Strength of Megathrust Faults: Insights from the 2011 M=9 Tohoku-oki Earthquake. Lonn Brown B.Sc., University of Alberta, 2011

Transcription

1 Strength of Megathrust Faults: Insights from the 2011 M=9 Tohoku-oki Earthquake by Lonn Brown B.Sc., University of Alberta, 2011 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in the School of Earth and Ocean Sciences Lonn Brown, 2015 University of Victoria All rights reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author.

2 ii Supervisory Committee Strength of Megathrust Faults: Insights from the 2011 M=9 Tohoku-oki Earthquake by Lonn Brown B.Sc., University of Alberta, 2011 Supervisory Committee Dr. Kelin Wang (School of Earth and Ocean Sciences) Supervisor Dr. George Spence (School of Earth and Ocean Sciences) Co-Supervisor Dr. Stan Dosso (School of Earth and Ocean Sciences) Departmental Member

3 iii Abstract Supervisory Committee Dr. Kelin Wang (School of Earth and Ocean Sciences) Supervisor Dr. George Spence (School of Earth and Ocean Sciences) Co-Supervisor Dr. Stan Dosso (School of Earth and Ocean Sciences) Departmental Member The state of stress in forearc regions depends on the balance of two competing factors: the plate coupling force that generates margin-normal compression, and the gravitational force, that generates margin-normal tension. Widespread reversal of the focal mechanisms of small earthquakes after the 2011 Tohoku-oki earthquake indicate a reversal in the dominant state of stress of the forearc, from compressive before the earthquake to tensional afterwards. This implies that the plate coupling force dominated before the earthquake, and that the coseismic weakening of the fault lowered the amount of stress exerted on the forearc, such that the gravitational force became dominant in the post-seismic period. This change requires that the average stress drop along the fault represents a significant portion of the fault strength. Two cases are possible: (1) The fault was strong and the stress drop was large or nearly-complete (e.g. from 50 MPa to 10 MPa), or (2) that the fault was weak and the stress drop was small (e.g. from 15 MPa to 10 MPa). The first option appears to be consistent with the dramatic weakening associated with high-rate rock friction experiments, while the second option is consistent with seismological observations that large earthquakes are characterized by low average stress drops. In this work, we demonstrate that the second option is correct. A very weak fault, represented by an apparent coefficient of friction of 0.032, is sufficient to put the Japan Trench forearc into margin-normal compression. Lowering this value by ~0.01 causes the reversal of the state of stress as observed after the earthquake. A slightly stronger fault, with a strength of 0.045, does not agree well with the observed spatial extent of normal faulting for the same coseismic reduction in strength. We also calculate distributions of stress change on the fault and average stress drop values for the Tohokuoki earthquake, as predicted from 20 published rupture models which were constrained by seismic, tsunami, and geodetic data. Our results reconcile seismic observations that average stress drops for large megathrust events are low with laboratory work on high-

4 iv rate weakening that predicts very high or complete stress drop. We find that, in all rupture models, regions of high stress drop (20 55 MPa) are probably indicative of dynamic weakening during seismic slip, but that the heterogeneous nature of fault slip does not allow these regions to become widespread. Also, coseismic stress increase on the fault occurs in many parts of the fault, including parts of the area that experienced high slip (> 30 m). These two factors ensure that the average stress drop remains low (< 5 MPa). The low average stress drop during the Tohoku earthquake, consistent with values reported for other large earthquakes, makes it unambiguous that the Japan Trench megathrust is very weak.

5 v Table of Contents Supervisory Committee... ii Abstract... iii Table of Contents... v List of Tables... vi List of Figures... vii Acknowledgments... ix Chapter 1. Introduction Strength of Megathrust Faults Thesis Objectives and Significance Thesis Organization... 5 Chapter 2. The Strength of Megathrust Faults Apparent Strength of Megathrust Faults Definition and Importance of Megathrust Strength Factors Affecting Apparent Strength Stresses in the Forearc Controlled by Megathrust Strength Gravitational and Plate Coupling Forces Modeling Stresses in the Forearc Other Constraints on Megathrust Strength Chapter 3. Stress Drop Theoretical Background Definition and Methods of Measurement Fault Friction During Earthquakes Observed Stress Drop in Great Earthquakes Average Stress Drops of Great Earthquakes Complete versus Partial Stress Drop Dislocation Modeling of Stress Changes on a Fault in an Earthquake Method Stress Drop of a Single Rupture Patch Dislocation Modeling of a Multi-Patch Rupture Field Chapter 4. Implications of Forearc Stress Changes Caused by the 2011 Great Tohoku-oki Earthquake The Tohoku-oki Earthquake Stress States Before and After Stress Drop Distributions from Rupture models FEM of Forearc Stresses After the Earthquake Chapter 5. Conclusion References Appendices A.1. Focal Mechanism Diagrams A.1. Stress Tensor B.1. Stress Change Distributions for Rupture Models from the 2011 Tohoku-oki Earthquake

6 vi List of Tables Table 3.1. A small survey of observationally determined average stress drop values for recent large megathrust earthquakes

7 vii List of Figures Figure 2.1. Byerlee s law as a general reference for fault strength Figure 2.2. Cartoon to clarify apparent strength as opposed to peak strength Figure 2.3. Cartoon of forces that act on a typical subduction zone forearc Figure 2.4. Finite-element modeling domain and parameters Figure 2.5. Finite-element modeling results of the effect of varying fault strength on the state of stress in the subduction zone forearc Figure 2.6. Finite-element modeling results of a subduction forearc, investigating importance of topography on the state of stress in the forearc Figure 2.7. Finite-element modeling results of a subduction forearc, investigating importance of the weight of the water column on the state of stress in the forearc Figure 3.1. A cartoon illustrating two cases of rate-and-state frictional behaviour Figure 3.2. Friction coefficient versus normalized slip, illustrating the magnitude of dynamic weakening for various rock types in rotary shear experiments Figure 3.3. Finite-element modeling stress profiles along the fault for three different fault strengths Figure 3.4. Schematic of model setup in Okada coordinate system Figure 3.5. Benchmarking dislocation code against Okada analytical solution Figure 3.6. Modeling results from Hu and Wang, (2008) comparing stress signals from different types of fault models Figure 3.7. Dislocation modeling results showing the stress drop distribution of a single Gaussian-distributed rupture patch Figure 3.8. Dislocation modeling results showing stress drop distribution as discrete rupture patches become more connected Figure 3.9. Dislocation modeling results showing stress drop distribution as discrete rupture patches become more connected Figure 4.1. Satellite image of Japan (left), with the main study area roughly outlined in red Figure 4.2. Focal mechanisms of small earthquakes (left) before and (right) after the Tohoku-oki earthquake Figure 4.3. An example of the fault mesh from out dislocation modeling, with the rupture model of Shao et al. (2012) applied to it Figure 4.4. The vectors of stress change produced by our method Figure 4.5. Stress perturbation at the ground surface calculated for the fault slip model of Shao et al. (2012) Figure 4.6. Stress distributions and average stress drop values for one rupture model Figure 4.7. Stress distributions and average stress drop values for several different rupture models Figure 4.8. FEM results simulating the pre-seismic strength of the megathrust fault in NE Japan for various effective coefficients of friction µ Figure 4.9. Three cases of Δµ along the fault Figure Stress drop along the fault, as well as stress perturbation for the small Δµ case... 78

8 Figure Six different cases of absolute deviatoric stress in the subduction forearc immediately after the Tohoku-oki earthquake Figure Stress drop along the fault, as well as stress perturbation for the intermediate Δµ case Figure Six different cases of absolute deviatoric stress in the subduction forearc immediately after the Tohoku-oki earthquake, for the intermediate Δµ case Figure Stress drop along the fault, as well as stress perturbation for the large Δµ case Figure Six different cases of absolute deviatoric stress in the subduction forearc immediately after the Tohoku-oki earthquake, for the large Δµ case Figure A.1. The nine couples represented in the moment tensor, and the generalized matrix form of the moment tensor...99 Figure A.2. Four examples of beachball diagrams and generalized moment tensors, and GMT plotting commands.102 Figure B.1. Stress drop distributions from various rupture models 103 Figure B.2. Stress drop distributions from various rupture models 104 Figure B.3. Stress drop distributions from various rupture models 105 viii

9 Acknowledgments I would like to give my sincere thanks to all those who helped out in one way or another to make this work possible. I would like to thank both the University of Victoria and the Pacific Geoscience Centre for providing both space and equipment to work with, and to the respective administrative staff who always cheerfully helped me with the dayto-day details of my program. I would like to give special thanks to: Kelin Wang, for being an excellent supervisor. Kelin s wealth of knowledge and tremendous experience guided me throughout my degree. Kelin always explained things clearly. His good nature and kind encouragement always motivated me to improve both my work and my work habits. George Spence, for helping me get started when I was a new graduate student, and for an excellent course on plate tectonics in which I had my first real practice writing in the academic style. George has a keen editorial eye, and it was a great benefit to this work. Stan Dosso, for being a great instructor and giving an excellent course on inversion, which I hope to put to good use someday. Jiangheng He, for taking me through the programs he wrote and for always helping me out when I couldn t quite figure out what I d done wrong with them. Tianhaozhe Sun, for his friendship and support, and for many thoughtful discussions about various aspects of what we were both working on. Yan Jiang, for graciously donating his time and efforts as my external examiner. Honn Kao, Earl Davis, and the other Weekly Group Meeting attendees, for providing feedback, asking questions, and making me think a little bit differently about the work I was doing. My friends and colleagues, namely: Jesse Hutchinson, Ayodeji Kuponiyi, and Dawei Gao, for their support, but for also making sure that I enjoyed my time at the University of Victoria. My Mother, Brother, Grandma, Grandpa, and my larger family for their endless love and support, and for encouraging me and supporting me throughout my academic career. My wife, my best friend, Jenni - to whom I would dedicate this work in its entirety. ix

10 Chapter 1. Introduction 1.1. Strength of Megathrust Faults Beginning with the discovery of plate motions, the strength of the great faults that serve as plate boundaries has been understood to be a fundamental factor in both the evolution of the plates themselves, and the deformation that takes place near their edges. The resistance these faults offer against motion or deformation must moderate the force that one plate exerts on its neighbouring plate; in this manner the tectonic stresses within the plates would be constrained by having knowledge of the forces that drive plate motion and the strength of the plate boundary faults. If we also know the physical properties of the crust and mantle, we can predict how the Earth responds to these forces and their change with time and space. Broadly speaking, there are two categories of rock deformation: (1) elastic deformation, which is not permanent and can be envisioned as storing energy in a spring by compressing it that energy may be released at a later time without any lasting deformation of the spring, and (2) permanent deformation, which cannot simply be undone, and is instead similar to compressing a spring past its yield limit and breaking it (or bending it permanently). On and around all kinds of faults, both types of deformation occur. Elastic deformation will store energy for future release, while permanent deformation may alter the landscape around the fault. If the stress on the fault becomes high enough to cause it to fail in a seismic event, the slip of the fault (one form of

11 2 permanent deformation) will allow the release of some of the stored elastic strain energy in the surrounding region. If we wish to understand why and how these events may occur and what the results may be, knowledge of the absolute strength of the faults are extremely important. If we understand the shear stress on a fault, we will be in a much better position to assess the seismic hazard the fault potentially represents and thus help society to mitigate the associated risk. On a global scale, the evolution of the plates themselves can be better understood if we know how much force they can exert on one another. With this knowledge and a history of plate deformation, we may also be able to better constrain the tectonic forces, such as basal drag, ridge slide, or slab pull. Thus, the strength of faults that make up plate boundaries is a fundamental question of geodynamics; however, any field with a strong relation to faults would benefit greatly as well (e.g. seismology, rupture mechanics, and seismic hazard assessment). A great deal of effort has already been put into determining the strength of large faults, from various works on the San Andreas Fault (Zoback et al., 1987; Rice, 1992; Lockner et al., 2010) to investigations into megathrust fault strength (Wang et al., 1995; Wang and He, 1999; Lamb, 2006; Seno, 2009; Gao and Wang, 2014). In general, it is very difficult to measure absolute stress; borehole breakouts and overcoring provide one method of doing so, but performing these measurements at depths that are relevant to large faults is extremely challenging, if not impossible. Determining what stress state currently dominates in a region is more readily accomplished, as focal mechanisms of earthquakes provide relevant information that can be inverted over a region to determine the state of stress which controls faulting style (thrust, normal, or strike-slip). This

12 information provides details on the orientation of the stress tensor. The magnitude, 3 however, is more difficult to determine. Studies of frictional heating do attempt to determine the absolute strength of the fault (Gao and Wang, 2014), but heat flow data that can adequately constrain frictional heating are not always available. Of further importance regarding fault strength is the character of how it varies spatially. There is much discussion within the literature about ideas of strong coupling leading to large earthquakes (Lay and Kanamori, 1981), or how the subduction of large geometrical incompatibilities such as seamounts should cause large earthquakes (Scholz and Small, 1997). Recent works instead highlight observations that the largest subduction earthquakes tend to occur where the downgoing plate is smooth (Wang and Bilek, 2011), and show that creeping rough faults (which are in a constant state of failure) are actually capable of offering more resistance to deformation than the faults that show stick-slip behavior to produce great earthquakes (Gao and Wang, 2014). These works highlight the very important questions: How does the stress field evolve towards a large earthquake? What must it look like just before failure occurs? While conclusive answers are not yet known, we hope to begin to shed some light on these questions with this work. Recent observations of the 2011 Tohoku-oki earthquake provide us an excellent and unique opportunity to investigate the strength of megathrust faults. Due to the earthquake, the regional stress field changed in such a way that we can use observations of the earthquake to constrain the absolute strength of the fault that produced it. The primary observation is that the state of stress in the continental forearc changed from horizontal deviatoric compression in the pre-seismic state, to horizontal deviatoric tension after the earthquake. Such a change is expected to only occur if the average stress drop of the

13 earthquake represents a significant portion of the pre-seismic fault strength. These 4 concepts will be discussed in detail in this dissertation. The process is analogous to a simple balance-weighing scale: if we have two weights, a heavier one of known mass and a lighter one of unknown mass, and removing a small (but known) portion of the heavier weight causes the balance to shift from the previously heavier side to the other, we should be able to constrain the mass of the lighter weight, and thus determine the relationship between the two. We will structure our argument in a similar way Thesis Objectives and Significance The primary aim of this work is to constrain the average absolute strength of the megathrust fault in NE Japan, as discussed above. However, we will also investigate the forces that affect the state of stress in the forearc above the megathrust fault, and how the balance of the relevant forces is affected by the fault strength. Improved knowledge of the fault strength and thus the stress field on the fault before an earthquake occurs could lead to improvements in the assessment of seismic hazard, as well as improve our understanding of the process of subduction. Another prime motivation for the work presented here is to investigate how stress change varies spatially on the fault. Results from the laboratory regarding the degree of coseismic weakening, coupled with the observations from the Tohoku-oki earthquake of large static slip, have led to predictions of complete stress relief over a wide area of the megathrust (Hasegawa et al., 2011; Hasegawa et al., 2012, Yagi et al., 2011a; Hardebeck, 2012, Lin et al., 2013; Obana et al., 2013). These predictions are currently at odds with a

14 5 wide body of seismic observations that indicate a much more modest decrease along the fault in large earthquakes. Our work will seek to reconcile the two ideas, by investigating the character of fault heterogeneity and how it affects the value of average stress drop. A good understanding of stress change on the fault will also be helpful in assessing seismic hazard, as well as improving tsunami modeling and studies of post-seismic deformation Thesis Organization In addition to the Introduction (Chapter 1) and Conclusions (Chapter 5), the rest of this dissertation is structured as follows. First, Chapter 2 provides a definition for what we really mean by fault strength, and discusses the theory that allows us to describe it mathematically. The most important forces acting upon the subduction zone forearc (i.e. the plate coupling force and the gravitational force) will be discussed, and their relative importance will be investigated through finite element modeling. These discussions will provide the backdrop within which we can discuss the strength of the fault and the constraining factors. Chapter 3 begins with a discussion of stress drop, which is the reduction in stress on the fault during an earthquake, and the main seismic source parameter that we discuss in this work. The two types of behaviours (i.e. weakening or strengthening) that a fault may show in response to slip will also be discussed, as they produce stress perturbations of opposing signs and thus are very important to the postseismic level of stress on the fault. Observationally constrained average stress drop will be briefly reviewed, which is an essential component of our study of the strength of the NE Japan megathrust fault. I will

15 6 use dislocation modeling to test different cases of heterogeneous fault slip and investigate heterogeneous stress change on the fault and the resulting implications for the average value of stress drop. Chapter 4 is devoted to the 2011 Tohoku-oki earthquake, which produced the change in the stress field mentioned above. I will discuss the state of stress in the forearc before and after the earthquake, and then use our dislocation modeling method to determine the average value of stress drop of the earthquake from a variety of published rupture models. We will then return to finite-element modeling to evaluate the strength of the megathrust, using as constraints the pre- and post-seismic states of stress in the forearc and the average stress drop along the megathrust.

16 7 Chapter 2. The Strength of Megathrust Faults 2.1. Apparent Strength of Megathrust Faults Definition and Importance of Megathrust Strength The static strength of megathrust faults is a fundamental parameter in the study of plate tectonics, as it governs the balance of forces that act upon the relevant plate margins. For each plate on either side of the megathrust, the fault strength defines the main boundary condition for the state of stress in the plate and moderates the degree of resistance to plate convergence. Thus, it strongly influences various intraplate geological deformation processes and is important in the study of global plate motions. The Coulomb definition of frictional resistance (or frictional strength) gives the following relation between shear stress ( ), cohesion ( ), coefficient of friction ( ), normal stress ( ), and pore fluid pressure ( ) for two surfaces in contact. (2.1) Given,,, and this relation describes the maximum resistance ( ) that the contact surface can offer against slip. If the shear stress reaches this level, then the strength of the fault has been overcome, and failure must occur. The actual mode of failure of real faults can be quite complex, ranging from steady creep to seismic slip; a full treatment requires dealing with rupture mechanics, geometrical effects, and considering the compositional nature of the real fault. For example, a large, welldeveloped fault (such as a megathrust fault) is always a zone with some measure of

17 thickness, and is comprised of gouge materials. Fault gouge is essentially the collection 8 of fine pieces of rock which have been produced by the fault-motion-driven gradual wearing down of the pre-existing fault material. Other processes may affect the composition of the fault materials, and in addition to the loss of cohesion due to fracture, the gouge region is often further weakened relative to intact rock by the presence of hydrous minerals. Thus, the frictional coefficient describes the shear deformation of a fault zone which is much weaker than the surrounding rock. Here, however, the fault is approximated as a mathematically flat plane between two regions in contact. The approximation is a good one if the dimensions of the fault surface of interest (such as the rupture zone of an earthquake) are many orders of magnitude larger than the thickness of the fault. Using the Coulomb definition, we may quantify the static strength of a fault in two ways, by stating either the coefficient of friction or the smallest magnitude of shear stress which would cause the fault to break. We can simplify (2.1) by combining the parameters of pore pressure and normal stress into an effective normal stress (2.2). We can then use the effective stress and the real friction coefficient to define the shear strength, that is,. (2.3) In practical use, we more often organize (2.1) so as to employ an effective coefficient of friction, which again includes the effects of along the fault. For example, (2.1) can be re-written as

18 . (2.4) 9 Further, we can define as a ratio of to the lithostatic pressure = gz, where is rock density, g is gravitational acceleration, and z is depth, (2.5). If we then recognize that for a gently dipping fault (such as a subduction megathrust) the normal stress on the fault is approximately the weight of the overlying rock column, that is,, we then can write (2.4) as. (2.6) It is common to define an effective coefficient of friction, (2.7) so that. (2.8) Pore pressure serves to lessen the effect of normal stress, and here the effect is included in the effective coefficient of friction, rather than in the effective stress as in equation (2.3). As will be explained later in this section, for actual geological faults such as the subduction megathrust, parameter is often further generalized to be the apparent (rather than just effective ) coefficient of friction, which includes not only the effects of, but also reflects the fact that spatial and temporal heterogeneities have been averaged out. One specific form of Coulomb friction that is widely used in geodynamic research is the following, known as Byerlee s law.

19 10 (2.9) We have altered the equation somewhat from the original form presented by Byerlee (1978), using the effective stress designated previously in (2.2) which includes the effects of pore pressure. This empirical relation comes from a large body of experimental laboratory data which shows that under intermediate to great normal stress, frictional strength is independent of rock type (Byerlee, 1978). Figure 2.1 gives an example of a strength-depth profile based on equation (2.8) but using a value of 0.7 similar to Byerlee s law. As implied by the figure, beyond some depth the mode of failure will change. At greater depths, increasing temperatures allow the activation of creep mechanisms which relieve stress more efficiently than brittle frictional failure at lower stresses, and allows rock to deform in a viscous manner. One example of a power law that is used to describe the viscous strength of olivine from Karato and Wu (1993), (2.10), gives the strain rate as it depends on the shear stress τ, grain size d, pressure p, and temperature T. In this relation, A is an experimentally determined constant, is the shear modulus, b is the length of the Burgers vector, n is the stress exponent, m is the grain-size exponent, is the gas constant, is the activation energy, and is the activation volume. Viscosity is often used to describe the strength of rocks at great depth. Viscous strength properly refers to the stress required to maintain a constant strain rate, rather than for brittle material where strength refers to the level of stress which would cause

20 11 material to fail in a brittle manner. As such, any value of viscous shear strength is only valid in relation to a specific strain rate. The definition of fault strength described by (2.8) does not consider failure processes once the strength limit has been reached, and thus describes strength only at one discrete point or in a long-term average sense. If the failure takes place in the form of seismic rupture, which may propagate some distance, the concept of strength is more complicated and requires further explanation. Figure 2.1. Byerlee s law as a general reference for fault strength. Not a true representation of the strength defined by Byerlee s law, as the kink implied in the brittle portion is absent. Light grey dashed lines represent the mathematical transition from brittle to ductile deformation, while the solid line represents some type of smooth transition between the two, as recent work by Shimamoto and Noda (2014) identifies in the mineral Halite. During the seismic cycle, strain accumulates and stress builds until the stress on some portion of the fault (which will become the location of rupture nucleation) reaches its peak strength that is roughly defined by (2.8). At that point in time other areas on the fault will be at varying levels of stress, but somewhere below the actual, peak strength of the fault. Due to the dynamic loading effects of the rupture, the shear stress at the edge of

21 12 the propagating rupture front will be increased. If this effect is able to raise the stress on the adjacent part of the fault up to its peak strength, then rupture will continue to propagate. The strength that we will be discussing is the pre-rupture condition, called the apparent strength of the fault. It is not the stress that will cause the fault to break at every location, but rather a snapshot of stress over the fault just before an earthquake occurs, except for at the point where rupture nucleates. This concept is illustrated in Figure 2.2. For the purpose of our study, we are less interested in the peak strength of the entire surface. The apparent strength we seek is closer to an average stress profile we might expect to observe over the fault prior to the earthquake, and is thus more relevant for both society and larger-scale geodynamics. As stated previously, the shear stress on a megathrust fault must influence the state of stress within both plates, and so prior to an earthquake the state of stress within the plates depends strongly on the apparent strength of the plate boundary fault. We discuss the nature of this dependence in section and employ simple finite-element-models to quantify the relation in section For frictional faults that creep aseismically, the situation is simpler: there is no distinction between apparent and peak strength, as the fault is slipping at its peak strength. We will now use the Coulomb friction law to describe the apparent strength, and use it simply to relate normal stress to predicted shear stress over the fault. Examples of other studies that take a similar approach to describing the strength of a fault in this manner include Wang and He (1999), Seno (2009), Yang et al. (2013), and Gao and Wang (2014).

22 13 Figure 2.2. Cartoon to clarify apparent strength (τas) as opposed to peak strength (τps). (Above) Peak and apparent strength over the profile of a fault. (Below) Schematic representation of stress on a fault during rupture, at the nucleation point (B) and away from it (A). Grey circle highlights area where complicated, poorly understood failure processes take place. Δσs is the static stress change at the point in question, and Dc is the critical distance over which the coseismic weakening takes place Factors Affecting Apparent Strength The Coulomb definition of frictional strength encompasses several parameters which may vary spatially and temporally, and we discuss some of those parameters here. Pressurized fluids within the pore spaces of rocks at depth bear some portion of the weight of the overburden, thus lowering the effective normal stress which acts upon the solid matrix (Terzaghi, 1924). A reduction in the effective normal stress reduces the shear strength as described by (2.3). The effect can be equivalently described by a reduction in the effective friction coefficient as described by (2.8). Pore pressures exceeding the

23 hydrostatic level may occur for various reasons, such as compaction or the release of 14 water in metamorphic reactions. If fluids cannot easily flow away due to low permeability, pore pressures will increase and may even begin to reach lithostatic levels in extreme cases. Compositional concerns must also play a role in the strength of a fault. For example, if weak materials such as certain types of clays are present in the fault zone, they will reduce the intrinsic coefficient of friction µ, and thus affect both the apparent and peak strengths of the fault. Clayey materials are understood to be an important source of weakness for certain sections of the San Andreas Fault in California, for which some laboratory experiments determined the intrinsic coefficient of friction to be 0.21, much lower than the value of the surrounding rock (Carpenter et al., 2011). The mechanisms by which failure occurs will further influence the strength of a fault. As mentioned in section and highlighted by the relevant equations (2.8) and (2.10), as burial depth increases, increasing temperature begins to exert a first-order control over the strength at depth, as it controls the manner in which materials ultimately fail that is, brittle failure at lower temperatures and different types of viscous flow at higher temperatures. Rock and faults will ultimately fail in the easiest way possible when the stress required to deform viscously becomes smaller than the stress required for brittle failure, then the dominant method of failure will change accordingly.

24 Stresses in the Forearc Controlled by Megathrust Strength Gravitational and Plate Coupling Forces As mentioned before, the apparent strength of a plate boundary fault is an important boundary condition for the state of stress in both plates. If the influence is important enough relative to the other active forces, then the apparent fault strength may also be constrained by the state of stress within the plates. For the ensuing discussion, we need to clarify that when we say the lithosphere or crust is experiencing tension or compression, we always mean deviatoric tension or compression in the horizontal direction, unless otherwise stated. Deviatoric stress refers to the non-isotropic portion of the stress tensor. Differential stress refers to a different concept. It is defined as σ1- σ3, the difference between the most compressive (σ1) and least compressive (σ3) principal stresses. Half of the differential stress is the maximum shear stress at the location in question, on an optimally-oriented plane. We now examine the factors that contribute to the state of stress in a subduction zone forearc, summarized schematically in Figure 2.3. Consider a vertical column of water held in place by glass, sitting on a table; if we were to remove the glass the water would fall, spreading out near the bottom where it meets the table. However, the force that causes the water to flow laterally exists even in the restrained standing column of water. Without any shear strength to hold itself together, the standing water tends to flow laterally to relieve the pressure of its own weight. This simple example occurs because the boundaries of the water column do not coincide with the gravitational equipotential surface near the level of the table top, and thus has some measure of topography relative to that surface. The tendency to collapse

25 16 also exists in the bodies of solid objects whose forms do not coincide fully with a single equipotential surface, but as solids have some shear strength, motion does not always occur. Instead internal stresses are generated that resist the gravitationally induced shearing forces. For example, if the glass of water in our example is replaced with a rock, the rock is not likely to break once the confining glass has been removed. The horizontal deviatoric tension generated in response to the vertical compressive force increases towards the bottom of the column; if the mass of the rock is large enough, at some point these stresses may overcome the strength of the rock and cause failure, similar to how a landslide occurs. In solid materials, the effect of gravity acting on elevated topography is always to induce deviatoric tension. On Earth, topographical and density gradients induce a lateral tensional force that tends to flatten out the topographical gradient relative to the lowest available equipotential surface. The magnitude of this force necessarily increases as topography or density increases. If we consider the continental forearc of a subduction system, every example on Earth shows elevated topography relative to the subduction trench and the ocean basin, and, for the reasons stated above we must expect some component of deviatoric tension to exist. Thus if we imagine a forearc infinitely long in the strike direction and in the absence of any other forces (plate coupling, back arc push, etc.), we would expect that it must be under strong deviatoric tension with the orientation of maximum tension being margin-normal. The second important force we consider, the plate coupling force, arises due to the push of the incoming plate being transmitted to the upper plate, because of the resistance offered by the megathrust fault. In effect, the subducting plate is squeezing the forearc

26 17 against the stable continental plate, and the magnitude of this compression is moderated by the integrated strength of the megathrust over the contact area. The resultant compressive force works against the gravitationally-induced tension. The magnitude of this force then increases as the strength of the fault increases, or as the size of the contact area increases. Again, if the megathrust fault had zero apparent strength, the forearc would be under strong deviatoric tension. However, if we observe a forearc that is experiencing deviatoric compression, we know that the plate coupling force must be large enough to overcome the effect of the gravitational force. Figure 2.3. Cartoon of forces that act on a typical subduction zone forearc. In the forearc, the state of stress will be determined by which of the forces is dominant: gravity in the presence of topography that causes tension, or the plate coupling force moderated by the strength of the fault and the length of the contact area that causes compression Modeling Stresses in the Forearc To illustrate the concepts described above, we use a finite element model of two converging elastic plates in frictional contact along their interface. We prescribe plane strain conditions to our model domain, that is, we assume no deformation in the strike direction. The model domain is 470 km wide, with a 60 km thick upper plate, which is

27 18 large enough to investigate both the effects of fault strength on the near trench and inland regions. We use a plate model rather than a half-space model; a plate model allows us to use an effectively inviscid substrate below the brittle upper plate, which simulates the long term behavior of the mantle. This allows us to model the state of stress after an interval has passed that allows the mantle to relax the stresses induced by the previous earthquake. As well, in half-space models at large depths, including gravity in the calculations would give rise to numerical instabilities because of the very large pressure and comparatively tiny differential stress. The finite element code we use was written by Jiangheng He, Pacific Geoscience Centre, Geological Survey of Canada. The geographical focus of this study is the Japan Trench subduction zone, and therefore our model setup, shown in Figure 2.4, follows that of Wang and Suyehiro (1999). The region of interest is the upper plate, representing the forearc. It is assigned a Young s modulus of 100 GPa, a value reasonable for continental plates. The downgoing plate itself is of no interest in this modeling but is needed for simulating the frictional plate interface. The downgoing plate is thus modelled effectively as a rigid body (Young s modulus several orders of magnitude higher than the upper plate), and its motion is assigned kinematically in the tangential direction of the plate interface. As is well known and explained in Wang and He (1999), because the boundary conditions for the upper plate are predominantly stress conditions, exactly what values for elastic moduli to use will have little impact on the stress solution. The plate interface has a circular shape, such that the motion of the downgoing plate in the tangential direction does not induce any stress due to geometrical incompatibility. The incurred stresses are then due purely to frictional resistance of the interface. Throughout the model domain,

28 we assume the material is incompressible (Poisson s ratio approaching 0.5) to avoid 19 gravitationally induced differential stresses. The part of the upper surface of the model that would be under the ocean is assigned normal force representing water pressure calculated from the water density and water depth. The base of the upper plate (60 km depth) is assigned the Winkler restoring force (Wang and He, 1999; Wang and Suyehiro, 1999), simulating the effect of an inviscid asthenosphere. The maximum depth of the elastically competent part of the upper plate is not uniform because of the presence of a cold nose of the mantle wedge that extends to greater depths (Figure 2.4) (Wada and Wang, 2009). Between the competent part of the upper plate and the base of the model, we introduce a cushion layer of highlycompliant (having an extremely small Young s modulus) material, so that the Winkler restoring force can be applied to the bottom of a flat boundary (the bottom of the Mantle Wedge in Figure 2.4). Compared to the earlier model of Wang and Suyehiro (1999), a slightly more realistic topography is used for the upper surface of the upper plate, and the fault geometry is also slightly refined, using the model outlined by Sun et. al (2014). To model a realistic topography, an average of 12 trench-normal profiles distributed over a span of 200 km along strike was used. With this method we intend to investigate the importance of the long-wavelength topography, while obscuring the details of local effects. The importance of using realistic long-wavelength topography will become obvious in subsequent discussions.

29 20 Figure 2.4. Finite-element modeling domain and parameters. Springs indicate a Winkler restoring force, while wheels indicated free movement in the vertical direction, but restricted in the horizontal direction. Movement of the rigid subducting plate is tangent to its circular geometry. For the frictional contact between the two plates, we prescribe an effective coefficient of friction (equation 2.8 with co = 0), and displace the subducting plate far enough to be sure that the maximum shear stress, that is, the strength of the fault as dictated by equation (2.8), is reached everywhere along the fault. If the fault is not brought to failure as described, the signature of a stress shadow can be seen in the results, where the seaward portions of the fault provide enough shallow resistance to prevent the breaking of the deeper portions of the fault, similar to what is discussed by Hu and Wang (2008). We begin with an illustrative model with =0, shown in Figure 2.5a. As the fault supports no shear stress, the forearc is under very strong deviatoric tension, on the order of ~70 MPa over most of the forearc. In this case, the only support for the upper plate offered by the subducting plate is the normal stress along the plate interface. Similar tests were run for weak ( = ) and strong ( = 0.4) megathrust faults. Faults that are stronger than = 0.04 transmit enough of the plate coupling force into the upper plate to produce a compressive state of stress throughout the entirety of the overriding plate. For the weaker case of = 0.03 (Figure 2.5b), most of the forearc is under deviatoric

30 compression, which agrees with results of a similar earlier work (Wang and Suyehiro, ) but some tensional regimes are visible near the regions with the greatest gradient of topography. The value of = 0.03 was proposed by Wang and Suyehiro (1999) to explain the margin-normal compressive stress observed in most of the Japan Trench forearc. They argued that because of the wide zone of frictional coupling along the plate interface as a result of the cold thermal regime, even a value as low as 0.03 could result in sufficient total plate coupling force to put the upper plate in compression. In other words, predominant compression of the forearc does not always require a strong megathrust. Values of 0.03 put the state of stress in the forearc in a subtle, nearly neutral state, where the opposite effects of plate coupling and gravitational forces are nearly equal in magnitude. A small change of boundary conditions can then result in a significant change in the stress state, as illustrated by the examples shown in Figures 2.6 and 2.7. For example, for = 0.025, a large portion of the forearc is in tension (Figure 2.6b), but for = 0.032, nearly the entire forearc is under compression (Figure 2.7a), with only a small portion being in neutral horizontal deviatoric tension. To examine the importance of using realistic topography, we compare the simplest possible upper-plate topography (with a height of 0 on land, and a constant slope from the coast to the trench) to one that is more realistic for the Japan Trench. The importance of using realistic topography is clearly shown in Figure 2.6. For = 0.025, the model of realistic topography (Figure 2.6b) features tension in a portion of the forearc (~ km from trench), but the model of simplified topography (Figure 2.6d) incorrectly shows compression in the same area. In fact, for the simplified topography, significant compression persists even for as small as (Figure 2.6c).

31 22 Figure 2.5. Finite-element modeling results of the effect of varying fault strength on the state of stress in the subduction zone forearc. Topography of the forearc represents an average profile of the Japan Trench. Four different strengths are investigated, from = Red crosses denote deviatoric compression relative to the horizontal, while blue crosses denote horizontal deviatoric tension. Purple crosses (seen only in a few spots in B, between regions of red and blue) indicates neutral horizontal tension, where neither tension nor compression dominates.

32 23 Figure 2.6. Finite-element modeling results of a subduction forearc, investigating importance of topography on the state of stress in the forearc. The average topography of the Japan Trench is used on the left, while the simplest topography is used on the right. The simplest topography uses the same horizontal and vertical location of the trench and coast, but otherwise has no detail for the slope and on land. Red crosses denote deviatoric compression relative to the horizontal, while blue crosses denote horizontal deviatoric tension. Purple crosses indicate neutral horizontal tension, where neither tension nor compression dominates. The results shown in Figure 2.6 are of great geodynamic importance. They show the fragile nature of the state of stresses in the forearc bounded by a very weak megathrust fault. A seemingly benign change in topography can cause a reversal of the stress regime in parts of the forearc, an effect not appreciated by previous studies. Models shown in Figure 2.7 further strengthen this notion. Figure 2.7 compares models with and without simulating the ocean water, both with the realistic topography. The ocean water compresses the ocean bottom in the normal direction, without a shear component. Without the compressive effect of the water, the forearc is under deviatoric tension until the frictional strength approaches ~0.065 (Figure 2.7d). Again, a seemingly benign factor produces a dramatic effect on the state of stress in the forearc.

33 24 Figure 2.7. Finite-element modeling results of a subduction forearc, investigating importance of the weight of the ocean water on the state of stress in the forearc. Both runs use a realistic average topography from the Japan Trench, but the model runs on the right neglect the weight of the water in their calculation. Red crosses denote deviatoric compression relative to the horizontal, while blue crosses denote horizontal deviatoric tension. Purple crosses indicate neutral horizontal tension, where neither tension nor compression dominates Other Constraints on Megathrust Strength Focal mechanisms of small earthquakes give information about the current state of stress near the fault they occur on. Thrust-style earthquakes indicate compression, while normal faulting indicates tension. Earthquakes that occur at similar locations reflect similar states of stress, and deviations from this pattern require unusual circumstances. The observation of a large aftershock of the 1968 M8.2 Tokachi-Oki earthquake having nearly the reverse focal mechanism from the main shock led Magee and Zoback (1993) to investigate the cause. Their inversion for the state of stress estimated that the axis of least compression (σ3) was oriented nearly perpendicular to the fault, which implies low shear stress on the fault to begin with, and that a small difference in the dip angles of the faults

34 25 producing the main shock and aftershock may have been responsible for the difference in their faulting regimes. As mentioned before, elevated pore pressure is one of the prime factors for weakening faults at depth. A survey of several subduction zones carried out by Seno (2009) compared observed stresses to predicted ones, and attributed the differences to elevated pore pressures, assuming the intrinsic fault fiction is 0.85 (Byerlee s law). The inferred values of ranged from , or megathrusts. Alternatively, the low of , representing weak can be due partially or entirely to a very low intrinsic friction, instead of a very high. Heat flow data help to constrain the strength of a fault. By numerically modelling the thermal regime of the Cascadia subduction zone and comparing the model predicted surface heat flow to heat flow observations, Wang et al. (1995) found the frictional heat from the megathrust to be negligibly small, indicating a very weak megathrust. Gao and Wang (2014) recently performed a similar analysis for a number of subduction zones where there are adequate heat flow measurements to constrain frictional heating, and found values of apparent coefficient of friction ranging from ~0.02 for smooth megathrust faults that produce great earthquakes, to ~0.15 for very rough megathrust faults that creep. In their analysis, the area of the Japan Trench megathrust that produced the Mw = 9 Tohoku-oki earthquake is considered to be relatively smooth because of lack of major subducting bathymetric features such as seamounts, and is thus expected to be relatively weak.

35 26 Chapter 3. Stress Drop 3.1. Theoretical Background Definition and Methods of Measurement Stress drop is the decrease in shear stress along the fault as a result of seismic or aseismic slip on the fault. The magnitude and direction of stress drop must vary spatially along the fault, and the slip zone often has a complex shape. However, an earthquake or other slip event can be simply characterized by an average stress drop over a slip zone of relatively simple shape. The scaling relation between average static stress drop ( ) of an earthquake and the rupture parameters is, (3.1) where is a non-dimensional constant which depends on the shape of the rupture surface and the type of faulting, is the shear modulus, is the average slip, and is a characteristic dimension of rupture. As described by Scholz (2002), this scaling law comes from static crack models. As we can see, for the same amount of slip a larger rupture zone undergoes a smaller stress drop, and a smaller rupture zone undergoes a larger stress drop. We can understand this by considering a small patch of a fault in relation to its neighbors. The farther a particle moves relative to its neighbors, the larger the stress drop. For a smaller patch with the same amount of slip as a larger patch, the gradient of slip will be steeper for the smaller patch, and thus a particle is moving farther relative to its neighbors.

36 Apart from determining the slip distribution on the fault as a fraction of fault 27 dimension, stress drop can also be measured seismologically. Brune (1970) showed that earthquake acceleration source spectra show an omega-squared shape, according to (3.2) where is the acceleration source spectrum, is the seismic moment, is frequency, and is the corner frequency of the spectrum. The corner frequency is the frequency below which the amplitudes of body and shear waves released due to an earthquake begin to decrease sharply (Brune, 1970). is inversely proportional to the source duration or fault dimension ( ), where r is the radius of a circular fault used to approximate the rupture area. By then recasting the fault dimension in terms of stress drop (Atkinson and Beresnev, 1997), we arrive at a scaling relation, where stress drop depends only on the seismic moment and the corner frequency. By this method, the measurement of the acceleration spectra and the identification of both the corner frequency and the seismic moment provide an estimate of the average stress drop on the fault. Hanks and McGuire (1981) extended this model by incorporating stochastic methods, which allowed them to effectively model the major features of highfrequency ground motion. Their work and subsequent studies confirming their findings resulted in stress drop becoming an important parameter of seismic hazard. Important differences exist between these two methods for estimating average stress drop. Estimating stress drop through the use of the scaling relation (3.1) has a clear physical meaning. The second method, while being a very useful approach for

37 seismological work, includes assumptions that may not always hold and are not 28 necessarily treated the same across different studies, as will be further discussed in section Fault Friction During Earthquakes The Coulomb friction law noted in section addresses static friction, that is, it deals only with the condition for the breaking of a frictional contact, and does not describe how the ensuing slip takes place. In order to better understand the nucleation and propagation of rupture and the incurred stress drop due to slip, we turn to two higherorder frictional responses, those of rate-and-state dependent friction and dynamic (or high-rate) weakening. These frictional responses will ultimately govern the change in stress on the fault due to an earthquake, as a fault patch may experience stress increase, partial stress drop, or complete stress drop. The rate-and-state friction law describes frictional response to changes in slip rate, and the change in strength as the frictional contacts evolve. It applies to faults that slip at very slow slip rates and therefore is relevant to rupture nucleation. Dynamic weakening, on the other hand, describes the dramatic weakening that is observed in many materials when slipping at much higher rates such as the seismic slip rate of ~1 m/s. The effect generates a large dynamic stress drop during rupture, and a large static stress drop remains after rupture has ceased. It also strongly influences the propagation of rupture. One equation, known as the Dieterich-Ruina or slowness law, describes rate-andstate friction according to (Marone, 1998)

38 ,, 29 (3.3) where µ is the final friction, is slip velocity, is the steady state friction for a surface slipping at reference velocity, is a critical distance over which a new steady-state friction evolves after a step change in, and is a state variable that evolves over time (relating to the sliding velocity and critical slip distance). Compared to the static friction law (2.8), equation (3.3) includes higher order behaviors of Coulomb friction. The variables of and are the physical parameters which ultimately determine the behaviour of the frictional contact. When the sliding velocity increases from to, the frictional response of the contact surface increases by, and over decreases again by. Thus, if the strength increase is greater than the strength decrease ( ), a net gain occurs, slip is retarded and the materials are said to exhibit velocity-strengthening behaviour. Similarly, if the strength decrease is larger ( ), a net strength loss occurs, the slip is unstable, and velocity should continue to increase; such a material is said to exhibit velocity-weakening behavior. Figure 3.1 summarizes these two behaviours. Velocity-weakening is the necessary condition for an earthquake to initiate. As the values of and are expected to vary over the surface of a fault, different portions of the fault may show different behaviours. Rupture must nucleate at a point within a velocity-weakening zone and would expand within its borders, provided the dynamic stresses generated are high enough to stress the fault to its peak strength level. If the propagating rupture pulse encounters a velocity-strengthening zone, the fault may still be brought to failure and slip (depending on the degree of strengthening), but a negative stress drop is incurred. The rupture must at some point

39 cease if a new velocity-weakening zone is not reached, or if rupture breaches the surface of the earth. 30 Figure 3.1. A cartoon illustrating two cases of rate-and-state frictional behaviour. The frictional (top) and stress responses (bottom) to an increase in slip rate for velocity-weakening and velocity-strengthening materials are shown. Equation (3.3) gives one empirical law which is currently used to describe this behavior. A more precise treatment of this behavior is shown by Marone (1999). Dynamic weakening (or high-rate weakening) has been primarily observed in laboratory experiments, and a large body of evidence shows that frictional contacts of many rock types undergo a dramatic weakening effect when slipping at high rates, regardless of their rate-and-state friction behaviour at low rates (Di Toro et al., 2011). Fault healing will bring the strength of the contacts up somewhat after rupture has ceased, but a large static stress drop is still incurred due to the very low resistance during rupture which encourages large static slip. The experiments are mainly carried out through the use of rotary shear apparatuses, where a strong weakening effect has been observed to reduce strength to about 10% to 30% of the initial value (Di Toro et al., 2011). Results from such experiments are shown in Figure 3.2. On a real fault, the onset of high-rate

40 weakening depends on whether other local conditions allow the slip to acquire a 31 sufficiently high rate. Clearer understanding of this process would provide better constraints on the stress drop (both dynamic and static), on mechanical work done, and on frictional heat arising from slip during earthquakes (Tullis et al., 2007). Several different mechanisms are theorized to allow this type of weakening, but are ultimately not critical to our work, as we are primarily interested only in the magnitude of the net weakening involved, or equivalently, the change in static stress on the fault. One important issue regarding the high-rate dynamic weakening effect is how it interacts with slow-rate regions exhibiting rate-and-state behaviors. For example, sections of a fault that previously exhibited velocity-strengthening behaviour may be induced to fail seismogenically by the dramatic dynamic weakening effect (Noda and Laputsta, 2013). Indeed, the peak slip of the 2011 Tohoku-oki earthquake occurred in an area that had previously been thought to be exhibiting velocity-strengthening behaviour. Velocitystrengthening sections of faults are often predicted to act as barriers to the propagation of rupture, but observations from Tohoku suggest this idea may need to be re-evaluated. The truly dynamic stress/strength drop that can occur during an earthquake is generally not the stress drop we investigate in this work. As discussed in section 2.1, and shown in Figure 2.2, while rupture is occurring the strength of the fault may be lower than the static strength afterwards; after rupture has ceased, fault healing brings the strength of the fault back up by some amount. The static stress drop, which we deal with in this work, is simply the difference between the pre-seismic apparent strength and the post-seismic post-healing strength. Nonetheless, the portion of dynamic weakening most relevant to our work remains: the large static stress change incurred. The static stress

41 drop observed in laboratory high-rate friction experiments is large, of the order of % of the prior static strength of the frictional contact (Di Toro et al., 2011). Figure 3.2. Friction coefficient versus normalized slip, illustrating the magnitude of dynamic weakening for various rock types in rotary shear experiments. Weakening mechanisms assumed are included as well. From (Di Toro et al., 2011). Final weakening values are shown within the red box marked SS (steady state), while the Dth arrow gives the extent of the critical distance over which weakening takes place from the peak value, shown by the red box marked P. References showing in the figure can be found in Di Toro et al. (2011). 3.2 Observed Stress Drop in Great Earthquakes In our work, we hope to provide some constraints on the strength of megathrust faults. As stress drop on the fault influences the state of stress in the surrounding plates, any measurable changes in the stress state of the plates before and after the earthquake can be weighed against the magnitude of the stress drop. If a stress drop for an earthquake produces no measurable effect on the stresses in the surrounding crust, one can conclude

42 33 that the average stress drop is too small a fraction of the strength of the fault. This may either indicate a very strong fault or a very small stress drop. Conversely, if the stress drop produces a pronounced effect on stresses in the surrounding crust, the stress drop must be a significant portion of the strength of a fault. This indicates either the fault is very weak or the stress drop is very large. Either way, if we can independently estimate the average stress drop due to a large earthquake, we can approach the strength of the fault itself by examining the stress changes in the surrounding crust caused by the earthquake. The following sections will review some observations of stress drop, and discuss the possibility of complete or near-complete stress drop on a fault as is often assumed in the literature, that is, the possibility that shear stress on the fault is mostly relieved during an earthquake Average Stress Drops of Great Earthquakes As discussed in section 3.1.1, and in strong relation to the manner in which stress drop is often estimated, stress drop is commonly treated as a macroscopic parameter, a value that describes the average change over the entire rupture patch of the fault. In reality, as both the strength and amount of slip vary along the fault, stress drop must also be heterogeneous and vary spatially as well. The nature of the heterogeneity will naturally affect the average value of stress drop. One example is illustrated by Madariaga (1979), who shows that for a fault that experiences higher stress drop near its edges, the seismically estimated average value will be different from the true average value, and the difference will increase as the stress drop near the edges increases. Section will further investigate the heterogeneity of stress drop, and how it influences the overall

43 34 average. We now delve a bit deeper into the details of these observations, as these are our primary source of data independent of our method. There are several seismic methods to estimate stress drop, as discussed in Boatwright (1984). In one method, discussed in section 3.1.1, the fault dimension is approximated using the corner frequency. As mentioned before, the different methods employ different assumptions and as such, produce different results, sometimes varying by orders of magnitude for the same event (Atkinson and Beresnev, 1997). These results are not from measurement error but arise due to differences in the underlying physics assumed by the different models. In addition to systematic differences, uncertainties within individual studies can be significant, as much as 15-25% (Boatwright, 1984) or higher. The largest uncertainties are likely due to the uncertainty in the fault dimension as estimated from the corner frequency. Chung and Kanamori (1980) estimated the uncertainty in the fault area to be within a factor of 1.5, corresponding to an average stress drop that is likely within a factor of 3 of the true value. Additional uncertainty arises from the assumptions made about the geometry of the fault or the physical parameters of the source region such as velocity of the seismic wave propagation. A further possible complication relating to seismic observations of stress drop is the treatment of the seismic moment,. is a scalar physical quantity, a measure of the energy released by an earthquake. Seismic moment is defined as the area of rupture times the rigidity times the average displacement on the fault (Aki and Richards, 2002). The moment can be estimated from the amplitude of the frequency spectra at some specific frequency; however, due to the saturation of the higher frequencies nearer to the source, far-field measurements have typically been used to estimate the seismic moment using

44 35 the low frequency portion of the spectrum. Seismic moments measured in this manner are considered to be influenced by a larger region (usually the entire fault) due to the longer wavelength of the seismic signal considered, and the total moment reported is one value that represents the entire ruptured area (note that more recently, with the advent of denser instrumentation, seismic moment is also estimated from the higher-frequency component of the frequency spectra; thus, the resolution of the seismic moment can be increased, which allows predictions of contributions to to vary spatially along the fault. The total moment, however, should theoretically remain the same as in the conventional method). The higher frequency details (which are also influenced by complex rupture) may thus be absent in the determination of, but are certainly important in determination of the corner frequency, as they affect the comparison of the real earthquake data against the f 2 decay model proposed by Brune (1970). Seismic measurements of the heterogeneity of stress drop that do not account for the heterogeneous contributions to the total would feel the averaging effect of the point-source treatment of. Average stress drop measurements for the entire fault that use the total should be well justified, but again, only as an average for the entire fault. As may be expected, complex rupture or source dynamics often make seismic determination of stress drop more difficult. Many scaling laws and other assumptions rely on self-similarity in order to be valid (Kanamori and Anderson, 1975), and the more complex a particular earthquake becomes, the less similar it is to other earthquakes. Indeed, even the relation between and stress drop depends on the geometry and the complexity of the source area, with simpler ruptures generating larger moments, but complex ruptures experiencing higher values of peak stress drop at some locations on the

45 fault (Madariaga, 1979). Nevertheless, many earthquakes over a wide range of 36 magnitudes do appear to be similar, primarily evidenced by the fact that stress drops are generally not correlated with earthquake magnitude (Aki, 1972; Kanamori and Anderson, 1975; Geller, 1976; Ide and Beroza, 2001; Shaw and Scholz, 2001; Allmann and Shearer, 2009, Oth et al., 2010; Baltay et al., 2011). Some statistically large (or small) values of stress drop are reported for smaller events, such as two examples shown by Baltay et al. (2010), but most important to the work being presented here is that larger earthquakes are almost always characterized by small average stress drops. Such studies as Hough (1996), Venkataraman and Kanamori (2004), and Allman and Shearer (2009) focus more on larger events, and support this statement. Average stress drop estimates for some recent large megathrust earthquakes reported in the literature are summarized in Table 1. These results show good agreement with the notion that large earthquakes should be characterized by low average stress drops. Within the broad observation that average stress drops are low, there are some systematic variations of stress drop, with the largest variations occurring between different types of faulting. Strike-slip faulting produces the largest average stress drop, on the order of 3-5 times that of reverse faulting for events with similar magnitudes (Allmann and Shearer, 2009). Reverse faulting events are almost always observed to have average stress drops of about 2-3 MPa, and Allmann and Shearer (2009) postulated that this may be due to the tendency for reverse faulting to occur in locations with lower rigidity. Interplate earthquakes in general were identified as having lower average stress drops, around ~3 MPa, by Kanamori and Anderson (1975), and the average value of over 400 subduction zone events was reported as 2.82±0.21 MPa by Allman and Shearer

46 (2009). Thus, while average stress drops are generally low for large earthquakes of 37 varying types and locations, there appear to be no exceptions to the observation that large megathrust earthquakes are characterized by low average stress drops. Event Stress drop (MPa) Derived from References Sumatran-Andaman (2004 M w = 9.2 ) Maule (2010 M w = 8.8 ) Tohoku-oki (2011 M w = 9.0) Iquique (2014 M w = 8.1 ) Seismic Waves Seismic waves Kanamori, (2006) Sorensen et al. (2007) 2.3 Seismic Waves Lay et al. (2010) 4 Static Slip Luttrell et al. (2011) Static Slip Huang et al. (2013) 2-10 Static Slip Simons et al. (2011) 7 Static Slip Lee et al. (2011) 4.8 Seismic Waves Koketsu et al. (2011) 6 Static Slip Yagi et al. (2011a) 2.5 Static Slip Lay et al. (2014) Table 3.1. A small survey of observationally determined average stress drop values for recent large megathrust earthquakes Complete versus Partial Stress Drop Having discussed the low static stress drops of great earthquakes in the previous section, and the large static stress drops predicted from dynamic weakening in section 3.1.2, we now address the question of whether faults experience complete (or very large) stress relief during earthquakes. If dynamic weakening is considered to be a first-order process, then one would likely expect near-complete stress relief on the fault, as several studies indeed proposed (Hasegawa et al., 2011; Hasegawa et al., 2012, Yagi et al.,

47 a; Hardebeck, 2012, Lin et al., 2013; Obana et al., 2013). In some cases, expectation of dynamic overshoot (i.e. inertial forces causing the final shear stress on the fault to be lower than the dynamic friction during rupture) and comparisons against results from much smaller scale earthquakes have led some researchers to speculate that the megathrust may temporarily become a normal fault at some locations (Ide et al, 2011); this requires that shear stress on the fault not only dropped to zero, but even built up in the opposite direction. However, the notion of very large average stress drops is at odds with the results presented in section 3.2.1, where we described the seismological observations of very low average stress drops. Using the FEM models discussed in Chapter 2, and somewhat crudely simulating the magnitude of complete stress drop by comparing the predicted shear-stress profiles along the fault (shown in Figure 3.3) to the zero-strength case, the average stress drops would be approximately 26 MPa, 53 MPa, and 322 MPa, for faults with strengths of = 0.03, 0.07, and 0.4, respectively. Complete stress relief on even the weakest fault we consider is then unreasonable when compared to the seismically determined stress drops, as they differ by an order of magnitude. The differences between what is determined seismically and what is inferred from dynamic weakening can be reconciled by considering the heterogeneity of stress change. When one patch of a large fault becomes dynamically weakened, it must affect (i.e. increase stress on) the neighboring patches around it as well. If they too become dynamically weakened, then the stress drop for that particular region of the fault will likely be large. However, if the neighboring patches do not experience dynamic weakening, and thus resist the rupture of the source region, they may actually experience

48 stress increase. Consider this same example in terms of slip; if a source region 39 experiences 20 m of static slip, but the neighboring regions resist rupture (i.e. exhibit strong velocity-strengthening behavior, as described in section 3.1.2) and experience only 2 m of static slip, then the shear stress on the fault may actually have increased for the neighboring regions. Since seismic observations and the scaling relation consider average stress drop, both positive and negative signals are included, and the average remains low, even though large static stress drop through dynamic weakening had occurred. One example of locally large stress drop during the 2011 Tohoku-oki earthquake is identified by Kumagai et al. (2012), which appeared to relieve ~40 MPa of stress on a small portion of the fault, at least an order of magnitude higher than the predicted average values for the entire rupture zone as reported in Table 1. Of course, this effect should be active at different scales, from small to large, and detailed knowledge of how the shear stress on the fault was relieved would allow us to investigate both the average stress drop of an earthquake, and how the average was affected by the heterogeneous pattern of stress change. To determine the distribution of stress change on the fault using estimates of static slip, we must be able to integrate the contributions to stress change over the entire fault at any one point; to do this, we turn to dislocation modeling.

49 40 Figure 3.3. Finite-element modeling stress profiles along the fault for three different fault strengths. The average shear stresses on the fault (and the average stress drop if the stresses were completely relieved due to an earthquake) are 26 MPa, 52 MPa, and 322 MPa for the µ = 0.03, 0.07, and 0.4 cases respectively. Blue lines show the frictional strength of each fault, while red shows the level of shear stress along the fault. Red asterisks show the average values given above on their respective faults Dislocation Modeling of Stress Changes on a Fault in an Earthquake Method Dislocation modeling involves calculations in an elastic half-space, where the effect of dislocation representing fault slip at one location can be predicted at another. The dislocation is prescribed and unrelated to any processes occurring in the background, and

50 thus dislocation modeling deals only with perturbations to the background and not the 41 absolute background state itself. Generally in the Earth sciences, dislocation modeling is used to predict surface deformation due to the rupture or creeping motion on some fault at depth, with an example for pure thrust faulting shown in Figure 3.4b. However, for our purposes we are less interested in what happens at the surface, but instead concerned with the internal deformation because it allows us to calculate the change in stress; that is, the incremental strain due to fault dislocation (i.e. a perturbation) can be constitutively related to the changes in stress through the use of Hooke s law. In our work, we want to investigate the change in shear stress on a fault associated with, or more precisely, that has led to the heterogeneous pattern of slip on that same fault. The resultant heterogeneous pattern of stress drop will then allow us to investigate the effect of averaging across the entire rupture zone. Here we use simple static slip models to gain some intuition about the problem of stress drop. In Chapter 4, we will use published static slip models, which are constrained by several types of data, to investigate stress drop in the 2011 Tohoku-oki earthquake. Okada (1992) provided an analytic solution for the internal deformation due to a point-source dislocation. We show the model setup in the Okada system in Figure 3.4a. By using the point source solution and integrating its Green s function over the surface of a fault, Okada was also able to derive an analytical solution for the deformation due to uniform slip of a buried planar rectangular fault. As we are interested in faults with arbitrary geometry and spatially variable slip, we choose to use the point-source solution.

51 42 Figure 3.4. A) Schematic of model setup in Okada coordinate system. Fault dips at 20 in the +y direction. Profiles of stress change are taken along the center of the fault, at varying distances from the fault as shown. Relevant benchmarking stress change data are shown in Figure 3.5) Cartoons of expected surface deformation along the line of symmetry due to purely thrust movement on a buried fault. U(x) is not shown, as it is uniformly zero in this case. Dislocation modeling is commonly used to predict surface deformation in this manner; however, our work uses dislocation modeling in a different manner, as described in the text. We create a mesh of triangles of roughly equal size, with each triangle containing a pointsource dislocation at its center (Flück et al., 1997; Wang et al., 2003). We use triangles because they allow for a simple method of creating a connected mesh to approximate the irregular fault surface. We can then map either an artificial slip distribution (for testing purposes) or a observationally based static slip distribution onto the mesh and then numerically integrate over the resulting fault surface to determine the final deformation

52 43 near the surface of the fault (or at any arbitrary location in the half-space). Using Hooke s law, we determine the change in the stress tensor from the output strain change tensor. The incremental shear stress along a given surface is the stress change vector that we are interested in at the queried location. The code Disl3d14 used in this work is developed by Kelin Wang, Pacific Geoscience Center, Geological Survey of Canada. Breaking a fault into many triangular pieces does come with some possible problems. In effect, we are simulating a larger coherent stress field by combining many smaller ones. Because we assign the value of a point-source dislocation to each triangle, there is no tapering off of slip at the edges of the triangular fault element, and these edges are singularities if neighbouring elements have different slip values. That is, between any two triangles in the mesh, the stress may be discontinuous, and calculations at or very near those points can produce erroneous results. Therefore, we cannot determine stress drop directly on the fault surface and must instead choose a surface parallel with (and some distance away from) the fault. The smaller the distance away from the fault, the better the approximation will be, but some finite distance away from the fault is required for the effects of the triangle-edge singularities to average out. Obviously, the larger the triangles in the fault mesh, the larger distance from the fault is needed to average out the singularity effects. However, if we move too far away from the fault, we cannot use the obtained results to approximate the real shear stress change on the fault. To produce meaningful results, we need to stay as close to the fault as possible, but not so close as to be affected by the singularity effects mentioned above. Thus, we need to determine the best compromise that allows us to derive accurate fault stress without having to use a

53 huge number of extremely tiny fault triangles which would slow down the numerical 44 integration. To determine how far away to take our measurements, we compare our numerical integration solution to Okada s (1992) analytical solution for a planar rectangular fault (shown in Figure 3.5). We assume dimensions of 200 km in the strike direction by 100 km in the downdip direction, a dip of 20 degrees, and a depth of 10 km to the shallowest portion of the fault. This solution, being a precise analytical solution, can be queried at distances infinitely near to the fault. We derive shear stress at 1 cm from the fault as our true fault stress (black line in Figure 3.5), but it makes no practical difference if we derive it at 1 mm or 10 cm from the fault. By comparing the stress solutions numerically derived at different distances from the fault with this analytically derived true fault stress, we can determine our best compromise distance for a given triangle size. For the numerical version of the same theoretical fault, we used a triangle size of ~0.5 km on average per side for our testing, which allows for reasonably quick calculation of the results. We then assign a purely thrust slip of 10 m to every triangle in the mesh to approximate a uniform 10 m thrust motion on the rectangular fault. We then query the stress field along the line of symmetry at different perpendicular distances to the fault surface, and compare the results to the analytically derived stress field. The model setup in the Okada coordinate system is illustrated in Figure 3.4a, which also shows the various distances from the fault that were tested. The numerical results are compared with the analytical results in two ways. (1) The accuracy of the numerical model is reflected by a comparison of the numerical results (the colored circles in Figure 3.5) with the analytical solutions for the same distances

54 45 from the fault (the colored curves in Figure 3.5). This helps us to determine the minimum distance from the fault where accurate results can be obtained. (2) A comparison of the numerical results with the analytically derived stresses along the fault (black curve in Figure 3.5) helps us to decide on the maximum distance from the fault where the stresses can be reasonably used to approximate fault stress. Figure 3.5. Benchmarking dislocation code against Okada analytical solution. The 0 km black line comes from Okada s solution, while the other colors are from our dislocation code. The data from Okada s algorithm highlights the stress singularities at the edges of the fault, which arise because slip stops immediately. The distance we use in our calculations, 2 km, adequately models the analytical solution over most of the fault. It is worth noting that for the purposes of these tests, a rectangular planar fault is a useful measuring stick; however, in comparison to real faults, it does a poor job of creating a realistic stress field near the edges of the fault. For a fault with no tapering-off of slip the stress field directly at the edge is a singularity, where the stress approaches

55 infinity. The signal created by this singularity is clearly visible in the results in Figure For real faults, slip must taper off over some distance, and the fault-edge singularity does not occur. Nonetheless, the rectangular fault solution suffices for our present purpose. The results show that, for fault triangles of roughly 0.5 km, at distances of ~1 km from the fault, the numerical solution diverges from the analytical solution. At distances around 0.7 km, the results are completely overblown by the singularities and no longer plot within the range of the graph. For a distance of 2 km or 5 km away from the fault, the results closely approximate the analytical solution. In the middle of the fault away from the fault-edge singularities, the numerical solutions at both these distances match the stresses on the fault (black line) very well. However, closer to the fault edges (less than 15% of the fault length), the 5 km solution is obviously a poor approximation of the stress change on the fault. Very near the fault edges (less than 5% of the fault length), the 2 km solution also fails to approximate the fault stress. Since severe slip and stress singularities like the fault edges in this model could never occur in real Earth, we can ignore the results that are less than 5% of fault length away from the edges. Thus, for ~500 m triangles, we take 2 km as our best compromise distance to derive stress changes on the fault. The rule of thumb is that the distance from the fault used for stress drop calculation should be about four times the length dimension of the fault triangles Stress Drop of a Single Rupture Patch The benchmarking tests done in the previous section use uniform static slip over the entire rupture patch with no tapering off. By comparison, a widely used crack model

56 47 shown in Figure 3.6 (dashed lines) assumes a uniform stress drop over the ruptured area as opposed to uniform slip. Such crack models still produce singular edge effects (continuous slip but discontinuous stress across the edge), albeit much less severe as the stress does not go to infinity at the fault edge. Figure 3.6. Modeling results from Hu and Wang, (2008) comparing stress signals from different types of fault models. The dotted line shows a uniformslip rectangular fault models similar to that in Fig The dashed line shows the constant-stress drop crack model, which does produce a stress singularity (a) near the edges of the fault, but does not produce a slip discontinuity (b). Both models were produced using the finite element method. In this work we use the above described numerically-integrated dislocation model to investigate the effects of more realistic slip distributions; that is, we generate a slip distribution that tapers towards the edges of the rupture patch more gradually than in the widely used crack model. Using this method, we aim to investigate the effects of heterogeneous static slip, both its effects on the heterogeneous distribution of stress change, and the resulting influences on the average stress drop value. We now investigate the simplest case of purely thrust slip in a patch with a Gaussian-distributed tapering of slip, according to (3.4) where a is the amplitude of peak slip (10 m in our test ), x is the distance away from the center of the rupture patch, and c is the standard deviation of the distribution (20 km in

57 the following tests cases). This 1D distribution is applied in both the updip and strike 48 directions, resulting in a circular patch. Our circular rupture patch is approximately km in diameter (depending on the cut-off value, as explained below) inside a larger fault mesh of approximately 200 x 200 km. The exact size and aspect ratio of the fault patch are not important, because most of the triangles away from the rupture patch have zero stress change and have no contribution to the calculation. The rupture patch is planar, with its center at a depth of 17 km, and dips towards the east at 10 degrees. Peak slip occurs in the center of the circle, is purely thrust motion (in this case, directed westwards) and is set to 10 m for these tests. The three tests shown are performed with different cut-off values of slip; that is, slip is set to 0 when its value is less than 1%, 10%, or 30% of the peak value (Fig. 3.7). As the generating functions that create the slip distribution are identical, increasing the cut-off value will decrease the size of the circular patch which is assigned non-zero slip. The results clearly indicate a smoothly changing stress perturbation due to the smoothly varying slip distribution, and a pronounced increase of stress outside the rupture patch as the cut-off value increases (Fig 3.7). One important point is the increase of stress outside the region that does not experience any slip at all. This effect occurs even for the model with the smallest cut-off value, which represents rupture that arrests smoothly. No slip means no release of seismic energy, and thus for a real earthquake, this region would not factor into seismologically determined stress drop (as described in section 3.1.1). Once our dislocation code has provided us with the distribution of stress change on the fault, we can then make observations about the pattern and heterogeneity of the

58 49 change in stress on the fault, such as we will do in section 4.3. However, as we will wish to compare our method and results against independent observations of average stress drop, we must first decide on a method to take an average of a vector field. For a large megathrust earthquake, the overall stress drop vector along the fault is primarily in the direction of plate convergence. However, because of the heterogeneities of the fault and static slip, stress change can be in various different directions with various magnitudes at different parts of the fault, resulting in a complex vector field on the fault surface, as is illustrated in Figure 3.7. Local convergence direction at subduction zone forearcs are usually not very oblique even though regional plate convergence can be rather oblique, with the margin-parallel component of relative plate motion accommodated by a strike-slip system in the upper plate (i.e. slip partitioning). For this reason, the overall direction of megathrust slip vectors and stress drop vectors is updip. The strike components of the various stress drop vectors will cancel one another when an average is taken. It is for these reasons that we choose to neglect the stress perturbations in the strike direction. This assumption allows us to calculate the average stress drop for the entire fault quite simply; by considering only the updip component of the stress change vector, we have defined a scalar variable with only positive (stress increase) and negative (stress decrease) values. A simple spatial averaging will then suffice. One could also pursue a vector summation of all the stress drop vectors weighted by their triangular areas to take a vector average. Unless the stress drop vectors are predominantly oblique to the dip direction, the results should be very similar to the above described scalar average.

59 50 To obtain an average stress drop, we must use some reasonable criteria to define the extent of the area involved. For the testing cases here, we will determine the ruptured area based on that rupture distribution; that is, areas that were assigned non-zero slip will be included in the calculation. The purpose here is to approach the theoretically true value of average stress drop within the ruptured patch, and thereby come closer to the fundamentals of the subject. Our averages will also include different values of areas outside the ruptured patch as well, where the stress on the fault increased. This will allow us to quantify an average stress change which is more relevant to the true state of stress on the fault; As described in section 2.1.1, this will be important towards constraining the state of stress in the forearc of the upper plate. As mentioned previously, including only the region that experienced slip would be similar to observations of stress drop estimated from seismic methods, which only reflect the seismic energy released due to rupture. The varying degrees of the abruptness of rupture arrest (modeled with different slip cut-off values) mentioned previously are intended to crudely approximate real-world examples. The most abrupt case corresponds to rupture encountering a strongly slipstrengthening barrier, where slip quickly arrests and the stress increase on the barrier is high (cut-off value of 30% shown in Figure 3.7c). Compared to the smoother cases (Figures 3.7a and 3.7b), the area of rupture is smaller, and has an overall higher average value of slip. This leads to a larger stress drop, as is expected from the scaling law (3.1).

60 51 Figure 3.7. Dislocation modeling results showing the stress drop distribution of a single Gaussian-distributed rupture patch. Rupture is set to 0 at three different percentages of the maximum value (10 m), with cutoff values of 1%, 10%, and 30% increasing from top to bottom. Δσav1 and Δσav2 are the two different averages that result from considering the areas within 10 and 20km of the ruptured patch, respectively. Stress change vectors shown in the lower portion of the figure represent case A and show the total stress change produced by our method, before our simplifying averaging procedure described in the text has been performed. Although slip is set to 0 at different cutoff values, stress vectors are still calculated outside of the region that slipped. Note the differing directions of stress change vectors, even though the slip is purely in the updip direction. We can then compare two values of average stress drop: one that includes the effects of the positive stress change within 10 km (Δσav1 in Figure 3.7) of the ruptured patch, and the other which includes the region out to 20 km (Δσav2 in Figure 3.7). For the most

61 abruptly arresting case (Figure 3.7c), including the extra distance lowers the average 52 stress drop by the largest amount within the three cases, (1.31 MPa for case C, versus 0.19 MPa for case A). This effect occurs due to two factors. First, the area of rupture is smallest for case C, and expanding the area of calculation for two differently sized circles by the same radial distance (10 km in this case) increases the area proportionally more for the smaller circle than for the larger one. Thus, the effects of the positive stress increase are larger for the smaller rupture patch. Secondly, the magnitude of the stress change within those areas is much higher for case C, which is quite visible in the stress change distribution modeling results in Figure 3.7. Both of these factors will lead to a larger difference between the different average values (i.e. between the ruptured area + 10 km and ruptured area + 20 km values) for case C relative to case A Dislocation Modeling of a Multi-Patch Rupture Field Once the simplest case of the single circular patch has been investigated, we can complicate the situation by adding more rupture patches, representing larger earthquakes in which slip varies heterogeneously over the fault surface. Regions of relatively high slip may be separated by some distance, and as the simplest case implies, they should both affect the region between them. Figure 3.8 uses the same fault domain as the simplest case model (with a middle degree of abruptness of rupture arrest, so as to facilitate comparisons of average stress drop values), but uses two copies of the circular slip distribution at varying distances from each other to model the interactions between the

62 separate patches. Figure 3.9 uses four copies of the circular slip distribution, to 53 investigate how the effects are compounded by increased complexity of slip. As the disconnected rupture patches become closer in Figure 3.8 (two patches), the stress change between them becomes increasingly positive (i.e. stress increase). So long as the discrete areas considered in the averaging calculation do not begin to overlap, this leads to smaller average stress drops, in agreement with results from Madariaga, (1979) and Rudnicki and Kanamori (1981). Figure 3.8. Dislocation modeling results showing stress drop distribution as discrete rupture patches become more connected. Rupture patches used here are identical to those used in Figure 3.7b, but duplicated to facilitate increasing slip complexity.

63 54 In the model with four patches of rupture (Figure 3.9), these effects are compounded, and the region between the discrete patches experiences a higher magnitude of stress increase as the patches grow closer. The implication here is that for a region that experienced little or no slip and is surrounded by regions that experienced high static slip, the magnitude of stress increase could be quite high, and it will certainly have a visible effect on the average stress drop value for the region which encompasses both the ruptured and unruptured portions of the fault. Both these results and the results from the previous section appear to predict that the degree of heterogeneity and the character of rupture arrest would be important factors to consider in determining an average stress drop for large and/or complex ruptures. The resolution of the estimated final static slip (or the seismic moment, if the average is to be determined seismologically), and the assumptions used in determining the area to be included in the average calculation, will have strong influences on the final value obtained. For ground-motion predictions, including the influence of the positive stress change outside the ruptured area may possibly not be useful, but for studies of postseismic deformation or for determining the state of stress in the forearc after an earthquake, such concepts are likely to prove beneficial to the work. In the next Chapter, we will employ these concepts in investigating the stress change of the 2011 Tohoku-oki earthquake.

64 Figure 3.9. Dislocation modeling results showing stress drop distribution as discrete rupture patches become more connected. Rupture patches used here are identical to those used in Figure 3.7b, but duplicated to facilitate increasing slip complexity. Stress vectors on the lower right are similar to those in 3.7, but represent the case shown here, and illustrate the complexity of stress change on the fault for a more complex rupture model. 55

65 Chapter 4. Implications of Forearc Stress Changes Caused by the 2011 Great Tohoku-oki Earthquake The Tohoku-oki Earthquake The Great Mw 9.0 Tohoku-oki earthquake occurred on March 11 th, 2011, and claimed over 15,000 lives, with more than an additional 5000 listed as missing (Mori et al., 2011). The event is much larger than what was expected for this region, and produced a very large and destructive tsunami which caused by far the greatest portion of the damage and nearly all the deaths. Recordings by the dense multidisciplinary networks of instrumentation in the region and by the global seismic and tsunami networks have made this earthquake the best recorded and studied earthquake in history. The Tohoku-oki earthquake is an interplate thrust event with a hypocentral depth of ~20 km. It began approximately 110 km off the northeastern coast of Japan, on the shallowly-dipping megathrust where the Pacific plate (aged ~ Ma in the region) is subducting approximately westwards underneath northeast Japan at a rate of ~9 cm/year, as seen in Figure 4.1. Many Mw 7-8 interplate events have occurred here in the past, but only one event catalogued so far, the 879 Jogan earthquake, is possibly of a similar magnitude to the 2011 event and produced a tsunami with similar severity of inundation (Minoura et al., 2001; Minoura and Nakaya, 1991). Peak slip was very high for the Tohoku-oki earthquake (as high as 80 m reported by some rupture models) while average slip was ~10-15 m over a fault area of roughly 200 km and 150 km wide in the strike and dip directions, respectively, and the maximum extent of the rupture patch was approximately 200 km downdip by 500 km along strike. Unusually, the largest slip has

66 57 been widely reported to have occurred on the shallower regions of the fault, previously believed to exhibit velocity strengthening behavior and thought to resist rupture (Noda and Laputsta, 2013). There is evidence in some areas (Fujiwara et al., 2011; Kodaira et al., 2012) that the coseismic slip extended to the trench, the depth of which averages ~7.5 km over its length. The large tsunami that was generated by the coseismic motion of the seafloor devastated approximately 2000 km of the Japan coast and affected other more remote locations within and around the Pacific Ocean. The average run-up heights along the coast of northern Honshu were reported to be ~10 m, but were higher than 20 m over a 260 km stretch of the coastline including the Sendai plains, where the waters surged up to 5 km inland (Mori et al., 2011). The large horizontal movements of the sloping seafloor contributed about 30-60% of the vertical uplift of the seafloor (Satake et al., 2013). Over 40 different published rupture models have been produced from different inversions, employing a wide array of different data types, including: geodetic data, both land and seafloor GPS (Iinuma et al., 2012; Ito et al., 2011; Ozawa et al., 2011; Pollitz et al., 2011; Simons et al., 2011), seismic waves (Hayes, 2011; Ide et al., 2011; Lay et al., 2011), strong motion data (Suzuki et al., 2011), tsunami wave measurements (Fujii et al., 2011; Satake et al., 2013), and joint inversions combining different types of data (Ammon et al., 2011; Gusman et al., 2012; Yue and Lay, 2013; Shao et al., 2012). Most models agree that the greatest peak slip occurred towards the shallowest region of the megathrust fault, and seismic observations of compressional geologic structures at the trench indicate that rupture did indeed continue to the trench (Kodaira et al., 2012). However, the reported peak slip ranges from ~30 m (Ide et al, 2011) to as high as 90 m

67 58 (Iinuma et al., 2012). The differences are due largely to the decreased availability of data in the near trench region. Compared to the large number of on-land GPS stations, only 7 seafloor stations exist to constrain the coseismic slip. One source of data that helps to constrain slip closer to the trench is a differential bathymetry profile taken by Fujiwara et al. (2011), near to the strike-normal corridor where many rupture models predict the largest peak slip. Figure 4.1. Satellite image of Japan (left), with the main study area roughly outlined in red. Approximate plate boundaries are shown in yellow, with approximate velocities shown in white. Location of profiles where focal mechanisms are located (right) in Figure 4.2.

68 4.2. Stress States Before and After 59 As discussed in section 2.2.1, the state of stress in the upper plate is the result of the compression due to plate coupling being balanced against the topographically induced tension. Several methods have been used to determine the state of stress in the upper plate. Firstly, as earthquakes occur to relieve accumulated strain, the state of stress in the upper plate may be constrained by the focal mechanisms of earthquakes that occur within it. A region where the focal mechanisms are mainly of the thrust-style must be in deviatoric compression with respect to the horizontal. For northeastern Japan, several focal mechanism studies concluded that prior to the Tohoku-oki earthquake, the majority of the region was indeed in compression with the maximum compressive stress σ1 being oriented roughly normal to the margin in the direction of plate convergence (Yoshi, 1979; Igarashi et al., 2001; Asano et al., 2011; Hasegawa et al., 2012; Yoshida et al., 2012, Kato et al., 2011). More recent studies of the pre-seismic state of stress by Yoshida et al. (2015) illuminate the spatial heterogeneity of the stress field in the forearc prior to the earthquake. They noted that while σ1 is oriented mostly WNW-ESE in the central regions of North Eastern Japan, it is instead oriented N-S and vertically in the northern and southern portions of the plate, respectively. Further, as our explanation of the forces involved in section predicts, regions of low topography are characterized by a compressive regime. However, in the backarc region, where topography is relatively high, the plate coupling force is balanced against an increasing gravitational force; we may then expect to observe a change in stress state from compression in the regions of low topography to neutral horizontal stress or tension in the regions of higher topography.

69 60 Yoshida et al., (2015) found that this was indeed the case and predicted that this level of stress heterogeneity is likely only possible for regions where differential stresses (σ1 σ3) are low, on the order of ~15-25 MPa for NE Japan prior to the Tohoku-oki earthquake. Another line of evidence comes from in-situ measurements of stress, specifically borehole breakout data. After a borehole is drilled, cracks and fractures may form in the walls of the borehole, the orientation of which gives information about the local state of stress, specifically the direction of the minimum and maximum horizontal stresses. The magnitude of the local stress can also be constrained as well, which is related to the shape of the breakouts, rather than the orientation. Prior to the Tohoku-oki earthquake, data from two ocean bottom boreholes showed the direction of σ1 to be mainly margin normal, which agrees well with earthquake focal mechanisms in that area (Lin et al., 2013). One of the most striking observations arising from the Tohoku-oki earthquake, and the main observation that allows this work to constrain the strength of the megathrust fault, was the dramatic changes in the focal mechanisms of small earthquakes observed after the event. In the study by Hasegawa et al. (2012), the observations indicated that before the earthquake σ1 was not only oriented in a roughly margin-normal direction, but also approximately oriented towards the regions where the largest static slip later occurred. However, focal mechanisms of the aftershocks occurring in the upper plate showed a widespread reversal throughout the ruptured area, from thrust events beforehand, to normal afterwards (Hasegawa et al., 2012, Asano et al., 2012, Yoshida et al., 2012, Kato et al., 2011). Thus, σ1 became oriented vertically after the earthquake (whereas it was roughly margin normal before), while σ3 was now oriented towards the region of largest

70 displacement (Hasegawa et al., 2012). Figure 4.2 shows the focal mechanisms of the 61 aftershocks located by Hasegawa et al. (2012). Figure 4.2. Focal mechanisms of small earthquakes (left) before and (right) after the Tohoku-oki earthquake (from Hasegawa et al. 2012). Normal faulting indicates horizontal deviatoric tension, and post-seismic plots of normal events are understood to show the extent of the reversal of stress after the earthquake. Red symbols represent normal faulting, and blue symbols represent thrust faulting. Locations of the six profiles are shown in Fig. 4.1.

71 4.3. Stress Drop Distributions from Rupture Models 62 As discussed in section 3.3.1, our Okada-based method of dislocation modeling produces vectors of shear stress change on the fault, using input rupture distributions that were either generated or observed. By simplifying the vector field (as discussed in section 3.3.1), we then produce a scalar version of the distributions of the shear stress change. Using the rupture models discussed in section 4.1.3, we now employ our method to investigate the resulting stress drop that each model produces on the megathrust, and obtain average values of stress change over the entire rupture patch. I will describe the process for one model in full first, and then discuss the broader results from the study. A typical model begins by preparing the mesh which approximates the irregular shape of the subduction interface and contains the slip information from whichever rupture model we are using; for this example we used the finite-fault model obtained by Shao et al. (2012), who jointly inverted land and marine GPS and seismic data. We created a high-resolution mesh consisting of triangular elements of about 500 m in dimension, following the actual geometry of the megathrust fault at the Japan Trench (using the geometry discussed by Sun et al., 2014). Next, we interpolated the published rupture model to produce a continuous and smooth distribution of slip vectors. We then mapped this continuous slip vector field onto our fault mesh. The original fault model used by Shao et al. (2012) has rather simple (piece-wise planar) geometry. When the slip vectors are mapped to our real fault geometry, the dip angles of the vectors are automatically adjusted to fit the new geometry. Figure 4.3 shows the resultant slip distribution, as well as the resolution of the fault mesh. The rupture models are quite smooth on the order of at least ~5 km and do not need a great number of grid points to be described. As discussed

72 in section 3.3.1, the reason we use higher resolution for the fault mesh is to make the 63 model smooth on the order of ~1 km (i.e. twice the element size), so that we can perform our stress change calculations close enough to the fault to produce accurate results. Figure 4.3. (a) An example of the fault mesh from out dislocation modeling, with the rupture model of Shao et al. (2012) applied to it. Inset shows the resolution of the mesh which is comprised of triangles approximately 500 m per side.(b) Slip vectors for the same rupture model. Red box in (a) shows map area in (b).

73 64 After the realistic fault geometry is assigned with the slip information, we then create a mesh of points where we calculate the stress drop vectors along the fault surface. This was done by using an evenly-spaced grid of points that is dense enough to adequately portray the distribution and variation of the stress drop field. The stress-calculation points are displaced downward a fault-perpendicular distance of 2 km, as explained and justified in section and in Figure 3.5. The calculated vectors of shear stress change on the fault are shown in Figure 4.4. From the results, we see that the pattern of stress change on the fault implied by the rupture model is indeed quite heterogeneous. Spatially varying amounts of slip imply stress change in varying directions. As discussed in section 3.3.1, the vectors of stress change that point updip represent stress drop. In some regions, vectors that point roughly downdip can be seen; these represent an increase in shear stress, also called negative stress drop. Figure 4.5 shows the stress perturbation at ground surface of seafloor associated with the rupture model of Shao et al. (2012) (Figures ). We use the graphic method most commonly used to plot focal mechanism to show the orientation and magnitude of the stress perturbation tensor. The beachball diagrams do not represent the focal mechanisms of earthquakes; we use them here to identify the directions of the three principal stresses and the magnitude of the differential stress associated with the coseismic stress perturbation at each location.

74 65 Figure 4.4. The vectors of stress change produced by our method, plotted on the fault surface. Due to spatially varying slip, vectors are not always pointing directly in the updip or downdip direction. As discussed in the text, vectors that point updip have some component of stress decrease, while vectors that point downdip have some component of stress increase on the fault. The center of the white quadrant, the direction of the P axis in a regular focal mechanism solution, is now the direction of 1. The center of the black quadrant and the direction the T axis in a regular focal mechanism solution, is then the direction of 3. 2 is orthogonal to both 1 and 3, and is thus located at the intersection point of the black and white quadrants. The size of the beachballs is determined by the magnitude of the differential stress (that is, 1-3). As the maximum shear stress possible (on an optimally-oriented plane) is half of the differential stress, then the size of the beachball is also a measure of the shear stress component of the co-seismically induced stress change. Further, as the magnitude of the signal varies over almost three orders of magnitude, the symbols are

75 66 logarithmically scaled. At the free surface, the shear stress is zero this requires that one of the principal stresses be oriented vertically, and thus we expect to see beach balls that represent stress signals either purely compressional or tensional or purely strike slip, but no combinations of the two. In this context, the beachballs give the direction of maximal tension in a simple way (usually oriented towards a nearby region with the largest slip, as expected), but also shows whether the rupture model implies stronger horizontal or vertical compression at a particular region. In the example shown in Figure 4.5, using the Shao et al. (2012) model, we do see a few beachballs that show the axis of compression pointing towards the nearest high-slip region; for the two large examples near the trench, this occurs due to the observation point actually being located on the other side of the trench (i.e. in the incoming plate), and thus are correct for that location. However, the other smaller examples of such compression occur within the overriding plate, at locations where slip is decreasing towards the trench. This corresponds to regions of stress increase on the fault, as is visible in Figure 4.6. Furthermore, we see that the stress perturbation west of the coast is very small, on the order of 0.1 MPa to a few MPa. The co-seismic signal does not have a large effect very far inland, and this will be an important point to consider for our finite-element method in the next section. Since the earthquake-induced stress perturbation in that far-field region is very small, it is very unlikely to cause a reversal of stress from pre-seismic compression to post-seismic tension.

76 67 Figure 4.5. Stress perturbation at the ground surface (beach balls) calculated for the fault slip model of Shao et al. (2012) (red-yellow color scale). Beachball symbols here do not represent earthquakes, but instead show the orientation of the three principal stresses of the induced stress perturbation, as discussed in the text. Beachballs are scaled according to the differential stress of the perturbation ( 1-3). More details of this method are shown in Appendix A. Lastly, we derive the scalar stress drop distribution from the stress change vectors (shown in Fig. 4.4) as discussed in section 3.3.1, which allows us to plot the results in simpler terms and also calculate an average value of stress change for a given rupture patch. Results are plotted in Figure 4.6. To highlight the importance of considering the heterogeneity of stress change, we plot the average values for three different fault sizes ; that is, we consider different rupture dimensions that are delineated by the contours of

77 68 slip for that particular model. We use four different contours, namely 30, 20, 10, and 5 m (only plotting three overtop of the slip and stress distributions, however) to define areas over which average stress change values are calculated. Obviously, we expect higher values for average stress drop for the patch that is predicted to have slipped 30 m or more, and the lowest value for the patch that encompasses almost the entire rupture model (i.e. within the 5 m contour region). For the Shao et al. (2012) model, the average stress drop for the area within the 5 m contour is ~3 MPa, which agrees with our earlier discussion on the low average value of stress drop for large megathrust earthquakes. However, even if we consider just the region that slipped in excess of 30 m, the average stress drop only reaches ~10 MPa, and while this value is certainly higher, this does not represent complete or near-complete stress drop over the region in question. We then perform the same procedure as above for other rupture models. Figure 4.7 shows both the rupture model used and the output stress change on the fault for five different models (a further fifteen models are shown in appendix B). Perhaps the most obvious observation in the results is the heterogeneous nature of the stress change on the fault. The different rupture models predict varying degrees of stress change complexity, but some degree of heterogeneity is present in each model.

78 69 Figure 4.6. Stress distributions and average stress drop values for one rupture model. Top: Slip model of Shao et al. (2012), with three contours plotted. Middle: The scalar stress change on the fault, produced by our averaging method as discussed in the text. Bottom: The average values of stress change for the area within the four different contour values considered. That is, the value of average stress drop for the area that slipped in excess of 30, 20, 10, and 5 m. A very important point to note is that even within the region that experienced the highest static slip and hence greatest stress drop, nearly all models predict some degree of stress increase in certain areas. If true, one explanation could be that these areas exhibit velocity-strengthening behaviour (discussed in section 3.1.2), and did not achieve a slip rate high enough to become dynamically weakened. Of course, the spatial variation of the

79 70 slip and hence stress drop distribution depends on these kinematic inversion models. It is important to remember that the rupture models do not consider the stress change implied. The models show very large stress drop in some regions, as high as ~55 MPa in the model with the most extreme slip. In this instance, we must take care to remember that Okada s solution assumes a homogenous half space. For shallow regions experiencing large static slip, some estimates of stress drop may be higher than what is likely possible for that region of the fault. From the scaling relation (3.1), we see that for materials of higher rigidity, the same amount of slip requires greater stress drop. As the near-trench regions of the forearc are understood to have relatively low rigidity, the magnitude of stress drop should be lower than predicted using the uniform half-space model. From equation 2.8, the strength of a fault (or the shear stress it can hold) increases with increasing normal stress. As normal stress increases with burial depth, we then expect that the shallow regions of the fault are much weaker than the deeper portions, and as a result must have less shear stress available to be released by rupture on the fault. However, equation 2.8 is a smooth representation of a complex situation. There can be small patches that are much stronger than described by this simple formula, due to factors such as locally lower pore fluid pressure. For a weak fault that can only resist shear stress that is a small fraction of the normal stress, we expect that the shallow regions of extremely large stress drop, such as 60 or 80 MPa, are unphysical; the calculated stress drop in those regions is likely higher than the pre-seismic shear stress on the fault. The unphysical stress drop shows the limitation of the inversion methods used to derive the rupture models. Because these regions would tend to over predict the stress drop in

80 general, we expect our average calculations (which are discussed below) to be higher 71 than the true value. Average calculations were performed, in the manner discussed in section 3.3.1, and results are included in the lower portion of Figure 4.7. Broadly speaking, the average stress drop of the Tohoku-oki earthquake agrees with previous results (seen in section 3.2.1) that the average stress drop was low. All of the predicted average stress drops are in the range of ~1-3 MPa for the region within the 5 m slip contour. Our average values for the high-slip regions (within the 30 m contour) are also generally fairly low, around ~10 MPa or less in most cases. Figure 4.7. Stress distributions and average stress drop values for several different rupture models. Identical to Figure 4.6, but considering different published rupture models. Results from the other 15 models tested are shown in Appendix B. How then, do we reconcile our results with the perception of complete stress drop? As discussed in section 3.2.2, several authors propose that both the occurrence of normal

81 72 faulting in the overriding plate and the lack of intraplate seismicity in the area of largest displacement indicate that nearly-complete stress drop occurred over a fairly large region of the fault (Hasegawa et al., 2011; Hasegawa et al., 2012, Yagi et al., 2011a; Hardebeck, 2012, Lin et al., 2013; Obana et al., 2013). Seismologically determined local stress drop in a small area is as high as ~40 MPa (Kumagai et al., 2012), likely representing complete stress drop or even dynamic overshoot in that area. In our results, we do see localized regions of high stress drop (likely representing complete stress relief), but they are surrounded by regions of moderate to low stress drop, or even stress increase. Our results would then suggest that the two differing predictions may be reconciled by considering the averaging nature of stress drop observations (discussed in section 3.2.1), and how the heterogeneous nature of stress change on the fault is not necessarily reflected in a single value. Predictions that dynamic weakening produces very large stress change and may result in complete stress drop are almost certainly valid, but our results show that dynamic weakening does not occur over a widespread region, even though a large area may undergo significant static slip. We have determined that the average stress drop remained low, despite some regions experiencing high or even complete stress drop. Hence, we can then determine which of the two cases introduced in section 3.2 is correct: whether the fault was very strong and required a large average stress drop in order to produce the observed reversal of stress, or if the fault could have been very weak and required only a small average stress drop. Because the average stress drop is low and produced a pronounced change in the state of stress in the forearc, we can conclude that the latter is true. The combination of the reversal of the state of stress and the small average stress drop provides a relatively

82 narrow constraint on the magnitude of the apparent strength of the fault, which we 73 investigate further in section FEM of Forearc Stresses After the Earthquake In order to quantify how the weakness of the megathrust and the stress drop in the Tohoku-oki earthquake gave rise to the reversal of stress in the overlying plate, we go back to the finite element modeling described in section We begin by modeling the pre-seismic state, where the forearc is characterized by horizontal strike-normal compression, as discussed in section 4.2. Pre-seismic stress-states are calculated for six different effective coefficients of friction which are held constant along the fault, and results are plotted in Figure 4.8. Some of the results were plotted earlier in section and are reproduced here for the reader s convenience. Effective coefficients of friction range from 0.025, which has already been determined to be too weak in section 2.2.2, to 0.032, which agrees with observations of pre-seismic stress states, to as high as 0.075, which shows strong deviatoric compression throughout the forearc. From our work, and very similar to the conclusion of Wang and Suyehiro (1999), we find that setting µ = is suitable to put most of the forearc into compression, with only a small region being in a horizontally neutral stress state (Fig. 4.8b). However, setting µ = produces a large area experiencing deviatoric tension, and thus we deem it too small to be appropriate for the state of stress in this region prior to the Tohoku-oki earthquake. From the pre-seismic case alone, friction coefficients of greater than cannot be differentiated, as we only have the predominantly margin-normal

83 compression state of stress as a constraining factor which can be produced by any µ 74 value greater than However, we can constrain the upper bound of µ using postseismic observations of forearc stresses and the stress drop on the fault. Figure 4.8. FEM results simulating the pre-seismic strength of the megathrust fault in NE Japan for various effective coefficients of friction µ. The different stress states will be the pre-conditions for post-seismic modeling runs plotted in figures 4.11, 4.13, and Blue crosses denote regions of horizontal deviatoric tension, while red crosses show regions of horizontal deviatoric compression. We simulate the change of stress in the forearc caused by the Tohoku-oki earthquake in a somewhat different manner from that of the pre-seismic case. We use any one of the

84 states of stress shown in Figure 4.8 as a pre-condition for the coseismic model. Elastic 75 strain energy is stored in this pre-stressed forearc. If the megathrust fault is weakened, the stored strain energy will be released in an earthquake, and the forearc stress will be relieved to some degree. In this static model, we do not model the resultant radiation of seismic waves but only the net stress changes. To simulate the earthquake, we assign a lower effective coefficient of friction (relative to the pre-seismic case) heterogeneously along the fault to represent coseismic weakening and hence stress drop. This results in the failure of the megathrust, with the amount of slip governed by the assigned fault stress drop. The deformation resulting from the coseismic slip is calculated for the model domain, and this also incurs an incremental change in stress. The stress-perturbation output of the earthquake run is then added back to the original pre-seismic stress state to result in the final prediction for the post-seismic state of stress in the forearc. By varying the pre-seismic strength and the magnitude of coseismic fault weakening (and hence stress drop), we determine how the state of stress in the forearc is affected by these two factors. Some model parameters for the earthquake models are different from the pre-seismic models, to more accurately reflect the coseismic behaviour of the real earth. First, the lower plate motion discussed in section is not needed. Second, the low-rigidity cushion that represents the fully relaxed mantle in the long-term is changed to be normal elastic mantle material to reflect the behavior of the mantle in the short term. We consider three different magnitudes of assigned fault weakening (Δµ ) (Figure 4.9) on six different pre-seismic constant strength faults on the state of stress in the forearc (Figures 4.8b - 4.8g). The three Δµ scenarios represent an intermediate case that incurs

85 an average stress drop of ~4-5 MPa, a smaller case that is simply 0.5 times the 76 intermediate case and results in an average stress drop of ~2-2.5 MPa, and a larger case that is 1.5 times the intermediate case and results in an average stress drop of ~6-7.5 MPa. Each Δµ case implies a nearly identical stress drop over the fault within model accuracy for any pre-seismic fault strength. Figures 4.10, 4.12, and 4.14 illustrate this result and the incremental stress change in the forearc caused by the stress drop on the fault. Figure 4.9. Three cases of Δµ along the fault. These values represent the coseismic weakening the fault experiences during rupture, and are tested on faults of varying strength. From the results of Hasegawa et al. (2012), shown in Figure 4.2, the width of the region that experienced a high degree of normal faulting varies along strike from approximately km, with the average being about 140 km. In our modeling, several different combinations of pre-seismic strengths and co-seismic stress drops result in forearc stresses that are consistent with these observations, ranging from the larger Δµ

86 77 cases on stronger faults, to lower Δµ cases on weaker faults. Figures 4.13a-b and 4.15a-c show states of stress that roughly agree with the observations of areas of normal faulting. None of the models in Figure 4.11 show deviatoric tension to 140 km, because the assumed stress drop of MPa is too small on faults of these strengths. Between the models shown in Figures 4.13 and 4.15, we prefer those in Figure The seismological results and our own dislocation modeling results from section 4.3 both predict that the average stress drop is low, and this suggests the stress drop of 4-5 MPa used to obtain the results in Figure 4.13a-b is a more logical choice. Our calculated slip is consistent with the slip seen in the published rupture models previously discussed. The peak slip of our model runs for the smallest Δµ, the intermediate Δµ, and the largest Δµ are 22 m, 42 m, and 60 m, respectively. While 60 m of slip is predicted by some models to occur over relatively small regions, 42 m is a better value to describe the average peak slip of the near-trench region in most rupture models.

87 Figure Stress drop along the fault, as well as stress perturbation for the small Δµ case (shown as blue line in Figure 4.9). Blue crosses denote regions of horizontal deviatoric tension, while red crosses show regions of horizontal deviatoric compression. 78

88 Figure Six different cases of absolute deviatoric stress in the subduction forearc immediately after the Tohoku-oki earthquake, for the small Δµ case. 79

89 Figure Stress drop along the fault, as well as stress perturbation for the intermediate Δµ case (black line in Figure 4.9). Blue crosses denote regions of horizontal deviatoric tension, while red crosses show regions of horizontal deviatoric compression. 80

90 Figure Six different cases of absolute deviatoric stress in the subduction forearc immediately after the Tohoku-oki earthquake, for the intermediate Δµ case. Blue crosses denote regions of horizontal deviatoric tension, while red crosses show regions of horizontal deviatoric compression. 81

91 Figure Stress drop along the fault, as well as stress perturbation for the large Δµ case (red line in Figure 4.9). Blue crosses denote regions of horizontal deviatoric tension, while red crosses show regions of horizontal deviatoric compression. 82

92 Figure Six different cases of absolute deviatoric stress in the subduction forearc immediately after the Tohoku-oki earthquake, for the large Δµ case. Blue crosses denote regions of horizontal deviatoric tension, while red crosses show regions of horizontal deviatoric compression. 83

93 84 One factor that affects all of the results presented above is the fact that coseismic stress change is truly a 3D problem, but we deal with only two dimensions in our finite element method, which assumes the rupture patch is infinitely long in the strike direction. The effect is such that our 2D results probably underestimate the stress change on the fault somewhat. But the effect should be small for the corridor where the largest slip occurred in the Tohoku-oki earthquake. Weighing the results of the different factors, and the slightly better agreement with the distribution of focal mechanisms seen in Figure 4.2, we determine a ~3-5 MPa average stress drop on a megathrust fault with a pre-seismic apparent strength of µ = (shown in Figure 4.13a) to be the result that best agrees with the observations of the state of stress before and after the 2011 Tohoku-oki earthquake.

94 85 Chapter 5. Conclusion In this dissertation, I constrained the average absolute strength of the megathrust fault in NE Japan by using the observations of stress change that occurred due to the 2011 Mw 9.0 Tohoku-oki earthquake. I also reconciled predictions that large coseismic weakening caused complete stress drop with observations of low average stress drop. Focal mechanisms of small earthquakes after the event showed that deviatoric stress had changed from margin-normal compression before the earthquake, to margin-normal tension afterwards. Such a reversal of stress requires the average stress change due to the earthquake to be a significant portion of the pre-seismic average shear stress on the fault, but not necessarily complete; by confirming that the average stress drop was small (as so far has always been the case for large megathrust earthquakes) through the use of published rupture models and our dislocation modeling method, I concluded that the preseismic fault must have been weak. In order to determine the degree of weakness, I employed finite-element methods to constrain the lowest effective coefficient of friction that would produce a forearc consistent with observations of pre-seismic margin-normal compression. I then used the observations of low average stress drop in conjunction with a co-seismic finite-element modeling method to determine which combination of factors (i.e. pre-seismic strength and co-seismic stress drop) would produce a forearc that showed deviatoric tension over the same region where extensional faulting was observed after the 2011 event. Although spatial variations in strength and stress drop are certain to occur (which can result in near-complete stress relief in isolated areas), by considering

95 86 only the average pre- and post-seismic cases, I was able to constrain the average absolute strength of the megathrust fault. Results from our 2D finite-element modeling discussed in Chapter 2 indicate that the state of stress in the continental forearc above a very weak megathrust fault is quite fragile. A realistic treatment of the important factors relating to the gravitational and plate coupling forces is critical to producing meaningful results. In our work, we showed that using the simplest topography (uniformly increasing water depth toward the trench) instead of a realistic one would cause an under-prediction, by roughly 33-50%, of the fault strength required to put the pre-seismic forearc into horizontal deviatoric compression. The extra topography of the realistic version enhances the effect of the gravitational force, requiring more compression from the plate coupling force to overcome it. The second factor we investigated was the presence or absence of the water column. By neglecting the weight of the water column (which has a very small compressive effect on the forearc), the minimum fault strength required to put the forearc into a compressive state doubled. These results indicate that for the forearc above a very weak megathrust fault, subtle changes in the balance of forces can cause large observable changes in the state of stress. The fragility of the stress state of forearcs above very weak megathrust faults is a new observation, not recognized by previous studies. Using both the realistic topography and including the presence of the water column, our finite element modeling showed that for NE Japan, an effective coefficient of friction of was sufficient to place the forearc into a horizontally compressive state, consistent with the results of Wang and Suyehiro (1999).

96 87 In Chapter 3, we investigated how the heterogeneous nature of stress drop on the fault arises from the heterogeneity of slip. While the scaling law given by equation (3.1) predicts the average stress drop of an event, it is important to understand that for real rupture on a real fault, stress drop will vary spatially, and will also increase for the regions neighboring the rupture patch (i.e. regions that did not experience slip). The character of rupture arrest will also affect the average value, as rupture that stops abruptly implies a higher degree of stress increase compared to rupture that arrests more smoothly. Furthermore, the complexity of rupture will have an additional effect: as initially distant rupture patches were moved closer together (but not overlapping), their effects upon the regions between them stacked together, resulting in larger stress increases calculated there. We recognize that this stress increase may not be highly relevant to seismic studies that try to predict ground motion, as no seismic energy is released from these areas, but studies of post-seismic deformation could be improved by considering these results. Our work discussed in Chapter 4 regarding the strength of the NE Japan megathrust fault conclusively shows that it must be weak; a small average stress drop (which is constrained both by our work here and by the independent observations of others) simply could not cause the forearc above a strong fault to undergo a reversal of the state of stress. However, the more detailed studies we performed indicate that the fault was actually very weak before the earthquake, with an average value for the effective coefficient being approximately 0.032, although definitely larger than The strength of the pre-seismic fault must also vary spatially, and our results indicate that any regions much stronger than µ > ~0.07 would require a large local stress drop in order to produce a large observable change in the local stress tensor large enough that detailed

97 observations of stress drop would identify the unusually large value in that region. 88 Related to our discussions in Chapter 3, we also found that even within regions that experienced large static slip (30 m), areas where the shear stress on the fault increased still exist. This is simply a result of the heterogeneity of fault slip, and the prime reason why although some regions of the fault experienced large (and likely complete ) stress drop during the earthquake, the average value of stress drop remains low. Complete stress drop is then understood to be able to occur in isolated regions, but does not occur over a large fraction of the fault. This is a new result which may help to reconcile conflicting predictions of complete stress drop based on laboratory experiments versus the general seismological observations of low average stress drops. It is also worth noting that our work considers areas of the fault that do not undergo significant rupture but which still experience stress increase. This is an effect that is not often considered in theoretical models of stress drop but certainly takes place in the real Earth. Future efforts that could be undertaken to continue this work could involve more detailed investigations into the variations of fault strength along strike. While in the finite element modeling in this work we pursue only a representative corridor, better alongstrike resolution of the absolute strength would be useful in further understanding the intricacies of the subduction process. A more detailed study may lead to greater improvement of the assessment of seismic hazard. If the character of the heterogeneity of fault strength was well understood, and the mechanisms that cause variation in strength (e.g. pore pressure, geometric incompatibilities, etc.) resolved, then predictions regarding the release of seismic energy and the tsunamigenic potential of a region could surely be improved.

98 The methods introduced in this work would also be well-suited towards making 89 predictions for whether large-scale stress change could occur within other subduction zones around the world. By modeling the topography and the regional state of stress, the magnitude of average stress drop that would be required to reverse the stress field could be constrained. In particular, if the same reversal of the state of stress that occurred in NE Japan was observed elsewhere, it could again be used to constrain the strength of the megathrust fault. This work also raises questions about (but does not deal with directly with) the possibilities of interactions between velocity-weakening and velocity-strengthening regions of the fault. Smaller regions showing velocity strengthening within larger patches of regions weakened by rate-and-state methods or by dynamic weakening obviously cannot completely resist being ruptured, but the stress on the fault will still increase for that region. If we understand this as a reorganization of the stress on the fault (i.e. the energy driving the stress increase is the release of energy from elsewhere on the fault), then it is easy to imagine that these smaller regions will be important to consider when investigating the dynamic evolution of slip on the fault, as they act as energy sinks. As we observe these regions even in areas that showed high static slip, then they would also be expected to affect the evolution of slip there. Better knowledge of the factors that control rate-and-state behaviors, as well as improved knowledge of the fault zone would thus be beneficial to studies of seismic hazard assessment for these reasons.

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107 98 Appendices A.1. Focal Mechanism Diagrams Focal mechanisms diagrams (or beachballs ) are graphical representations of earthquake moment tensors. The beach ball diagrams are a lower-hemisphere stereographic projection of the shortening (inward motion, represented by the white areas including the P axis) and lengthening (outward motion, represented by black areas including the T axis) deformation at the source region. In our work, we use focal mechanism diagrams to describe the stress tensor at a point, rather than the moment tensor. The elements of the moment tensor represent nine generalized couples (shown in Figure A.1) and describe the deformation of the source region during rupture. Any real moment tensor may contain signals of three kinds of deformation: isotropic, CLVD, and double-couple components. The isotropic and CLVD portions of the tensor are often removed to facilitate clearer focal mechanism diagrams.

108 99 Figure A.1. The nine couples represented in the moment tensor (left, from Aki and Richards, 1980), and the generalized matrix form of the moment tensor (bottom left). A.1. Stress Tensor In continuum mechanics, the stress tensor (or Cauchy stress tensor) is used to describe the state of stress at a point for a material body experiencing small deformations. Although there are an infinite number of planes that pass through any given point in a material body and each feels the force of a different stress vector, Cauchy s stress

109 100 theorem states that by knowing the stress vectors that act on three mutually orthogonal planes, we can calculate the stress on any other plane. Thus these three stress vectors can be grouped together and decomposed into one normal and two shear components to form a second-order tensor, which completely describes the state of stress at one point. Figure A.1 shows nine stress vectors acting on a small volume element, and the generalized form of the stress tensor. While the moment tensor Mij and stress tensor ij possess different physical meanings, they share some characteristics which will be useful for our purposes here. They are both symmetric second-order tensors. Thus, they both have nine components with six being independent. Figure A.1 shows the common graphical definition and notation for each tensor. The double-coupled component of the moment tensor corresponds to the pureshear component of the stress tensor, that is, the deviatoric state of stress where σ1 = -σ3 after other components are subtracted. With the beachball, we wish to show the directions of the three principal stresses, not their magnitudes, as if we were looking only at the pure-shear portion. A purely double couple beach ball diagram portrays four parameters: strike, rake, dip, plus magnitude (or scalar moment), or equivalently, the direction of the three principal axes (P, B, T) of the moment tensor, plus the magnitude. For the state of stress, the three directional parameters are the directions of the three principal stresses 1, 2, and 3. We are left to choose the fourth, a scalar measure of the size of the shear stresses represented by the diameter of the beach ball. The most commonly used scalar measures of the non-isotropic component of a stress tensor are its second and third invariants or the

110 differential stress (σ1 - σ3), which directly measures to the maximum shear stress as 101. Here we use the differential stress because it is the simplest of the three. The generic mapping toolset (GMT) creates the beachballs from the raw stress tensor Mij in the geographic frame of reference. We pipe the raw stress tensor ij in the geographic coordinate system into the generic mapping toolset (GMT), with the option enabled to draw the best-fitting double-couple solution. GMT will then remove what it considers the isotropic and CLVD components and plot only what it considers the double-couple component. The size of the plotted beach balls is determined by using the magnitude of the differential stress σ1 - σ3 in the place of the exponent for a moment tensor. The plotting command in GMT requires that the moment exponent be provided separately (e.g. for an earthquake with a moment of 5e25 dynes/cm, 25 should be provided). The beach balls are then plotted according to their moment magnitude, using the relations, where the argument of the log function will be the determinant of the moment tensor multiplied by ten to the exponent power. For our stress tensors, we take the logarithm of the differential stress and pipe it into GMT as the exponent of the stress tensor. Note that because of the constant offset of in the above relation, if the logarithm of the differential stress is less than six, negative magnitudes will result. We overcome this problem by adding a small constant after taking the logarithm, to ensure all magnitudes have the correct sign.

111 Figure A.2 shows some example plots off beach ball diagrams, coming from both 102 moment and stress tensors, and provides the generic mapping tools (GMT) commands that generate each diagram. gmt psmeca -R239/240/34/35.2 -Jm4c -Sm0.4 -h1 << END > test.ps e END gmt psmeca -R239/240/34/35.2 -Jm4c -Sm0.4 -h1 << END > test.ps e END gmt psmeca -R239/240/34/35.2 -Jm4c -Sm0.4 -h1 << END > test.ps e END gmt psmeca -R239/240/34/35.2 -Jm4c -Sm0.4 -h1 << END > test.ps e END Figure A.2. Four example beach ball diagrams, with generalized tensors (stress or moment tensor), and the plotting commands for use with GMT. From top to bottom: Isotropic compression (explosion), compensated linear vector dipole (tensile failure), thrust fault (double couple), strike-slip fault (double couple). Note that in GMT, the tensor elements must be entered in the following order mrr, mtt, mff, mrt, mrf, mtf, which is slightly different than the coordinate system used in the previous examples matrices.

112 B.1. Stress Change Distributions for Rupture Models from the 2011 Tohoku-oki Earthquake 103 As discussed in Chapter 4, many different rupture models were proposed for the 2011 Tohoku-oki earthquake. We discussed the stress drop distribution as calculated from the rupture models in section 4.3. Average results were discussed, and were taken from a wider variety of models than was shown in Chapter 4. We present the rest of our results here for review. Figure B.1. Stress distributions and average stress drop values for several different rupture models. Generated in the same manner as results shown in Figure 4.7 and 4.8, but using different rupture models as input.

113 104 Figure B.2 Stress distributions and average stress drop values for several different rupture models. Generated in the same manner as results shown in Figure 4.7 and 4.8, but using different rupture models as input.

114 105 Figure B.3. Stress distributions and average stress drop values for several different rupture models. Generated in the same manner as results shown in Figure 4.7 and 4.8, but using different rupture models as input.