and regional dip, such that the difference between model and observed topography is minimized. Because the regional dip is small (<5°), the horizontal component of rotation may be neglected. After optimizing each model, that which most nearly reproduces topography is selected as "best." Results suggest non-listric faults dipping 50° and extending uniformly to 750m depth. RMS residual values for each trial fault configuration are presented in Table 2.1. Because FSF is roughly three times larger than any other fault on the Tableland, it seems appropriate to treat its depth independently. Being more than 10km long and exhibiting some 150m throw, it likely extends deeper than 750m. Evaluating all permutations of FSF and smaller fault geometries would have required meshing and evaluating thousands of models. Therefore, interaction with FSF was considered only for those 17 of the 96 geometric configurations for smaller faults that independently generate RMS misfit less than 6m. Only permutations in which the FSF has the same down-dip profile and extends to a greater depth than smaller faults were investigated. There is no reason to believe the dip of FSF would be different from that of the smaller faults or that its depth would be less. This second optimization procedure maintains the 50° fault dip and 750m depth of Tableland

Figure 2.4: Modeled and observed topography along two east-west transects, A-A' and B-B' identified in Figure 2.2. The vertical datum is arbitrary and the model topography has been scaled, rotated, and shifted vertically to minimized the RMS difference between model topography and observed topography across the area

Table 2.1 (opposite): RMS difference between vertical displacement of earth's surface from Poly3D models and topography for various down-dip projections of fault surface traces not including the Fish Slough fault. "Profile" represents the down-dip geometry. LUD represents "linear unscaled depth," meaning that the down-dip extrapolation of fault traces is linear (not listric) and each fault surface extends to the same depth. LSD represents "linear scaled depth," meaning that the down-dip extrapolation of the fault trace is linear and the maximum depth of each fault surface scales linearly with trace length. The depth value identified for such models is the depth of the longest fault trace. For listric faults, dip is measured where the fault intersects the surface, and decreases with depth. All listric faults have uniform depths. Only a single listric profile is investigated for each model. The optimal model, which minimizes RMS difference, is highlighted gray.

Dip Depth Profile RMS Residual Dip Depth Profile RMS Residual (°)* (m) (m) (°)* (m) 50 100 LUD 7.9019 70 100 LSD 8.1357 50 300 LUD 6.1517 70 300 LSD 7.2907 50 500 LUD 5.4045 70 500 LSD 7.2036 50 750 LUD 5.1653 70 750 LSD 6.9835 50 1000 LUD 5.5211 70 1000 LSD 6.7730 50 1500 LUD 5.8097 70 1500 LSD 6.7259 50 2000 LUD 6.0220 70 2000 LSD 6.4082 50 3000 LUD 6.0990 70 3000 LSD 6.4793 60 100 LUD 7.8929 80 100 LSD 8.1697 60 300 LUD 6.2624 80 300 LSD 7.2407 60 500 LUD 5.8459 80 500 LSD 7.0885 60 750 LUD 5.6710 80 750 LSD 7.3863 60 1000 LUD 5.6657 80 1000 LSD 7.4089 60 1500 LUD 6.0207 80 1500 LSD 7.2808 60 2000 LUD 6.2132 80 2000 LSD 7.1636 60 3000 LUD 6.2983 80 3000 LSD 7.0474 70 100 LUD 7.9078 50 250 Listric 6.6191 70 300 LUD 6.4800 50 400 Listric 5.5686 70 500 LUD 6.6579 50 600 Listric 5.8562 70 750 LUD 6.3246 50 750 Listric 6.0337 70 1000 LUD 6.3756 50 1000 Listric 6.3120 70 2000 LUD 6.3740 50 1500 Listric 6.5113 70 3000 LUD 6.4566 50 2000 Listric 6.4978 80 100 LUD 7.8950 50 3000 Listric 6.6149 80 300 LUD 6.5339 60 250 Listric 6.7133 80 500 LUD 6.3876 60 400 Listric 5.6392 80 750 LUD 7.0491 60 600 Listric 5.7094 80 1000 LUD 7.2474 60 750 Listric 5.9174 80 1500 LUD 7.0773 60 1000 Listric 6.2391 80 2000 LUD 7.0546 60 1500 Listric 6.5348 80 3000 LUD 6.8412 60 2000 Listric 6.5365 50 100 LSD 8.1307 60 3000 Listric 6.6441 50 300 LSD 7.2031 70 250 Listric 6.8904 50 500 LSD 6.6685 70 400 Listric 6.0219 50 750 LSD 6.2553 70 600 Listric 5.6467 50 1000 LSD 6.1416 70 750 Listric 5.7690 50 1500 LSD 6.1343 70 1000 Listric 6.3191 50 2000 LSD 6.2748 70 1500 Listric 6.5590 50 3000 LSD 6.4222 70 2000 Listric 6.7055 60 100 LSD 8.1282 70 3000 Listric 6.7557 60 300 LSD 7.2447 80 250 Listric 7.0116 60 500 LSD 6.8903 80 400 Listric 5.9118 60 750 LSD 6.4844 80 600 Listric 5.4297 60 1000 LSD 6.2046 80 750 Listric 5.7297 60 1500 LSD 6.2854 80 1000 Listric 5.9762 60 2000 LSD 6.2047 80 1500 Listric 6.5513 60 3000 LSD 6.3707 80 2000 Listric 6.9283 80 3000 Listric 7.0957

Table 2.2 (opposite): RMS difference between vertical displacement of earth's surface from Poly3D models and topography for various down-dip projections of fault surface traces including the Fish Slough fault. LUD means "linear uniform dip," as in Table 2.1. Primary depth is the depth of standard faults, as in Table 2.1, in contrast to the independent depth of FSF. The optimal model which minimizes RMS difference is highlighted gray.

Dip Primary FSF Profile RMS Dip Primary FSF Profile RMS (°)* Depth Depth Residual (°)* Depth Depth Residual (m) (m) (m) (m) 50 500 500 LUD 5.3589 50 600 1500 Listric 6.1616 50 500 750 LUD 5.4078 50 600 3000 Listric 6.3004 50 500 1000 LUD 5.3934 50 600 4000 Listric 6.3319 50 500 1500 LUD 5.3450 60 400 500 Listric 5.6733 50 500 2000 LUD 5.4630 60 400 750 Listric 5.6713 50 500 3000 LUD 5.7800 60 400 1000 Listric 5.6565 50 500 4000 LUD 5.9985 60 400 1500 Listric 6.4587 50 500 5000 LUD 6.1390 60 400 2000 Listric 6.4320 50 750 750 LUD 5.1741 60 400 3000 Listric 6.2744 50 750 1000 LUD 5.1746 60 600 1000 Listric 5.8467 50 750 1500 LUD 5.0651 60 600 2000 Listric 6.3713 50 750 2000 LUD 5.0610 60 600 3000 Listric 6.3478 50 750 3000 LUD 5.2839 60 600 4000 Listric 6.3402 50 750 4000 LUD 5.3627 60 750 750 Listric 6.0101 50 750 5000 LUD 5.4075 60 750 1500 Listric 6.2934 50 1000 1000 LUD 5.5110 60 750 2000 Listric 6.2824 50 1000 1500 LUD 5.3540 60 750 3000 Listric 6.1333 50 1000 2000 LUD 5.4340 60 750 4000 Listric 6.2536 50 1000 3000 LUD 5.5955 70 600 750 Listric 5.6282 50 1000 4000 LUD 5.7219 70 600 1000 Listric 5.7614 50 1000 5000 LUD 5.8202 70 600 1500 Listric 5.8764 50 1500 1500 LUD 5.7663 70 600 2000 Listric 6.1479 50 1500 2000 LUD 5.8868 70 600 3000 Listric 6.1763 50 1500 3000 LUD 6.0528 70 600 4000 Listric 5.9340 50 1500 4000 LUD 6.1444 70 750 750 Listric 6.0190 50 1500 5000 LUD 6.2718 70 750 1000 Listric 5.8393 60 500 500 LUD 5.8315 70 750 1500 Listric 5.9921 60 500 750 LUD 5.8313 70 750 2000 Listric 6.3400 60 500 1000 LUD 5.8411 70 750 3000 Listric 6.2709 60 500 1500 LUD 5.7915 70 750 4000 Listric 6.1635 60 500 2000 LUD 5.8391 80 400 500 Listric 5.9428 60 500 3000 LUD 5.9638 80 400 750 Listric 5.9928 60 500 4000 LUD 6.1622 80 400 1000 Listric 6.1061 60 500 5000 LUD 6.1689 80 400 1500 Listric 6.3746 60 750 750 LUD 5.7596 80 400 2000 Listric 6.6947 60 750 1000 LUD 5.7419 80 400 3000 Listric 6.5436 60 750 1500 LUD 5.7342 80 600 750 Listric 5.7289 60 750 2000 LUD 5.7268 80 600 1000 Listric 5.8165 60 750 3000 LUD 5.8123 80 600 1500 Listric 6.0287 60 750 4000 LUD 5.8957 80 600 2000 Listric 5.8642 60 750 5000 LUD 5.9466 80 600 3000 Listric 6.1630 60 1000 1000 LUD 5.6893 80 600 4000 Listric 6.0819 60 1000 1500 LUD 5.6200 80 750 750 Listric 5.6866 60 1000 2000 LUD 5.6152 80 750 1000 Listric 5.8392 60 1000 3000 LUD 5.7017 80 750 1500 Listric 5.8758 60 1000 4000 LUD 5.7153 80 750 2000 Listric 6.1598 60 1000 5000 LUD 5.7427 80 750 3000 Listric 6.0150 50 400 500 Listric 5.7119 80 750 4000 Listric 6.1002 50 400 750 Listric 5.7405 80 1000 1500 Listric 6.0190 50 400 1000 Listric 6.5294 80 1000 2000 Listric 6.2341 50 400 2000 Listric 6.4150 80 1000 3000 Listric 6.3146 50 400 3000 Listric 6.3303 80 1000 4000 Listric 6.5307 50 600 750 Listric 5.9406 80 1000 5000 Listric 6.2312 50 600 1000 Listric 5.9703

71 faults, but suggests that FSF extends to 2000m depth. RMS residual values for fault configurations including FSF are presented in Table 2.2. Optimized models and observed topography are presented along two east-west transects in Figure 2.4. On the northern transect (A-A', Figure 2.2), the model reproduces observed topography almost perfectly, except at the ends, where observed topography is likely influenced by faults that are beyond the extent of the model. Along the southern transect, the model reproduces topographic trends well, but almost universally underestimates topography. This is the result of regional flexure/oscillation in the topography that is not accounted for by the model. The origin of such regional topography is unknown, but may be structural, depositional, or of some other nature. Nonetheless, these two transects indicate the success with which elastic models are capable of reproducing the topography observed across the Tableland, supporting the inversion technique employed. Basin bounding faults are represented by down-dip projection of surface traces at 60°, which is within the range of published estimates [Berry, 1997; Kirby et al., 2006] and is a typical dip angle for normal faults. These faults were not considered when optimizing the subsurface geometry of smaller faults on the Tableland. Lacking strong evidence for listric faulting, we assume that dip remains constant at depth. Traces are extrapolated 15km down, assuming an approximate depth of crustal-scale faulting based on the local seismogenic thickness [Nazareth and Hauksson, 2004]. The WMF trace is derived from Lee et al. [2001] and traces of the RVF and HCF are adapted from Berry [1997].

Evaluation of elastic models Rock deformation characterized by large strain and accumulated over geologic time scales is not elastic. Such deformation is permanent and may be accommodated by inelastic processes including faulting [e.g. Scholz, 1990] and fracturing[e.g. Atkinson, 1987; Pollard and Aydin, 1988], dissolution and precipitation [e.g. Stockdale, 1922; Baron and Parnell, 2007], granular flow [e.g. Borradaile, 1981; Goldsmith, 1989], and pore collapse [e.g. Aydin, 1978; Mollema and Antonellini, 1996]. Elastic models of the Tableland (Figure 2.5a) suggest stress magnitudes more

Figure 2.5: Two dimensional models representing a vertical transect across the Tableland at 4,146,500m northing are used to evaluate the implications of assuming a simplified linear elastic constitutive model, rather than a more sophisticated, and probably more appropriate, Drucker-Prager elasto-plastic model. Results represent a) maximum principal stress distribution from linear elastic model, b) maximum principal stress distribution from elasto-plastic model, c) maximum principal strain distribution from elastic model, and d) maximum principal strain distribution from elasto-plastic model. Black contours define the boundary between regions of positive and negative perturbation, relative to the background stress/strain values in a comparable model without faults

73 than an order of magnitude greater than the tensile strength of rock, which is approximately 10 MPa for intact rock at the scale of laboratory samples [Bieniawski, 1984] and potentially much less for large, fractured rock masses such as the Bishop Tuff. In situ measurements, as well as theoretical limits based on the finite strength of rock, indicate that stress in the earth is predominantly compressive below a few hundred meters (less for weak, unconsolidated sediments) [e.g. Hubbert, 1951; Zoback, 2007], but elastic models indicate tensile horizontal stress at 3-4km depth. Elasticity may provide a reasonable approximation of rock deformation given small strain and a short time period, for processes including earthquake rupture [e.g. King et al., 1994], aseismic creep [e.g. Lohman and McGuire, 2007], and dynamic, or even quasi-static, fracture propagation [e.g. Pollard and Segall, 1987], but it is of questionable use for modeling the complex, nonlinear material behavior of rock undergoing large strain over geologic time. A variety of past studies have investigated fault-related deformation using elasto-plastic material models and nonlinear numerical methods in two dimensions [Erickson and Jamison, 1995; Salvini et al., 2001; Cardozo et al., 2003; Hardy and Finch, 2005; Crook et al., 2006b; Crook et al., 2006a; Sanz et al., 2007; Sanz et al., 2008; Smart et al., 2010]. For 2D studies, it may be unnecessary to resort to elastic models because computational power and numerical methods for more advanced models are readily available. However, in many cases, a fully three-dimensional model may be required in order to appropriately simulate the effects of geometrically irregular (non-planar) fault surfaces and lateral and down-dip fault tip-lines , which play a critical role in governing the patterns of fault-related deformation [Willemse et al., 1996; Crider and Pollard, 1998; Kattenhorn and Pollard, 1999; Maerten et al., 1999; Bourne et al., 2000; Kattenhorn et al., 2000; Maerten et al., 2000; Crider et al., 2001; Maerten et al., 2005]. Kaven and Martel [2007] emphasize the importance of considering normal fault growth as a 3D process. Inelastic numerical models that best represent rock deformation accumulated over hundreds of thousands or millions of years and that include numerous frictional contact surfaces are highly nonlinear, and when extended to three dimensions computational cost may be prohibitive, but elastic

74 methods such as the boundary element code Poly3D [Thomas, 1993; Maerten et al., 2009] enable computationally efficient implementation of a fully three dimensional mechanical model. Thus, elastic methods, despite limitations, are attractive. But what about the issue that elastic stress magnitudes may be unrealistic? The success of previous studies [e.g. Cowie et al., 2000; Maerten et al., 2002] suggests that elastic models may provide valuable geological insight. Lovely and Pollard [in prep] present results that are generic (not site-specific), but a comparison of 2D models governed by elastic and inelastic constitutive laws illustrates the capabilities and limitations of elastic models to simulate fault-related deformation in a fully mechanical and three dimensional framework. To evaluate the site-specific limitations of applying elastic models to the Tableland field site, we consider a two dimensional elasto-plastic model that represents an east-west cross-section at 4,146,500m northing. In the manner of Lovely, et al. [in prep], stress and strain fields are compared with those calculated from an elastic model with the same initial geometry, gravitational loading, and far- field strain boundary conditions (Figure 2.5). The model characterized by elasto- plastic continuum deformation and frictional fault slip is itself an idealization of rock behavior, but the nonlinear material models may provide reasonable representations of cumulative deformation [e.g. Yu, 2006; Jaeger et al., 2007]. Perhaps most importantly, the elasto-plastic model returns stress magnitudes that do not exceed rock strength. In regions where elastic and elasto-plastic models consistently differ one can infer the limitations of elasticity. Consistent similarities between certain aspects of elastic and elasto-plastic models point to the efficacy of elastic models. In the 2D elastic model, continuum deformation is governed by Hooke's law for an isotropic, homogeneous linear solid, with Young's modulus, E = 2.5 GPa, and Poisson's ratio, υ = 0.25, the same elastic moduli used for three-dimensional Tableland models; faults are treated as frictionless sliding surfaces that are not permitted to separate or interpenetrate. The resulting numerical system is completely linear. In the nonlinear model, continuum deformation is governed by a Drucker-Prager elasto- plastic constitutive law, and fault slip is governed by a Coulomb friction model [e.g.

Jaeger et al., 2007], with a coefficient of friction, μ = 0.4. In the elastic domain, the Drucker-Prager model is identical to the linear elastic model, but this model incorporates a circular-conical yield surface, parameterized by an angle of internal friction of 41° and a yield strength (σy) of 3 MPa. Plastic deformation follows a non- associative flow rule, with a dilation angle of 21° [Lin et al., 1993]. Models are subject to no strain hardening or softening (i.e. perfect plasticity), but we acknowledge that rock mechanics tests indicate hardening/softening behavior that may be dependent upon confining stress and other factors [e.g. Jaeger et al., 2007]. Models are restricted to plane-strain; thus, faults behave as though they are infinitely long perpendicular to the plane of the model. This results in significantly greater slip, particularly on the two central faults, than is observed in 3D models because in 3D, slip is restricted by nearby lateral tip-lines of each fault. Such effects emphasize the importance of a fully 3D analysis. However, otherwise equivalent 2D models provide a valid comparison of constitutive laws. Initially stress-free models are first subjected to gravitational loading, which is applied as a vertical body force while the lateral and basal boundaries are constrained by rollers (shear traction-free and zero face-perpendicular displacement). Subsequently, models are subjected to 1.9% horizontal far-field extension [Pinter, 1995]. Linear and nonlinear 2D models are implemented in the commercial finite element package Abaqus®, version 6.9EF (http://www.simulia.com). Results, presented with a tension-positive sign convention, indicate that the stress state of elastic (Figure 2.5a) and elasto-plastic (Figure 2.5b) models differ dramatically. The predominantly horizontal maximum principal stress component from elasto-plastic models is compressive (negative), except in the very near-surface, and the distribution throughout the model space is dominated by a vertical gradient. Contours of constant principal stress are generally drawn up in the hanging wall and drawn down in the footwall, but significant stress concentrations near fault tips are not evident. The elasto-plastic stress field presents a dramatic contrast to the elastic model in which maximum principal stress is tensile almost everywhere and locally is an order of magnitude or more greater than the strength of rock. The elastic stress field is

76 dominated by fault-tip perturbations and tensile stresses are sufficiently large throughout that the lithostatic gradient is obscured. The orientation of principal stress axes in elastic and elasto-plastic models is broadly similar, but beyond this the two stress fields have little in common. We conclude that this type of elastic model provides a poor representation of stress magnitude and the distribution of stress perturbations beneath the Tableland. The strain fields from elastic (Figure 2.5c) and elasto-plastic (Figure 2.5d) models have more in common than the respective stress fields, particularly in the near surface, at depths less than that of fault tips, which is the model space relevant to this study. Strain tensor orientation is broadly similar in both models, although the frictionless, elastic model has exaggerated rotation near fault surfaces because stress and strain axes must be perpendicular to the traction-free surface. In the near-surface, elasto-plastic strain perturbations (deviation from the far-field 1.9% extension) are somewhat larger in magnitude and extent than elastic perturbations, but the distribution is broadly similar. Moving from west to east (left to right) across the model space in the near-surface, both models are characterized by dominant extensional strain shadows in the vicinity of the westernmost fault, followed by a region of relatively greater extension centered about 1km to the east. Continuing east, extension gradually decreases to a minimum in the vicinity of the two faults that are separated by only a few hundred meters. About 1km east of these faults is another region of relatively greater strain, followed by a stress shadow near where the FSF (easternmost fault on the cross-section) intersects the earth's surface. In the vicinity of fault tips, elastic and elasto-plastic models differ more significantly. Elastic models are characterized by locally concentrated strain perturbations, in contrast to more extensive plastic shearing that extends down-dip, subparallel to the fault surface. The differences in stress fields between elastic, frictionless models and elasto- plastic models with frictional fault slip develop because plastic yielding effectively suppresses large stress perturbations. Finite rock strength may have a similar effect in nature [e.g. Hubbert, 1951]. As extension boundary conditions are applied incrementally, the horizontal component of stress in elasto-plastic models evolves

77 such that the local stress state nearly everywhere in the model is on the yield surface. This is in agreement with geophysical studies that suggest the earth's crust is critically stressed (i.e. at the point of failure) nearly everywhere [e.g. Townend and Zoback, 2000]. Because the models assume perfect plasticity and the yield surface does not evolve in space or time, once the critically stressed state is reached, further strain is almost exclusively plastic. Locally, the stress state may evolve by migrating along the yield surface, primarily in response to geometric evolution of the model. Such effects tend to be small, and the stress field changes little with boundary condition increments. In contrast, stress increases indefinitely in elastic models, linearly proportional to far-field strain, irrespective of material strength. Previous studies of subsidiary faulting have evaluated elastic stress perturbations as a predictor of deformation resulting from slip on primary faults [e.g. Maerten et al., 2002]; however, we suggest that the final stress state suggested by numerical models may not be the best predictor of cumulative local deformation and damage, including subsidiary faulting and fracturing. Elastic stress may be an appropriate predictor of fracturing in models that consider fracture initiation and propagation at every increment of the model's evolution [e.g. Olson and Pollard, 1991; Thomas and Pollard, 1993; Renshaw and Pollard, 1994]. In such models that account for incremental evolution if inelastic deformation, a similar critically stressed state could evolve, in which the suppression of large stress perturbations is accommodated by brittle deformation. In models like those evaluated in this manuscript, which do not account for fracture initiation or propagation, plastic strain is the only means available to maintain stress states within the yield surface, and therefore brittle deformation beyond that accommodated by the discretized faults must be represented by continuum strain. Because evolution of the stress field is minimal once the model reaches a critically stressed state (it is after this point that most yielding and failure occur), we suggest that regions of enhanced strain, rather than model stress, may be a significantly better predictor of geologic deformation, including subsidiary faulting.

The similarities demonstrated between strain fields calculated by elastic and inelastic models suggest that, while inelastic models may be a more appropriate representation of geologic deformation, elastic models may provide first-order insight about strain perturbations in the near-surface, resulting from cumulative fault slip. Therefore, strain fields calculated from elastic models may be an appropriate predictor of subsidiary fault development on the Tableland. Maximum principal strain may be regarded as a proxy for regions of enhanced deformation without consideration of the dominant failure mechanism in the evolution of the Tableland fault system. Whether faults initiated and propagated predominantly by linkage and shearing of opening mode joints [e.g. Davatzes and Aydin, 2003; Kaven and Martel, 2007] or by shear failure, fault development should be most favorable in regions of greatest extension.

Role of subsidiary faulting Using the optimized fault geometry determined previously, numerical models are evaluated to calculate deformation fields resulting from slip on large faults on and around the Tableland, in order to discriminate among different mechanisms of fault evolution. Models are considered that represent faulting at three different scales, enabling us to investigate the plausibility of secondary faulting at corresponding scales. The first model includes three basin bounding faults (WMF, RVF & HCF), FSF, and the 12 largest faults scattered across the Tableland (Figure 2.1). These faults were selected by a simple computer algorithm that scans the DEM and identifies scarps with consistently greater than 20m throw. Because Tableland fault growth appears to be a scale-independent process [Dawers et al., 1993], it should not be necessary to consider additional models with more, smaller faults. Beside, such models become highly susceptible to numerical instabilities as the number of fault elements increases. The second model represents FSF and the basin bounding faults, and the third model represents the three basin bounding faults only. By comparing maximum principal strain distributions with density of smaller faults, we infer whether smaller faults are likely to have formed in response to slip on the larger faults, or if the larger faults more likely inhibited slip on smaller faults. Locations of greatest extension are used to identify regions of potential subsidiary

79 normal faulting. Smaller faults may be subsidiary to, and younger than, larger structures if they strike parallel to ε2 (perpendicular to ε1) and occur in the vicinity of positive strain perturbations. On the contrary, slip should be inhibited on normal faults in the extensional strain shadows of larger faults. Such faults may be older and inactive or less active. Our analysis focuses on strain perturbations at the earth's surface. If the faults initiated by shearing and linkage of opening joints [e.g. Davatzes and Aydin, 2003; Kaven and Martel, 2007], it is reasonable to assume they initiated from the cooling joints within the Bishop Tuff. If they initiated in shear, failure would be most favorable near the earth's surface, where confining stress is least. It is possible that many Tableland faults pre-date the deposition of the Bishop Tuff, in which case they would have propagated up from its base [Bateman et al., 1965]. If faults have propagated up from depth, it might be more appropriate to consider subsidiary faulting in the context of strain perturbations at the base of the tuff. However, in cross-sections (Figure 2.5c & d; Figure 2.6) the lateral distribution of strain perturbations is broadly similar throughout the upper few hundred meters, particularly in the elasto-plastic model. This suggests that conclusions based on comparison of observed fault density with model strain perturbations at the earth's surface would hold whether faulting initiates at the surface or anywhere within the upper few hundred meters. Because relative ages of individual Tableland faults are unavailable, we consider a model in which all faults slip simultaneously. The model does not address the propagation of faults, but starts with present day fault geometry.

Paleostress analysis Far-field boundary conditions for the three forward models used for analysis of strain perturbations are derived from a mechanics-based paleostress inversion [Kaven, 2009]. Fault slip distributions may be determined from scarp offsets at the earth's surface, but subsurface slip distributions are unknown. A complete slip distribution is necessary to determine strain perturbations throughout the continuum. We use elastic models, assuming complete shear stress drop across all elements, to calculate a

Figure 2.6: Maximum principal strain for the model representing Tableland faults, including FSF and basin-bounding faults, on cross-section A-A' (Figure 2.7a). Vectors represent principal strain orientations: ε1 = white; ε2 = blue; ε3 = red. Black lines indicate faults included in the model.

81 reasonable, though estimated, slip distribution in the subsurface [Crouch and Starfield, 1983]. Models are governed by homogeneous far-field stress, which resolves tractions on each fault element. Paleostress analysis determines this far-field stress to optimize model slip where throw may be measured in the field. The paleostress analysis method is based on the principles of Poly3D. Faults are discretized as triangular elements of constant displacement discontinuity in a homogeneous linear elastic half-space, and least squares optimization procedure is implemented to solve for the three horizontal components of the far-field stress tensor that minimize the residual difference between modeled and observed slip on elements where slip is known. "Modeled slip" is the slip on each element calculated by the Poly3D forward model, given far-field stress boundary conditions and assuming complete shear stress drop on every element. Fault surface elements on which slip is unknown are not directly included in the optimization procedure, but are allowed to slip such that their mechanical influence is felt throughout the model. One principal stress component is assumed vertical because the earth's surface is effectively traction- free (σxz = σzx = σyz = σzy = 0), and this vertical stress may be assumed zero (σzz = 0) because models are linear elastic and frictionless. It is unnecessary to account for lithostatic loading because only the deviatoric component of the stress tensor influences slip magnitude and distribution [Kaven, 2009]. It is important to recognize that paleostress analysis is simply a means of determining optimal far-field boundary conditions to generate appropriate slip distributions in 3D elastic models. Just as elastic stress magnitudes in Figure 2.5a are an order of magnitude greater than the tensile strength of rock, the magnitude of far- field tension suggested by paleostress analysis is similarly large and unphysical (77 MPa, to be specific). Since far-field stress and strain may be equated by Hooke's law in elastic models, the application of paleostress analysis to cumulative fault slip in regions such as the Volcanic Tableland might be more appropriately thought of as 'paleo-strain' analysis because the equivalent far-field strains are much more geologically appropriate. However, to avoid confusion we adhere to traditional usage of the term "paleostress" inversion.

The fault model used for paleostress inversion includes the three basin bounding faults, FSF, the 12 other large Tableland faults, and one additional fault, which has slip locally, but not consistently, greater than 20m. This fault, located in the southwest corner of the Tableland (Figure 2.1) strikes NE-SW, roughly 45° oblique to nearby north-south striking faults, and therefore provides constraint on the north-south component of the far-field boundary conditions. Such constraint is otherwise lacking because the component of north-south stress resolved as traction on subparallel elements is inconsequential. Therefore, unacceptably large magnitude of σyy and, correspondingly, εyy (note: positive x is east and positive y is north) does not significantly influence model slip magnitude. Paleostress inversions that do not include this fault are characterized by ill-conditioned matrix equations and optimal far- field stress and strain may be characterized by maximum tension/extension oriented north-south, subparallel to fault strike. Assuming slip on the faults is predominantly dip-slip, maximum extension should be nearly perpendicular to fault strike [Anderson, 1951], and under no circumstances should the maximum tension/extension be subparallel to the average fault strike. There are about five other faults of comparable size across the study area that are not considered in any numerical analysis. Slip distributions used for the paleostress optimization procedure are prescribed on those elements of the Tableland faults that intersect the earth's surface and are within the lateral extent of the ALSM data (Figure 2.1). Dip-slip is equated to throw, which may be readily measured as the vertical offset of topography across a fault scarp between footwall and hanging wall, and strike-slip is prescribed as zero on the same elements, as suggested by field data [Bateman et al., 1965; Dawers et al., 1993; Pinter and Keller, 1995]. Slip is not prescribed on fault surface elements beyond the extent of the ALSM data and they are not directly included in the optimization procedure, but their mechanical influence affects slip on all elements. The one exception is the extra, northwest-southeast striking fault (gray on Figure 2.1) that is beyond the extent of the ALSM data, but for which it is necessary to prescribe slip. If slip were not prescribed, this fault would not directly influence the optimization procedure, and therefore would not have the desired effect of

83 constraining the north-south component of the stress tensor. Slip on this fault is derived from throw measured from the National Elevation Dataset (NED) 1/3 arc- second (~10m) DEM, which lacks the decimeter precision and 2m resolution of ALSM, but nonetheless may adequately constrains throw. The NED DEM has local RMS accuracy 1.64m [Maune, 2007]. Due to limitations of the numerical model, perhaps the elastic constitutive law or simplified fault geometry, a homogeneous far-field stress tensor that generates appropriate slip magnitudes on Tableland faults, including FSF, results in slip magnitudes roughly three times greater than those suggested by geologic evidence on the basin bounding faults since the deposition of the Bishop Tuff. Thus, for paleostress inversion and subsequent models investigating strain perturbations resulting from fault slip, we found it necessary to prescribe slip on the basin bounding faults, rather than allowing them to slip freely with complete shear stress drop. For paleostress analysis, this prescribed slip, unlike prescribed slip on Tableland faults, is used only to calculate the magnitude of stress perturbations associated with slip on basin bounding faults and the resulting slip induced on the Tableland faults that slip freely. Slip magnitude on basin bounding faults is not part of the optimization criteria. Since measurements of slip magnitude are available only at a few points where the faults intersect the earth's surface [Berry, 1997; Kirby et al., 2006], we estimate three dimensional slip distributions by prescribing a preliminary paleostress tensor to a forward model with only the basin bounding faults (stress perturbations due to slip on the much smaller Tableland faults are relatively inconsequential) and then applying a constant scaling factor (determined independently for dip-slip and strike-slip components on each of the three faults) to adjust slip magnitudes appropriately. Far-field principal stress components (tension positive) suggested by Paleostress analysis are 77.5 MPa, oriented N59.8°W and 40.5 MPa, oriented N30.2°E, which equate to much more reasonable far-field strain magnitudes of 2.7% and 0.8% in the same respective directions.

Forward model results Maximum principal strain magnitude at the earth’s surface, as calculated by Poly3D forward models, is presented in Figure 2.7a-c. Results represent effects of slip on: a) large (consistently greater than 20m throw) Tableland faults, including FSF, and basin bounding faults (WMF, RVF and HCF); b) FSF and basin bounding faults; and c) basin bounding faults only. From here onward, these three models will be referred to as models a-c, respectively. In all three models, significant extensional strain shadows are observed (extension is inhibited) in the near-surface in the footwalls and hanging walls of all faults, and strain magnitudes across the Tableland are generally less than the applied far-field strain. Because of its size (length and depth), slip on FSF is greater and its strain shadow is larger in both magnitude and area relative to smaller faults in model (a). Positive strain perturbations are observed almost exclusively in the vicinity of fault segment tips. The only other positive strain perturbation in any of the three models is observed in the vicinity of point 1, in model (a). It is subtle, and its extent is limited. Strain perturbations due to slip on FSF and basin bounding faults are negative (extensional strain shadow) throughout the study area (models b & c). Maximum principal strain on an east-west striking vertical cross section from model (a) is presented in Figure 2.6. We note that the strain perturbations in the vicinity of the two faults between 373,000 and 374,000m easting is significantly different from that suggested by the 2D elastic model (Figure 2.5c). This occurs because the cross-section intersects these faults very near their lateral tip-lines, and emphasizes the importance of considering a 3D model. In all models, maximum principal strain is generally oriented ESE-WNW, subparallel the far-field strain; however, some counterclockwise rotation is evident, particularly in the northern portion of the study area and in the vicinity of FSF. The orientation of maximum extension is generally perpendicular and intermediate principal strain parallel to fault traces in the southern portion of the study area, where faults strike slightly east of north. This conforms to Andersonian fault theory [Anderson, 1951], according to which faults should strike parallel to the intermediate

Figure 2.7 (opposite): (a-c) represent maximum tensile stress magnitude at the earth’s surface. Bold black lines represent faults included in each model and thin black lines represent smaller fault traces. Vectors represent principal stress orientations: σ1=white; σ2=blue; σ3=red. Models represent stress associated with slip on a) all seismic-scale Tableland faults and basin bounding faults (not on map), b) FSF and basin bounding r faults, and c) basin bounding faults only. The sign convention is tension-positive. ε1 on the colorbar represents the far-field maximum extension. Larger values are positive perturbations (enhanced extension) and lesser values represent negative perturbations. (d-f) represent spatially averaged fault density, excluding bold faults that are included in models (a-c), respectively. White shaded region in (d) represents area where Fish Slough has eroded the welded cap of the Bishop Tuff and fault scarps may be obscured.

87 principal stress. In elastic models, stress and strain tensors have the same orientation. This need not be the case in nonlinear materials; however, it is a close approximation if the orientation of far-field boundary conditions is constant throughout a simulation. However, to the north, scarps generally strike slightly west of north. Counterclockwise rotation of the model strain tensor is insufficient to account for the change of fault strike and model stress is 15-30° oblique to fault traces. Perhaps right lateral strike-slip is greater on Tableland faults to the north; conclusive evidence either way is lacking. Spatially averaged fault density representing faults identifiable in hillshade images generated from LiDAR data is presented in Figure 2.7d-f. Each figure represents the density of those smaller (potentially subsidiary) faults that are not represented in corresponding numerical models (a-c), respectively. Fault density is calculated by the method of Wu and Pollard [1995], using a 1000m by 1000m moving window. Along the southern margin of the study area (Figure 2.7) the Bishop Tuff has been eroded and fault scarps are not preserved. The three plots of fault density are generally difficult to distinguish because the majority of faults are smaller than those included in models; however, density plotted in Figure 2.7(d-f) represent respectively increasing numbers of faults, as fewer faults are represented in each of the corresponding numerical models. Noteworthy features of the fault density plots are the lack of small faults in the hanging wall of FSF, particularly in the northern portion of the study area, and the greatest density of small faults in the south-central portion of the study area. In the footwall, east of FSF, we observe no faults in the north, but a dense cluster of faults in the south. The absence of observable fault traces immediately west of FSF may be partially attributable to greater fluvial weathering rates in Fish Slough (white shaded area in Figure 2.7d), but because the fault system remains active [Lienkaemper et al., 1987], we might expect to see some evidence if significant faulting were active in this region.

Discussion Maximum principal strain perturbations across the Tableland are generally negative (i.e. model strain is generally less than far-field strain). The region identified

88 by the number 1 in model (a), which is the only positive strain perturbation not associated with a fault tip in any of the three models discussed, correlates with relatively low fault density (Figure 2.7d). This observation, as well as the ubiquitous extensional strain shadow in models (b) and (c), suggests that secondary faulting, in a strict sense, may not be a dominant player in the evolution of the Tableland fault system. By a rigorous definition, subsidiary faulting should be expected only where tension/extension are enhanced by slip on pre-existing faults; thus, subsidiary faulting cannot be rigorously supported by elastic models representing the Tableland. However, as discussed in the comparison of 2D elastic and elasto-plastic models, regions of enhanced extension tend to be more abundant and larger in magnitude and extent when plastic yielding is allowed. Regions of enhanced extension may actually be more pervasive than suggested by our elastic models of the Tableland, and subsidiary faulting may not be suppressed to the extent that models suggest. Nonetheless, the primary effect of fault slip is to accommodate extension, and therefore reduced extension should be the primary effect on continuum strain. It is unlikely that regions of enhanced extension are widespread across the Tableland, or that the presence of a few such regions would significantly change the conclusions of this study. We have demonstrated that elastic models may provide valuable insight to relative strain magnitude in the near-surface. We may make important inferences regarding the mechanical influence of larger faults on the distribution of smaller by using elastic models, regardless of whether the larger faults actually enhance the development of smaller faults in some areas or if their sole effect is to inhibit the development of small faults more in some regions than in others. Correlations are evident between fault density and strain intensity. Such correlations indicate mechanical interaction between faults at many scales and may provide insights to the processes of fault initiation and growth. A significant strain shadow associated with slip on the basin bounding faults is clearly visible in results of model (c), despite being almost entirely obscured by local

89 perturbations due to slip on smaller faults in models (a) and (b). Its shape and extent, particularly the way in which it bulges westward in the north, correlates remarkably well with the region of low fault density west of FSF, and to the east of FSF in the northern part of the study area. It seems likely that slip on the basin bounding faults has inhibited the initiation and growth of smaller faults in much of this area. However, this strain shadow is at odds with the location and size of FSF and with the abundant faulting in southern portion of the footwall of FSF. Slip on FSF generates a large (magnitude) extensional strain shadow (clearly identifiable in models (a) and (b)) that extends roughly 1km west in the hanging wall and east to the edge of the study area in the footwall. This strain shadow correlates well with the low fault density in the immediate hanging wall of FSF, and with low fault density in the northern part of the footwall, suggesting that perhaps slip on FSF inhibits growth and initiation of smaller faults in these regions. Noteworthy correlations between strain shadows and reduced fault density suggest that the combined effects of slip on the basin bounding faults and FSF may inhibit initiation and growth of small faults throughout much of the eastern portion of the Tableland; however, exceptions are significant. The dense cluster of faults in the southern footwall of FSF is even more anomalous in the context of this stress shadow than in the context of the stress shadow of the basin bounding faults. The cluster of small faults suggests that east-west extension in this region should be relatively great in all models, but particularly model (b). Trace geometry clearly indicates that the southern portion of FSF has developed by coalescence of echelon segments [Ferrill et al., 1999]. Perhaps the cluster of faults to its east developed prior to coalescence of FSF, when the strain shadow was smaller, and these faults are less active today, or perhaps the linked echelon segments suggest geometric complexity at depth, which might be omitted from our models and could enhance near-surface strain perturbations. This fault cluster could also indicate a limitation of the elastic model; however, the 2D elasto-plastic model indicates a similar strain shadow in the footwall of FSF.

Additionally, slip on FSF appears to inhibit the westward bulge of the strain shadow of the basin bounding faults that is prominent in model (c) and that correlates well with low fault density several kilometers west of FSF in the northern portion of the study area. Extension in this region is greater in the model that includes FSF than in the model with basin bounding faults only; the effect of FSF is to enhance extension here. This is puzzling because FSF must be among the oldest structures on the Tableland in order to have grown so large, yet its strain perturbation (Figure 2.7b) here seems to favor the growth and initiation of smaller faults where clearly there are few. Locally low fault density is observed around most of the smaller (but >20m throw) Tableland faults included in model (a) (Figure 2.7d). Though less pronounced, these regions of low fault density persist in Figure 2.7e, indicating that they are not solely a result of excluding all modeled faults from the density calculation in Figure 2.7d. These regions of low fault density correlate relatively well with the strain shadows associated with those Tableland faults with consistently greater than 20m throw, which are included in model (a). Although their strain shadows are much smaller than those associated with FSF and basin bounding faults, these faults do appear to influence the locations of smaller, subseismic-scale structures by inhibiting their initiation and growth. Because fault growth on the Tableland appears to be a scale-independent process [Dawers et al., 1993], and because the laws of mechanics dictate that smaller faults will generate similar, but spatially smaller, strain shadows (assuming the same down-dip profile) [e.g. Pollard and Segall, 1987], we may conclude that similar processes of fault inhibition occur at smaller scales, as at the scale of the faults with greater than 20m throw represented in model (a). It appears that extensional strain shadows in footwalls and hanging walls of Tableland faults at all scales inhibit the growth and initiation of relatively smaller faults to a significant degree, despite noted exceptions. In parallel with the rough correlation between strain shadows and regions of low fault density, regions of greatest fault density (Figure 2.7d-f) correlate loosely with the regions of greatest extension (Figure 2.7a-c). Particularly in the south-central portion of the study area, both fault density and maximum principal strain are near

91 maximum values in all three models. Maximum strain in all three models is similarly large, though generally slightly less, further north where fault density also is only slightly less. However, as noted previously, these regions are generally characterized by reduced extensional strain shadows, as opposed to enhanced extension. Given the abundant evidence for fault growth by coalescence, as well as clear evidence that slip on large faults effects smaller faults primarily through resulting strain shadows that inhibit growth and initiation of smaller structures, we suggest that the Tableland fault system initiated as a broad array of small faults, but that the distribution was likely not homogeneous. The basin bounding faults have been active since long before the deposition of the Bishop Tuff, and its strain shadow, as observed in Figure 2.7c, has likely inhibited fault initiation and growth in the vicinity of Fish Slough and FSF, particularly in the northern part of the study area, since the deposition of the tuff and inception of the fault system. The absence of faults further south in the hanging wall of FSF correlates with the strain shadow of FSF. We suggest that FSF, too, may pre-date the deposition of the Bishop Tuff, in which case it likely propagated quickly through the young tuff, and its strain shadow, like that of the basin bounding faults, has inhibited the growth of smaller faults immediately to its west. It may, however, have been more segmented and discontinuous soon after the tuff's deposition, considering the breached relays evident in the trace geometry [Ferrill et al., 1999] and that its strain shadow seems not to have inhibited fault development in its footwall in the southern portion of the study area. Both FSF and the basin bounding faults have probably remained active through the past 750 ka, and their strain shadows have continued to inhibit the initiation of smaller faults. Across the rest of the study area, away from the primary strain shadows of basin bounding faults and FSF, early fault initiation was likely relatively homogeneous, though it may always have been slightly greater to the south where models indicate the strain shadow of the basin bounding faults is slightly less influential. As these faults grew, they interacted mechanically [Dawers et al., 1993; Willemse et al., 1996] and began to coalesce [Dawers and Anders, 1995; Ferrill et al., 1999]. With the progression of coalescence, larger faults developed larger strain

92 shadows, which inhibit fault growth over greater areas. Thus, as the fault system has matured faults have continued to grow and perhaps initiate due to tectonic loading where they are not inhibited by larger faults, but growth and initiation of smaller faults is inhibited in the immediate footwalls and hanging walls of newly coalesced larger faults. However, our models consider the fault system only in its present state. Some, though few, small faults are observed clearly within strain shadows of larger faults. These small faults are likely old, having developed prior to the maturation of the faults whose strain shadows they are in today. Such small faults are likely inactive or less active today. Because relatively few small faults are observed within the calculated strain shadows of larger faults, we hypothesize that most of Tableland faults are not inhibited by larger faults and remain active today, but that as faults continue to coalesce and strain shadows continue to grow, more and more faults will see their growth inhibited by a few dominant large faults. Basin evolution will likely be characterized by progressive strain localization, in a manner similar to that suggested by Cowie, et al. [2000].

Conclusions The conceptual model of fault growth by coalescence has been well- documented on the Volcanic Tableland; however, the means to distinguish between fault growth by coalescence and fault growth by subsidiary faulting is poorly understood. These two conceptual models have distinct implications for the relative ages of faults and for mechanisms of basin evolution. Coalescence implies that small faults are older structures whose activity is inhibited by stress shadows of larger faults, and that deformation progressively localizes on a few large faults. Subsidiary faulting, on the other hand, implies that small faults are younger, and that deformation may become more diffuse with time. We examine slip distributions on long, linear, continuous faults to demonstrate that coalescence is a dominant mechanism of fault growth even when trace maps may not provide clear evidence. We then use a rudimentary linear elastic inversion to constrain subsurface fault geometry from faulted topography and use linear elastic

93 models to calculate 3D strain perturbations associated with slip on large faults across the Volcanic Tableland. Previous studies of subsidiary faulting have focused on elastic stress perturbations as an indicator of failure; however, elasticity is a poor representation of rock deformation accumulated over geologic time scales. Elastic models generate tensile stresses in the area of interest an order of magnitude greater than the tensile strength of rock, whereas geological stress is primarily compressive. A comparison of 2D elastic and elasto-plastic models demonstrates that elastic stress distributions may bear little resemblance to those modulated by plasticity because inelastic deformation suppresses large stress perturbations in the earth. Ideally, we could use 3D inelastic mechanical models for this study, but such methods are computationally prohibitive, while elastic models are relatively efficient and stable. Comparison of strain distributions from 2D models demonstrates that elastic models may provide valuable geological insights in the near surface, at depths shallower than fault tips, which is the primary area of interest for this study. Strain perturbations suggested by elastic models are almost entirely negative (extensional strain shadows), except in the immediate vicinity of fault tips. This may be partially a limitation of the elastic models, but suggests that subsidiary faulting, in a rigorous sense, is likely not a dominant mechanism of fault growth on the Tableland. However, numerical models clearly demonstrate mechanical interaction between fault segments at all scales investigated. Correlation between regions of greater extension and greater fault density are significant. Similarly, strain shadows correlate with low fault density, despite a few noted exceptions. Evolution of the fault array is likely driven by tectonic loading, as opposed to subsidiary faulting, and the strain shadows of larger faults appear to inhibit the initiation and growth of smaller faults nearby. FSF, however, being the largest fault in the study area, is a puzzling case, as it is located in an apparently large strain shadow of the basin bounding faults. We suggest that the Tableland fault system initiated as a broad array of small faults early after the deposition of the Bishop Tuff, but that since its inception, the distribution of faulting has likely been influenced by strain shadows of basin bounding

94 faults and perhaps FSF. Strain shadows associated with the largest faults across the rest of the Tableland remain relatively small, but smaller-scale faulting appears to be inhibited in their vicinity. Processes of fault inhibition by neighboring larger faults probably occurs at all scales on the Tableland. As faults across the Tableland continue to coalesce, more small faults will fall into a few larger strain shadows, and the system will likely come to be dominated by a few large faults.

Acknowledgements The authors thank Betsy Madden for assistance with field mapping, Franz Maerten and IGEOSS for sharing their PaleoStress code and for assistance with paleostress inversion, and Chevron for permission to publish this work. Airborne LiDAR data courtesy of Chevron Corporation. PL has been supported by Chevron through a summer internship, the National Science Foundation’s Collaborations in Mathematical Geosciences grant no. CMG-0417521, and the Robert and Marvel Kirby Stanford Graduate Fellowship.

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Lohman, R. B., and J. J. McGuire (2007), Earthquake swarms driven by aseismic creep in the Salton Trough, California, Journal of Geophysical Research-Solid Earth, 112(B4), -.

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Chapter 3

Pitfalls among the promises of mechanics-based restoration: addressing implications of unphysical boundary conditions

Abstract Mechanics-based restoration has been seen by some in the structural geology community as a panacea -- a new technology that melds the retrodeformational merits of traditional kinematic restoration and geometric balancing techniques with principles of continuum mechanics including conservation of mass, momentum and energy, and implementation of the well-established linear elastic constitutive law to relate stress and strain. The method has been touted for its ability to simulate complex 3D deformational systems with no assumptions of plane strain, and allowing for heterogeneous fault slip distributions and mechanical interaction of fault segments. Further, mechanics-based restoration has been suggested as a means to predict distributions of geologic strain and associated small-scale structures (e.g. joints, veins, compaction bands, solution seams or small faults); however, we demonstrate theoretically and through analysis of synthetic models and a field based example from the extensional fault system in the Volcanic Tableland, Bishop, CA, that the kinematics of restoration models may differ significantly from forward deformation. Restoration models are governed by boundary conditions that are fundamentally different from the forces driving forward geologic deformation. These models may be improved by supplementing the restoration boundary condition with loads that attempt to reverse tectonic forces, which are determined by paleostress analysis, but unphysical artifacts persist. Mechanics-based restoration may be an appropriate tool for traditional applications of kinematic models including validation of structural interpretation and schematic illustration of the evolution of complicated structures; however, more subtle kinematic features, particularly the distribution of strain perturbations, should be treated with skepticism. Restoration models may provide insights to the initial configuration of forward mechanical models that are governed by physically appropriate boundary conditions and that may incorporate non-linear

107 material behavior. Such forward models provide the best means for simulating deformation and predicting associated small structures.

Introduction Structural geologists are commonly faced with the challenge of interpreting the present day geometry of complex structures from sparse (e.g. well logs and outcrops) or low resolution (e.g. seismic imaging) data, and with the more daunting task of understanding the evolution of the structure through geologic time from data that is limited to the present. Hence comes a strong appeal for retrodeformational modeling techniques that provide a means to back-step (restore) from present geometry through the structure's evolution, and a means to evaluate the present-state geometric interpretation, by ensuring that the model may be restored to a reasonable initial configuration. The concept of structural restoration was developed along with the concept of kinematically balanced (fully restorable) cross sections during the 1950's and 1960's [Hunt, 1957; Carey, 1962; Bally et al., 1966], and was formalized by Dahlstrom [1969]. Dahlstrom's method, designed to model compressional tectonics in the Alberta, Canada foothills, assumes that deformation may be modeled as plane-strain, that cross-sectional area within strata and line length of contacts between strata are conserved throughout deformation, and that all off-fault strain is accommodated entirely by bed-parallel slip. In the decades since Dahlstrom's seminal paper, a plethora of ad hoc, empirically derived kinematic mechanisms (e.g. vertical shear [Gibbs, 1983; Williams and Vann, 1987], inclined shear [White et al., 1986; Dula, 1991], and tri-shear [Erslev, 1991; Hardy and Ford, 1997]) have been developed that may explain nearly any structure in any geologic environment, worldwide. Kinematic methods have become a standard tool for retrodeformational and forward modeling in structural geology, particularly at reservoir and larger scales [e.g. Woodward et al., 1989], and have proven invaluable to structural analysis in the hydrocarbon industry; however, they invoke no causative physical principles and therefore cannot explain how or why a structure evolved as it did [Fletcher and Pollard, 1999].

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Additionally, the simplified, ad hoc nature of the kinematic mechanisms governing internal deformation suggest strain fields that may not accurately reflect the physical processes that govern rock deformation. For example, kinematic models assume rock is incompressible (conservation of area in cross-section) and therefore cannot accommodate volumetric strain that is pervasive in nature, as evidenced by the abundance of contractional and dilational structures including joints, veins, pressure solution seams, and shear and compaction bands [e.g. Pollard and Fletcher, 2005]. As exploration moves to increasingly complicated and hard-to-reach structures, interest in subsurface stress and strain history has piqued, with a growing desire to predict small faults, fractures, and other structural features that might influence fluid flow. We seem to have reached the limits of what we may learn from purely kinematic methods. Forward modeling approaches based on the principles of continuum mechanics overcome these limitations. Mechanical models, commonly implemented numerically in finite element, boundary element or distinct element formulations, honor conservation of mass, momentum and energy, and employ a constitutive law to relate stress to strain or rate of deformation. Such methods have been applied to predict displacement, strain and stress fields associated with reservoir scale fault-related folding [Erickson and Jamison, 1995; Hardy and Finch, 2005; Morgan and McGovern, 2005; Maerten et al., 2006; Benesh et al., 2007; Sanz et al., 2007; Sanz et al., 2008], but have the disadvantage that present-day geometric observation may not be explicitly incorporated into the model. One must begin with estimates of initial geometric configuration and loading path, and it may be necessary to iteratively adjust in order to obtain a final configuration that resembles geometry observed in the field. Mechanics-based restoration methods have been developed by structural geologists over the past decade as a new tool in the arsenal for structural interpretation and evaluation [Muron, 2005; Maerten and Maerten, 2006; Moretti et al., 2006; Plesch et al., 2007; Moretti, 2008; Guzofski et al., 2009]. The method, which is described in detail by Maerten and Maerten [2006] and Guzofski et al. [2009], involves generation of a finite element (FEM) volume, composed of discrete

109 tetrahedral elements, which honors the present-day observed geometry, followed by restoration, implemented by boundary conditions that flatten one or more deformed stratigraphic horizons to an assumed initial (unfolded and unfaulted) configuration. Those faults explicitly incorporated in the mesh are allowed to slip during restoration such that shear traction is relaxed completely as deformation is propagated through the continuum according to a linear elastic constitutive law. Elasticity, while a great simplification of real rock rheology, lends itself to restoration because it is entirely reversible. The application of mechanics to restoration represents a novel and necessary innovation in a field traditionally dominated by kinematic methods and seems, on the surface, to be a panacea, coupling the retrodeformational benefits of kinematic models with the physical backbone of continuum mechanics. Mechanics-based methods offer improvement over traditional kinematic methods by (1) honoring conservation of mass and momentum and employing a constitutive law to derive the kinematic variables, (2) incorporating material heterogeneity in the form of spatially variable elastic moduli, (3) admitting non- uniform slip on faults, and (4) accounting for mechanical interaction of fault segments. Particularly for 3D applications in structurally complex regions, the method appears to offer better constrained solutions than kinematic methods. Because these models employ a constitutive law to define a stress-strain relation, theoretical stress fields may be derived from restoration methods. If restoration derived stress, strain, and displacement fields, each of which is mathematically dependent upon the others [Chou and Pagano, 1967], accurately reverse forward deformation, it should be possible to identify regions of intensified stress and strain, as well as principal stress and strain orientations, from mechanics-based restoration models. To date, it has been assumed that the kinematic properties (strain and displacement) predicted by mechanics-based restoration models are equal in magnitude and opposite in sign to forward deformation. However, despite the mechanical basis of the new restoration methods, unphysical boundary conditions and the limitations of linear elasticity to model rock deformation prove detrimental. We use a series of synthetic and field based examples from the Volcanic Tableland,

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Bishop, CA, to illustrate the theoretical and practical limitations of boundary conditions applied to mechanics-based restoration models. We suggest several ways to improve linear elastic retrodeformational models, but demonstrate that large, unphysical strain perturbations may be unaviodable when restoration boundary conditions are prescribed and strain predicted by restoration models should be treated with skepticism. Appropriately sophisticated forward mechanical models, driven by realistic tectonic loading paths, are the best way to predict geologic stress and strain. Mechanics-based restoration, for the reasons described previously, provides improvement over kinematic methods, and should be used to validate structural interpretations and as a guide to infer appropriate initial geometric configuration for forward models. A subsequent manuscript will address the limitations of the linear elastic constitutive law. Elasticity results in unphysical stress magnitudes in many geologic models, and we therefore address only strain and displacement fields in the remainder of this study. Before continuing, it is necessary to clarify terminology used throughout the remainder of this manuscript. "Restoration" is used to describe kinematic (displacement) boundary conditions applied directly to the earth's surface or some deformed horizon, which force that surface to conform to a presumed undeformed state, and to models governed by such boundary conditions. "Reverse" refers to boundary conditions (displacement or traction) that attempt to invert tectonic loading, and are applied to the vertical boundaries of the finite element volume. "Retrodeformation" is used to refer generically to any model (restoration or reverse) that simulates deformation from the present state to some previous state.

Theory Linear elastic deformation is, by definition, perfectly reversible. The principle of superposition, as applied to linear elasticity, states that "two (or more) stress [or strain or displacement] fields may be superposed to yield the results for combined loads" [Chou and Pagano, 1967, p. 82]. Because all of the governing equations are linear, stress, strain and displacement fields are path-independent in linear elastic systems. Therefore, an elastic body deformed by an arbitrary load configuration will

111 return to its initial configuration if those loads are exactly reversed (equal magnitude, opposite orientation). Effectively, this is equivalent to unloading, but it also implies that if one begins with a body geometrically equivalent to the deformed configuration, but free of stress, and applies the reverse load configuration, that body will have the same shape as the original stress-free body, and will be characterized by stress, strain and displacement fields with identical magnitude but opposite orientation to those characterizing the forward deformation. Any non-linear properties of a mechanical system, including material models (non-linear stress-strain relation) and large deformation (non-linear strain-displacement relation) introduce load path dependency, and therefore forward and retrodeformation will differ. Additionally, linear elastic deformation, being reversible, does not increase the entropy of a closed system [e.g. Wegner and Haddow, 2009, p. 119]. In other words, none of the input work is lost to heat or other irreversible processes such as damage or plastic strain that increase the disorder of the system. Because entropy may not be removed from a closed system, it is assumed that retrodeformation is best performed by a model that results in no additional increase of entropy. Faults are treated as frictionless in all reverse models because frictionless sliding, like linear elasticity, is reversible. Even the simplest Mohr-Coulomb friction law [e.g. Byerlee, 1978; Jaeger et al., 2007, p. 69] is non-linear and therefore irreversible. The issues relating to friction in reverse models may be understood by considering the stress state of compressional vs. extensional tectonic settings. While vertical stress is always approximately lithostatic, horizontal compressive stress may be nearly twice as large in a compressional tectonic setting as in an extensional setting [Zoback, 2007]. Normal tractions on fault surfaces are greater, and therefore slip is inhibited by friction in a compressional setting. A reverse tectonic model of an extensional system effectively simulates compressional deformation; thus, equivalent reverse tectonic stress or strain will result in less slip. The opposite holds true for reverse models of compressional tectonics. While real faults are not actually frictionless, previous studies [e.g. Aydin and Schultz, 1990; Cooke and Marshall, 2006] suggest that the effects of uniform friction do not have a strong influence on

112 mechanical interaction of fault segments, but only on the magnitude of slip. Hence, most aspects of deformation associated with frictional fault slip may be approximated by models that treat faults in a simplified manner as frictionless surfaces. Previous authors considering mechanics-based restoration have acknowledged that such models cannot be used to derive geologic stress, primarily because the elastic constitutive law results in stress magnitudes far greater than those that may be sustained by rock [e.g. Maerten and Maerten, 2006; Guzofski et al., 2009]. However, limitations of elasticity aside, restoration models are further plagued by loading that does not exactly reverse the tectonic loads that drove forward deformation. Unphysical boundary conditions applied to restoration models have considerable implications for both stress-state and kinematics. The fundamental restoration boundary condition consists of vertical displacement imposed on the stratigraphic horizon of interest, which restore it to some presumed undeformed datum [Maerten and Maerten, 2006; Plesch et al., 2007; Moretti, 2008; Guzofski et al., 2009]. Displacement boundary conditions imply surface tractions that are mechanically necessary to achieve the displacements. Additional boundary conditions are necessary at least at one point in order to prevent rigid-body translation and rotation, and to define a stable, well-posed boundary value problem. To date, most mechanical restoration efforts have kept additional boundary conditions to a minimum [Maerten and Maerten, 2006; Plesch et al., 2007] or constrained some bounding walls of the FEM mesh to in-plane displacement [Guzofski et al., 2009], while leaving other vertical and horizontal walls of the model as traction-free surfaces. In contrast, forward geologic deformation is driven by tectonic forces in the subsurface, while the earth's surface is generally treated as traction-free. Thus, the issues concerning boundary conditions in current mechanics-based restoration methods are two-fold. First, the tractions implied by the restoration boundary condition are a direct violation of the traction-free nature of the earth's surface. Second, treating other surfaces of the finite element model as traction-free fails to reverse the tectonic loading which drove forward geologic deformation.

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Mechanics-based restoration models force the kinematic principle of surface restoration, but simulate a different mechanical system from forward deformation. Because the boundary value problem is solved such that the equation of static equilibrium

∑Fi = 0 (1) is satisfied at all points in the volume, the restoration model, characterized by different surface loads than the true reverse problem, will be characterized by a different stress field throughout the volume. Because stress is related to strain by Hooke's law, resulting strains are not calculated correctly, and because strain is the derivative of displacement, calculated displacement is correspondingly erroneous.

Synthetic Example The reversible nature of elasticity given precise reversal of forward loads and the qualitative implications of unphysical restoration boundary conditions are illustrated by a 2D synthetic example with a single normal fault (Figure 3.1) in a homogeneous elastic body. Deformation is driven by 10% horizontal extension. The model base is fixed vertically and the upper surface is traction-free. The fault surface is allowed to slip without frictional resistance, but separation and interpenetration are prohibited. 2D models are performed using the finite element code ANSYS® (http://www.ansys.com). It is important to note that here, and throughout this manuscript, strain from retrodeformational models is calculated in a forward deformational sense. The mesh is retrodeformed according to the calculated displacement vectors and then the sign of displacement vectors is changed to reflect forward deformation prior to calculating displacement gradient and strain components. Magnitude and orientation of maximum principal strain calculated from forward and reverse models are presented in Figure 3.1a & b, respectively. As predicted by the linear elastic principle of superposition, deformation fields predicted by forward and reverse elastic models are very similar, although not identical due to numerical imprecision and non-linear geometric effects. The only significant difference in either magnitude or orientation is in the footwall, near where the fault

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Figure 3.1: Model configuration and results representing slip on an idealized frictionless normal fault in a two-dimensional elastic body for a) forward model, b) reverse model, c) restoration model with vertically constrained base, and d) restoration model with unconstrained base. Top: schematic illustrations of the initial model configuration and boundary conditions (left) and the deformed model configuration (right). The initial geometric configuration of retrodeformational models (b, c & d) is equivalent to the deformed configuration of the forward model (a). Bottom: Maximum principle strain magnitude and orientation calculated for forward deformation. The gray dashed box in b, c & d represents the outline of the undeformed forward model.

115 reaches the surface, where notably larger strain magnitude is suggested by the reverse model. The gray dashed line in Figure 3.1b represents the outline of the original forward model, prior to deformation. The deformed geometry, subjected to reverse boundary conditions, matches the initial undeformed geometry almost perfectly. In contrast to the similarity between strain distributions calculated from forward and reverse models, strain distributions calculated from restoration models (Figure 3.1c & d) differ significantly. We present two different restoration models, one in which vertical displacement constraints are imposed on the model base, and one in which the base is treated as a free surface. In the case of these synthetic examples, we are fortunate to know the initial thickness of the undeformed model exactly, which enables us to impose constraints on the model base that accurately restore the initial thickness. When dealing with real geologic data, the undeformed thickness of a single stratigraphic unit or package of units is probably difficult to constrain. For this reason, published examples of mechanics-based restoration [e.g. Maerten and Maerten, 2006; Plesch et al., 2007; Guzofski et al., 2009] have treated the model base as a traction- free surface. Field examples investigated later in this study do the same. Although neither restoration model accurately represents the strain distributions associated with forward deformation, imposing thickness constraint does provide a slightly improved result, particularly regarding principal strain orientations. In the restoration model with a free base, neither orientation nor magnitude of model strain represents forward deformation; however, when the base is constrained to enforce appropriate thickness in the restored model, maximum principal strain is generally horizontal, in agreement with the forward model. As in Figure 3.1b, the gray dashed lines in Figure 3.1c & d represents the outline of the initial model, prior to forward deformation. Both models driven by restoration boundary conditions restore to geometric configurations that differ markedly from the initial geometry of the forward model. Specifically, the horizontal dimension of each restoration model exceeds the horizontal dimension of the initial geometry by nearly 2km (10%), the magnitude of extension applied to the forward model. Additionally, the thickness of the restored geometry in the model with the

116 unconstrained base is several hundred meters less than the initial geometry. In restoration models, the horizontal component of the strain tensor and, correspondingly, the displacement of the free vertical boundary do not reflect the forward model because the restoration model fails to capture the extensional tectonic deformation that drove fault slip and forward deformation. This failure is a direct result of treating vertical model boundaries as traction-free surfaces.

Analysis of field data In the following section, we consider the challenges of retrodeformational modeling in the context of 3D field data from the extensional fault system of the Volcanic Tableland, Bishop, CA. The approach parallels that presented previously for synthetic models; however, we focus on addressing the complications of retrodeformational modeling that arise due to imperfect models and real geologic data. While the synthetic model could be retrodeformed almost perfectly by reversing the boundary conditions of the forward model, there are two primary challenges that arise when dealing with a real geologic system. First, appropriate reverse boundary conditions are usually poorly constrained. In most geologic applications, strain history is not well known. Poor constraint on regional geologic strain is a primary reason why previous mechanics-based restoration efforts have treated vertical bounding surfaces as traction-free ‒ it has been considered best to omit constraint entirely where accurate constraint is unavailable. We advocate using a mechanics-based paleostress analysis to help overcome this limitation, but acknowledge that this technique may be inadequate in some cases. Second, even with optimized far-field loads, reverse boundary conditions fall short of restoring the horizon of interest to its assumed undeformed state. Far-field stress heterogeneities due to structures beyond the extent of the model, inelastic continuum deformation, and deformation associated with geometric complexities of structures beyond model resolution, structures omitted from the model for simplicity, and structures below model resolution are just a few of the numerous limitations that may prevent a reverse elastic model from perfectly restoring a faulted and folded horizon. Thus, we consider models that couple reverse tectonic loads and restoration boundary conditions, and demonstrate that such models offer

117 significant improvement over traditional mechanics-based restoration, but that they continue to suffer from the shortcomings of unphysical restoration boundary conditions. Ultimately, we conclude that forward mechanical models are the best path forward to simulate geological stress, strain and displacement, but we suggest that pure reverse elastic models or mechanics-based restoration models supplemented by reverse tectonic boundary conditions may help to develop a more appropriate forward model, and to understand the limitations thereof.

Geologic Setting The Volcanic Tableland, north of Bishop, CA, in the Owens River Valley, boasts one of the world’s premier field exposures of a young, evolving extensional fault system. Pinter [1995] mapped the fault system in great detail, and the site has become a classical field laboratory for studies investigating normal fault slip distributions and growth [Dawers et al., 1993; Dawers and Anders, 1995; Willemse et al., 1996; Ferrill et al., 1999]. It has also been the field site of studies investigating fault kinematics [Ferrill et al., 2000; Ferrill and Morris, 2001], micro-scale deformation mechanisms [Evans and Bradbury, 2004] and fault zone permeability [Dinwiddie et al., 2006]. The Volcanic Tableland is a plateau, formed by the weathering resistant surface of the Bishop Tuff, a welded ash flow that erupted 757,000 +/-9000 years ago [Izett and Obradovich, 1994], from the Long Valley Caldera, roughly 40km to the northwest [Bailey et al., 1976]. The tuff consists of a thin (7-8 m, 23-26 ft), densely welded unit that caps the sequence and defines the topography, with deeper layers that are characterized by lesser degrees of welding, to reach a total thickness of 70-150m in the study area [Wilson and Hildreth, 1997; Evans and Bradbury, 2004]. The tuff is underlain by 1-2 km (3300-6600 ft) of volcanic, alluvial and lacustrine sediments, themselves underlain by crystalline basement rock [Hollett et al., 1991]. Gilbert [1938] and Bateman [1965] present a comprehensive overview of the Tableland and surrounding geology. The Tableland fault system consists of over 100 distinct scarps, of which trace lengths sum to several hundred kilometers. Most of the faults strike roughly north- south. Some dip to the east, and others west. Scarps range from tens of meters to

118 greater than ten kilometers in length, and maximum throw ranges from less than 1m to about 150m. The Fish Slough fault (FSF), on the eastern edge of the Tableland, stands out among the faults, as it is about three times larger than any other fault as measured by length or maximum throw. The Owens River Valley is located at the western edge of the Basin and Range province, and at the northern end of the Eastern California Shear Zone. The large basin-bounding faults, including the White Mountain fault to the east of the Tableland (Figure 3.2), accommodate kilometers of extension and right- lateral strike slip [Stockli et al., 2003; Kirby et al., 2006]. Geologic and geophysical evidence suggests, however, that Tableland extension is primarily east-west, and the faults accommodate primarily dip slip [Bateman et al., 1965; Dawers et al., 1993; Dawers and Anders, 1995]. The Volcanic Tableland is an ideal site for a 3D field based study of fault mechanics using linear elastic theory because infinitesimal strain is a realistic approximation. Lovely et al. [in prep] demonstrate that, despite relatively small strain, inelastic deformation is prevalent, but that simplified elastic models capture the kinematics relatively well. Fault scarp geometries and slip distributions suggest substantial mechanical interaction between nearby faults [Dawers et al., 1993; Dawers and Anders, 1995; Willemse et al., 1996], behavior that cannot be modeled in 2D or by kinematic methods. The weathering resistant surface of the tuff represents a time- stratigraphic marker [Ferrill et al., 1999] and comparable recent tuff ash flow deposits such as that deposited in the Valley of Ten Thousand Smokes during the 1912 eruption of Katmai Volcano, Alaska [Griggs, 1922] suggest that the top of the Bishop tuff may be approximated as planar at the time of deposition. Erosion and deposition are thought to be less than 1-2m across the Tableland [Goethals et al., 2009], much less than fault-related topography (10's of meters, up to 150m). Thus, surface topography and small faults may be resolved more precisely than in subsurface seismic imaging data, and kinematically appropriate restoration boundary conditions may be defined with ease.

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Figure 3.2: Map of the Volcanic Tableland study area. Hillshade image represents topography from 2m (6.6 ft) DEM acquired by ALSM. Thin lines represent fault scarps identifiable from ALSM data. Bold solid lines represent west-dipping seismic- scale faults and bold dashed lines represent east-dipping seismic-scale faults. The bold grey line in the southwest corner represents a fault slightly below the seismic- scale threshold, which is used only for paleostress analysis. The dotted line represents the extent of the FEM mesh used for Dynel3D models and the bold dotted line A-A' represents the location of cross sections presented in figures 3.4, 3.5 & 3.6. Stippled region represents the extent of alluvium in Fish Slough, where the upper surface of the Bishop Tuff has been eroded and fault scarps may be obscured. Inset: regional map Tableland vicinity. Abbreviations: WMF – White Mountain Fault; IF – Inyo Fault; OVF – Owens Valley Fault.

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Geologic model Three dimensional numerical models of fault-related deformational systems require 3D surface representations of fault geometry. Unfortunately, subsurface data is not available at the Tableland, so we use a coarse inversion to infer fault geometry from topography and fault traces extrapolated to depth. 96 trial fault geometries with variable dip and depth of faulting were implemented in a linear-elastic half-space in the boundary element code Poly3D [Thomas, 1993; Maerten et al., 2009], and subjected to uniaxial east-west (fault-perpendicular) extension. Strain magnitude was optimized independently for each trial fault geometry, following which the optimal geometric configuration was selected as that which minimized the difference between topography and vertical displacement of the model earth surface. Results suggest that fault dip is relatively shallow, approximately 50°, and faults extend to about 750m depth. The Fish Slough fault was treated independently due to its greater length and throw, and its inferred depth is 2000m. Details of the modeling process are described by Lovely et al. [in prep]. The fault model used for numerical simulations represents only seismic-scale faults - that is, faults that would likely be easily discernible in typical industry quality seismic imaging data. Those faults with consistently greater than 20m throw are identified in an unbiased manner by a simple algorithm that scans the DEM for topographic features that are sufficiently large and continuous. Fourteen seismic-scale faults were identified, including two distinct segments of the Fish Slough fault, and are implemented as sliding contact surfaces in models. Seismic-scale faults are identified by bold lines in Figure 3.2. This limited fault selection (as opposed to including all of the hundreds of faults discernible in the DEM or in the field) minimizes numerical instabilities, maintains a computationally feasible problem, and reduces complications inherent to 3D finite element mesh generation. This selection of faults offers the additional benefit of providing a direct analog to the fault model that might be generated from interpretation of seismic reflection data, but the outcrop exposure provides an opportunity to map smaller structures as well.

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All numerical models of the Tableland, forward and retrodeformational, are implemented in a homogeneous elastic medium. The homogeneous elastic model is a significant simplification of the complicated local geology. However, stratigraphy beneath the tuff is not well known and the depth of sediments overlying crystalline basement rocks is poorly constrained, so these heterogeneities cannot be accurately defined. The Bishop Tuff and underlying units do not deform as a perfectly elastic material, as evidenced in the field by abundant fractures and by the unrealistic tensile stress magnitudes suggested by the paleostress analysis (next section). However, linear elasticity is necessary for mechanics-based retrodeformational models and inelastic behavior is not the focus of this study. We elect to implement a homogeneous material model for several reasons. The stratigraphic heterogeneity of the tuff is well below the resolution of the finite element mesh, and therefore cannot be accommodated. Elastic moduli are selected from laboratory tests investigating properties of tuff found at Yucca Mountain, Nevada, which have a similar composition to the Bishop Tuff. Laboratory measurements are scaled to account for the damage inherent to large rock masses and observed (e.g. fractures) in the tuff. Laboratory measurements suggest Young's moduli ranging from 2-33 GPa, depending on the degree of welding and porosity [Lin et al., 1993; Avar et al., 2003]. Although the uppermost welded layer would undoubtedly be quite stiff at the scale of a laboratory sample, the densely welded unit is thin (7-8m) and highly fractured. We use a Young's modulus of 2.5 GPa, considering that most of the tuff is only slightly welded and that the effective stiffness of rock at outcrop and larger scales is approximately 20-60% that of a laboratory sample [Bieniawski, 1984]. Poisson's ratio for numerical models is selected as 0.25 [Lin et al., 1993].

Paleostress analysis using mechanics Paleostress analysis provides a means to define far-field boundary conditions for elastic models that accurately reproduce fault slip distributions observed in the field. In many geologic settings, total regional strain throughout the deformational history is poorly known. Restorations are, in fact, commonly used to estimate tectonic

122 extension or contraction [e.g. Allmendinger et al., 1990], despite the fact that the kinematic methods generally employed account only for strain accommodated by fault slip and assume that no volumetric strain is accommodated by internal rock deformation. We suggest using a mechanics-based paleostress analysis to determine far-field loads for elastic models, though we acknowledge that such analysis may be difficult or impossible to perform in the cases of some particularly complex or highly deformed structures. In such situations, however, even a coarse estimate of regional loading should provide a more sound retrodeformational model than one in which vertical boundaries are treated as traction-free. The Volcanic Tableland, with abundant faults spanning a broad range of size, all of which appear to have been active simultaneously during the Tableland's geologically brief history, is an ideal setting to implement paleostress analysis. Unlike the well-known method of paleostress analysis that is based on the Wallace-Bott hypothesis [Wallace, 1951; Bott, 1959; Ramsay and Lisle, 2000], mechanics-based paleostress analysis implements a least squares inversion to calculate the components of the far-field stress tensor that best reproduce observed slip on faults modeled in an elastic continuum. Thus, it provides a means of defining tectonic boundary conditions for forward and reverse elastic models that most accurately reproduce fault slip everywhere that slip measurements may be made. We use a code developed by IGEOSS, which is based on the forward problem solved by Poly3D [Thomas, 1993]. Faults are discretized as triangular elements of constant displacement discontinuity implemented in a linear elastic half-space. In Poly3D, far-field stress may be prescribed, and slip is calculated such that the shear traction resolved on each triangular element is relaxed completely. The paleostress inversion calculates the three horizontal components of the far-field stress (σxx, σyy and σxy; vertical components σxz = σyz = σzz = 0 are prescribed) in order to minimize the difference between observed slip and model slip on all elements where slip is known. Prescribing the vertical component of the stress tensor as zero is justified because continuum deformation is elastic and faults are frictionless; therefore, only the deviatoric component of the stress tensor influences fault slip and it is unnecessary to

123 account for the lithostatic stress gradient. It is generally accepted that a principal stress is vertical because the earth's surface is traction-free [e.g. Lisle et al., 2006; Zoback, 2007]. The paleostress inversion implementation is described in detail by Kaven [2009]. The paleostress tensor is calculated using the seismic-scale fault model described previously. Dip-slip is derived from profiles of throw across fault scarps, which are extracted from a digital elevation model (DEM) with 2m resolution generated from airborne laser swath mapping (ALSM, also known as airborne LiDAR) data (Figure 3.2). The ALSM survey was acquired by Chevron through a commercial vendor. ALSM methods are capable of mapping topography with decimeter vertical precision [Shrestha et al., 1999]. Dip-slip constraint is prescribed for the paleostress inversion on all fault-surface elements that intersect the earth's surface within the coverage area of the ALSM data, although the fault model extends beyond the ALSM data to the north and west, and all faults are allowed to slip in order to account for their interaction in a more complete mechanical system. Strike-slip is constrained to zero on these same elements in light of field evidence that strike-slip is insignificant [Bateman et al., 1965; Dawers et al., 1993; Dawers and Anders, 1995]. Initial paleostress calculations suggested most tensile stress in excess of 100MPa and oriented north-south, parallel to the average fault strike. Such an unphysical result (most tensile stress should be perpendicular to fault strike in a normal fault regime [Anderson, 1951]) is the result of an ill-conditioned inverse problem [e.g. Aster et al., 2005] in which all fault elements are nearly parallel to each other. Because the north-south (fault-parallel) component of the stress tensor resolves little traction on fault surface elements, it has little effect on fault slip, and this component of the paleostress tensor is essentially unconstrained. Maerten [2010] introduces a Monte-Carlo approach to solving the paleostress problem which may incorporate additional data types, including fault and fracture orientation and displacement fields (e.g. GPS, InSAR), which could help to overcome the ill-conditioning of this initial model. However, we address ill-conditioning by including one additional fault in the paleostress inversion (Figure 3.2). This fault,

124 located in the southwest corner of the Tableland, is unusual because it strikes northeast-southwest, and therefore the north-south component of the stress tensor resolves significant traction on its surface. The fault is relatively large, with throw locally exceeding 20m, although not consistently. It alone is added to the set of faults used for paleostress inversion, although there are about five other faults of comparable size that are omitted. Because this fault is beyond the extent of the ALSM data, throw is mapped from the coarser and less precise National Elevation Dataset 1/3 arc-second DEM [Maune, 2007], the same DEM used to represent topography in FEM models. The improved paleostress tensor suggests maximum tension oriented N72°W, with a magnitude of 62 MPa, and an intermediate principal stress of 51 MPa oriented N18°E. Root mean square residuals are 19.6m for dip-slip and 7.5m for strike slip. Although paleostress magnitudes do not account for lithostatic stress gradient, suggested stresses are nearly an order of magnitude greater than the tensile strength of rock, which is generally about 10 MPa [Bieniawski, 1984]. Considering our choice of a minimal elastic stiffness (2.5 GPa, Geologic model section), it is clear that inelastic deformation must play a significant role in Tableland deformation. Nonetheless, such large stresses are necessary to drive appropriate magnitudes of slip in this elastic body. 3D slip distributions are reasonable and the kinematics (strain and displacement fields) suggested by elastic models do not differ dramatically from those suggested by more realistic elasto-plastic models [Lovely et al., in prep].

Numerical Models Retrodeformational models of the Volcanic Tableland are peformed using Dynel3D©, a commercially available finite element (FEM) package from IGEOSS (http://www.igeoss.com), which has been developed with mechanics-based restoration in mind. Forward models are performed using Poly3D, which is based on a boundary element formulation. The FEM lends itself to retrodeformational modeling because models may accommodate geometric complexities such as faulted and folded stratigraphic horizons, and particularly to restoration boundary conditions because displacement may be prescribed at nodes on the boundaries of, or within, the elastic continuum. However, finite element models require that the entire volume be

125 discritized by 3D tetrahedral or hexahedral elements, which poses challenges for mesh generation and leads to computationally cumbersome solutions as the number of elements grows rapidly. As a result, relatively coarse mesh discretization is necessary to maintain a computationally feasible solution. Ideally, model boundaries must extend significantly beyond the primary area of interest in order to minimize edge effects due to idealized boundary conditions, further contributing to computational cost. Boundary elements are computationally advantageous because only fault surfaces need to be discretized as triangular elements of displacement discontinuity in 3D space; however, initial model geometry must be simple, either a whole- or half- space in Poly3D, and boundary conditions may be prescribed only on the faults as either displacement discontinuity (slip or opening) or traction. Thus, it is difficult to incorporate complicated 3D surface topography or to prescribe restoration boundary conditions in the boundary element model. The FEM mesh representing the Volcanic Tableland, which is used for all Dynel3D retrodeformational models, includes the seismic-scale fault model introduced previously (each fault is represented as frictionless sliding contact surfaces) and the upper surface of the volume conforms to topography as defined by the National Elevation Dataset 1/3 arc-second (~10m) DEM. ALSM coverage is not sufficient to define the entire extent of the restoration model (Figure 3.2), and because mesh resolution is only 250m, the 10m DEM is sufficient. Mean relative error of the DEM at a kilometer scale is 1.64m, with a standard deviation of 2.08m [Maune, 2007], which is significantly better than precision of seismic reflection images [Yilmaz and Doherty, 2001]. As discussed in the following section, it is essential that the finite element mesh have a rectangular shape in map view, with edges perpendicular to far-field principal stress trajectories, in order to define appropriate boundary conditions. Therefore, it was necessary to extrapolate topography south of the Tableland where the Bishop Tuff has been eroded. Similarly, it was necessary to extrapolate the surface of the tuff from west-to-east across Fish Slough and into the footwall of the Fish Slough fault, because the welded tuff has been eroded or is covered by alluvium

126 in the slough. Analysis focuses on an east-west cross-section through the 3D model at 4,147,000m northing (Figure 3.2), in a region where the implications of extrapolating topography should be minimal. The mesh, shown in Figure 3.3, covers an area of roughly 11x11km (Figure 3.2), extends to 4km depth, and consists of 339,816 tetrahedral elements. In contrast, the Poly3D forward models consist of only 3,603 triangular elements, and resolution is improved, with each element roughly 180m on a side. Four different models of the Tableland are considered, representing (1) forward deformation, (2) restoration boundary conditions only, (3) restoration boundary conditions supplemented by optimized (paleostress) reverse tectonic boundary conditions, and (4) optimized, though imperfect, reverse tectonic boundary conditions alone. Forward models performed using Poly3D may be governed by far- field boundary conditions directly from paleostress analysis. In the model with restoration boundary conditions only, vertical displacement is prescribed such that topography is flattened to Z=0 and all other boundaries of the FEM mesh are treated as traction-free. Restoration boundary conditions may be supplemented by additional constraints that reverse the tectonic loading that drove forward deformation. In finite element models, it is impossible to prescribe true far-field stress or strain boundary conditions, and the Dynel3D interface does not accommodate traction boundary conditions. Thus, it is necessary to simulate reverse tectonic boundary conditions by prescribing on vertical model boundaries displacements that best reproduce far-field paleostress conditions. This is most easily done by defining a model volume that is rectangular in map view with vertical edges perpendicular to horizontal principal stress trajectories (Figure 3.2). Face-perpendicular displacement is prescribed on the four vertical model boundaries (A, B, C & D as identified in Figure 3.3), but nodes on these boundaries are free to displace within the plane of the boundary such that the surface is shear traction-free. Face-perpendicular displacement on boundaries A & B is zero and appropriate displacements are calculated for boundaries C & D such that the net strain of the volume is equivalent to far-field strain, which may be derived from far-field paleostress by Hooke's law. Far-field stress magnitudes of σ1 = 62MPa,

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Figure 3.3: Undeformed FEM mesh representing the Volcanic Tableland system in its present state

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σ2 = 51MPa, and σ3 = σzz = 0 correspond to ε1 = 0.0199, ε2 = 0.0143, and ε3 = εzz = - 0.0114, respectively. Stress and strain tensors have the same orientation. Reverse boundary conditions for the Tableland model consist of displacing boundary C 229m toward boundary A, and boundary D 162m toward boundary B. These displacements imply normal tractions of approximately the same magnitude as principal components of the paleostress tensor, and vertical model boundaries are shear traction-free in accordance with the paleostress tensor orientation. Such displacement boundary conditions do not exactly reproduce far-field stress conditions, which undoubtedly would not exactly result in homogeneous displacement along FEM model boundaries due to fault slip within the volume, but both types of boundary conditions are idealized and one is not necessarily superior to the other.

εzz is never imposed, but rather either the upper or lower model boundary is always treated as traction free. The model base is treated as traction free in both restoration models because there is no way to provide improved constraint without prior knowledge of the initial (undeformed) thickness of those units represented by the model. The reverse model is governed by the same reverse tectonic boundary conditions as described above, but unlike the previously described model in which restoration boundary conditions are applied to the earth's surface, the earth's surface is traction-free and zero vertical displacement is prescribed on the model base. Such vertical boundary conditions result in zero average vertical stress, which is appropriate for a model that does not account for lithostatic load, though vertical stress may be locally large near where displacements are prescribed (hence the issues with restoration boundary conditions).

Results Disscussion of mechanics-based retrodeformational models of the Volcanic Tableland focuses on plots of maximum principal strain (Figure 3.4), vertical strain (Figure 3.5) and displacement vector fields (Figure 3.6) from the four models described previously. Maximum principal strain is generally horizontal in the forward model (a), and magnitude is relatively uniform (~0.02) across most of the cross

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Figure 3.4: Maximum principal strain magnitude and orientation on cross-section A-A' (Figure 3.2) as calculated by a) forward model in Poly3D, b) restoration boundary conditions alone, c) restoration boundary conditions supplemented by reverse tectonic boundary conditions, and d) reverse tectonic boundary conditions alone. Note that principal strain orientation vectors are plotted in 3D. Thus, short vectors indicate that a large component of ε1 is out of the plane of the cross section, and long vectors indicate that ε1 is almost entirely in the cross section plane.

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Figure 3.5: Vertical strain magnitude on cross-section A-A' (Figure 3.2) as calculated by a) forward model in Poly3D, b) restoration boundary conditions alone, c) restoration boundary conditions supplemented by reverse tectonic boundary conditions, and d) reverse tectonic boundary conditions alone.

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Figure 3.6: In-plane displacement on cross-section A-A' (Figure 3.2) as calculated by a) forward model in Poly3D, b) restoration boundary conditions alone, c) restoration boundary conditions supplemented by reverse tectonic boundary conditions, and d) reverse tectonic boundary conditions alone. Displacement is referenced to zero at 4km depth, on the eastern edge of the cross-section. All displacement vectors are exaggerated by a factor of two, except (b), which is exaggerated by a factor of eight.

132 section. Strain is somewhat relaxed in the vicinity of fault surfaces, and we note in the vicinity of the Fish Slough fault that east-west strain is reduced sufficiently that maximum principal strain is oriented north-south, parallel to the fault and perpendicular to the plane of the cross-section. Extensional and contractional quadrants are observed around fault tips, as expected in a linear elastic material [e.g. Pollard and Segall, 1987]. Principal strain, as modeled by restoration boundary conditions, bears no resemblance to the forward model in either orientation, magnitude, or distribution of perturbations. Orientation is highly heterogeneous and maximum principal strain is near zero throughout much of the model, except for a few regions of localized but intense extension, which are generally vertical and tend to be concentrated near the earth's surface in the footwall of modeled faults. These regions of localized extension in fact coincide with regions of relatively less extension near fault surfaces in the forward model. The quadrants of extension and contraction predicted around fault tips by fracture mechanics are not clearly evident. Principal strains in the restoration model supplemented by reverse tectonic boundary conditions resemble the forward model much more closely than the model with restoration boundary conditions alone. Average strain magnitudes are appropriate (~0.02) and orientation is nearly horizontal, similar to the forward model. The extensional perturbation in the footwall at the tip of the Fish Slough fault is evident, but other fault-tip related stress perturbations are not apparent. Large positive perturbations of ε1 persist in the footwall of the Fish Slough fault and in the footwall of the westernmost fault on the cross section, near the earth's surface. These features are absent from the forward model. Particularly in the center of the cross-section, orientation vectors are short, suggesting a much greater north-south component of ε1 than suggested by paleostress analysis or the forward model. Of the three retrodeformational models of the Tableland, the pure reverse model most nearly resembles the strain distribution of the forward model. Magnitude and orientation are consistent, with a few exceptions. One such exception is in the footwall of the Fish Slough fault, where the reverse model suggests a more extensive

133 region in which maximum principal strain is oriented north-south, perpendicular to the section orientation. This may be an edge effect because the perturbation is very near a model boundary on which constant displacement is prescribed. Also, quadrants of extension and contraction around fault tips are not as evident as in the forward model, with the exception of the extensional quadrant at the tip of Fish Slough fault. It is worth noting the geometry of the pure reverse model, in which topography is flattened only as much as reverse boundary conditions and resulting back-slip on modeled faults may do. Though subtle, topographic heterogeneity that persists after reverse deformation is apparent in cross section shape (Figures 3.4d, 3.5d, 3.6d). Topography of the undeformed model and of the deformed model after application of reverse tectonic boundary conditions is shown in Figure 3.7. Fault back-slipping that results from retrodeformation closes most fault gaps, but much topographic heterogeneity remains. In fact, if topographic heterogeneity is measured as the root mean square of vertical nodal coordinates after rotating and translating the surface such that the best-fitting plane is horizontal and passes through the origin, reverse boundary conditions reduce heterogeneity from 31.8m to 28.3m. Only 11% of topography is restored. Vertical strain distributions (Figure 3.5) emphasize the effects of the unphysical restoration boundary condition. Both forward and reverse models suggest relatively uniform vertical strain magnitude, generally somewhat less than -0.01. Despite zero vertical stress, vertical contraction follows from regional horizontal extension, due to Poisson's effect. In forward and reverse models, relatively greater (near-zero) vertical strain is observed in the immediate vicinity of fault surfaces. The pure restoration model differs in two obvious ways: average vertical strain is more nearly zero because there is no Poisson effect in the absence of reverse tectonic boundary conditions, and large vertical strain perturbations (both positive and negative) are pervasive in the near surface. The largest perturbations are found near fault scarps, but others are isolated from modeled faults. The model coupling restoration and reverse tectonic boundary conditions demonstrates vertical strain

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Figure 3.7: Topography of the FEM model's surface in a) undeformed (present day) state and b) deformed by reverse tectonic boundary conditions.

135 magnitude in line with forward and reverse models, but vertical strain perturbations remain prevalent in the near surface. Displacement vector fields (Figure 3.6) from forward and reverse models, as well as restoration models with reverse tectonic boundary conditions, are very similar. Regional extensional strain is the dominant feature. Relative to a fixed reference point at the eastern edge of the cross section at 4000m depth, downward displacement increases upward and westward displacement increases to the west. A discrete step in displacement magnitude may be observed from east-to-west across the Fish Slough fault, but is less apparent across the smaller faults. The displacement field suggested by restoration boundary conditions alone differs dramatically. Displacement magnitude is generally much smaller (note that displacement vectors are exaggerated by a factor of 8 for this figure, versus exaggeration by only a factor of 2 for all other plots) and displacement direction is predominantly vertical. The dominant component of the displacement field in all other models, which results from tectonic extension and vertical contraction due to Poisson's effect, is absent. Predominantly vertical displacement results from the vertical displacement boundary condition. Underlying material is dragged upward along with the restored surface.

Discussion Comparison of forward and retrodeformational models presented in figures 3.4-3.6 highlights the importance of carefully considering the implications of boundary conditions in mechanics-based restoration models. We select the forward elastic model as a basis for comparison and evaluation of models with restoration boundary conditions not because the forward model is necessarily the best representation of Volcanic Tableland deformation (the elastic constitutive model and limited seismic-scale fault configuration, among other features, make this a highly simplified model), but because it is honors principles of continuum mechanics, including conservation of mass, momentum and energy, the kinematic variables are derived through implementation of an established, though simplified, constitutive law relating stress and strain, and deformation is driven by boundary conditions that resemble tectonic load in orientation, distribution, and strain magnitude, if not stress

136 magnitude. We do not consider a more appropriate inelastic constitutive law for forward models in order to maintain our focus on evaluating the implications of boundary conditions and to enable a direct comparison with retrodeformational models. Additionally, an appropriate, representative 3D elasto-plastic model with frictional faults would require huge computational cost and likely be impeded by numerical stability issues. Such models are unfeasible for routine analysis today, though they will undoubtedly become more commonplace in the future. Similar to the synthetic example discussed previously, restoration models in which all boundaries, except the horizon to be restored, are treated as traction-free (figures 3.4b, 3.5b, 3.6b) result in strain and displacement fields that bear little semblance to the kinematic behavior that might be expected from forward deformation (figures 3.4a, 3.5a, 3.6a). However, unlike the ideal synthetic example, in which geometric configuration is known precisely and forward deformation is perfectly linear, and thus perfectly retrodeformed by reverse boundary conditions, reverse tectonic boundary conditions must be estimated for a real geologic setting. Realistically applicable (homogeneous) boundary conditions are unlikely to be capable of perfectly undoing observed deformation because of inherent non-linearity and because current models are greatly simplified representations of geology. When considering restoration boundary conditions supplemented by reverse tectonic loads, the principal of superposition enables us to consider the loads applied in any order. It is constructive to consider the reverse tectonic load applied first. This restores, in a mechanically admissible way, what topography it may, and then the restoration boundary condition restores the rest. Although these models result in kinematics (figures 3.4c, 3.5c, 3.6c) that match forward models better than models with restoration boundary conditions alone, we must question if the restoration boundary conditions are valid. Because vertical displacement boundary conditions on the earth's surface are unphysical and the structures that cause topography may be either improperly represented or entirely absent from the model, strain perturbations and displacement fields produced by the restoration boundary condition provide little, if any, geological insight.

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The magnitude and orientation of strain fields calculated from forward models and from restoration models supplemented by reverse tectonic boundary conditions are similar in much of the cross section, but there are locally significant differences. The models differ most in the near-surface. Large strain perturbations are a direct result of the unphysical restoration boundary condition that violates the traction-free nature of the earth's surface. Thus, they appear most strongly in the vertical component of strain, parallel to the driving force, and tend to fade in magnitude radially, away from a focal point at the surface. Perturbations represent topography (hence, presumably, some form of geologic strain) that is not flattened by reverse boundary conditions, perhaps because of inelastic deformation, or association with structures that are inappropriately represented or absent from the simplified model. But the strain perturbations resulting from restoration boundary conditions do not indicate the nature or orientation of the geologic structure that produced the topographic feature. Positive vertical strain perturbations, as are evident in Figure 3.5c, might suggest horizontal opening-mode fractures, for which field evidence is lacking in the Tableland. In the Tableland, we know from field observations that much of this topography that is not restored by reverse boundary conditions and that generates unphysical restoration strain perturbations is associated with smaller normal faults that are omitted from the simplified seismic-scale fault configuration used for models. Given seismic data in which we could not see these smaller faults, we might infer the same conclusion that small topographic features are associated with small normal faults. However, the diffuse nature of restoration strain perturbations bears no resemblance to fault-related strain perturbations, and therefore should not be used to predict either fault geometry (perturbations for east or west dipping faults are difficult to differentiate) or the potential extent or characteristics of fault-related damage. In other words, a critical examination of topography of the surface to be restored, in the context of the regional tectonic setting, provides as much, if not more information than may be gleaned from analyzing strain perturbations in restoration models.

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Kinematics of the pure reverse model (Figures 3.4d, 3.5d, 3.6d) are quite similar to the forward model. This result is expected, given that elastic models with frictionless faults are reversible, the two models have the same fault configuration, and the boundary conditions of the reverse model very nearly invert those of the forward model despite subtle differences in application of boundary conditions in finite element and boundary element methods. Under the assumptions of elasticity, neither forward nor reverse models are subject to the effects of unphysical boundary conditions, and therefore, neither is susceptible to the limitations of restoration boundary conditions. Forward and reverse models each have benefits and drawbacks. In the case of the elastic models of the Tablelands, forward models provide improved resolution with the boundary element method. Reverse models do not capture the strain perturbations around fault tips as well as forward models, in part because the coarse mesh discretization required for finite element simulations is unable to capture details on such a fine scale. Forward model results demonstrate that the spatial extent of fault-tip strain perturbations is very small. Reverse models offer the benefit of incorporating present-day topography of the deformed horizon - the top of the Bishop Tuff in the Tableland case. Thus, by exporting the topography after retrodeformation (Figure 3.7), it is possible to observe what topography can and cannot be explained by elastic deformation and slip on those faults represented in the model. In the case of the Tableland, we see that most topography (89%) is in fact not accounted for by our simplified models. The retrodeformed topography could, perhaps, be used to help identify important structures that are absent from the model, though smoothing of topography in the coarse FEM mesh may limit such prospects, or to identify regions where the shortcomings of elasticity are most evident. That the reverse model restores only 11% of topography demonstrates the limitations of this method, and motivates the imposition of restoration boundary conditions. However, we must still ask what we might learn from forcing this topography to flatten, and if we might be better off working to generate an improved forward or reverse model.

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Perhaps the greatest advantage to forward models is that they allow one to incorporate non-linear elasto-plastic and visco-plastic constitutive laws, which are more appropriate for rock deformation over geologic time, and frictional sliding on fault surfaces. Increasingly sophisticated numerical models, if applied appropriately, will allow structural geologists in the future to simulate kinematics with increasing accuracy and to better predict not only regions of localized strain, but also in-situ geological stress distributions. Because entropy is a one way street, and because nonlinear constitutive laws are highly sensitive to load path and therefore do not behave in the same way for forward deformation and retrodeformation, forward mechanical models are the only way to capture the true nature of rock deformation without employing nonlinear inversion techniques that would require currently unavailable computational power. A more practical example of the limitations of restoration strain to predict smaller structures is provided in Figure 3.8. Having demonstrated the clear benefits of supplementing restoration boundary conditions with appropriate reverse tectonic boundary conditions, we compare maximum principal stress perturbations at the earth's surface from this model with smaller faults mapped in the field (Figure 3.8a).

Black vectors represent the local orientation of ε2. If restoration strain were an accurate predictor of subseismic-scale faulting, we would expect subseismic-scale faulting to be densest where strain perturbations are greatest, and we might expect that faults would strike parallel to ε2. However, given the limitations of restoration boundary conditions, it is not altogether surprising to find that the correlation between restoration strain magnitude or orientation and subseismic-scale faulting is weak at best. The greatest density of subseismic-scale faulting would be expected in the vicinity of large positive perturbations of ε1, which are found in the vicinity of point A and around Fish Slough fault (e.g. points E and H), where positive strain perturbations are observed. The only one of these areas which corresponds to a large density of faults is around point E, in the footwall of the southern half of Fish Slough fault. There are no subseismic faults observed near the northern portion of Fish Slough fault,

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Figure 3.8: Maps of maximum principle strain (ε1) magnitude for a) restoration model with reverse boundary conditions and b) FEM model with reverse tectonic loads only. Bold lines represent faults included in models as sliding contact surfaces and thin lines represent other faults mapped in the field. Short, intermediate weight lines represent intermediate principle strain orientation, which is the expected strike of small faults or joints if strain is an appropriate predictor of small structures.

141 where the strain perturbation is equally large, and around point A subseismic fault density is average, but not unusually large. However, there is system of relatively large echelon faults running through this corridor, which were studied by Dawers et al. [1995]. The flattening of topography associated with these scarps is the source of strain in the restoration model. Some, though not all, of these faults are large enough to be represented in the seismic-scale fault model. Thus, a relatively large amount of strain may be accommodated by the faults in this region. The discrete fault scarps responsible for the distributed zone of strain may be easily identified in the 1/3 arc- second DEM, and the restoration model provides no additional geologic insight than could be extracted by careful analysis of raw topography. Subseismic fault density is greatest around points C and D, where large strain perturbations do not appear in the restoration results. Low fault density predictions would be appropriate near points F and G, where a significant strain shadow is observed in the vicinity of Fish Slough. However, there are a few faults, near F and G, and predicted orientation would be erroneous. Maximum principal strain distributions predicted by the restoration model seem to correlate with subseismic-scale fault density in some regions of the model; however, the approach is not successful everywhere. Similarly, strain orientations agree with subseismic fault orientations in parts of the model, but the heterogeneity of strain orientation and relative homogeneity of fault orientation mean that they do not agree everywhere. Particularly in the vicinity of points C, F, and E, we might predict east-west striking faults, rather than the north-south striking faults that are observed. The limited nature of the correlations between restoration strain and subseismic fault density and between strain orientation and fault orientation provide direct field evidence to supplement our theoretical argument why restoration strain should not be used to predict subseismic-scale structures. The same fault map is overlain on the strain distribution predicted by the model with reverse tectonic boundary conditions in Figure 3.8b. This model is governed by mechanically appropriate boundary conditions, and represents appropriate strain distributions that would result from tectonically driven slip on the

142 seismic-scale faults represented in models. Strain distributions represented by this model are very different from those suggested by the restoration model. Interestingly, this model suggests opposite results of the restoration model in a couple areas where the restoration model correlates well with subseismic fault density. We see a strain shadow, rather than concentration, around Fish Slough fault, including in the vicinity of point E, where subseismic faulting is intense. In the vicinity of point B, where no subseismic faults are observed, we see a positive strain perturbation (faulting would be expected here), where we see negative perturbations around point F in the model with restoration boundary conditions. These features demonstrate the unphysical nature of the strain perturbations in the restoration model. The perturbations are not a result of slip on the modeled faults. Surprisingly, the reverse model, despite being more mechanically sound, does no better predicting subseismic-scale faulting than does the restoration model. It might do a bit better regarding fault strike, although it predicts east-west striking faults around point C and east of Fish Slough fault, but the correlation between subseismic fault density and ε1 magnitude is, in fact, probably worse. This occurs because subseismic faults on the Tableland are not secondary faults. They did not develop in response to stress/strain perturbations due to slip on larger faults, but rather through coalescence of smaller faults [Dawers and Anders, 1995; Ferrill et al., 1999]. The small faults are old faults that never coalesced, and today their growth is inhibited by the stress shadows of other faults [Lovely et al., in prep]. Geomechanical models representing seismic-scale structures are probably not an appropriate means for predicting subseismic-scale structures everywhere, the Volcanic Tableland being a prime example, despite success of this approach in other geological settings [Maerten et al., 2006; Dee et al., 2007; Wilkins, 2007]. The future of restoration, whether mechanics-based or not, seems limited to first-order kinematic analysis. Many of the touted benefits of mechanics-based restoration are real, including conservation of mass and momentum and the ability to simulate mechanical interaction of fault segments, heterogeneous slip distributions in three dimensions, out-of-plane strain, and displacement and strike-slip on faults. In

143 complicated three-dimensional systems, mechanics-based restoration provides distinct benefits over kinematic methods for validation of structural interpretation (e.g. verifying that back-slipping a fault on a young horizon results in appropriate back-slip on the same fault in the context of older, deeper horizons), for reconstructing initial (undeformed) configuration of complicated faulted and folded systems, and for visualizing the 3D evolution of such structures. However, strain fields and subtle features of displacement fields from mechanics-based restoration, even when supplemented by appropriate reverse tectonic boundary conditions, should be treated with skepticism. The perturbations observed in restoration models of the Tableland are unlikely to result from mechanically valid tectonic processes. Mechanics-based restoration models, because of their unphysical boundary conditions, should not be confused with true mechanical models. Elastic models with reverse boundary conditions, although they may restore only a fraction of topography, are the only type of retrodeformational model that represents a deformational system with mechanically valid boundary conditions; however, increasingly sophisticated forward models, whose construction (e.g. initial geometric configuration) might be guided by the kinematic behavior of retrodeformational models, are the most appropriate way to simulate geologic deformation and to predict stress, strain, and associated small-scale features. Nonetheless, we acknowledge that in some tectonic settings, the Tableland being a prime example, where there is not a causative link from large to smaller structures, it is never appropriate to predict smaller structures from strain (or stress) due to slip on larger faults.

Conclusions Mechanics-based restoration methods provide significant improvement over traditional kinematic methods - improvements that have been touted as providing a means to predicting geologic strain and displacement fields from numerical models that do not require assumptions about the initial state of the deformational system, but rather begin with the present day, observed geometric configuration and work backwards. Despite the reversible nature of the assumed (albeit simplified) linear elastic constitutive law, restoration boundary conditions are mechanically

144 inappropriate. They violate the traction-free nature of the earth's surface, and restoration models as performed to date do not account for tectonic loads that are presumed as the driving mechanism behind forward deformation. Synthetic examples and analysis of field data from the Volcanic Tableland extensional fault system demonstrate that the potential implications of boundary conditions must be carefully considered before interpreting geologic deformation from mechanics-based restoration models. We suggest that mechanics-based restoration models are a valid tool for verification of structural interpretation and for first order illustration of the kinematic evolution of a 3D structure, but even when supplemented with appropriate reverse tectonic boundary conditions, restoration models may not accurately predict geologic strain and should not be used to infer small-scale structures that might be associated with such perturbations. We suggest that elastic reverse models governed only by physically appropriate reverse tectonic boundary conditions may be used to extract geologic strain and displacement, although one must be cognizant that elasticity is a dramatic simplification of the complicated constitutive laws that govern rock deformation on a geologic time scale. The kinematics suggested by reverse models are nearly identical to the kinematics of forward elastic models, and enable visualization of the retrodeformed topography and comparison with present day topography in order to understand what may and what may not be accounted for by the simplified elastic model. Forward models, however, are clearly the best route forward to predict geologic deformation because, in addition to physically plausible boundary conditions that simulate tectonic loads and do not violate the earth's traction free surface, they may account for inelastic deformation and other non-linear material behaviors that characterize rock deformation over geologic time scales. Although retrodeformational models may capture strain and displacement fields with variable degrees of accuracy, well designed forward models with appropriate material properties should provide increasingly accurate kinematic results and such models are the only way to simulate geologically appropriate stress distributions.

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Acknowledgements Bishop Tableland LiDAR data courtesy of Chevron Corporation. IGEOSS provided access and support for the Poly3D and Dynel software. The authors thank Chevron for permission to publish this work. P. Lovely has received financial support from the Robert and Marvel Kirby Stanford Graduate Fellowship Fund, the National Science Foundation's Collaborations in Mathematical Geosciences grant no. CMG- 0417521, and the Stanford Rock Fracture Project.

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Chapter 4

Implications of assuming a linear, frictionless, elastic constitutive model for simulation of fault-related deformation

Abstract Linear elastic models with frictionless faults are an imperfect representation of rock deformation characterized by large strain that has accumulated over geologic time scales. However, such models appeal to structural geologists for two reasons. First, they provide computationally efficient and numerically stable simulations that honor a complete mechanics, including field equations and a constitutive law relating stress and strain. Second, the linear nature of such models makes deformation reversible; hence elastic frictionless models have been used for mechanics-based restoration applications and other mechanics-based retrodeformational methods. We evaluate the implications of assuming frictionless, elastic deformation in forward models representing a single crustal-scale fault in regions with extensional and contractional tectonic strain magnitude less than 5%, through comparison of results with comparable models (same geometry and loading) characterized by an elasto-plastic constitutive law and frictional fault slip. These models account for some of the inelastic and nonlinear behaviors of rock as revealed in laboratory rock mechanics tests. Elastic models may capture the first-order kinematic patterns of fault-related deformation, including tensor orientation, slip distribution, and locations of positive and negative strain perturbations, distal from fault tips, but linear and nonlinear models deviate near fault tips. The magnitude of strain perturbations and slip distributions can differ significantly, and elastic models may simulate patterns and magnitudes of stress distributions poorly. Solutions to the elasto-plastic constitutive law may be subject to numerical instabilities associated with the physical phenomenon of strain localization. Instability results in non-unique, mesh-dependent numerical solutions. Implications are discussed, but we avoid numerical instabilities and suppress localization by reducing the plastic dilatancy and prescribing modest strain hardening.

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Models with multiple faults indicate that, if the fault network is extensive, fault slip may accommodate most deformation and elastic models may be appropriate up to relatively large far-field strain. However, elastic models will fall short if folding is a dominant mechanism of deformation. In linear and nonlinear models, it is important that the subsurface fault geometry be accurate and its extent adequate.

Introduction In the past few decades, with the rapid growth of computational capabilities and advancements in numerical methods, efforts have gained momentum to move beyond traditional kinematic modeling approaches, and to simulate structural evolution using geomechanical tools. In order to simulate heterogeneous slip distributions over non-planar faults, linear elastic models based on the displacement discontinuity method [e.g. Jeyakumaran et al., 1992; Thomas, 1993] sum the effect of many planar, rectangular or triangular elements of constant displacement discontinuity (dislocations) within a homogeneous elastic whole- or half-space. More recent advances enable the simulation of frictional resistance to fault slip, as well as elastic bodies with topographic and material heterogeneity [e.g. Williams and Wadge, 2000; Maerten, 2010]. Such 3D elastic models have been applied to investigate earthquake ruptures [King et al., 1994; Wald and Heaton, 1994; Crider et al., 2001; e.g. Johnson et al., 2001; Healy et al., 2004; Maerten et al., 2005] and to studies of cumulative deformation, more traditionally considered within the realm of structural geology [e.g. Bourne and Willemse, 2001; Dee et al., 2007; Wilkins, 2007; Hilley et al., 2010]. It has long been acknowledged that, while elastic models may represent geologic deformation characterized by small strain and brief time (e.g. earthquakes, creep events, and dynamic or quasi-static fracture propagation) with a reasonable degree of accuracy, the large strain associated with geologic deformation accumulated over hundreds of thousands or millions of years is not elastic. Some deformational processes, particularly folding and the development of shear zones at elevated temperature and pressure, may be characterized by viscous or visco-elastic models [e.g. Biot, 1961; Johnson and Fletcher, 1994]. Shallow crustal deformation is typically brittle and may be characterized by irreversible (inelastic) processes

156 including fracturing [e.g. Atkinson, 1987; Pollard and Aydin, 1988], faulting [e.g. Scholz, 1990], intergranular slip [e.g. Borradaile, 1981; Goldsmith, 1989], pore collapse [e.g. Aydin, 1978; Mollema and Antonellini, 1996], and dissolution and precipitation [e.g. Stockdale, 1922; Fletcher and Pollard, 1981; Baron and Parnell, 2007]. Permanent deformation may be appropriately represented in a continuum framework by elasto-plastic or visco-plastic models [e.g. Brown and Bray, 1987; Cristescu and Gioda, 1994; Paterson and Wong, 2005]. Numerous studies over the past two decades have made significant progress in elasto-plastic simulation of faulting, including fault-related folding [Erickson and Jamison, 1995; Salvini et al., 2001; Cardozo et al., 2003; Hardy and Finch, 2005; Crook et al., 2006a; Crook et al., 2006b; Sanz et al., 2007; Sanz et al., 2008; Smart et al., 2010]. Such models have the potential to accurately represent in situ stress and large strain accumulated on geologic time scales. Despite the strides made, the highly nonlinear nature of inelastic models, particularly those with frictional contact, makes them computationally expensive and in some cases numerically unstable. Increasing model complexity (e.g. 3D simulation) generally increases the likelihood of numerical instability in addition to increasing computational cost. For this reason, most geomechanical studies involving nonlinear material behavior have been limited to two dimensional simulations. However, most geologic structures are inherently 3D in nature [e.g. Kaven and Martel, 2007; Mynatt et al., 2007; Sagy and Brodsky, 2009] and 2D approximations are limiting. Additionally, most nonlinear models require numerical parameters (e.g. penalty parameters and mass scaling constants) that may not influence the physics of the simulation, but the selection of which is critical to obtain a stable, physically meaningful solution [e.g. Wriggers, 2008]. Thus, in addition to computational requirements, execution may be a tedious process. Defining numerous material parameters necessary to characterize complicated plastic yield surfaces (e.g. SR3 [Crook et al., 2008]) as they evolve with strain and time can be a daunting task. In this context, the computational efficiency and numerical stability of linear

157 geomechanical models, such as those governed by linear elasticity and frictionless faults [e.g. Maerten et al., 2006; Lovely et al., in prep-a] have great appeal. We investigate the limitations of assuming frictionless faults and elastic continuum behavior by investigating two dimensional models representing a single fault that intersects the earth's surface in a homogeneous medium. The kinematics and stress fields suggested by models governed by an idealized elastic material and a more realistic, but still simplified, Drucker-Prager elasto-plastic material are compared, as are the results of models with faults represented by frictionless and frictional contact. Because mean normal stress dictates the cross sectional radius of the conical Drucker- Prager yield criteria [e.g. Yu, 2006] and influences frictional resistance to slip as defined by a Coulomb criteria [e.g. Jaeger et al., 2007], we investigate extensional and contractional tectonic settings, which represent the least and greatest horizontal compressive stress, respectively, that the crust may support. Subsequently, brief consideration is given to models of extensional tectonics representing more extensive fault networks, which may represent geological systems more realistically, and to the physical implications of plastic strain localization phenomena, which are suppressed in most analyses.

Model description Numerical models presented in this study were evaluated using the commercially available finite element package Abaqus®, version 6.9EF (http://www.simulia.com). Each model consists of a homogeneous elastic or elasto- plastic body containing one or more discretized fault surfaces. Faults are modeled as contact surfaces, which are not allowed to interpenetrate during simulations. In frictionless models, tangential motion (slip) between contact surfaces is calculated such that the surfaces are shear traction-free, and in models with friction slip is governed by a Coulomb friction law [e.g. Byerlee, 1978; Jaeger et al., 2007].

τ s ≤ µτ n (1)

The fault slips such that the shear traction (τs) resolved on the fault surface never exceeds the product of the normal traction (τn) and coefficient of friction (μ). A coefficient of friction μ = 0.6 [e.g. Byerlee, 1978] is used for frictional models, unless

158 otherwise specified. For simplicity, the models do not account for bedding plane slip or other effects of stratigraphic layering that probably are an important mechanism in the deformation of many reservoir scale structures [e.g. Sanz et al., 2008; Smart et al., 2010]. In elastic models, continuum deformation is governed by Hooke's law for a linear, isotropic material 1 ε = [(1+ν )σ −νδ ε ] (2) ij E ij ij kk Young's modulus, E = 15 GPa, and Poisson's ratio, ν = 0.30, are selected [e.g. Ord et al., 1991; Pollard and Fletcher, 2005]. The same parameters apply to the elastic domain of elasto-plastic models in which linear elasticity is supplemented by a Drucker-Prager plastic yield criterion [Drucker and Prager, 1952]. The Drucker- Prager yield surface is a smooth approximation of the Mohr-Coulomb surface and is represented by a circular cone in 3D stress space, centered about the line σ1 = σ2 = σ3, such that the deviatoric stress that a Drucker-Prager material may support increases linearly with compressive mean normal stress. The Drucker-Prager yield criterion is a simplified representation of rock deformation that does not account for the evolution of elastic properties as a result of brittle deformation, a process that may be accommodated by incorporating a damage rheology [e.g. Lyakhovsky et al., 1997], Plasticity models dependent on all three stress invariants are characterized by non- circular cross-sections in the deviatoric plane and may offer a better representation of rock [e.g. Colmenares and Zoback, 2002; Yu, 2006]; however, the Drucker-Prager model offers significant improvement over linear elasticity. The elastic domain, which comprises stress space inside and including the yield surface, is defined by

f = J 2 − aI1 − k ≤ 0 (3) where I1 = σkk is the first invariant of the stress tensor, J2 = SijSji is the second invariant of the deviatoric component of the stress tensor, and a and k are material constants defined by

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2sinϕ a = (4) 3(3 − sinϕ) and 6c cosϕ k = (5) 3(3 − sinϕ) where φ is the angle of internal friction, and c is the cohesion [e.g. Yu, 2006]. Although sedimentary rocks may exhibit a broad range of plastic properties, values used in this study are φ = 42° and c = 8.33 MPa. The elastic moduli and yield surface are characteristic of a competent sedimentary rock (e.g. sandstone [Ord et al., 1991]). All models are constrained to plane-strain deformation. Plastic flow is governed by a non-associative flow rule with a dilation angle, ψ = 38°, and modest isotropic strain hardening is imposed for stability. This dilation angle is significantly greater than values typically suggested by rock mechanics tests (ψ = 20° [Ord et al., 1991]), and may exaggerate the volumetric component of plastic strain; however, we use this value because smaller values lead to instability of the constitutive law that results in strain localization [e.g. Rudnicki and Rice, 1975] and mesh-dependent deformation [e.g. Deborst, 1988] when implemented in a dynamic formulation (see following paragraphs). Rudnicki and Rice [1975] identify a critical hardening modulus for the Drucker-Prager model, dependent upon elastic moduli, dilation angle, internal friction, and the local stress state, above which localization cannot occur and stability is guaranteed. However, the dependency on local stress state prevents straightforward calculation of parameters that guarantee stability throughout the simulation. The selection of a hardening modulus, h = 100 MPa, in conjunction with the greater dilation angle, prevents strain localization in the continuum, distal from faults. We do not guarantee the complete absence of localization in the vicinity of fault tips; however, none of the solutions presented exhibit mesh dependent behavior. Constant strain hardening is a significant simplification; however, it ensures numerical stability and strain hardening (or softening) is a particularly difficult property to characterize in rock. Rock mechanics testing indicates that rocks generally

160 undergo initial strain hardening, followed by softening; however, even for samples of the same composition, the stress-strain path may be highly dependent upon other variables such as confining pressure [e.g. Jaeger et al., 2007, p. 86]. Strain hardening may be a product of granular plastic deformation, including compaction, whereas softening often occurs due to cataclastic processes and fracturing, which decrease cohesion. Many mechanical properties of rock, including strain hardening and dilatancy, are likely scale dependent. Testing apparatuses are limited to small samples that, for example, do not include the influence of macro-scale fractures. How rock properties scale to the 100 meter size of the smallest elements in the mesh used here is poorly understood. Implicit integration of a quasi-static formulation would be the numerical implementation because solutions are guaranteed to be unique and stable if they are obtained. However, convergent solutions are difficult to obtain when establishing mechanical equilibrium after prescribing an initial lithostatic stress state in a faulted model, and due to numerical complexities associated with large deformation of the plastic continuum accompanied by frictional contact. The onset of strain localization may also result in model failure. Some of these numerical challenges may be circumvented by using the Abaqus/Explicit® dynamic formulation. The explicit time integration scheme is only conditionally stable, with a maximum time step dependent upon elastic wave speed and the smallest element size [e.g. Belytschko, 1983; Hughes, 1987]. Care must be taken to ensure that explicit solutions are physically valid, particularly if plastic strain localization occurs; however, it offers simplified treatment of frictional contact, and may provide accurate solutions to problems that the implicit procedure fails to solve [e.g. Prior, 1994]. Despite the dynamic nature of the formulation, quasi-static processes may be simulated by ensuring that kinetic energy is always much less than mechanical strain energy, thus negating the significance of inertial forces. Such applications of the explicit FEM formulation to the solution of quasi-static problems involving complex frictional contact conditions were first introduced in the metal forming industry [e.g. Prior, 1994; Taylor et al., 1995], and have subsequently been used in geologic studies [e.g. Peric and Crook, 2004].

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Single fault models In an effort to independently evaluate the kinematic implications of two fundamental simplifications commonly made to linearize mechanical models of fault- related deformation, we investigate all four permutations of models governed by elastic and elasto-plastic continuum deformation and by frictional and frictionless faults. We begin the analysis by considering simple models representing a single, linear fault in a homogeneous, two-dimensional body, in an effort to characterize the mechanical behavior and kinematics of fault-related deformation given various constitutive laws, while avoiding complicating factors such as mechanical interaction of multiple faults [e.g. Willemse et al., 1996] and slip on non-planar surfaces [e.g. Saucier et al., 1992; Chester and Chester, 2000]. The initial geometric configuration consists of a rectangular body 60km long by 40km deep, with a fault at the top center that extends to 5km depth, dipping to the right at 60° in models representing an extensional tectonic regime and at 30° in models representing contractional tectonics (Figure 4.1). Thus, in an effort to minimize edge effects, the fault is at least 25km from all model boundaries except for the free surface, and the analysis focuses on the region near to the fault. The finite element meshes (Figure 4.1) consist of linear elements, primarily four-node quadrilaterals with some three-node triangular elements. The meshes are refined in the primary area of interest, near the fault surface (100m element edge length along the fault), but are coarser (2000m element edge length) near the model periphery where high resolution is unnecessary. Extensional models consists of 4,785 elements and contractional models consist of 6,527 elements. Models are initially loaded with a lithostatic stress state, which is maintained throughout the simulation by gravitational body force. In the initial state, the base of the model is fixed to zero vertical displacement and the lateral model boundaries are fixed to zero horizontal displacement. All boundaries are treated as shear traction- free, and the upper boundary, corresponding to the earth's surface, is normal traction- free as well. After allowing the model to equilibrate with gravitational loads (displacements are minimal, much less than 1m), tectonic loading is applied by

162

Figure 4.1: Schematic illustration of model configuration and mesh geometry for single fault models representing a) extensional tectonics and b) contractional tectonics. In all models, the left boundary is constrained to zero horizontal displacement and the model base is fixed to zero vertical displacement. The fault is indicated by a solid black line and black arrows on the right boundary indicate displacements prescribed to impose 5% far-field strain. The black box around the fault indicates the primary area of interest, in which stress and strain are plotted in subsequent figures.

163 incrementally prescribing horizontal displacement on the right boundary of the model. Incremental application of loads enables analysis of results throughout evolution of the system. For contractional models, the boundary is ultimately shifted 3,000m to the left (-5% strain) and in extensional models it is shifted 3,000m to the right (+5% strain). Fault opening is prohibited in those models where it would otherwise occur (elastic extensional models). Because the non-linear behavior of rock is dependent on compressive stress magnitude (e.g. propensity for frictional sliding depends on normal traction and propensity for plastic strain depends upon mean normal stress), we consider extensional and contractional tectonic regimes. Assuming vertical stress to be a principle stress that depends only on the product of depth, overlying rock density and the acceleration due to gravity, and that the crust is critically stressed (i.e. at failure), contractional and extensional tectonics may represent end-members of the range of stress states that may exist in the brittle crust [e.g. Zoback, 2007]. Thus, generalizations that apply to contractional and extensional tectonics concerning the treatment of faults as frictionless sliding surfaces and the treatment of rock as a linear elastic material are likely to apply to intermediate stress states and other tectonic regimes.

Extensional model results Results are presented for only three extensional models (Figures 4.2-4.4) because, in fact, only three of the four models evaluated are unique. Under extensional loading in models governed by elastic continuum deformation, the fault surface is universally subjected to tensile normal traction. If permitted, the fault would open, forming a large chasm. Because friction resists sliding and allows for the accumulation of shear traction only when normal traction is compressive, faults in elastic extensional models with and without friction are shear traction free and therefore both elastic models behave identically. The slip distribution for the fault in elastic extensional models (Figure 4.2) is semi-elliptical, with maximum slip of 626m at the model's free surface. Slip distributions derived from models governed by Drucker-Prager continuum

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Figure 4.2: Fault slip distributions for extensional single fault models with various constitutive laws.

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Figure 4.3: Maximum principle strain (ε1) for single fault extensional models governed by various constitutive laws. a) elastic, b) Drucker-Prager with frictionless fault and c) Drucker-Prager with frictional fault (μ = 0.6).

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Figure 4.4: Maximum principle stress (σ1, tension positive convention) for single fault extensional models governed by various constitutive laws. a) elastic, b) Drucker- Prager with frictionless fault and c) Drucker-Prager with frictional fault (μ = 0.6).

167 deformation differ significantly from the elastic distribution. Each elasto-plastic model is characterized by relatively constant slip along the upper portion of the fault, with significant slip gradients only relatively close to the fault tip at depth. The frictionless elasto-plastic model has uniform slip of 485m from the surface down to nearly 4000m depth. The slip profile is rounded over the deepest 1000m, and a huge slip gradient (300m / 100m) is necessary across the single element adjacent to the fault tip in order for slip to go to zero. Maximum slip is significantly less (229m) in the Drucker-Prager model with friction. The region of nearly homogeneous slip is shorter, extending to only about 2500m, at which point slip begins decreasing gradually toward the fault tip. The slip gradient at the fault tip is steep (60m / 100m), but not as steep as the frictionless model. All three extensional models exhibit homogeneous maximum principle strain

(ε1, Figure 4.3) distal from the fault, with magnitude roughly equal to the far-field 5% extension. Strain perturbations near the fault tip, however, differ significantly, the largest difference being significantly greater (factor of 50% or more) magnitude of perturbations in elasto-plastic models. Elastic extensional models are characterized by a broad extensional strain shadow in the footwall, and a smaller strain shadow in the hanging wall, focused where the fault intersects the model's free surface (Figure 4.3a). Further to the right, above the fault tip, there is a subtle region of enhanced extension in the near surface. A small strain shadow develops up-to-the-right from the fault tip, but the dominant enhanced extension is below and to the left of the tip. ε1 is nearly horizontal throughout most of the model, but rotates near the fault such that the two in-plane principle strain components are perpendicular and parallel to the shear traction free fault surface. Elasto-plastic models exhibit a similar strain shadow adjacent to the fault in the near surface, though the extent is somewhat greater, particularly in the hanging wall. Fault-tip related strain perturbations are significantly greater in magnitude, and may be better characterized as a single lobe that is an extension of the fault surface, as opposed to the elastic perturbations that are two lobes, one adjacent to the fault and the other a down-dip extension of the fault. The largest strain perturbation occurs in the

168 frictionless elasto-plastic model (Figure 4.3b). A region of intensified strain extends down-to-the-right from the fault tip, effectively simulating downward propagation of the fault as a plastic shear zone, and a smaller, but still significant, perturbation extends up-to-the-right, which becomes broader as it approaches the model's free surface. A very subtle positive strain perturbation extends some distance down-to-the- left from the fault tip, in a similar location to the dominant elastic extensional perturbation, but its shape is not similar. In addition to the strain shadows adjacent to the fault surface, subtle strain shadows also occur to the left and right of the dominant perturbation extending down-dip from the fault tip. Significant strain rotation occurs near the fault, such that in-plane principle strain components are nearly perpendicular and parallel to the shear traction-free fault surface. The distribution of perturbations to the maximum principle strain field in the Drucker-Prager model with a frictional fault (Figure 4.3c) is very similar to that in the elasto-plastic frictionless model (Figure 4.3b), except that perturbations are significantly smaller in magnitude and extent. The dominant positive perturbation extends down-to-the-right from the fault tip, and a subtler perturbation broadens up-to- the-right toward the model's free surface. No perturbation is evident extending down- to-the-left. Similar, though subtler and smaller, strain shadows develop adjacent to the fault surface and on the periphery of the perturbation that trends downward from the fault tip. ε1 is generally subhorizontal, as in other extensional models, and exhibits significantly less rotation in the vicinity of the fault surface ‒ a result of non-zero shear traction related to the frictional strength. Stress magnitudes present the most dramatic differences between elastic and elasto-plastic extensional models. Although the orientation of maximum principle stress (σ1, tension positive convention) is generally similar in all three models (Figure 4.4), maximum principle stress is tensile throughout nearly all the model space in elastic models, but compressive almost everywhere in elasto-plastic models. The tensile stresses suggested by elastic models are an order of magnitude greater than elasto-plastic compressive stresses. Throughout the continuum, distal from the fault, elastic tensile stress exceeds 600 MPa. Large magnitude, though relatively local,

169 stress perturbations are observed in the vicinity of the fault tip (tensile to the left, compressive to the right), and a tensile stress shadow develops near the fault intersection with the model's free surface. This stress shadow is much more extensive in the footwall, as a broad, though subtle, tensile stress perturbation develops in the near surface of the hanging wall, above the fault tip. Tensile stress due to far-field extension is so great that the lithostatic gradient is difficult to discern in elastic models. Tensile stresses are nearly two orders of magnitude greater than the tensile strength of rock (~10 MPa for intact rock [Bieniawski, 1984] and likely less for fractured rock masses), and therefore could not realistically be achieved.

In contrast, the σ1 distribution in elasto-plastic models (Figure 4.4b & c) is compressive everywhere below about 1km depth, and the effect of the lithostatic stress gradient is dominant. In heavily fractured rock with no cohesion, stress would be compressive everywhere. The vertical gradient of this predominantly horizontal principal stress component is 6-7 MPa/km, only a fraction of the lithostatic gradient (25 MPa/km), because the applied extension reduces horizontal stress to the point at which plastic yielding occurs throughout. Perturbations around the fault, including the fault tip, have small magnitude compared to elastic perturbations. In the frictionless elasto-plastic model (Figure 4.4b), a positive (less compressive) perturbation develops to the left of the fault tip. It has two lobes, evidenced by sizable dips in contours of constant stress beneath the fault tip. A negative (compressional) perturbation develops to the right of the fault tip, which is evidenced by a narrow but intense up-to-the-left bulge in contours of constant stress. Stress perturbations are qualitatively similar in the elasto-plastic model with friction (μ = 0.6, Figure 4.4c), but are smaller in magnitude and area. Maximum principle stress is horizontal distal from the fault in all three extensional models; however, frictionless models exhibit significantly greater rotation of in-plane principle stress components near the fault surface. In both frictionless models, the traction-free nature of the fault requires that one in-plane principle stress component be parallel, and the other perpendicular, to the surface. In contrast, the

170 frictional fault in the elasto-plastic model can support shear traction, and therefore rotation of principle stresses is significantly less. Elastic, frictionless material models are linear, and therefore displacement, strain and stress perturbations are self-similar throughout evolution of a structure. As loads are applied incrementally, the magnitude of perturbations increases with linear proportionality and the geometry of distributions does not change, assuming that nonlinear geometric effects are small, as they are in models we consider. Models governed by inelastic material properties do not necessarily evolve in a self-similar manner. Therefore, it is important to consider the evolution of strain and stress fields; the final (5% far-field strain) state of non-linear models may not reveal important aspects of complex model behavior. Maximum principle strain distributions for elasto-plastic models with frictional faults at 0.5%, 1%, 2%, and the final 5% far-field strain are presented in Figure 4.5. The inelastic strain distribution at any stage of evolution may be compared with the elastic strain distribution in Figure 4.3a. We focus on the incremental evolution of this model because, of the models considered, elasto-plastic continuum behavior and frictional sliding on fault surfaces may be considered the most realistic representation of faulting in the brittle crust. Comparison with the results of this model should provide the most effective analysis of potential negative implications of simulating cumulative fault-related deformation with frictionless, linear elastic models at progressive stages of structural evolution that are represented by increasing far-field strain magnitude. At 0.5% far-field strain, the elasto-plastic extensional model is characterized by a strain shadow in the immediate vicinity of the fault that is broadest and most intense at the model's free surface and diminishes downward. The strain shadow is difficult to discern along the lower third of the fault surface. A very small extensional perturbation is evident at the fault tip, though it does not yet indicate down-dip propagation of the fault, and a triangular zone of enhanced extension develops in the hanging wall above the fault tip. This extensional perturbation coincides with a broad region of enhanced tension/extension in the near-surface of the hanging wall of the

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Figure 4.5: Evolution of maximum principle strain for single fault models of extensional tectonics characterized by elasto-plastic continuum deformation and frictional fault slip. Plots represent a) 0.5% far-field strain, b) 1% far-field strain, c) 2% far-field strain, and d) the final state, with 5% far-field strain.

172 elastic model (Figure 4.3a & 4.4a), but is significantly more intense relative to far- field extension, is narrower, and extends deeper than the elastic perturbation. At 1% far-field extension, the strain perturbation extending down-to-the-right from the fault tip has become the dominant feature. The strain shadows on either side of the fault surface extend down nearly to the tip, and the triangular region of enhanced extension has extended downward and linked with the fault tip, becoming narrower and more intense with depth. At 2% far-field strain, the perturbation extending down from the fault tip is longer and broader, but the distribution of perturbations is otherwise similar. The distribution does not change significantly (though perturbation magnitudes increase) as far-field extension increases to 5%. Stress distributions at the same stages of evolution are presented in Figure 4.6. At 0.5% far-field extension, stress contours are nearly linear and horizontal. The vertical gradient of the horizontal stress component is significantly less than the initial lithostatic gradient, due to extension, but fault-related perturbations are difficult to discern. By 1% strain, the vertical gradient of the predominantly horizontal σ1 is further reduced, and a stress perturbation is evident in the vicinity of the fault tip. Stresses are less compressive to the left of the fault tip, evidence by a downward dip in contours of uniform stress, and compression is enhanced to the right of the fault tip. Further changes in maximum principle stress distribution, magnitude, and orientation as far-field strain grows beyond 1% are remarkably subtle because the stress state nearly everywhere in the model is on the yield surface and further strain is plastic. Strain perturbations may be compared directly with the elastic model in Figure 4.4a; however, it is important to note that the elastic stress distribution is not entirely self- similar throughout the structure's evolution. While the stress perturbations grow in a self-similar manner, they are superimposed on a lithostatic stress gradient that does not change with time. A dramatic contrast between elastic and elasto-plastic stress fields develops quickly because elastic perturbations continue to grow linearly with incremental application of far-field strain, while stress changes are minimal in elasto- plastic models beyond 1% far-field strain.

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Figure 4.6: Evolution of maximum principle stress for single fault models of extensional tectonics characterized by elasto-plastic continuum deformation and frictional fault slip. Plots represent a) 0.5% far-field strain, b) 1% far-field strain, c) 2% far-field strain, and d) the final state, with 5% far-field strain.

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Contractional model results For models simulating contractional tectonics, the elastic, frictionless model produces a semi-elliptical slip distribution (Figure 4.7), similar to the extensional model with elastic material and frictionless slip. However, slip in the contractional model is greater (maximum slip is 1017m) because the fault is longer (same depth, shallower dip). It is important to note that slip distributions plotted throughout this manuscript represent only magnitude ‒ displacement discontinuity is always presented as positive. Thus, the sign of extensional normal faulting and contractional thrust faulting is not distinguished. In the elastic model in which fault slip is subject to frictional resistance (μ = 0.6), the slip distribution differs subtly from the semi-ellipse of elastic models. It is characterized by a rounded slip gradient near the fault tip at depth, and a nearly constant gradient (linear slip profile) from more than 4000m depth to the model's free surface. This fault has significantly less slip than the frictionless elastic model, with a maximum of 698m. The slip distribution from the elasto-plastic model without friction is similar to that of the extensional model with the same constitutive properties: slip is nearly constant (maximum slip of 1296m) from the model's free surface to 4000m depth, below which the slip profile is gently curved, and there is an extremely large slip gradient (nearly 800m / 100m) across the deepest element on the discretized fault. Slip magnitude nearly everywhere on the fault is greater than the maximum slip in the other models. Irregular displacement discontinuity near where the fault intersects the model's free surface results from gravitational collapse of the hanging wall where thrusting has pushed it up and it loses contact with the footwall. The Drucker-Prager model with a frictional fault accommodates less slip than any other contractional model, with a maximum magnitude of 487m. The slip distribution on this fault is similar to that of the elastic model with friction ‒ in fact, they are nearly identical within 500m of the fault tip and slip on the upper portion of the fault is characterized by a constant gradient ‒ but the slip gradient along the upper portion of the fault is somewhat less.

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Figure 4.7: Fault slip distributions for contractional models with various constitutive laws.

176

For contractional tectonics, we consider the minimum in-plane principle strain and stress (ε2, σ2) because they represent the predominantly horizontal (contraction- parallel) component of deformation. All models are characterized by relatively homogeneous minimum principle strain distant from the fault (Figure 4.8), approximately equal to the far-field contraction (-5%). In elastic models, the most prominent strain perturbation is a region of enhanced contraction below and to the left of the fault tip. This perturbation is geometrically similar in both elastic models, though significantly larger in the model without friction (Figure 4.8a). It is geometrically similar, though opposite in sign, to the elastic strain perturbation in extensional models. In both elastic models, ε2 is near-zero in the hanging wall near where the fault intersects the free surface, though the strain shadow is more extensive in the frictionless model, and in both models significant rotation of principle strain is observed near the fault. In the frictionless model, strain rotation is prominent throughout much of the hanging wall, and reaches 60° (ε2 is approximately fault- perpendicular) at the fault surface. Rotation is somewhat less in the model with friction. Rotation of the strain tensor in the elasto-plastic frictionless model (Figure 4.8c) is similar to that observed in the elastic frictionless model (Figure 4.8a), but in the elasto-plastic model with friction (Figure 4.8d), very little strain rotation occurs. In both elasto-plastic models, the most prominent strain perturbation is a region of enhanced contraction associated with the fault tip. In the frictionless model, this perturbation is quite large, in size and magnitude. It is relatively broad, and extends beyond the lower right corner of the area in which strain is plotted (Figure 4.8c). The very large magnitude of this perturbation at the fault tip is necessary to accommodate the extremely large slip gradient (Figure 4.7). A subtle region of enhanced contraction extends down-to-the-left from the fault tip, as well as one that extends nearly straight up, which separates two regions where contraction is inhibited. To the left, ε2 is near zero throughout much of the hanging wall, except very near where the fault intersects the model's free surface, where the hanging wall loses contact with the footwall and

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Figure 4.8 (opposite): Minimum principle strain (ε2) for single fault contractional models governed by various constitutive laws. a) elastic with frictionless fault, b) elastic with frictional fault, c) Drucker-Prager with frictionless fault and d) Drucker- Prager with frictional fault.

178

179

Figure 4.9 (opposite): Minimum principle stress (σ2) for single fault contractional models governed by various constitutive laws. a) elastic with frictionless fault b) elastic with frictional fault, c) Drucker-Prager with frictionless fault and d) Drucker- Prager with frictional fault .

180

181 collapses under gravitational load. To the right, a region of inhibited contraction evolves above the region of enhanced contraction associated with the fault tip. In the elasto-plastic model with a frictional fault (Figure 4.8d), strain perturbations are much smaller and the magnitude and orientation are more homogeneous than in the elasto-plastic model without friction. The contractional strain shadow in the hanging wall, near where the fault intersects the model's free surface, is much smaller than in any other model, and the region of enhanced contraction at the fault tip is much smaller. This region is relatively narrow, and extends down-to-the-right from the fault tip, effectively simulating fault propagation. This presents a significant contrast to the elastic strain concentration that is centered below and left of the fault tip. Other strain perturbations evident in the frictionless elasto-plastic model are absent or too small to identify in the model with friction. As in extension, elastic models of contractional tectonics generate extremely large stress magnitudes (Figure 4.9a & b). Most compressive stress (σ2) magnitudes exceed 2 GPa in the region beneath the fault tip. Both elastic models exhibit similar distributions, including vertical stress ranging from approximately -700 MPa in the near surface to -1 GPa at 10km depth, significant positive and negative perturbations above and below the fault tip, respectively, and greatly reduced compression in the hanging wall, particularly near where the fault intersects the model's free surface. Perturbations in the hanging wall and associated with the fault tip are significantly greater in the frictionless model (Figure 4.8a). Although rock is much stronger in compression than in tension and compressive strength typically increases with confining stress, deviatoric stresses in elastic contractional models are significantly greater than permitted by shear failure criteria such as Mohr-Coulomb [e.g. Jaeger et al., 2007] or Drucker-Prager. Stress rotation in the vicinity of the fault surface is greater in the frictionless model, though significant rotation occurs in the frictional model as well. σ2 is consistently horizontal elsewhere in both models. The stress distribution in elasto-plastic contractional models (Figure 4.9c & d) is dominated by a vertical gradient, though in contractional models the vertical gradient of the predominantly horizontal σ2 (nearly 100 MPa/km) is several times

182 greater than lithostatic because horizontal compression increases elastically with contractional loading until plastic yielding occurs. In both elasto-plastic models, stress is perturbed near the fault, as evidenced by contours of constant stress, which are horizontal in the far-field, being drawn down toward the fault in the hanging wall and up toward the fault in the footwall. Compressive stress is enhanced beneath the fault tip in both models, and slightly reduced above the tip. All stress perturbations are more pronounced in the model without friction. As in any model in which the fault is frictionless (shear traction free), in-plane principle stress components rotate such that they are parallel and perpendicular at the fault surface. This results in particularly large stress rotation throughout much of the hanging wall. On the contrary, stress rotation is minimal in the elasto-plastic model with friction. As with extensional models, it is important to consider the evolution of nonlinear models because kinematic and mechanical fields do not evolve in a self- similar manner. We only consider evolution of the most realistic, fully nonlinear model with elasto-plastic continuum deformation and frictional resistance to fault slip.

At 0.5% far-field strain (Figure 4.10a), ε2 is homogeneous (-0.5%) throughout the model, except for a small positive perturbation (contractional strain shadow) near the fault where it intersects the model's free surface. This strain shadow is more extensive in the hanging wall, but evident in the footwall, as well. These strain shadows grow through 1% and 2% far-field strain (Figure 4.10b & c), extending downward, along the fault. Additional perturbations do not appear until 2% far-field strain, at which point contractional strain has begun to concentrate in the vicinity of the fault tip. By 5% far-field strain (Figure 4.10d), the fault tip strain concentration has grown significantly, contractional strain shadows extend along the entire length of the fault in the footwall and hanging wall, and another region of reduced contraction extends up- to-the-right from the fault tip. Throughout the evolution of this model, the stress distribution is dominated by a vertical gradient. Perturbations are relatively small (Figure 4.9d) and the evolution of the stress field is not shown because it provides little insight.

183

Figure 4.10: Evolution of minimum principle strain for single fault models of contractional tectonics characterized by elasto-plastic continuum deformation and frictional fault slip. Plots represent a) 0.5% far-field strain, b) 1% far-field strain, c) 2% far-field strain, and d) the final state, with 5% far-field strain.

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Implications of a linearized constitutive law A variety of previous studies relating models to field observations suggest that elastic models with frictionless faults may simulate the kinematics of fault-related deformation surprisingly well [e.g. Maerten et al., 2006; Resor, 2008; Lovely et al., in prep-a]. The success of these studies coupled with the linear nature of the elastic, frictionless boundary value problem, which provides for stable and computationally efficient numerical implementation, make such models very appealing for mechanical simulation of fault-related processes. However, it is well known that rock deformation accompanying multiple fault slip events over geologic time scales is permanent and inelastic. The premise of the research described here is that elasto-plastic (Drucker- Prager) models with frictional faults provide an improved representation of the kinematic and mechanical behavior of faults and the surrounding rock. We suggest that elasto-plastic continuum deformation and frictional sliding are the most realistic behaviors of the four combinations investigated; however, we acknowledge that the Drucker-Prager material model remains a significant simplification of rock deformation.

Elastic, frictionless models There are issues with elastic simulations of cumulative fault-related deformation that are readily apparent without in-depth comparison with inelastic models. One such limitation is the large tensile normal traction necessary to maintain contact between fault surfaces in models of extensional tectonics characterized by elastic deformation. In elastic extensional models, the fault would open forming a large chasm if tensile tractions were not applied to prevent such behavior, but there is no geological or mechanical means by which such a force might be applied in nature. Although fissures may form at shallow depths when normal faults breach the surface [e.g. Gudmundsson, 1987; Grant and Kattenhorn, 2004], rock is not sufficiently strong to support open chasms along dipping fault surfaces extending more than tens of meters depth. Gravitational force and inelastic deformation lead to hanging wall collapse, which naturally maintains contact compliance across the fault.

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Perhaps the most glaring limitation of elastic models simulating extensional tectonics (Figure 4.4a) is that the horizontal component of stress is almost everywhere tensile. This contradicts in situ stress measurements that typically are compressive in any tectonic regime [e.g. Zoback, 2007], and clearly is unphysical where the tensile stress is greater than the strength of rock, which typically is less than 10MPa in tension [Bieniawski, 1984]. In contractional models (Figure 4.9a & b) the stress state on the compressive side of the fault tip, and in fact throughout most of the model space, exceeds typical shear failure strengths for rock (e.g. Coulomb failure, [Jaeger et al., 2007] or the Drucker-Prager yield criteria [Drucker and Prager, 1952]). Comparison of elastic stress fields with those of elasto-plastic models with friction (Figures 4.4c & 4.9d) provides an explanation for how excessive elastic stress may be reduced. Stress magnitude, and even sign, may differ dramatically between geometrically identical models subjected to the same far-field strain. Clearly, any mechanical model that might be used to infer in situ stress magnitude in a highly deformed region must account for material strength and appropriate yield criteria. Despite differences in magnitude, do distributions of stress and strain perturbations and the orientation of the tensor fields suggested by elastic models accurately reflect those features of inelastic models? In extension, the elasto-plastic model with friction (Figure 4.4c) does exhibit a tensile stress perturbation to the left of the fault tip , and a compressive perturbation to the right, similar to elastic models (Figure 4.4a); however, the spatial extent of these perturbations, in addition to their magnitude, is much less. Both models suggest negative perturbations in the vicinity of the fault surface in the hanging wall and in the footwall near where the fault intersects the model's free surface, as well as a broad tensile perturbation in the near surface above the fault tip. However, all perturbations are much more subtle in the elasto- plastic model, and despite the similarities suggested, the overall geometry of stress contours is remarkably different. At the 5% far-field strain on which our analysis focuses, results suggest that elastic models have a very limited ability to simulate the first order distribution of stress perturbations due to cumulative fault-related deformation. Plastic deformation prevents the development of large stress

186 perturbations. Further, frictionless models suggest too much rotation of the stress tensor near the fault surface. One might expect that elastic stress perturbations would better reflect elasto- plastic deformation at small far-field strain, because plastic strain accumulation is minimal; however, such does not appear to be the case. The evolution of the elasto- plastic stress field up to 5% far-field strain (Figure 4.6) demonstrates that elastic models do no better at simulating the distribution or orientation of stress distributions at lesser strain. At very small (0.5%) far-field extension, elasto-plastic stress perturbations are negligible, and as far-field strain grows beyond 1%, stress distributions change remarkably little, even in magnitude. On the contrary, elastic perturbations, which are superimposed upon the initial lithostatic gradient, grow linearly in magnitude and in a geometrically self-similar manner. Elastic, frictionless models representing contractional tectonics exhibit a similarly limited resemblance to the elasto-plastic model with frictional slip. The elastic, frictionless model (Figure 4.9a) exhibits significantly greater rotation of the stress tensor near the fault, particularly in the hanging wall, and produces a much larger stress shadow near the fault. In the elastic model, fault-tip perturbations form distinct lobes, whereas in the non-linear model perturbations are limited to drawn-up contours of constant stress in the footwall and draw-down contours in the hanging wall (Figure 4.9c & d). As in extension, plastic deformation suppresses elastic stress perturbations. Elastic, frictionless models are characterized by two to three times more slip than elasto-plastic models with frictional faults (Figures 4.2 & 4.7). An important implication to consider is that any study based on elastic models that attempts to reproduce measured fault slip may dramatically under-estimate far-field strain magnitude because the linear model does not account for frictional resistance to slip or plastic continuum deformation. However, the differences in the down-dip slip profiles are more subtle. Inelastic models tend to exhibit relatively uniform slip gradients in the near surface (nearly constant slip on the upper half of the fault in extension), versus the semi-elliptical slip distributions of linear elastic models. Because cross-

187 sectional exposures of crustal-scale faults are difficult or impossible to find in the field, it is difficult to determine if the slip distributions of one material model are significantly better than the other [Resor, 2008]. However, the nearly constant slip on the upper half of the frictional fault in the elasto-plastic model presents an interesting comparison with kinematic models, which typically assume slip is constant everywhere on a fault [e.g. Woodward et al., 1989; Shaw et al., 2005]. The greatest difference between strain fields in elastic (Figure 4.3a) and elasto- plastic models (Figure 4.3c) is the size and nature of the fault-tip strain perturbation. In extension, the elastic perturbation is dominated by two lobes of enhanced extension to the left of the fault tip, in contrast to the single band of extension in the elasto- plastic model that extends down-to-the-right, in line with the fault surface. At 5% far- field extension, these simplified models suggest that elastic models could provide reasonable insight to the distribution of geologic strain perturbations in the near- surface, although absolute magnitudes may differ significantly, but in the vicinity of the fault tip, the limitations of elasticity are significantly greater. In extension, the limitations of elasticity are somewhat less at very small far-field strain (<1%), as demonstrated by the evolution of the elasto-plastic strain field (Figure 4.5). However, as soon as the fault tip perturbation becomes a dominant feature, significant differences in this region of the model become apparent. In contraction, the implications of using elasticity are similar. In the near surface, the elastic, frictionless model (Figure 4.8a) suggests a contractional strain shadow in the hanging wall, near where the fault intersects the surface, that is similar in nature to the elasto-plastic model with friction (Figure 4.8d), but the fault-tip stress perturbation is significantly different. The elastic strain perturbation at the fault tip is essentially opposite that of the extensional elastic model, with two lobes of enhanced contraction below/left of the fault tip, in contrast to an elasto-plastic perturbation characterized by enhanced contraction in an elongated region extending down-dip in the line of the fault surface. Comparison of strain distributions at 5% far-field strain in extensional and contractional tectonic environments suggests that elastic models may provide reasonable insight to the location and extent of regions of enhanced strain and

188 relative strain shadows associated with fault slip distal from the fault tip, but the differences between models are much greater in the vicinity of the fault tip. Unlike extensional tectonic models, the strain field of elasto-plastic contractional models at earlier stages of evolution (Figure 4.10) is not similar to the elastic distribution. Although intuition suggests that the linearized model might better represent the nonlinear model when plastic strain is minimal, a pronounced difference is apparent between linear and nonlinear models when far-field contraction is small because of the effect of friction. Frictional strength prevents the entire fault from slipping at far-field strain less than 2%. The model geometry may be inappropriate if the fault did not extend to 5km depth at this stage of tectonic evolution or if it propagated up from depth and did not yet breach the surface. The ability of linear models to simulate earlier stages of fault system evolution may be significantly restricted in models that do not explicitly represent fault propagation because the entire fault slips, even at the earliest stages of model evolution. Frictional strength may effectively simulate fault growth at small strain, and models could probably be further enhanced by applying some initial cohesive strength to the fault surface in an effort to simulate initially intact rock. By 2% far-field strain, at which point the entire fault has slipped, the differences between the elasto-plastic strain distribution and that suggested by the linear model for contractional tectonics are similar to the previously discussed differences at 5% strain. Discussion of fault propagation and the strain distribution around the fault tip raises an important issue concerning initial model geometry and fault interpretation. In elasto-plastic models with a single fault representing 5% extension or contraction, strain is concentrated in a narrow zone extending down-dip along the line of the discretized fault trace that effectively simulates down-dip propagation of the fault tip, and indicates that models may be overly simplified. Although the models demonstrate significant value for illustrating the effects of frictional slip and plastic continuum deformation, they may provide a poor representation of any real geological system because a single fault extending to 5km depth is not sufficient to accommodate the far- field strain that is applied. A more appropriate model should be defined by an initial

189 geometry that contains more faults and/or faults that extend to greater depth. Faults often form in systems, rather than as isolated structures, and it may be important, particularly in inelastic models, to represent the entire system, and not just a single structure, in order to capture appropriate mechanical interaction [e.g. Willemse et al., 1996]. Similarly, when tectonic strain is very small, continuous faults likely have not formed to 5km depth, and fault represented in models should be appropriately smaller. Frictional strength in nonlinear models may help alleviate the implications by preventing fault slip at depth when far-field strain is small. In elasto-plastic models with frictional fault slip, under representation of the fault geometry (i.e. too small or too few faults) may have significant implications not only for the location of fault-related strain perturbations, but also for their extent and shape, particularly in the vicinity of the fault tip. Strain distributions in linear models, too, are dictated largely by fault geometry. Geologic data, particularly in the subsurface, is almost always limited, and precise characterization of fault geometry, tip locations in particularly, is rarely possible. Although strain magnitudes may differ significantly, linear and nonlinear models suggest similar distributions of strain perturbations distal from the fault tip. Up to the 5% far-field strain we investigate, elastic models are likely appropriate for simulating many first-order characteristics of fault-related deformation, despite significantly different stress distributions. We do not question that appropriate inelastic models generally offer an improvement over linearized elastic simulations, however, the value of the improvement is undoubtedly greatest in situations where all aspects of the deformational system (fault geometry, material properties and heterogeneity, etc.) are particularly well constrained. In a situation where any model would be poorly constrained and first order accuracy is all that can be hoped for, nonlinear models may offer limited additional value for relatively great expense. We return to this topic of fault geometry, and down-dip extent in particular, later, in the context of strain localization and multi-fault models.

Partially linearized constitutive models We consider two additional models, one with an elastic continuum and frictional faults and another with elasto-plastic continuum and frictionless faults, in an

190 effort to independently evaluate the implications of neglecting each of the two dominant nonlinear material effects concerning cumulative fault-related deformation. Elastic models with frictional fault slip are commonly, and appropriately, applied to studies of earthquake ruptures [e.g. Aochi and Fukuyama, 2002; Lovely et al., 2009]. More advanced dynamic friction laws are often used [e.g. Dieterich, 1978], but friction is the primary nonlinear process in most seismic events where continuum deformation may be effectively approximated as elastic. Elastic models with frictional contact have also been used to represent cumulative deformation, more traditionally considered in the realm of structural geology [e.g. Cooke and Pollard, 1997]. Several studies suggest that the primary effect of including homogeneous Coulomb friction in quasi-static elastic models is to reduce slip magnitude, and that slip distributions are relatively unaffected. [e.g. Aydin and Schultz, 1990; Cooke and Marshall, 2006]. The results presented here confirm that the primary effect of friction is to reduce slip magnitude, but also demonstrate that the influence of friction in elastic models depends on the tectonic regime, and calls into question elastic models of processes dominated by inelastic continuum deformation. For example, a direct implication of the tensile normal traction across the fault in the extensional elastic models is that friction has no influence; however, normal faults do accumulate shear stress, as demonstrated by fault gouge and the occurrence of earthquakes. Contractional elastic models impose large compressive normal stresses on the fault surface that result in significant resistance to slip. Maximum slip in the elastic model with friction is approximately 30% less than in the frictionless model; however, this is only half the approximately 60% reduction in fault slip due to friction in elasto- plastic models (Figure 4.7). Slip distributions and other fields affected by friction in elastic models of cumulative fault-related deformation may not be any more appropriate than models without friction. In both extensional and contractional settings, the slip distribution for elasto- plastic models without friction is characterized by a large slip gradient near the fault tip and nearly constant slip elsewhere. In extensional and contractional models, the slip gradient is associated with strain perturbations that are probably unphysically

191 large in magnitude (>300% in extensional models and >800% in contractional models) near the fault tip, but that are geometrically similar to those of inelastic models with friction, though significantly greater in spatial extent (Figures 4.3b & 4.8c). In light of the results for frictionless elasto-plastic models, we must ask if it is ever appropriate to treat one important form of nonlinearity while neglecting another?

Strain localization A variety of elasto-plastic constitutive models, including Drucker-Prager, can be subject to instabilities that lead to plastic strain localization [Hill, 1952; Thomas, 1961; Rudnicki and Rice, 1975; Rice, 1976]. Rudnicki and Rice [1975] demonstrated that strain localization may develop in Drucker-Prager materials even in the presence of modest strain hardening, and identified a critical hardening modulus above which localization cannot occur. Typical elasto-plastic properties from rock mechanics tests on sedimentary rocks are subject to localization, which may result in non-convergence of quasi-static, implicit FEM analyses, and may lead to the development of non- unique, mesh-dependent shear bands in explicit, dynamic analyses [e.g. Deborst, 1988]. Strain localization may be likened to real geologic processes, including the formation of faults or deformation bands [e.g. Borja and Aydin, 2004], and laboratory tests have demonstrated that strain localization, even in the presence of hardening processes, is a real phenomenon in rock [e.g. Ord et al., 1991]; however, physical inferences may not be made from models that suffer from numerical instabilities and do not have unique solutions. A localization limiting procedure [e.g. Leroy and Ortiz, 1989], or some other method to regularize localization (e.g. visco-plasticity [Loret and Prevost, 1990]) is necessary to obtain a unique, convergent solution, but such procedures are not implemented in Abaqus®. Initial analyses for this study were performed assuming perfect plasticity (h = 0) and a dilation angle, ψ = 21°, in accordance with rock mechanics tests [e.g. Ord et al., 1991], and these models were subject to mesh dependent localization. Templeton

192

Figure 4.11: Maximum principle strain (ε1) distributions for the entire 40x60 km extent of models in which continuum deformation is characterized by a more appropriate dilation angle (ψ = 21°) and perfect plasticity (h = 0). Deformation is subject to strain localization, but solutions are mesh-dependent and non-unique. Results represent a) continuous (faultless) elasto-plastic extensional model and b) elasto-plastic extensional model with a single frictional fault. The black box indicates the extent of the primary area of interest on which analysis of single fault models focuses.

193 and Rice [2008] encounter a similar, unstable localization phenomenon in an analysis of elasto-plastic effects on simulations of earthquake rupture dynamics using Abaqus/Explicit®, and provide a thorough explanation of the numerical source of the instability. In all single fault models presented previously, we circumvent such instabilities and the associated numerical challenges by adjusting material properties (dilation angle, ψ, and hardening modulus, h) such that localization is suppressed throughout the continuum, distal from fault-tips. By suppressing localization we force deformation to be more distributed. If localization processes occur at a small scale (e.g. distributed networks of deformation bands), much less than the 100+ meter element size in the FEM mesh, to suppress localization may be appropriate. On the contrary, if localization is characterized by the development of a system of large faults, suppression may be inappropriate. Abaqus/Explicit® simulations in which unstable localization occurs clearly lean toward the latter process of fault system development (Figure 4.11). Models in which elements range in size from 100-2000m are characterized by dominant shear bands with kilometer scale spacing, between which extension is relatively small. A continuous (faultless) model is presented (Figure 4.11a), as well as a model with a single, frictional fault (Figure 4.11b), and patterns of strain localization are similar. The greatest difference is the predominant shear band dip, which is an arbitrary function of mesh geometry. The influence of mesh density on shear band formation is apparent: bands tend to be narrower and more closely spaced in the top center of each model where the mesh is most highly refined. Apparently, a single fault extending to 5km depth is insufficient to accommodate the far-field strain. We suggest that a natural system with similar material properties and 5% extension would be characterized by a system of faults extending greater than 5km in depth, perhaps with a configuration similar to the shear bands of Figure 4.11. Shear bands in contractional models dip ~30°, typical of thrust faults, whereas extensional shear bands dip ~60°, typical of normal faults as predicted by Anderson's theory of faulting [Anderson, 1905].

194

Multi-fault models We suggest that the single fault models presented here probably represent only an early stage in the development of a fault system because every elasto-plastic model is characterized by a single, narrow band of strain localization extending down from the fault tip in the line of the fault, effectively simulating fault propagation. To simulate the evolution of a fault system, an elasto-plastic model prone to localization (e.g. Figure 4.11) may be appropriate, though it is critical that an appropriately calibrated numerical mechanism to stabilize the simulation and regularize localization (eliminate mesh dependency) be used. In such a study, it would be critical to consider the effects of boundary conditions applied to the model base and sides. The constant displacement boundary conditions applied in these models result in reflection of shear bands and apparently do not simulate boundaries that are likely to occur in the earth. To simulate the behavior of a specific fault system, it would be appropriate to construct a model geometry that represents the complete fault system, or at least the dominant faults within it. By prescribing appropriate frictional strength and cohesion, it may even be possible to effective simulate fault propagation along these discretized structures as far-field strain is applied incrementally. Although some geometric simplification is probably necessary, for every fault that is omitted from a model more continuum deformation in that location would be necessary to accommodate the far- field strain. We simulate such a fault system with a model in which the dominant bands of strain localization from Figure 4.11b are represented as nine fault surfaces (Figure 4.12). We acknowledge that the distribution of faults is arbitrary and that structures extend significantly deeper than typical seismogenic depth of the brittle crust [e.g. Nazareth and Hauksson, 2004], but suggest that the model is indicative of processes that might occur in a well-developed extensional fault system. The relatively small strain between the shear bands in Figure 4.11 indicates that they represent a fault system that is sufficient to accommodate the applied far-field strain. The interconnected nature of the fault system prevents fault-tip stress and strain concentrations, and, as long as frictional strength of the faults is moderate (μ≤0.4), fault slip may accommodate most of the applied strain, such that continuum

195

Figure 4.12: Schematic illustration of model configuration and mesh geometry for densely faulted model representing extensional tectonics. The bold black box represents the area of interested for which results are presented in figures 4.13 and 4.14.

196 deformation throughout much of the area of interest (<10km depth) is almost entirely elastic. As a result, elastic and elasto-plastic models have remarkably similar strain (Figure 4.13a, b, d & e) and even stress (Figure 4.14a, b, d & e) distributions. Elastic stress magnitudes typically are less than rock strength and, as a result, elastic models with moderate friction may be useful approximations. However, if frictional strength is large (μ = 0.6), slip is less and more deformation must be accommodated in the continuum; plastic strain becomes greater, and the differences between elastic (Figures 4.13c & 4.14c) and elasto-plastic (Figures 4.13f & 4.14f) models increase. In other words, the negative implications of using elasticity become greater. The inelastic models presented in Figures 4.13 (d-f) and 4.14 (d-f) are governed by the adjusted elasto-plastic material model with increased dilatancy angle and hardening modulus. Thus, strain localization is not observed; however, localization would be abundant in the elasto-plastic model with μ = 0.6 (Figure 4.13f) if it were not suppressed. The effect of modifying the Drucker-Prager properties is minimal in densely faulted elasto-plastic models with lesser friction (Figure 4.13d & e) because deformation is almost entirely elastic. Many laboratory tests suggest μ ≈ 0.6 [e.g. Byerlee, 1978]; however, field examples suggest that some crustal scale faults are significantly weaker (e.g. California's San Andreas Fault [Zoback et al., 1987]). μ ≤ 0.4 may be a reasonable estimate in some cases. The results of multi-fault simulations indicate that elastic models with modest friction may be quite appropriate for simulating the behavior of well-developed fault systems. As fault systems evolve, existing faults will grow and link together and new faults might form until a well-connected fault system has developed [e.g. Cowie et al., 2000]. With a well-developed fault network that may accommodate nearly all tectonic strain, elastic models can provide reasonable results up to relatively large far-field strain, providing that folding is not a dominant part of the deformational process and that the fault network is well-connected in the subsurface. The multi-fault models presented in this manuscript consider only extensional tectonics, and the observations presented may be more pertinent to extensional regimes. Kinematic studies suggest that extensional environments such as the Basin

197

Figure 4.13: Maximum principle strain (ε1) for extensional models with multiple intersecting faults, governed by various constitutive laws. a) elastic with frictionless fault, b) elastic with fault friction μ = 0.4, c) elastic with fault friction μ = 0.6, d) Drucker-Prager with frictionless fault, e) Drucker-Prager with fault friction μ = 0.4, and e) Drucker-Prager with fault friction μ = 0.6.

198

Figure 4.14: Maximum principle stress (σ1) for extensional models with multiple intersecting faults, governed by various constitutive laws. a) elastic with frictionless fault, b) elastic with fault friction μ = 0.4, c) elastic with fault friction μ = 0.6, d) Drucker-Prager with frictionless fault, e) Drucker-Prager with fault friction μ = 0.4, and c) Drucker-Prager with fault friction μ = 0.6.

199 and Range of the Western United States are dominated by horst and graben structures, and that the crustal blocks between faults predominantly undergo rigid body rotation [e.g. Proffett, 1977]. In contrast, contractional tectonics tend to be characterized by highly deformed fold and thrust belts [e.g. Shaw et al., 2005]. Elastic deformation is a much more reasonable representation of the relatively undeformed blocks between normal faults than of the folded rocks characteristic of contraction. Mechanical representation of fault-related folds requires some inelastic mechanism to accommodate folding processes. Bedding-parallel slip, which could be represented by numerous contact surfaces or by some form of inelastic continuum deformation, is one such mechanism that may accommodate folding. In any situation where folding is a dominant process, the fault network is not adequately represented (e.g. critical faults are omitted), or fault tips accommodate significant slip gradients in the subsurface, inelastic continuum deformation is likely to play a relatively greater role in deformational processes, and the implications of elastic models may be significantly greater.

Evaluating the implications of elastic models on a site-specific basis Elastic deformation may be a reasonable assumption for systems consisting of well-connected fault networks that are fully represented in the model framework, but if fault-tips and large slip gradients are prevalent elastic models have greater limitations. However, even in such cases, elastic models may provide reasonable first order insight to the distribution of strain perturbations distal from the fault tip. We also point out that elastic models are likely limited in systems where deformation is characterized by significant folding. Linear elastic models with frictionless faults are of particular interest to structural geologists because of their numerical stability and computational efficiency for simulating 3D deformational systems with non-planar faults and other geometric complexities. The application of finite element analysis to simulate elasto-plastic (or other nonlinear constitutive model) continuum deformation of 3D models with numerous frictional contact surfaces requires tremendous computational expense and is prone to numerical instabilities. Dense, 3D finite element meshes with embedded

200 faults are difficult to construct and lead to large computational cost, even for linear analysis, making boundary element methods, which require discretization of fault surfaces only, particularly appealing [e.g. Thomas, 1993; Maerten, 2010]. Lovely, et al. [in prep-a] use a linear elastic boundary element model for 3D analysis of deformation associated with slip on normal faults in the Volcanic Tableland, Bishop, CA, but compare results from preliminary linear and nonlinear 2D models to evaluate the implications of assuming elastic deformation and frictionless faults. We recommend using a similar, preliminary, 2D, site-specific analysis to evaluate the negative implications prior to interpreting geological meaning from 3D elastic models.

Conclusions Linear elastic models with frictionless faults are of significant interest to structural geologists for two primary reasons. First, they are computationally efficient and numerically stable, enabling straightforward and rapid simulation of slip distributions and off-fault deformation in geometrically complicated fault systems, particularly in three dimensions [e.g. Maerten, 2010]. Second, the linear nature of these models makes them reversible, and applicable to retrodeformational modeling techniques [Lovely et al., in prep-b], including mechanics-based restoration [Muron, 2005; Maerten and Maerten, 2006; Plesch et al., 2007; Moretti, 2008; Guzofski et al., 2009]. However, fault-related deformation accumulated over geologic time scales is largely inelastic in nature. A comparison of simple 2D models representing a single fault in a homogeneous body indicates that elastic, frictionless models may capture kinematic patterns of fault-related deformation to first order, particularly distal from the fault tip, as demonstrated by comparison of slip and strain distributions, but tip-related strain perturbations may differ significantly. The magnitude of strain perturbations in elasto- plastic models tends to be greater but slip magnitude is less when fault friction is used. Elastic models suffer much greater shortcomings for predicting geologic stress distributions, and may err by orders of magnitude.

201

Simple models with a single fault are valuable for illustrative purposes, but elasto-plastic strain concentrations around fault tips, and plastic strain localization patterns in models where such behavior is not suppressed, indicate that the models with a single fault are overly simplified. We consider an extensional model representing a more extensive fault system. Results indicate that, given a sufficiently dense and interconnected fault network, most strain may be accommodated by fault slip and elastic models may be appropriate, even with relatively large far-field strain. Stress magnitudes in such models may be geologically realistic, in which case elastic models with friction may be appropriate. Site-specific evaluation of the negative implications of using elastic models may be performed by comparing deformation fields from linear and nonlinear 2D models of a representative cross-section [Lovely et al., in prep-a], and is recommended before using elastic models to study fault-related deformation in 3D.

Acknowledgements This work has been supported by the National Science Foundation's Collaborations in Mathematical Geosciences grant no. CMG-0417521. P. Lovely has been supported by the Robert and Marvel Kirby Stanford Graduate Fellowship Fund.

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Appendix

Regions of reduced static stress drop near fault tips for large strike slip earthquakes

Abstract

We present a case study of slip distributions for the 1992 Landers (MW 7.3) and the 1999 Hector Mine (MW 7.1) earthquakes in California’s Mojave Desert. Slip distributions, as determined from geophysical inversion of geodetic, strong ground motion, and teleseismic data, are complex, heterogeneous, and often exhibit linearly tapering or concave upward patterns toward segment tips, distinctly different from the elliptical slip distributions characteristic of uniform stress drop. Mechanical interaction of discontinuous fault segments fully explains the reduced slip near the southern termination of the Camp Rock / Emerson fault segment of the Landers rupture; however, numerical models demonstrate that such interactions are insufficient to explain slip distributions observed at other segment terminations. Numerical models demonstrate that long (5-25 km) zones of reduced stress drop in the vicinity of some rupture segment terminations can explain the slip distributions for these large earthquakes. Zones of reduced stress drop are implemented as regions of increased Coulomb strength. Slip distributions are improved 30-70% relative to models with uniform stress drop. Regions of reduced stress drop appear to play a relatively greater role near segment tips at which rupture terminates than near segment tips at which rupture jumps to a nearby fault segment. Similar results are obtained implementing discrete, stepwise and spatially linear reductions of stress drop. Plausible mechanical explanations for such zones of reduced stress drop include heterogeneous fault strength or friction, spatial or temporal changes in pore pressure, geometric complexity of the fault surface, heterogeneity of normal tractions resolved on the fault surface, inelastic deformation, and dynamic rupture effects.

Introduction Given a uniform stress drop along a linear fault trace, slip should follow an elliptical distribution, with no slip at the fault tips and a maximum value at the 213 midpoint [Eshelby, 1957; Pollard and Segall, 1987]. Field observations of earthquake surface ruptures as well as geophysical inversions for slip distributions, however, suggest that such an elliptical distribution does not accurately describe most earthquake ruptures. Manighetti, et al. [2004, 2005] suggest that slip follows a triangular distribution not only when measuring cumulative slip distributions on well- established faults, but also for individual earthquake ruptures. Manighetti, et al. [2005] document such triangular slip distributions along both strike and dip on most of the 120 surface ruptures and geophysical inversions they investigated, with slip tapering roughly linearly toward either tip from a maximum which usually does not coincide with the fault midpoint. Martel and Shacat [2006] attribute such slip distributions to mechanical interaction of faults and fault segments. Local asperities and the non-elliptical nature of observed slip distributions suggest that earthquake ruptures do not, in fact, produce uniform stress drops along strike [e.g. Beroza and Mikumo, 1996; Peyrat, 2001]. Ward [1997] describes slip distributions near rupture terminations, in the context of earthquakes in the Imperial Valley, CA, as ‘rainbows’ (concave downward) associated with near uniform stress drop, and ‘dogtails’ (concave upward) associated with reduced stress drop. Ward suggests that, while concave downward terminations occur at strong fault tips, kinks, and segment boundaries, concave upward terminations can occur anywhere on a fault, in regions of low or moderate strength due to changes in the traction on the fault surface. Ward models concave upward rupture terminations by imposing optimal, but arbitrary, pre-stress on fault surfaces with homogeneous strength, and his pre-stress distributions are highly heterogeneous. Cohesive end-zone theory [Dugdale, 1960; Barenblatt, 1962], as applied to faults [Ida, 1972; Palmer and Rice, 1973; Rudnicki, 1980; Li, 1987; Burgmann, et al., 1994; Martel, 1997; Wilkins and Schultz, 2005] suggests that a short zone of increased friction in the immediate vicinity of the fault tip causes slip to taper, produces a negative stress drop in the end-zone, and eliminates the stress singularity associated with the fault tip. Cohesive end zones are used to approximate a region of inelastic deformation near the tip, while maintaining an elastic solution to the problem.

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Cohesive end zones are generally understood to be very short relative to fault length [Okubo and Dieterich, 1984; Cowie and Scholz, 1992; Martel, 1997], and hence are not appropriate to explain the long, linear tapering and concave upward slip distributions observed after earthquake ruptures. We adapt this concept, however, to suggest an alternative explanation of slip distributions near rupture segment terminations indicative of reduced stress drop, as regions of increased fault strength in a homogeneous stress field. Tractions on a fault surface in a given stress field are not arbitrary [Pollard and Fletcher, 2005], and while tectonic stress is heterogeneous, it too is not arbitrary, and Hauksson [1994] shows that in the decade leading up to the Landers rupture, spatial stress heterogeneity along the rupture trace was small.

Focusing on the June 28, 1992 Landers earthquake (MW 7.3) and the October

16, 1999 Hector Mine earthquake (MW 7.1), we apply a numerical model to investigate the implications of regions of reduced or negative stress drop over unrestricted lengths extending to rupture segment terminations on these large, multi- segment strike-slip faults. Both the Landers and Hector Mine events were right- lateral, primarily strike-slip events which occurred in California’s Mojave Desert on sub-parallel, multi-segment faults separated by only 20 km (figure A.1a). Each event ruptured three nearly linear fault segments and each event produced maximum local slip in excess of 700 cm [Wald and Heaton, 1994; Ji, et al., 2002]. Figure A.1b-c presents maps of the region, with the two rupture traces idealized as in various numerical models. The Landers and Hector Mine events were selected for this study because they are among the best documented strike-slip events of MW > 7 on record due to their recent occurrence and the abundant, high-quality geophysical monitoring and documentation in the region [e.g. Campbell and Bozorgnia, 1994; Wald and Heaton, 1994; Hudnut, et al., 1994; Ji, et al., 2002; Agnew, et al., 2002; Graizer, et al., 2002]. Slip distributions and the associated static and dynamic stress drop have been calculated for both of these events using combinations of geodetic, strong ground motion, and teleseismic data [Wald and Heaton, 1994; Cotton and Campillo, 1995; Hernandez, et al., 1999; Zeng and Anderson, 1999; Peyrat, et al., 2001; Ji, et al.,

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Figure A.1: Map of the Mojave Desert, CA region, showing principal stress orientations from the World Stress Map database [Reinecker, et al., 2005], and epicenter locations and rupture model geometries for the Landers and Hector Mine earthquakes. (a) Surface rupture traces of the Landers and Hector Mine earthquakes, from Treiman, et al. [2002]. (b) Multi-segment rupture model geometries derived from Wald and Heaton [1994] for the Landers event and Ji, et al. [2002] for the Hector Mine event; each rupture is modeled as 3 intersecting straight fault segments. (c) Single-segment rupture model; each rupture is modeled as 1 kinked fault segment. (d) Curvilinear fault geometries based on surface rupture maps, modified from Aochi, et al. [2003] for the Landers event and from Maerten, et al. [2005] for the Hector Mine event. Abbreviations: CRE – Camp Rock / Emerson faults; HV – Homestead Valley fault; LJV – Landers / Johnson Valley faults; LLW – Lavic Lake fault, west segment; LLE – Lavic Lake fault, east segment; LL – Lavic Lake fault; B – Bullion fault.

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2002; Jonsson, et al., 2002; Kaverina, et al., 2002; Salichon, et al., 2003, Maerten, et al., 2005]. Each of these studies assumes a rupture geometry consisting of rectangular, planar fault segments, discretized into rectangular dislocations, with the exception of Maerten, et al. [2005], who study a non-planar fault model for the Hector Mine event, using triangular dislocations. Stress drop distributions calculated for each event are heterogeneous. Subsequent studies have investigated the total static stress drop and the evolution of dynamic stress drop, particularly for the Landers earthquake. Bouchon, et al. [1998] calculate static and dynamic stress drop associated with the Landers rupture and show that each follows a similar, heterogeneous spatial distribution, with static stress drop roughly 40% less in magnitude. They use the dynamic evolution of the stress field during rupture to explain how the rupture jumps onto the northern Camp Rock / Emerson fault segment despite the segment’s unfavorable orientation relative to the regional stress state. Day, et al. [1998] study the evolution of shear traction on fault surfaces for the Landers event, as well as the

1994 Northridge, CA (MW 6.7) and 1995 Kobe, Japan (MW 6.9) events, in the context of friction laws, drawing the conclusion that a simple slip- or velocity-weakening friction law can explain the evolution of surface traction resolved on the fault surface, without needing a more complicated, dynamic ‘self-healing,’ or velocity / slip strengthening model [Heaton, 1990]. Aochi and Fukuyama [2002] investigate 3D dynamic rupture simulations on a non-planar representation of the Landers fault geometry. Implementing a spatially uniform dynamic friction law and a remote stress tensor of uniform magnitude, but rotating 18 degrees along the length of the fault, they obtain, to first order, a realistic slip distribution and appropriately heterogeneous stress drop. Mechanical interactions of discrete fault segments in these multi-segment ruptures, and stress heterogeneity lingering from nearby historic fault slip, lead to non- elliptical slip distributions and non-uniform normal stresses across fault planes, which result in non-uniform stress drop; however, as we will demonstrate, mechanical interactions alone can not explain the observed slip distributions. We compare depth- averaged slip distributions derived from geophysical inversions of geodetic data,

217 strong ground motion, and teleseismic energy [Wald and Heaton, 1994; Ji, et al., 2002] with numerically modeled slip distributions parameterized by uniform fault strength and with slip distributions modeled with zones of increased strength in the broad vicinity of rupture segment terminations. In the context of earthquake dynamics, these regions of increased fault strength are associated with ruptures which cannot release enough energy to sustain themselves. These regions are characterized by negative static stress drop. We use these zones of increased strength as a proxy for regions of increased frictional strength, or geometric complexities beyond the resolution of discretized models, or changes in pore water pressure, or traction heterogeneity acting on the fault surface, or inelastic deformation, or dynamic rupture effects.

Methods We utilize a quasi-static, two-dimensional, plane-strain boundary element method (BEM) based on Crouch and Starfield [1983]. The model uses the displacement discontinuity method in an infinite linearly elastic body and has been expanded to incorporate Coulomb strength using inequality constraints and principles of complementarity [Pang and Trinkle, 1996; DeBremaecker and Ferris, 2004] to solve the resulting nonlinear set of equations for displacements and tractions on each element of the discretized fault model. The numerical implementation, which enables the assignment of a distinct quasi-static coefficient of friction (μ) for each element of the fault model, is described by Mutlu and Pollard [2008]. We use μ as a proxy for fault strength and associated regions of reduced stress but do not envision such a simple frictional sliding mechanism, as explained previously. Many researchers today would approach the problem addressed here with a dynamic rupture model [Harris, 2004]; however, we select a quasi-static model because the simplified friction model, governed by a single parameter, minimizes the degrees of freedom when optimizing regions of increased fault strength. While we acknowledge that the quasi-static model is a substantial simplification of the complex dynamic rupture process, we contend that the implications do not substantially compromise the results of this study. It is widely accepted that friction is dependent 218 on sliding distance or velocity. Numerous sophisticated dynamic friction laws have been proposed [e.g. Dieterich, 1978], the simplest of which involves a stepwise weakening from static friction (µs) to dynamic friction (µd) with the onset of sliding. Day, et al. [1998] demonstrated that stress drop distributions inferred from geophysical inversions for evolution of slip distribution do not require self-healing dynamic friction models, as suggested by Heaton [1990], and that a relatively simple slip- or velocity-weakening law is adequate. The quasi-static friction model is a further simplification; however, because µ is only a proxy for fault strength, the quasi- static friction model is sufficient. The value of µ is somewhere between µs and µd, which represent the maximum static strength and minimum dynamic strength, respectively. Two other features differentiate the quasi-static model from dynamic models: neglect of time-dependent dynamic rupture propagation and neglect of the inertia terms in the equations of motion. Ward [1997] suggests through comparison of quasi- static and dynamic rupture simulations that the inertial effects of propagating seismic waves play only a secondary role in controlling slip distributions. While dynamic rupture propagation can be iteratively implemented in quasi-static models if one assumes a constant rupture propagation velocity [e.g. Ward, 1997], such an approach requires at least two friction coefficients (static and dynamic). This approach would add an additional degree of freedom for each region of increased fault strength. The minimal number of degrees of freedom permitted by the relatively simple model presented here contributes to the statistical validity of the conclusions. A thorough study of the implications of ignoring dynamic propagation of the rupture pulse and the inertial terms of the equations of motion would make an interesting and valuable project, but it is beyond the scope of this study. Two rupture model geometries are investigated in detail. In the first, which we refer to as “multi-segment” models, each fault segment trace is represented as a straight line, and discretized to 198 elements of equal length (figure A.1b). Each event is modeled with three distinct fault segments. At segment intersections, a common node is shared by elements of both segments, enabling a complete transfer of slip

219 between segments. Such a perfect transfer of slip is probably not realistic; however, geophysical inversions for slip distributions [Wald and Heaton, 1994; Ji, et al., 2001] suggest some transfer of slip between adjoining segments, and models better match the geophysical inversions if slip is transferred than if fault segments are modeled as discrete entities separated by intact rock that would require slip to go to zero at all segment tips. In these models, the effects on slip distribution of regions of reduced stress drop are considered at all rupture segment terminations, including those at which rupture ceases on that fault segment but continues along another nearby. In the second model geometry, which we refer to as “single-segment” models, the multi segment rupture models are simplified into a single kinked segment (figure A.1c). In these models, the effects of regions of reduced stress drop are considered only at rupture extremities, and not at segment tips on the interior of the complete rupture. These models, when compared with the multi-segment models, provide insight concerning the importance of regions of reduced stress drop at absolute rupture terminations relative to segment terminations where rupture continues along a nearby fault. For both geometries, the two earthquake ruptures were modeled independently. If the two events were modeled simultaneously, stress perturbations due to slip on the Hector Mine faults would influence slip on the Landers faults, a scenario which does not make sense considering the chronological sequence of the events. Additionally, inelastic relaxation of stress perturbations due to the Landers event, prior to the Hector Mine event, cannot be quantified. Young’s modulus and Poisson’s ratio were selected as 20 GPa and 0.2, respectively, typical elastic moduli of crystalline rock at the kilometer scale [Pollard and Fletcher, 2005]. Remote compressive stress magnitudes were selected based upon typical effective stresses for a strike-slip faulting regime at 7.5 km depth, roughly half the seismogenic thickness in the Mojave Desert [Nazareth and Hauksson, 2004], assuming hydrostatic pore pressure, σHmax = 125 MPa and σhmin = 50 MPa [Lund and Zoback, 1999]. We assume one principal stress to be vertical; however, horizontal principal stress orientations described in the literature for the Mojave Desert region are not sufficiently well constrained for the purposes of this study because modeled slip

220 distributions are highly sensitive to the orientation of the remote principal stress. King, et al. [1994] suggest that the most compressive stress for Landers is oriented approximately N7˚E. Because of proximity, we assume a similar orientation for Hector Mine. Focal mechanism inversions provided by the World Stress Map (WSM) database [Reinecker, et al., 2005] suggest most compressive stress orientations locally varying from north to N45˚E. WSM data in the region are presented in figure A.1. Principal stress orientations were selected by running 2D models for each multi- segment rupture geometry parameterized with a uniform strength of μ = 0.4 on all fault segments for a range of principal stress orientations. Most compressive stress orientations of N7˚E for Hector Mine and N14.5˚E for Landers were selected because they produced the most balanced distributions of slip between all fault segments and are within the range of documented stress orientations. Deviations of just a few degrees from these orientations result in zero, or near-zero, slip on certain fault segments. No single remote stress orientation produced uniform slip distributions on all segments of both rupture models. We suggest that insisting upon a uniform remote stress for both events is neither necessary or appropriate because WSM data suggest local heterogeneity of principal stress orientations greater than the 7.5˚ difference between models, and because a variety of models show that the Landers rupture could have significantly altered local stresses in an area large enough to include the Hector Mine event [King, et al., 1994; Deng and Sykes, 1997; Freed and Lin, 2001; Masterlark and Wang, 2002; Price and Burgmann, 2002; Harris and Simpson, 2002; Pollitz and Sacks, 2002]. Aochi and Fukuyama [2002] suggest that the stress tensor rotates along the length of the Landers rupture. Fault strengths were varied systematically by varying μ on each fault segment to determine the optimal model slip distribution, assuming uniform strength on each segment, but allowing for variations in strength between segments. Optimal model slip distributions are those that produce the least RMS error relative to depth averaged slip distributions determined by geophysical inversion [Wald and Heaton, 1994; Ji, et al., 2002]. Allowing for variation of strength between fault segments was particularly important for the Landers rupture, where dissimilar segment orientations result in

221 much greater slip on the central Homestead Valley segment than on the Camp Rock / Emerson or Landers / Johnson Valley segments under uniform strength conditions. By imposing a greater strength on the Homestead Valley segment, the slip distributions are brought into line with those determined using geophysical data [Wald and Heaton, 1994]. We are unaware of data suggesting variation of fault strength due to material heterogeneity, fault zone maturity, or other factors, between segments on either rupture. Aochi and Fukuyama [2002] suggest a rotation of the remote stress tensor along the length of the Landers rupture. We suggest that such local heterogeneity of stresses throughout the region, as illustrated in figure A.1 [Reinecker, et al., 2005], is a more likely explanation than heterogeneous fault strength; however, the quasi-static model is limited to a uniform remote stress tensor and the focus on stress drop is not compromised. Another potential explanation suggests dynamic stresses associated with propagating seismic waves, and neglected in the quasi-static model, drive rupture on the northern Camp Rock and Emerson fault segments, in which case absolute fault strength may be greater than represented in numerical models, and more comparable to the strength we attribute to the central Homestead Valley segment [Bouchon, et al., 1998; Harris, et al., 2002]. This effect should not significantly influence conclusions related to stress drop and strength variation within fault segments. The effect of regions of reduced stress drop in the vicinity of fault segment tips is studied by implementing regions of increased fault strength. For an isolated, straight fault segment in a perfectly elastic material, uniform strength along the fault would result in an elliptical slip distribution and a uniform stress drop [Eshelby, 1957; Pollard and Segall, 1987]. For Landers and Hector Mine multi-segment rupture models, calculated stress drops, illustrated in figure A.2, are spatially heterogeneous even with uniform fault strength, because of variations in tractions induced by mechanical interaction with other segments. These interactions, however, are not sufficient to explain the tapered and concave-upward slip distributions observed on rupture segments. Regions of reduced or negative stress drop in the vicinity of fault

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Figure A.2: Plots illustrating slip distribution (top) and stress drop (middle) from multi-segment rupture models parameterized by frictional strength profiles (bottom) for (a) Landers and (b) Hector Mine earthquakes. Results are shown for models incorporating linear and stepwise reductions of stress drop as well as uniform fault strength. Slip distributions from depth-averaged geophysical inversions for Landers [Wald and Heaton, 1994] and Hector Mine [Ji, et al., 2002] are shown by stars. Rupture segments are plotted from north to south and segments are labeled on the map in figure A.1. 223 segment tips, implemented as regions of increased Coulomb fault strength, produce slip distributions which match geophysical inversions much better. Models were run in which strength increases toward the segment tip in a discrete, stepwise manner, as well as those in which strength increases in a gradual, linear manner relative to a base value. While a wide variety of reduced stress drop distributions could be possible, depending on the geological mechanism of causation, we analyze linear and stepwise strength increases for their simplicity and because they are representative of a variety of geological mechanisms. Dimensions and strength magnitudes for regions of reduced stress drop are selected to optimize in a least squares sense the fit of model slip distributions to depth-averaged slip distributions from geophysical inversions. Generally, if a concave upward slip distribution exists near a segment tip, the zone of increased strength begins near the inflection point in the slip distribution as measured by geophysical inversion. Because the geologic mechanism producing the region of reduced stress drop is not known (several were proposed in the introduction), we are unable to constrain the locations of strength increase or their magnitudes better than can be done by minimizing the deviation of model slip from the geophysical inversions. Not all segment tips require regions of reduced stress drop to produce accurate model slip distributions. Slip distributions at these segment tips are generally elliptical in character, suggestive of uniform stress drop. Regions of reduced stress drop are implemented on an “as needed” basis, meaning that they are applied only in the vicinity of segment tips where they can produce a substantially better fitting slip distribution to the geophysical inversions. No limits are placed on the length of these zones of reduced stress drop, and some are greater than 75% of segment length. A similar procedure was followed for the single-segment rupture models, illustrated in figure A.1c. In these models, because only one fault segment is implied and there are no segment terminations along the rupture, zones of reduced stress drop are considered only at distal rupture extremities, and slip must approach zero only at these absolute rupture extremities. As in the case of the multi-segment models, the central Homestead Valley fault is more optimally oriented according to the Coulomb

224 strength criteria employed, than the distal faults. Thus, for the Landers single-segment rupture geometry, a “uniform strength” model produces unrealistic slip distributions. To account for this, we apply greater strength to the central portion of the rupture, representing the Homestead Valley fault, to produce a best fitting model without regions of reduced stress drop. “Uniform strength” does not apply along the entire rupture. Such measures were not necessary for the Hector Mine model; hence, uniform strength is represented by μ = 0.425 everywhere. The rupture models involve substantial geometric simplification because they are 2D and because we assume linear segment geometries. Slip inversions by both Wald and Heaton [1994] and Ji, et al. [2001] resolve slip on a planar surface in 3D model space and suggest substantial depth dependence of slip magnitude, which suggests that 2D models may be an oversimplification of the rupture mechanics, which could lead to incorrect conclusions. To address this concern, we briefly analyze slip distributions and the effects of regions of increased fault strength and reduced stress drop near segment tips, both laterally and vertically, using the 3D boundary element code Poly3D [Thomas, 1993]. We utilize rectangular, planar fault geometries with identical dimensions to those used by Wald and Heaton [1994] and Ji, et al. [2001] for their inversions. Fault planes for the Landers rupture are vertical and those for the Hector Mine event dip 75º - 80º northeast. Models assume a linear elastic half-space with the same elastic moduli as the 2D models. We assume a vertical effective stress gradient with the intermediate principal stress oriented vertically [Hauksson, 1994] and similar stress magnitudes and orientations in the horizontal plane to those used for the 2D models, such that stress magnitudes at half the seismogenic depth are equivalent to stress magnitudes in 2D models. The vertical effective stress gradient is

σHmax = 16.667 MPa / km, σV = 15 MPa / km, and σhmin = 6.667 MPa / km. All stress components are zero at the free surface. Models with uniform fault strength are compared with models with lateral regions of increased fault strength and reduced stress drop of the same dimensions as those used in the 2D models. Because faults in a 3D model have a tip-line at their base, a region of increased fault strength and reduced stress drop is considered in association with this tip-line as well.

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Recently, researchers have expressed concern over the implications of the simplified fault geometries used for many of the early studies of the Landers and Hector Mine events. Surface rupture maps show that planar dislocation models are a substantial simplification of complicated 3D surfaces. Maerten, et al. [2005] use geodetic data to invert for fault slip on non-planar fault models constructed from triangular dislocations. Aochi and Fukuyama [2002] and Aochi, et al. [2003] perform 3D dynamic rupture simulations of the Landers earthquake using a non-planar fault model. These studies use fault surface models produced by extrapolating surface rupture trace maps to depth assuming constant dip. Fliss, et al. [2005] use the Landers rupture as a field example for a numerical investigation of dynamic rupture propagation and directivity at fault segment boundaries and triple junctions. While they consider several detailed segment boundaries within the rupture, they do not consider the complete rupture geometry. We address concern over use of simplified fault geometries with a series of 2D models using curvilinear fault geometries (figure A.1d) derived from the models of Maerten, et al. [2005] and Aochi and Fukuyama [2003], which were themselves derived from surface rupture maps. We do not consider regions of reduced stress drop on these models, but compare slip distributions resulting from homogeneous strength along each segment with slip distributions generated by the simplified fault geometries also assuming homogeneous strength, in order to assess the influence of geometric complexity on slip distributions. Ideally, this analysis would have been performed in 3D, however, surface traces provide little insight to fault connectivity and geometry at depth. Felzer and Beroza [1999] use aftershock relocations from the Landers event to suggest that fault connectivity at depth is substantially different from its surface representation, specifically at the junction of the Homestead Valley fault and the Emerson fault. While it is theoretically possible to constrain complex subsurface fault geometry using precise aftershock relocations and focal mechanisms, such an effort is beyond the scope of this study. Because fault connectivity plays a substantial role in quasi-static model results, 3D models with complex, non-planar fault geometry are unlikely to provide meaningful insight due to the poor geometric constraint in the

226 subsurface. Additionally, direct, quantitative comparison of slip modeled on complex 3D surfaces with inversions on planar surfaces posed a problem.

Results Figure A.2 shows slip distributions, stress drops, and fault strength profiles for each fault segment of the Landers and Hector Mine multi-segment rupture models, using uniform fault strength, as well as regions of stepwise and linearly increasing strength of optimal size and magnitude near segment tips. Slip distributions derived from depth-averaged geophysical inversions are plotted for comparison. Table A.1 presents RMS error values comparing all model slip distributions to those measured by geophysical inversion and presents percent improvement for stepwise and linear reduced stress drop models relative to uniform strength models. The implementation of regions of reduced stress drop in the vicinity of segment tips enables multi-segment models to produce substantially improved slip distributions relative to models without such reductions. For the Landers event, improvements are about 35 % and for Hector Mine improvement is 65-70%, as measured by RMS error. For both events, the magnitude of improvement enabled by linear and stepwise zones of reduced stress drop near segment tips is similar. Linear zones of reduced stress drop generally need to be longer than stepwise zones in order to have the optimal effect on slip distributions, and the maximum reduction of stress drop (at the segment tip) for the linear zones is generally greater than that needed in optimal stepwise zones of reduced stress drop. Regions of reduced stress drop are necessary at almost all absolute rupture extremities; however, only for the Hector Mine model with linear reduction of stress drop are they necessary at all segment terminations on the interior of the rupture. For Landers models with stepwise and linear zones of reduced stress drop, only three zones of reduced stress drop are necessary. They are located at the absolute rupture extremities, and at the southern extremity of the central Homestead Valley segment. All of the zones are at least one third the length of the associated rupture segment. The size of the zones of reduced stress drop is best illustrated in the third row of figures A.2 & A.3, where fault strength can be seen to shift in an abrupt step 227

Model RMS error (cm) RMS error (cm) Percent with reduced without reduced improvement stress drop stress drop Landers, multi-segment, stepwise 65.56 99.29 34.0 Landers, multi-segment, linear 65.04 99.29 34.5 Hector Mine, multi-segment, stepwise 21.29 74.09 71.3 Hector Mine, multi-segment, linear 26.21 74.09 64.6 Landers, single-segment, stepwise 84.11 90.07 6.6 Landers, single-segment, linear 85.08 90.07 5.5 Hector Mine, single-segment, stepwise 31.96 126.3 74.7 Hector Mine, single-segment, linear 30.62 126.3 75.8 Landers, 3D, stepwise 85.6 164.7 48.0 Hector Mine, 3D, stepwise 63.5 160.1 60.3

Table A.1: RMS error for modeled slip distributions relative to depth-averaged slip distributions from geophysical inversions. Results are presented for models with uniform strength and with linear and stepwise regions of reduced stress drop toward rupture segment terminations for multi-segment and single-segment 2D rupture models. Results are also presented for 3D rupture models with stepwise regions of reduced stress drop. Percent improvement is calculated from the reduction of RMS error by models with regions of reduced stress drop relative to models without such zones.

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Figure A.3: Plots illustrating slip distribution (top) and stress drop (middle) from single-segment rupture models parameterized by frictional strength profiles (bottom) for (a) Landers and (b) Hector Mine earthquakes. Results are shown for models incorporating linear and stepwise reductions of stress drop as well as models without regions of reduced stress drop. Slip distributions from depth-averaged geophysical inversions for Landers [Wald and Heaton, 1994] and Hector Mine [Ji, et al., 2002] are shown by stars. Ruptures are plotted from north to south.

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(stepwise models) or to shift from a constant value to a uniform non-zero slope (linear models). It is interesting to note that the southern end of the Camp Rock / Emerson segment, despite dramatically reduced slip relative to the rest of the fault, does not require a zone of reduced stress drop implemented as increased fault strength. The reduced slip here is entirely a product of mechanical interactions of fault segments. The Hector Mine rupture is modeled with three absolute rupture extremities at the northern tips of both Lavic Lake fault segments and at the southern end of the Bullion fault. The northern end of the relatively smaller Lavic Lake fault, west segment, in the stepwise model is the only absolute rupture extremity in any of the multi-segmented models at which reduced stress drop does not significantly improve the slip distribution. In the linear model, a zone of reduced stress drop is implemented, but it is small in size and magnitude. Zones of reduced stress drop are implemented effectively at all segment terminations in the linear model. For the stepwise model, no zone of reduced stress drop is necessary at the northern ends of the Lavic Lake fault, west segment, or the Bullion fault. With the exception of the Lavic Lake fault, west segment, zones of reduced stress drop at distal rupture extremities are longer than those between segments along the rupture. For the single-segment rupture models, in which zones of reduced stress drop are implemented only at absolute rupture terminations, these regions of reduced stress drop do very little to improve modeled slip distributions for the Landers earthquake. RMS error is minimized by broad zones of reduced stress drop of 15-20 km at each end, but error magnitude is reduced by only 5-6%. For the Hector Mine earthquake, however, improvement of 74-75% can be obtained. Interestingly, because the slip distribution, as measured by Ji, et al. [2002], is quite asymmetric, the region of reduced stress drop at the southern end of the rupture is far longer than that at the northern extremity. This zone is roughly one third the rupture length for a stepwise reduction of stress drop and greater than half the rupture length for a linear reduction. Figure A.4 presents contour plots of model slip distributions with and without zones of reduced stress drop, geophysical inversion slip distributions, model error distributions, and the optimal extent of regions of reduced stress drop for 3D analysis

230

Figure A.4: Contour plots of 3D model results for the Bullion Fault segment of the Hector Mine earthquake. The right-hand edge of each plot corresponds to the southeast tip of the fault. a) Model results with uniform fault strength. b) Model results with stepwise regions of reduced stress drop near the northern and basal fault tips. c) Slip distribution derived from geophysical data (Ji, et al., 2001). d) Error distribution between model without regions of reduced stress drop and geophysical inversion. e) Error distribution between model with regions of reduced stress drop and geophysical inversion. f) Gray shading represents the extent of the region of reduced stress drop.

Figure A.5: Slip distributions for the (a) Landers and (b) Hector Mine earthquakes from curvilinear fault models with uniform fault strength (solid line), linear, multi- segment fault models with uniform strength (dashed line) and depth-averaged geophysical inversions (stars). Slip calculations for the non-linear fault models are projected onto the linear fault trace for comparison.

231 of the Bullion fault segment of the Hector Mine event. This fault segment is presented as a typical example of 3D analysis for each of the three fault segments in the Landers and Hector Mine events. Uniform strength model results (figure A.4a) concentrate peak slip at a much greater depth than suggested by geophysical inversion, and permit too much slip near fault segment tips. Stepwise zones of reduced stress drop along strike, with identical dimensions to those used in the 2D analysis, accompanied by a stepwise reduction of stress drop at half the fault’s depth, result in slip distributions quite similar to those suggested by geophysical inversion. As seen in table A.1, implementing regions of reduced stress drop of equivalent lateral extent to those used in 2D models, as well as regions of reduced stress drop associated with the tip line at the base of the fault, model slip distributions can be improved by 48% and 60% for the Landers and Hector Mine events, respectively, as compared with geophysical inversion results. Figure A.5 presents slip distributions computed from uniform strength models on curvilinear fault geometries, and for comparison, slip distributions computed from uniform strength on linear, multi-segment models and slip from depth averaged geophysical inversions. Slip calculated on curvilinear faults is not compared quantitatively with the depth-averaged geophysical inversions because of complications introduced by the need to project slip measurements onto a different fault geometry; however, qualitative analysis of figure A.5 shows that, assuming homogeneous strength, slip computed on the curvilinear fault geometries derived from surface ruptures does not correlate with slip calculated by geophysical inversion better than does slip computed on linear faults. Model slip is not presented for the northern 10 km of the Bullion Fault because this fault segment was not included in the curvilinear fault model from Maerten, et al [2005]. For the Landers event, the high resolution (500 m) of model elements, as derived from Aochi and Fukuyama [2002] results in high frequency slip heterogeneity beyond inversion resolution (3 km). These oscillations may or may not be geologically realistic.

232

Discussion The substantially improved fits of modeled slip distributions to those measured by geophysical inversions when regions of reduced stress drop are applied in the vicinity of rupture segment terminations suggests that such zones play an important role in earthquake slip distributions. That such zones have a greater impact and are generally longer at distal rupture terminations of multi-segment models suggests that reduced stress drop plays a relatively greater role there than at intermediate segment terminations along the rupture. Lengths of zones of reduced stress drop generally exceed 5 km and some are greater than 20 km. Shorter zones likely exist at some or all of the segment terminations at which we have not applied them; however, because slip distributions derived from geophysical inversions have a 3 km resolution, we are unlikely to be able to discern small zones of reduced stress drop. With rupture segments having lengths between 20 and 35 km, zones of reduced stress drop are often a substantial fraction of segment lengths, too long to fit the common interpretation of cohesive end-zones. Similarly, in the single-segment models, zones of reduced stress drop range in length from 5 to greater than 30 km, relative to rupture lengths of 55 km and 75 km for Hector Mine and Landers, respectively. For the Hector Mine event, regions of reduced stress drop in the vicinity of absolute rupture terminations in single-segment models produced slip distributions with 65-70% reduction of deviation from geophysical inversion results. This significant improvement suggests that regions of reduced stress drop are most important at absolute rupture terminations. Results of the Landers event, however, are very different. In single-segment models, regions of reduced stress drop in the vicinity of absolute rupture terminations enabled only 5-6% improvement of computed slip distributions, an insignificant factor. Our inability to explain slip distributions in the Landers event with regions of reduced stress drop only at absolute rupture terminations in single-segment model relates to greater lateral heterogeneity of slip suggested by geophysical inversion for the Landers event [Wald and Heaton, 1994] than for the Hector Mine event [Ji, et al., 2001]. Differences in spatial heterogeneity of slip distributions between the events could reflect geological

233 heterogeneities or could relate to numerical methods used in the inversion process. If the contrast is geological, multi-segment models would suggest that regions of reduced stress drop at segment boundaries at which rupture jumps to another segment play a greater role in the Landers event than in the Hector Mine event. Perhaps the Landers event ruptured a less developed fault system in which individual fault segments are linked by stronger, less mature fault zones than those linking fault segments ruptured during the Hector Mine event. Reduced stress drop could be attributed to one or more of many possible factors, including geometric complexity of the rupture surface [Aochi, et al., 2003], perhaps associated with reduced maturity of fault zone development, material heterogeneity [Byerlee, 1978; Pollard and Fletcher, 2005], stress heterogeneity [Madariaga, 1979], inelastic deformation [Manighetti, et al., 2005], or dynamic rupture effects including velocity dependence of friction [Burridge and Knopoff, 1967; Dieterich, 1972; Carlson and Langer, 1989; Cochard and Madariaga, 1994] and dynamic stress perturbations [e.g. Bhat, et al., 2004]. This study does not suggest or negate the validity of any of these potential explanations. Regardless of the mechanical mechanism (or mechanisms) which produce these regions of reduced stress drop, mechanical models support the hypotheses of Ward [1997] and Manighetti, et al. [2005] that concave upward and linearly tapering slip gradients associated with most large earthquakes may be due to relatively large regions near segment tips where rupture conditions are energetically unfavorable, or at least less favorable, and reduced stress drop results. Stepwise and linear models of reduced stress drop in the vicinity of segment tips produce noticeably different distributions of slip and stress drop (see figures A.2 & A.3) but the reduction of error is indistinguishable between the two models. Patterns of reduced stress drop and increased fault strength could be complex, and given that there is substantial error in the slip distribution model derived by geophysical inversion, it seems unlikely that we have sufficient data to determine the best fitting mechanism of reduced stress drop. That maximum local slip on the central Homestead Valley segment of the Landers rupture is only slightly greater than on the distal segments required

234 substantially greater fault strength on this segment to achieve realistic slip distributions in models that are limited to a uniform remote stress tensor. Because we are unaware of geologic evidence for such heterogeneous fault strength between segments, and because comparable slip on segments of comparable length requires a similar stress drop, we concur with Aochi and Fukuyama [2002] that there may be some rotation of the local stress field along the length of the Landers rupture. Bouchon, et al. [1998] suggests that dynamic stress perturbations drive rupture propagation along the Camp Rock and Emerson fault segments; however, Aochi and Fukuyama [2001] demonstrate that this alone is insufficient. Using Coulomb strength, comparable shear traction magnitudes are necessary on all fault segments in order to achieve similar magnitude slip, assuming similar strength. World Stress Map data [Reinecker, et al., 2005] (figure A.1), however, provides only weak support for this conclusion. On figures A.2 & A.3, discrete jumps in slip magnitude, as well as spikes (both positive and negative) in stress drop, occur at segment intersections in multi-segment models and at kinks in the single-segment models. These regions near fault intersections are likely to be characterized by damage zones with significant inelastic deformation. The likelihood of inelastic deformation, accentuated by three- dimensional geometric complexities and presumably imperfect linkage of fault segments, suggests that model slip magnitudes and stress drop in these regions must be interpreted carefully. It is interesting to note that all models are highly sensitive to subtle changes in fault strength. Strength variations of order Δμ = 10-2 result in order of magnitude changes of model slip magnitude. For example, on the Homestead Valley fault (central segment) of the Landers rupture, changing the coefficient of friction from a uniform value of μ = 0.41 to μ = 0.45 results in maximum displacement dropping from nearly 1300 cm to roughly 300 cm. Although we do not believe these models are well enough constrained to make claims about precise values of fault strength, and we acknowledge that the quasi-static friction model is overly simplified, we suggest that, given a well-constrained slip distribution, remote stress tensor, and fault geometry,

235 models such as this, particularly if extended to 3 dimensions, could provide valuable insights about fault strength and its spatial variation. The analysis of 3D and curvilinear fault models adds validity to the conclusions drawn from the 2D linear fault models derived from planar 3D geometries used by Wald and Heaton [1994] and Ji, et al. [2001]. The 3D models demonstrate that laterally, regions of increased fault strength of equivalent magnitude and dimension have a similar ability to reduce deviation between model slip distribution and geophysical inversions. While it is necessary, as expected, to add a zone of increased strength at the base of the fault in 3D, 2D models represent depth-averaged stress drop; because vertical reduction of stress drop in models is laterally homogeneous, the effects of simplifying the problem to 2D are small. Because faults are nearly vertical and displacement discontinuity is almost entirely strike-slip, a 2D plane-strain model is an adequate approximation of the mechanics. Slip distributions computed from curvilinear fault models with homogeneous strength on each fault segment are substantially different from slip distributions computed for simplified linear models; however, they match geophysical inversions no better. Regions of reduced stress drop in the vicinity of segment terminations remain necessary in order to explain tapered and concave upward slip distributions. Although these regions would undoubtedly have different sizes and magnitudes when optimized to minimize deviation of slip on curvilinear faults from geophysical inversions, the results would not necessarily be more meaningful. There is little doubt that the complexities of fault geometry at depth are very different from those evidenced by the 2D rupture trace map [Felzer and Beroza, 1999], and to place much weight on such complexities evidenced by the surface trace alone is artificial and could lead to incorrect conclusions. The curviplanar fault geometries used by Maerten, et al. [2005] and Aochi and Fukuyama [2001], while more complex than planar models, may not actually be an improvement when considering forward models. Because subsurface complexities of fault geometry are currently unknown, we contend that a simple fault geometry is adequate. This study suggests several clear directions for future work. Additional studies of other strike-slip systems in geologically diverse regions could solidify, or possibly

236 refute, the conclusions of this study; however, potential error of both slip inversions and remote stress tensor orientation would likely be even greater in areas which are less thoroughly instrumented and studied. The implementation of a dynamic rupture propagation model could provide insights about the mechanism producing zones of lesser stress drop, particularly if causation relates to dynamic stresses or the velocity dependence of friction. Potentially, the velocity dependence of frictional strength could be inferred also from a comparison of these model results with inversion- derived maps of rupture propagation, from which slip velocity can be estimated. Demonstration of a correlation between reduced propagation velocity and reduced stress drop would be evidence for velocity dependence of friction causing zones of reduced stress drop in the vicinity of rupture segment terminations.

Conclusions Model results suggest that long regions (5-30 km) of reduced stress drop in the vicinity of rupture segment terminations play a significant role in slip distributions in large strike-slip earthquakes which involve rupture of multiple fault segments. These zones of reduced stress drop, modeled by increased fault strength, are required to reproduce the asymmetric, linearly tapered, and locally concave upward slip distributions of the Landers and Hector Mine events, and presumably those commonly observed on other strike-slip faults [Manighetti, et al., 2005], more effectively than uniform strength models. We suggest that geometrically constrained regions of reduced stress drop in the immediate vicinity of rupture segment terminations offer a better explanation for non-elliptical slip distributions than the arbitrarily optimized stress distributions suggested by Ward [1997]. Results also suggest that such zones play a relatively greater role at distal segment terminations than at segment terminations between ruptured fault segments. As demonstrated by the southern extremity of the Camp Rock / Emerson segment of the Landers rupture, where dramatically reduced slip relative to elsewhere on the segment can be explained entirely by mechanical interaction of fault segments, mechanical interactions of adjacent rupture segments must be understood before assuming some other mechanism to explain regions of small slip indicative of reduced stress drop. 237

Acknowledgements This work was supported by the Stanford Rock Fracture Project and the Stanford Graduate Fellowship program. The manuscript benefited greatly from a constructive review by Ruth Harris, and comments from editor Fred Pollitz. We would like to thank Mark Zoback for helping to inspire this work as P. Lovely’s term project in his tectonophysics course, and Greg Beroza for helping to evaluate the implications of our quasi-static modeling approach.

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