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.
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
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 Tohoku-oki 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-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
1.2. Thesis Objectives and Significance ... 4
1.3. Thesis Organization ... 5
Chapter 2. The Strength of Megathrust Faults ... 7
2.1. Apparent Strength of Megathrust Faults ... 7
2.1.1. Definition and Importance of Megathrust Strength ... 7
2.1.2. Factors Affecting Apparent Strength ... 13
2.2. Stresses in the Forearc Controlled by Megathrust Strength... 15
2.2.1. Gravitational and Plate Coupling Forces ... 15
2.2.2. Modeling Stresses in the Forearc ... 17
2.3. Other Constraints on Megathrust Strength ... 24
Chapter 3. Stress Drop ... 26
3.1. Theoretical Background ... 26
3.1.1. Definition and Methods of Measurement ... 26
3.1.2. Fault Friction During Earthquakes... 28
3.2 Observed Stress Drop in Great Earthquakes ... 32
3.2.1. Average Stress Drops of Great Earthquakes ... 33
3.2.2. Complete versus Partial Stress Drop... 37
3.3. Dislocation Modeling of Stress Changes on a Fault in an Earthquake ... 40
3.3.1. Method ... 40
3.3.2. Stress Drop of a Single Rupture Patch ... 46
3.3.3. Dislocation Modeling of a Multi-Patch Rupture Field ... 52
Chapter 4. Implications of Forearc Stress Changes Caused by the 2011 Great Tohoku-oki Earthquake ... 56
4.1. The Tohoku-oki Earthquake ... 56
4.2. Stress States Before and After ... 59
4.3. Stress Drop Distributions from Rupture models ... 62
4.4. FEM of Forearc Stresses After the Earthquake ... 73
Chapter 5. Conclusion ... 85
References ... 90
Appendices ... 98
A.1. Focal Mechanism Diagrams... 98
A.1. Stress Tensor ... 99
B.1. Stress Change Distributions for Rupture Models from the 2011 Tohoku-oki Earthquake ... 103
List of Tables
Table 3.1. A small survey of observationally determined average stress drop values for recent large megathrust earthquakes. ... 37
importance of topography on the state of stress in the forearc ... 23 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 ... 24 Figure 3.1. A cartoon illustrating two cases of rate-and-state frictional behaviour ... 30 Figure 3.2. Friction coefficient versus normalized slip, illustrating the magnitude of dynamic weakening for various rock types in rotary shear experiments ... 32 Figure 3.3. Finite-element modeling stress profiles along the fault for three different fault strengths ... 40 Figure 3.4. Schematic of model setup in Okada coordinate system ... 42 Figure 3.5. Benchmarking dislocation code against Okada analytical solution. ... 45 Figure 3.6. Modeling results from Hu and Wang, (2008) comparing stress signals from different types of fault models ... 47 Figure 3.7. Dislocation modeling results showing the stress drop distribution of a single Gaussian-distributed rupture patch ... 51 Figure 3.8. Dislocation modeling results showing stress drop distribution as discrete rupture patches become more connected ... 53 Figure 3.9. Dislocation modeling results showing stress drop distribution as discrete rupture patches become more connected ... 55 Figure 4.1. Satellite image of Japan (left), with the main study area roughly outlined in red ... 58 Figure 4.2. Focal mechanisms of small earthquakes (left) before and (right) after the Tohoku-oki earthquake ... 61 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 ... 63 Figure 4.4. The vectors of stress change produced by our method ... 65 Figure 4.5. Stress perturbation at the ground surface calculated for the fault slip model of Shao et al. (2012) ... 67 Figure 4.6. Stress distributions and average stress drop values for one rupture model ... 69 Figure 4.7. Stress distributions and average stress drop values for several different
rupture models. ... 71 Figure 4.8. FEM results simulating the pre-seismic strength of the megathrust fault in NE Japan for various effective coefficients of friction µ ... 74 Figure 4.9. Three cases of Δµ along the fault ... 76 Figure 4.10. Stress drop along the fault, as well as stress perturbation for the small Δµ case ... 78
Figure 4.11. Six different cases of absolute deviatoric stress in the subduction forearc
immediately after the Tohoku-oki earthquake ... 79
Figure 4.12. Stress drop along the fault, as well as stress perturbation for the intermediate Δµ case ... 80
Figure 4.13. Six different cases of absolute deviatoric stress in the subduction forearc immediately after the Tohoku-oki earthquake, for the intermediate Δµ case. ... 81
Figure 4.14. Stress drop along the fault, as well as stress perturbation for the large Δµ case ... 82
Figure 4.15. Six different cases of absolute deviatoric stress in the subduction forearc immediately after the Tohoku-oki earthquake, for the large Δµ case ... 83
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
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.
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
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
information provides details on the orientation of the stress tensor. The magnitude, 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
between the two. We will structure our argument in a similar way.
1.2. 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
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.
1.3. 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
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.
Chapter 2. The Strength of Megathrust Faults
2.1. Apparent Strength of Megathrust Faults
2.1.1. 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, well- developed fault (such as a megathrust fault) is always a zone with some measure of
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
.
(2.4)
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.
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 causematerial 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.
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
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.
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 2.2.1 and employ simple finite-element-models to quantify the relation in section 2.2.2. 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).
2.1.2. 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
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
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 2.1.1 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.
2.2. Stresses in the Forearc Controlled by Megathrust Strength
2.2.1. 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
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
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.
2.2.2. 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
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.
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,
we assume the material is incompressible (Poisson’s ratio approaching 0.5) to avoid 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
highly-compliant (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.
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 ( = 0.03 – 0.07) 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
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.
compression, which agrees with results of a similar earlier work (Wang and Suyehiro, 1999) 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 (~ 40 – 100 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
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 = 0 – 0.4. 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.
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.
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.
2.3. 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
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.
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 0.90-0.98, or of 0.017 – 0.089, representing weak
megathrusts. Alternatively, the low 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.
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.
Apart from determining the slip distribution on the fault as a fraction of fault
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 high-frequency 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
The Coulomb friction law noted in section 2.1.1 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 higher-order 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-and-state friction according to (Marone, 1998)
, ,
(3.3)
where µis 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
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
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).
weakening depends on whether other local conditions allow the slip to acquire a 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. Velocity-strengthening 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
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
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).
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.
3.2.1. 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 3.3.4 will further investigate the heterogeneity of stress drop, and how it influences the overall
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
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
(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
(2009). Thus, while average stress drops are generally low for large earthquakes of 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 Mw = 9.2 ) 3.8 6 Seismic Waves Seismic waves Kanamori, (2006) Sorensen et al. (2007) Maule (2010 Mw = 8.8 ) 2.3 4 Seismic Waves Static Slip Lay et al. (2010) Luttrell et al. (2011) Tohoku-oki (2011 Mw = 9.0) 4.5-7 2-10 7 4.8 6 Static Slip Static Slip Static Slip Seismic Waves Static Slip Huang et al. (2013) Simons et al. (2011) Lee et al. (2011) Koketsu et al. (2011) Yagi et al. (2011a) Iquique
(2014 Mw = 8.1 )
2.5 Static Slip Lay et al. (2014)
3.2.2. 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.,
Table 3.1. A small survey of observationally determined average stress drop values for recent large megathrust earthquakes.
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
stress increase. Consider this same example in terms of slip; if a source region 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.
3.3. Dislocation Modeling of Stress Changes on a Fault in an Earthquake
3.3.1. 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
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.
thus dislocation modeling deals only with perturbations to the background and not the “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
We create a mesh of triangles of roughly equal size, with each triangle containing a point-source 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 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.
