Complex states of stress during the normal faulting seismic cycle: Role of midcrustal postseismic creep

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2010jb007557, 2010 Complex states of stress during the normal faulting seismic cycle: Role of midcrustal postseismic creep Jens Alexander Nüchter 1,2 and Susan Ellis 1 Received 15 March 2010; revised 2 September 2010; accepted 8 September 2010; published 14 December [1] Numerical models are used to analyze coseismic and postseismic change in the state of stress near the lower tip of a normal fault directly below the brittle ductile transition. The models are compared with information obtained from the geological record of exhumed metamorphic rocks regarding the magnitude and the geometry of coseismic stress increase close to the lower fault termination and the mechanisms involved during postseismic stress relaxation. The numerical results predict a drop in differential and fault parallel shear stress in the upper crust and a stress increase in the lower crust due to the tapering off in fault slip. The coseismic deformation field in the crust causes significant deflection of the principal stresses from the horizontal and the vertical. During the postseismic period, the recovery of the crustal stress field remains incomplete, even for long recurrence intervals. Instead, the stress field evolves in self similar cycles, controlled by repeated earthquake activity. The numerical models show that postseismic viscous creep localized beneath the fault contributes to the recovery of all stress tensor components in the upper crust close to the fault. This result implies that a detailed understanding of the processes and conditions active during postseismic stress relaxation in the middle and lower crust is essential in order to estimate reloading rates in the upper crust surrounding the fault. Citation: Nüchter, J. A., and S. Ellis (2010), Complex states of stress during the normal faulting seismic cycle: Role of midcrustal postseismic creep, J. Geophys. Res., 115,, doi: /2010jb Introduction [2] In seismically active regions, the crust and the mantle are subjected to stress cycles controlled by repeated earthquake activity. Determining spatial and temporal changes in stress field magnitude and geometry during the seismic cycle is essential in assessing seismic risk and the stress buildup on a particular fault. However, most estimates for the magnitude of stress disturbance during the seismic cycle have been based upon seismological observations, which cannot constrain stress changes from regions that do not radiate seismic energy (e.g., the parts of the fault that slide stably by afterslip). The objective of this paper is to address the critical information gap regarding stress changes near the lower tip of an active fault, using constraints provided by the geological record for rocks thought to have deformed near active faults in the midcrust. We use the rock record to constrain numerical models of cyclic crustal stress changes during normal fault slip. We show that our numerical models are consistent with the magnitude and geometry of midcrustal coseismic stress changes from the rock record, and plausibly match the typical range of seismological data 1 GNS Science, Lower Hutt, New Zealand. 2 Now at Institut für Geologie, Mineralogie und Geophysik, Ruhr Universität Bochum, Bochum, Germany. Copyright 2010 by the American Geophysical Union /10/2010JB for coseismic fault parallel shear stress drops. We demonstrate that the coseismic stress change is accompanied by a significant deflection in principal stresses close to the fault in the upper crust, and in a wide region below the brittle ductile transition (BDT). Near a fault, our results imply that stress orientations strongly depart from the uniform stress field geometry predicted from regional tectonic stresses [e.g., Anderson, 1905, 1951]. 2. Background: Theory and Observations Concerning Stress Near Faults [3] In the brittle upper crust, coseismic slip relieves elastic strain and fault parallel shear stress that has accumulated in the surrounding crust during the previous interseismic period [Reid, 1910]. Simultaneously, the dropoff in slip below the BDT causes significant loading of the middle and lower crust and the upper mantle. Stresses are likely to rotate due to coseismic fault parallel shear stress drop [Yin and Rogers, 1995]. During the subsequent interseismic period, the state of stress in the upper crust recovers until the fault parallel shear stress exceeds the fault strength and the next earthquake is triggered, while the stress peak below the BDT relaxes. Reloading of the fault requires the replacement of the released elastic strain and the backward rotation of the principal stress toward its pre earthquake orientation in the surrounding crust. Measurements of the coseismic fault parallel shear stress drop in the upper crust have been reported for numerous recent earthquakes [e.g., 1of15

2 Keilis Borok, 1959; Bouchon, 1997; Day et al., 1998; McGarr and Fletcher, 2002; Fletcher and McGarr, 2006]. [4] Similar information about coseismic stress changes below the BDT is not available for recent earthquakes, as inversion of postseismic surface deformation does not provide unique information about the magnitude, location, and geometry of deep seated stress cycles [e.g., Bürgmann and Dresen, 2008]. It has been demonstrated that postseismic viscous creep at depth due to relaxation of the coseismic stress peak below the BDT contributes significantly to reloading of the upper crust [Savage and Prescott, 1978; Thatcher, 1983; Li and Rice, 1987; Huc et al., 1998; Kenner, 2004; Lin and Freed, 2004; DiCaprio et al., 2008]. This suggests that detailed constraints on the geometry and magnitude of the coseismic loading below the brittle ductile transition (BDT) as well as the processes and conditions active during postseismic relaxation are critical for the understanding of the postseismic stress field recovery in the upper crust close to the fault. Unfortunately, seismology cannot provide such constraints since no seismic energy is radiated from the lowest tip of faults. Such information is available for the middle and lower crust, provided by the geological record of exhumed metamorphic rocks, for example from the Sesia Zone, European Alps [Küster and Stöckhert, 1999; Trepmann and Stöckhert, 2001, 2002, 2003], south of Evia island, Greece [Nüchter and Stöckhert, 2007, 2008], and on Rugsundøya Island, Norway [Birtel and Stöckhert, 2008]. 3. Constraints From the Geological Record on Midcrustal Stress Cycling Near Normal Faults [5] Nüchter and Stöckhert [2007, 2008] described low aspect ratio quartz veins crosscutting the high pressure lowtemperature metamorphic rocks of the Styra Ochi Unit exposed in the south of Evia island, Greece. The Styra Ochi Unit is exhumed in the footwall of a metamorphic core complex within the internal Cyclades belt [Gautier and Brun, 1994]. Here, the structural development has been governed by extensional tectonics since the Early Miocene [Le Pichon and Angelier, 1981; Jolivet and Faccenna, 2000]. The following features are important. First, crosscut relationships between the veins and all synmetamorphic structures in the host rock, and the prevalence of microfabrics indicating thermally activated crystal plastic deformation in the vein quartz, imply that the veins formed during exhumation, but still below the BDT. Second, the shape of the veins and the microfabrics related to sealing indicate that the veins originated as tensile fractures with a single increment of opening and syntaxial sealing of a fluidfilled cavity. Third, the low aspect ratios of the veins preclude purely elastic deformation of the host rock during the stage of cavity opening. Rather, the systematic decrease in the intensity of heterogeneous crystal plastic deformation of the vein quartz from the vein margins toward the centers is compatible with deformation of the host rock by an episode of viscous creep to low strain during the stage of cavity formation and coeval sealing [Nüchter and Stöckhert, 2007]. Fourth, a systematic increase in the density of primary fluid inclusions trapped in the vein quartz toward the vein centers reflects an increase of the fluid pressure by MPa in the residual cavity during the stage of sealing [Nüchter and Stöckhert, 2008]. Due to the limited fracture toughness of rock [Atkinson and Meredith, 1987], the increase in fluid pressure must have been counteracted by a simultaneous increase in the cavity normal stress s 3 by about the same magnitude. The increase in the fluid pressure is interpreted to reflect the relaxation of a major stress peak in the host rock during the stage of vein formation. [6] Based on these features, Nüchter and Stöckhert [2007, 2008] interpret these quartz veins to represent mesoscopic features of damage in the middle crust, resulting from stress cycles in the host rock related to seismic activity in the formerly overlying, now eroded upper crust. Tensile fracture propagation and the associated drop in the pore fluid pressure are related to rapid coseismic loading to high differential stress by a major drop in s 3. After fracture arrest, the residual stress relaxes and s 3 increases during an episode of accelerated postseismic viscous creep distributed in the host rock. Progressive shortening of the host rock parallel to the fracture and concomitant recovery of the fluid pressure lead to buckling of the fracture walls. The evolving cavities transform into veins by mineral precipitation from the instreaming pore fluid. [7] Although the position of the veins at their stage of formation with respect to the conjectured fault in the eroded upper crust is not exactly known, the structural position of the Styra Ochi Unit in the footwall of a metamorphic core complex [Gautier and Brun, 1994] implies that the veins were formed below the crustal scale BDT in the deep footwall of a normal fault. If so, the geological record of the Evia veins sets the following constraints for the numerical models: (1) coseismic loading and postseismic viscoelastic stress relaxation control time dependent stress field magnitudes and geometries in the middle crust close to the tip of a fault; (2) in the deep footwall of a normal fault, coseismic loading is the result of a major drop in s 3 ; and (3) locally, the magnitude of s 3 drop can be as high as MPa. During the postseismic period, s 3 increases gradually, and the coseismic stress relaxes with time. 4. Numerical Model Experiments 4.1. A Numerical Model Including an Embedded Fault [8] In the following, all symbols for parameters used in the paper and the stress convention are introduced in Table 1. To investigate the influence of fault movement on the stress state in the lithosphere, we use a numerical model including a combination of brittle, ductile, and elastic rheologies based on rock mechanics (section 4.2). We embed a fault, represented by a contact surface with a Coulomb friction criterion [Ellis et al., 2006]. Since our goal is to investigate stress cycling over the entire seismic cycle to match the constraints of the Evia veins, we keep our fault friction parameters deliberately simple; that is, we represent the effects of dynamic weakening of faults during an earthquake by reducing the friction coefficient along the fault m f for a small prescribed time interval. There is no need to include wave propagation or to distinguish between velocity weakening and velocity strengthening behavior in our model. This means that the results for coseismic fault slip during a model earthquake already include all afterslip as well. [9] Despite this simplification, the fault model is more realistic for interseismic timescales than rate state models studying fault rupture events because we include inelastic 2of15

3 Table 1. Symbol and Sign Convention Used in the Text Symbol Explanation Rheological Parameters E, n Young s modulus, Poisson s ratio C Cohesion, m, m c, m f Angle of friction, friction coefficient, coefficient of internal friction of the crust, friction coefficient of the fault A, Q, R, n Pre exponential constant, activation energy, gas constant, stress exponent G Geometric factor T Temperature _" Strain rate BDT Upper crust Middle crust Lower crust Crustal Sections and Boundaries Brittle ductile transition Surface to BDT BDT to termination depth of seismic slip Termination depth of seismic slip to bottom of crust ( Moho ) Stress and Displacement Field Parameters T (equation (4)) Stress tensor s xx, s yy, s xy = Cartesian stress components s yx (equation (4)) s 1, s 3 (equation (5)) Principal stresses (eigenvalues of T) s d (equation (7)) Differential stress t, t f (equation (8)) Shear stress, fault parallel shear stress Angle between s 1 and the fault normal direction P, P f Pressure, pore fluid pressure 8 (equation (6)) Angle between the principal stresses and the model coordinate system D (equation (10)) Prefix to indicate stress change CED Contour lines of equal displacement Stresses Rotation Fault parallel shear stress drop Sign Convention Compressive stresses denoted by positive values Angles of rotations in clockwise direction denoted by positive values Fault parallel shear stress drops are denoted by negative values a a We acknowledge that it is commonplace to denote coseismic fault parallel shear stress drops by positive values. We do not adopt this convention. rheologies in the surrounding crust, and we load the fault in a self consistent manner from far field tectonic boundaries and from creeping below the fault. That is, the model selfdetermines: (1) the depth of the brittle ductile transition as a function of ambient stress, strain rates, and prescribed temperature; (2) the depth to which the contact surface slips (which is generally less than the maximum extent of the contact surface); and (3) the manner of fault loading, which results from a combination of far field stress loading from the boundaries of the model and a readjustment of stresses near the fault caused by inelastic effects, including elevated creeping below the fault [e.g., Ellis et al., 2006]. Note that we allow postseismic viscoelastic stress readjustment in both crust (surrounding the fault) and in the mantle below Geometrical Setup and Mechanical Parameters of the Model Section [10] The numerical modeling is performed using the finite element software Abaqus 6.7. The modeled section is 300 km wide and consists of a 30 km thick crustal layer, underlain by a 70 km thick mantle section (Figure 1a). The crust and the mantle are considered homogeneous and isotropic. The initial temperature field is described by a linear geothermal gradient of 20 C km 1 in the crust, and of 5.5 C km 1 in the mantle (Figure 1a). We assume that the frictional yield strength in the brittle crust is described by the Mohr Coulomb criterion for optimally oriented slip: ¼ P P f sinðþþc cosðþ In Abaqus, we represent this frictional yield condition using a pressure dependent cohesion. As a result, extensional shear angles away from the imposed fault (for a uniform horizontal extension) are predicted to form at 45 to s 1, i.e., shallower than optimally oriented faults in a true, dilational frictional crust. [11] Thermally activated creep relieves stresses to below frictional yield at temperatures above about 300 C (depending on ambient strain rates). In the model, this transition is calculated to occur when predicted creep stress is less than or equal to frictional yield stress, at which point the model material passes from the brittle to the ductile regime [e.g., Fullsack, 1995]. In the ductile region, maximum shear stress is a function of temperature and strain rate: ¼ G _" 1=n Q= e ð nrt Þ ð2þ A For the description of creep in the crust, we use values derived from laboratory creep experiments on wet quartzite [Paterson and Luan, 1990]. For the mantle, we use the laboratory derived values for creep of wet olivine [Hirth and Kohlstedt, 2004]. The latter values were found to best explain deep seated postseismic surface deformation measured after the 1992 M7.3 Landers and 1999 M7.1 Hector Mine earthquake [Freed and Bürgmann, 2004]. The applied elastic and rheological parameters of the crust and the mantle are listed in Figure 1a. The geometric factor G is necessary for conversion from triaxial creep laboratory data to shear strain [Ranalli, 1987]: G ¼ 3 ð nþ1 Þ=2n 2 1 n ð1þ ð Þ=n ð3þ We assume a hydrostatic pore fluid pressure P f throughout the experiments, although we acknowledge that variations in P f have an important role in the seismic cycle [Nur and Booker, 1972]. On the other hand, coseismic permeability changes are expected to cause an immediate drop in pore fluid pressure [Miller and Nur, 2000] so that a nearhydrostatic pore fluid pressure may prevail for most of the postseismic period. For the middle and lower crust, this simplification is supported by (1) the geological record of microfabrics in metamorphic rocks exposed in the Sesia Zone, European Alps [Küster and Stöckhert, 1999], (2) the geological record of the Evia veins [Nüchter and Stöckhert, 2008], (3) fluid inclusions trapped in back shear arrays in the hanging wall of the Alpine Fault, New Zealand [Wightman and Little, 2007], and (4) the record of veins formed in the lower crust exposed on Rugsundøya Island, Norway [Birtel and Stöckhert, 2008]. Evidence for meteoric water trapped close to and below the BDT from rocks exposed on Tinos Island, Greece [Famin et al., 2004], and 3of15

4 Figure 1. (a) Two dimensional model setup. Boundary conditions are applied to represent extension. Varied extension rates (V x ) used in the experiments are listed in Table 2. The grey square represents the analyzed section of the crust. Inelastic material behavior is either frictional or power law creep with parameters corresponding to extrapolations from laboratory tests on wet quartzite [Paterson and Luan, 1990] in the crust and for wet olivine in the mantle [Hirth and Kohlstedt, 2004]. Left profile shows increase of temperature with depth. (b) Flowchart displaying the model stages of the experiments. Analysis of the steps in grey boxes is presented in this study. at Benmore Dam, New Zealand [DeRonde et al., 2001] imply the temporal existence of even subhydrostatic pore fluid pressure gradients, possibly related to rapid coseismic stress redistribution Experiments [12] The general procedure of the experiments is described in the flowchart in Figure 1b. The applied boundary conditions for the four experiments are outlined in Table 2. In the initial step, we allow the model section to reach geostatic equilibrium. Throughout the setup stages of all experiments, the model section is extended for 20 kyr by a bulk strain rate of s 1 to simulate the long term stress built up before the nucleation of the fault. During this setup stage, the right hand boundary is displaced at a constant rate of 9.5 cm/yr, while the left hand boundary remains fixed. In the next steps, 5 seismic cycles with constant recurrence intervals are simulated. Since no information about the recurrence interval is available from the record of the Evia veins, we choose a long recurrence interval of 1000 years [Yeats et al., 1997] to investigate the long term recovery of the crustal stress field during the postseismic periods. The fault is represented by a straight contact surface dipping at y = 60, reaching from the surface down to 20 km depth (Figure 1a). Along this model fault, earthquakes are triggered by a short term imposition (less than 1 day) of the fault frictional coefficient to m f < m c (Table 2), where m f and m c denote the coefficients of friction of the fault and of the internal friction of the crust, respectively. During the seismic steps, we apply a constant m f = 0.1 in experiments 1 and 2, m f =0.12inexperiment3,and m f = 0.2 in experiment 4 along the fault down to 10 km depth (Table 2). Below 10 km, m f increases linearly to 0.3 at 20 km depth in all experiments. A m f = 0.1 is used to represent a weak fault, consistent with inferences of low fault strength from heat flow observations along major faults [e.g., Lachenbruch and Sass, 1980]. The postseismic periods are simulated by setting m f > m c to lock the fault. This method 4of15

5 models static stress changes associated with fault slip, but we do not model dynamic rupture propagation. It is important to note that we do not prescribe the distribution of slip along the fault, but allow the contact algorithm to determine the slip as a function of depth. Finally, one additional seismic cycle is simulated and analyzed in detail (Figure 1b). In experiment 1, the boundary conditions applied during the previous steps remain unchanged. Experiment 2 was designed to simulate the effect of a reduced extension rate on the restoration of the stress field geometry during the postseismic period. This is achieved by the reduction of the extension rate applied during the sixth postseismic period to 3 cm/yr (Table 2). The purpose of experiments 3 and 4 is to deduce the effects of slightly (experiment 3) and significantly (experiment 4) reduced fault slip on the model results. This is achieved by the application of higher values for m f during the seismic steps (Table 2). 5. Results 5.1. Determination of Stresses and Stress Changes [13] For stress analysis the components of the planar stress tensor, T, T ¼ xx xy ð4þ yx yy are extracted for each element in the crustal section (Figure 1). s xx and s yy represent the stresses in x and y direction of the model coordinate system and s xy = s yx represent shear stresses. The magnitude of the maximum and minimum compressive principal stresses s 1 and s 3 and their inclination with respect to the model coordinate system, 8, are calculated using equations (5) and (6) [Jaeger et al., 2007]: ( 1;3 ¼ xx þ yy 2 ) 1=2 xx yy þ 2 xy ð5þ ¼ xy tan 1 xx yy Equations (7) and (8) determine the differential stress s d and the fault parallel shear stress t f : d ¼ 1 3 f ¼ ð6þ ð7þ sinð2þ ð8þ where denotes the angle between the fault normal direction and s 1. [14] The change in the state of stress resulting from incremental deformation at a single point is regarded as the result of the superposition of a preexisting stress tensor T P by a stress tensor DT that describes the change in stress. Therefore, the stress tensor T R prevailing after the deformation increment can be described as 0 P T R ¼ T P xx þ D xx P xy þ D 1 xy þ DT A ð9þ P yx þ D yx P yy þ D yy [15] For the analysis of the changes in the magnitude of the principal stresses, the differential stress and the faultparallel shear stress, we solve equations (5) (8) individually for T R and T P. The changes in s 1, s 3, s d, and t f are indicated by the prefix D and calculated using equation (10): D i ¼ R i P i ð10þ 5.2. Initial Stress Field Geometry Resulting From the Locked Fault Stage (No Earthquakes) [16] The uniform far field extension applied during the 20 kyr locked fault setup stage (Figure 1b) results in continuous horizontal extension and vertical thinning of the model crust. Since the strain axes at this stage are controlled by the geometry of far field extension, s xy remains 0, and s yy and s xx represent the maximum and minimum principal stresses s 1 and s 3 throughout the crust (Figure 2b). Initially, the imposed elastic strain causes an increase in differential stress throughout the crust. When the yield strength of the crust is reached or ductile creep becomes significant, inelastic deformation cause the stresses to reach a steady state. At the end of the locked fault stage, the differential stress increases linearly with depth to 140 MPa at 12 km depth (Figure 2a). Below 12 km, thermally activated viscous creep causes the magnitude of differential stress to decay nonlinearly with depth. The distribution of the state of stress attained at this stage is in agreement with predictions for the strength envelope of the crust, obtained from laboratory experiments [Brace and Kohlstedt, 1980; Kirby, 1980], observations in nature [Sibson, 1982; Meissner and Strehlau, 1982], and in situ stress measurements in deep bore holes [Townend and Zoback, 2000]. For his seminal theory of formation of new faults, Anderson [1905, 1951] assumed this type of stress orientation in the crust Stress Redistribution During the Seismic Cycle [17] Coseismic fault slip and elastic rebound in either fault wall impose a displacement field on the crust (Figures 3a and 3b). Gradients in the displacement field reflect deformation resulting from the coseismic stress field perturbation in the crust. During the postseismic interval, the interaction of far field extension and postseismic creep below the BDT controls the restoration of the crustal stress field. In all experiments, we observe a gradual spin up of the stress field during the first seismic cycles. After the fourth earthquake step, the stress field evolves in self similar cycles, by which we mean that a given state of stress is similar when compared at the same time after each seismic event. Similar spin ups to self similar stress cycles have been observed by Ellis and Stöckhert [2004a, 2004b], Ellis et al. [2006], Hetland and Hager [2006], and DiCaprio et al. [2008]. In the following, we analyze the stress evolution during the sixth seismic cycle. [18] During the seismic step, the results for experiments 1 and 2 are identical as the same boundary conditions are applied (Table 2). The results for experiments 3 and 4 show differences in the magnitudes of stress change, but are comparable to the results of experiments 1 and 2 regarding the geometry of stress change. Hence, in sections and 5.3.2, we analyze the coseismic stress perturbation in experiments 1 and 2 in detail, and compare these to the stress perturbation in experiments 3 and 4. Then, in section 5.3.3, 5of15

6 Table 2. Applied Elastic and Rheological Parameters in the Numerical Experiments Experiment Young s Modulus of the Crust (GPa) Coseismic Fault Friction m f a Setup Extension Rate b (cm/yr), (s 1 ) Analysis Extension Rate c (cm/yr), (s 1 ) , , , , , , , , a Applied for the upper 10 km. Below 10 km, m f increases to 0.3 at 20 km depth. b Extension rate applied throughout the setup stage and the first five seismic cycles. c Extension rate during the sixth seismic cycle. we analyze the mechanisms that control the recovery of the stress field during the postseismic period Coseismic Displacement Field [19] Coseismic displacement is highest in the upper crust along the fault (Figure 3). Fault slip tapers off below the BDT over a depth range dictated by the dropoff in stress (Figure 3a) [Ellis et al., 2006]. In experiments 3 and 4, fault slip is reduced compared to the experiments 1 and 2 as a result of the higher coefficients of friction (Figure 3a). In the upper crust close to the fault, coseismic deformation is indicated by the lateral decrease in near fault parallel displacement. Here, the low angles between the displacement vectors and the contour lines of equal displacement (CED) render a steep simple shear component the dominant deformation mechanism (Figure 3b). Most intense deformation is located in a narrow region at the lower tip of the fault, associated with the fault slip taper below the BDT (Figure 3b). Here, the high angles between the displacement vectors and the CED render the contribution of pure shear deformation important (Figure 3b). Because of the opposite directions of displacement, the deep hanging wall and the footwall are subjected to fault parallel compression and extension, respectively (Figure 3b) Coseismic Changes in the Magnitude and the Geometry of the Crustal Stress Field [20] Figure 4 shows the coseismic changes in the crustal stress field, described by Ds d (Figure 4a), Dt f (Figure 4b), Ds 1 (Figure 4c), Ds 3 (Figure 4d), and the orientation of s 1 with respect to the vertical (Figure 4e). The coseismic deformation field causes a decrease in both differential stress s d and fault parallel shear t f stress in the upper crust, whereas both increase in the middle and lower crust (Figures 4a, 4b, and 5). Differential stress relief is the result of an increase in s 3 (Figure 4d). At depth, coseismic loading is predominantly caused by a major increase in s 1 in the deep hanging wall (Figure 4c), and a major decrease in s 3 in the deep footwall (Figure 4d). The increase in the strain rates associated with coseismic loading cause a downward deflection of the BDT (Figure 5) [e.g., Rolandone et al., 2004; Ellis and Stöckhert, 2004a]. The salient results for the magnitudes of stress changes in all experiments are listed in Table 3. Coseismic stress redistribution is accompanied by a deflection in principal stresses throughout the crust (Figure 4e) due to oblique deformation (Figure 3b). In the upper crust, stress deflection is restricted to a zone of some kilometers width around the fault (Figure 4e). In the middle and lower crust where preexisting stresses are low due to viscous creep, stress deflection is significantly more pronounced (Figure 4e). Due to coseismic principal stress deflection, the relationship between Dt f and Ds d is nonlinear (Figures 4a and 4b). In particular, a depth interval exists in which t f decreases while s d increases (Figures 4a and 4b) Postseismic Recovery of the Crustal Stress Field [21] During the postseismic period, all stress tensor components and the differential stress peak relax below the BDT during an episode of viscous postseismic creep at initially high, but decreasing strain rates (Figure 5). Due to the Figure 2. Initial state of stress prevailing at the end of the 20 ka fault locked stage. (a) The profile shows a linear increase of differential stress down to 12 km depth. Below 12 km, thermally activated creep relieves stress below the frictional yield. (b) Orientation distribution of the maximum principal stress s 1 with respect to the vertical. Shading shows the magnitude of the inclination of the principal stresses. Black lines are parallel to s 1. Scatter in stress orientations in the lower crust is numerical. The analyzed crustal section is indicated by the grey box in Figure 1a. 6of15

7 Figure 3. Coseismic displacement during the sixth earthquake step (Figure 1b). (a) Slip along the contact surface as a function of depth for experiments 1 4. (b) Magnitude and orientation of coseismic off fault displacement in experiments 1 and 2. Shading indicates the magnitude of displacement. Arrows point toward the direction of displacement (the length of the arrows does not indicate the magnitude of displacement). Bold dashed grey line indicates the extent of the contact surface that does not slip. The analyzed crustal section is indicated by the grey box in Figure 1a. distribution of coseismic differential stress increase (Figure 4a), postseismic creep is localized in a downward broadening shear zone below the fault. The BDT gradually migrates toward shallower depth levels due to the decay in the creep strain rates [Rolandone et al., 2004] (Figure 5). At the same time, the upper crust containing the fault is gradually reloaded by far field extension, and by stress transfer due to postseismic creep below the BDT [Savage and Prescott, 1978; Thatcher, 1983; Li and Rice, 1987; Huc et al., 1998; Kenner, 2004; Lin and Freed, 2004; DiCaprio et al., 2008]. Postseismic stress field recovery is accompanied by a progressive rotation of the principal stresses toward their pre earthquake orientation (Figure 6). The rotation rate decreases with time and considerable deflection of the principal stresses away from the vertical and the horizontal persists at the end of the simulated postseismic periods (Figure 6). [22] Postseismic stress transfer from the middle and lower crust and the mantle into the upper crust results from deformation within a complex displacement field generated by postseismic creep (Figure 7). When postseismic creep is active in the crust below the BDT and the mantle, significant gradients in the magnitudes and the orientations of the displacement fields in the footwall and the hanging wall focus simple shear deformation in the crust near the fault (Figure 7a). The sense of shear resembles the sense of fault slip. When viscous creep is suppressed in the middle and lower crust, and postseismic creep is exclusively located in the mantle, significant gradients in displacement are absent (Figure 7b). In this case, postseismic stress transfer into the upper crust close to the fault is less effective. The comparison implies that localized postseismic creep in the crust below the BDT focuses stress transfer to the upper crust close to the fault. [23] To estimate the contribution of far field extension and of postseismic stress transfer to the recovery of the stress field in the upper crust close to the fault, we compare the results for stress recovery obtained from experiments 1 and 2. Both experiments have the same stress evolution until the sixth seismic step (Table 2). Hence, any difference in the stress field evolutions between the experiments during the sixth postseismic interval is attributed to the reduced far field extension rate in experiment 2 (Table 2). Comparison between experiments 1 and 2 shows that a reduction in the far field extension rate causes (1) a decrease in the efficiency of principal stress reorientation (Figures 6a and 6b) and (2) a reduction in the reloading rate in the upper crust (Figures 5a and 5b). The contrast between experiments 1 and 2 is based on differences in the recovery rates of the stress tensor components, as shown in Figure 8. Throughout the crust, the recovery of s xx shows the highest sensitivity to the far field extension rate due to the related deformation geometry. Differences in the recovery rate of s yy (Figure 8b) and s xy (Figure 8c) are restricted to the upper part of the crust and only become evident 200 years after the seismic step. The differences in s yy and s xy (Figures 8b and 8c) do not reflect a dependence on the far field extension rate but are caused by bumps in the stress profile related to inelastic deformation close to the coseismic BDT where the crust is at frictional yield (Figure 5). For all stress tensor components, the differences between the experiments is less than the factor 3 difference in the far field extension rate (Table 2). This demonstrates that the importance of the contribution of stress transfer due to postseismic creep is comparable to the contribution of far field extension. Far field extension is significant only for the recovery of s xx, while postseismic stress transfer is the predominant factor that controls the relief of s xy. [24] According to equation (6) the rotation rate of the principal stresses depends on the recovery rate of s xy, and the changes in the difference between s xx and s yy. The 7of15

8 Figure 4. Distribution of the coseismic stress field perturbation. Changes in (a) differential stress, Ds d, (b) fault parallel shear stress, Dt f, (c) maximum principal stress, Ds 1, and (d) minimum principal stress, Ds 3. (e) Deflection of the maximum principal stress, Ds 1, from the vertical. Short black lines indicate the orientation of Ds 1. Bold dashed grey line indicates the extent of the contact surface that does not slip. The analyzed crustal section is indicated by the grey box in Figure 1a. following reasons cause the decrease in the principal stress reorientation rate with time (Figure 6): (1) the contribution of postseismic stress transfer to the recovery of the Cartesian stress components fades with time due to stress relaxation and the decrease in the postseismic creep rate; and (2) in the upper crust, the gradual increase and decrease in s xx (Figure 6a) and s yy (Figure 6b), respectively, cause 1/(s xx s yy ) (equation (6)) to converge to zero. Due to both effects, less principal stress rotation is predicted for deformation increments late in the postseismic period compared to early deformation increments. This allows a complex stress field to persist long into the interseismic period (Figure 6). 6. Discussion [25] In the models, the crustal stress field spins up during the first seismic cycles, and then evolves in self similar cycles, as reported for previous numerical experiments [Ellis and Stöckhert, 2004a, 2004b; Ellis et al., 2006; Hetland and Hager, 2006; DiCaprio et al., 2008]. During the earthquake step, stress redistribution is associated with a major deflection of the principal stresses throughout the crust. Although the model fault is locked during the postseismic period, the recovery of the stress field geometry remains incomplete, and complex states of stress prevail in the crust throughout the seismic cycle (Figure 6). The stress field recovery significantly depends on stress transfer by postseismic creep within a shear zone localized directly below the lower fault tip, while the contribution of stress transfer from the mantle is low. The effects of localized postseismic creep in crustal shear zones and the implications for stress transfer were also explored in previous studies by Montesi [2004] and Savage et al. [2007]. [26] Assuming that the numerical results are comparable to natural processes, understanding the mechanisms and conditions that control postseismic stress relaxation directly below the lower fault tip is crucial for determining on the postseismic reloading rates of faults. In order to judge 8of15

9 Figure 5. Profiles of differential stress, s d, along the fault in the hanging wall immediately after the earthquake (black lines) and in 100 year steps during the postseismic period. Differential stress is calculated for the average of the fault elements, i.e., represents the stress at approximately 150 m distance to the fault. Results for (a) experiment 1, (b) experiment 2, (c) experiment 3, and (d) experiment 4. whether the numerical results are a good representation of natural processes, we compare the numerical predictions for coseismic loading at the lower fault tip to the constraints obtained from the geological record of the Evia veins [Nüchter and Stöckhert, 2007, 2008] (section 3). Then, we compare the associated results for fault parallel shear stress drops and average fault slips to published data reported for recent earthquakes to determine if the model results for the coseismic displacement and deformation in the upper crust are within a relevant range for natural earthquakes. Throughout the discussion, we caution that the comparison of the numerical results to data obtained from seismological and geological sources must be general, because of the simplifications in rheology and fault mechanics used in modeling outlined at the start of section 4. [27] 1. The power law rheology used to simulate creep below the BDT (equation (2)) is calibrated for steady state deformation in the regime of dislocation creep [Paterson and Luan, 1990; Hirth and Kohlstedt, 2004]. Although frequently used, the relevance of such creep laws for describing fast and transient postseismic creep is questionable. The microfabric evolution in exhumed metamorphic rocks indicates a shift in the deformation regime from initially low temperature plasticity toward dislocation creep, followed by static grain growth when stresses are relaxed to a large extent [Trepmann and Stöckhert, 2003]. [28] 2. The applied rheology inherently prescribes a weak middle and lower crust between a strong upper crust and a strong upper mantle, in accordance with the classical jelly sandwich model of lithospheric strength distribution [Brace Table 3. Salient Results of the Numerical Models and Information Obtained From the Geological Record a Experiment Average Fault Slip (m) Maximum Decrease in s d (MPa) Stress Decrease in Upper Crust Average Decrease in s d (MPa) Maximum Drop in t f (MPa) Average Drop in t f (MPa) Maximum Increase in s d (MPa) Stress Increase in Lower Crust Maximum Increase in t f (MPa) Maximum Increase in s 1 (MPa) Maximum Decrease in s 3 (MPa) 1, b 17 b 33 c 17 c 86 b 52 c 101 b 99 b b 16 b 28 c 15 c 77 b 47 c 77 b 83 b b 9 b 11 c 7 c 41 b 25 c 37 b 39 b a The parameters used in the experiments are listed in Table 2. Maximum decrease in s 3 obtained from the Evia veins is 80 to 120 MPa, depending on the vein and the assumed trapping temperature of the fluid inclusions. b Average values for the fault elements, i.e., within 150 m distance of the fault. c Measured directly along the fault. 9of15

10 Figure 6. Residual deflection of the maximum principal stress, s 1, from the vertical 100 and 1000 years after the sixth earthquake. Short black lines indicate the orientation of s 1. Results for (a) experiment 1, (b) experiment 2, (c) experiment 3, and (d) experiment 4. Bold dashed grey lines indicate the extent of the contact surface that does not slip. The analyzed crustal section is indicated by the grey box in Figure 1a. 10 of 15

11 Figure 7. Displacement accumulated during the first year after the earthquake exclusively generated by postseismic creep. (a) Postseismic creep is located in the middle and lower crust and in the mantle section. (b) Postseismic creep is exclusively located in the mantle section (viscous creep in the crust suppressed by decreasing the temperature here to T = 0 C). Bold dashed line indicates the extent of the contact surface that does not slip. The analyzed crustal section is indicated by the grey box in Figure 1a. and Kohlstedt, 1980; Goetze and Evans, 1979]. This assumption is justified by evidence from the geological record for significant postseismic viscous creep in the middle and lower crust. [29] 3. The postseismic stress evolution also depends on the strength of the fault. In the models, we assume infinite fault strength during the postseismic period, although we acknowledge that natural fault strength should be below the strength of the surrounding crust. [30] 4. There is no discrimination between velocity strengthening and velocity weakening behavior of the model fault. This means that fault slip (Figure 3a) already includes all possible afterslip. [31] 5. The model is 2 D, while faults in nature reveal large variations along strike. [32] Despite these caveats, the numerical experiments (Figure 4d) and the geological record of the Evia veins [Nüchter and Stöckhert, 2008] (section 3) consistently predict coseismic loading in the deep footwall of a normal fault by a major decrease in minimum compressive stress, s 3. Furthermore, the magnitude of the changes in s 3 during seismic cycle predicted by the experiments 1 3 matches the magnitude range obtained from the Evia veins [Nüchter and Stöckhert, 2008] (section 3, Table 3, and Figure 4d). The numerical results indicate that, for the assumed rheology and temperature profile (Figure 1a), stress relaxation by postseismic creep occurs on timescales of 10 2 to 10 3 years, i.e., on timescales similar to recurrence intervals in nature [e.g., Yeats et al., 1997]. Although the timescale of stress relaxation during the formation of the Evia veins is not constrained, the viscous deformation of the host rock during vein formation evident from the vein shapes and the systematic gradients in the intensity of crystal plastic deformation in the vein quartz [Nüchter and Stöckhert, 2007] (section 3) suggests that the numerical predictions for the timescale of stress relaxation are in rough agreement with the record of the Evia veins. [33] The average fault parallel shear stress drops in the experiments are between 7 and 17 MPa (Table 3), consistent with the range for natural fault parallel shear stress drops of between 0.03 to 30 MPa suggested by a compilation of seismological data from 390 earthquakes including data from earthquakes in southern California/USA and along the Tonga Kermadec Island arc [Hanks, 1977]. Data from selected recent normal fault earthquakes (Table 4) predicts a similar range for average fault parallel shear stress drops between about 2.2 MPa (1983, M6.9 Borah Peak earthquake [Barrientos et al., 1987]) and 18 MPa (1959, M7.3 Hebgen Lake and 1972, M6.0 Aleutian Arc earthquake [Barrientos et al., 1987; Mori, 1983]). Fletcher and McGarr [2006] demonstrated for the 1992, M6.9 Landers strike slip and the 1994, M6.7 Northridge reverse fault earthquakes that the magnitude of local fault parallel shear stress drops tends to increase with depth and is a maximum close to the hypocenters. A comparable increase in the faultparallel shear stress drops is observed in the experiments (Figure 4b). For experiments 1 3, the average fault slip range ( m, Table 3) agrees with m of average fault slip during the 1959 Mw7.3, Hebgen Lake normal fault earthquake [Barrientos et al., 1987]. [34] This comparison supports the conclusion that the numerical results are consistent with the mechanisms that control the stress changes during the seismic cycle in major normal fault earthquakes. This means that the numerical models and the geological record of the Evia veins consistently predict coseismic loading of the middle and lower crust to high differential stress followed by postseismic relaxation during episodes of viscous creep localized to a narrow region close to the lower tip of the fault [Ellis and Stöckhert, 2004b]. The results suggest that the geological record of the Evia veins reflects coseismic loading of the middle crust during a major normal fault earthquake in the geological past, which possibly was comparable or somewhat stronger than the Hebgen Lake earthquake. [35] Intense coseismic loading followed by viscoelastic stress relaxation is predicted in a depth interval for which geodetic inversion models frequently imply afterslip as the 11 of 15

12 main source for postseismic surface deformation [e.g., Savage and Svarc, 1997; Hearn et al., 2002; Hsu et al., 2002, 2006]. However, it has been shown that, with the exception of subduction interface earthquakes, the importance of afterslip is restricted to the early phase of the postseismic interval, while viscoelastic relaxation is important on significantly longer timescales [e.g., Wang, 2007]. For instance, Hearn et al. [2009] demonstrated for the 1999 Izmit and Düzce earthquakes a rapid decrease in the contribution of afterslip to the near fault surface deformation during the first month, while viscoelastic relaxation became increasingly important during the following years. Data comprising decades of repeated geodetic surveys has been used to demonstrate that on decadal and longer timescales, viscoelastic processes in the lower crust and the mantle are predominant [Nishimura and Thatcher, 2003; Kenner and Segall, 2003]. Therefore, we conclude that the prediction of postseismic creep located close to the lower fault tip is not in contradiction with recent interpretations of early postseismic surface deformation, but represent postseismic processes at depth on longer timescales. [36] Deflections in principal stresses have been observed during recent earthquakes, including the 1983 Coalinga earthquake [Michael, 1987], the 1986 Oceanside earthquake [Hauksson and Jones, 1988], the 1989 Loma Prieta earthquake [Michael et al., 1990], and the 1992 Joshua Tree and Landers earthquakes [Hauksson, 1994; King et al., 1994; Hardebeck and Hauksson, 2001]. Coseismic deflection of the principal stresses means that the fault is no longer in a favorable orientation with respect to the stress field after an earthquake in terms of the Coulomb failure criterion [Coulomb, 1773]. Incomplete postseismic recovery of the principal stress orientation implies that faults with orientations adjusted to the general trend of the deflected stress field would generally experience higher Coulomb stresses throughout the seismic cycle than faults with dip angles predicted for new formed faults by Anderson s theory of faulting [Anderson, 1905, 1951]. In the case of normal faulting, the results of the numerical experiments (Figures 4e and 6) suggest that normal faults dipping at a lower angle would be in a more favorable orientation with respect to the general trend of the stress field. The importance of lower normal fault dips becomes evident from a histogram of global active normal fault dips derived from focal mechanisms of earthquakes with a magnitude >5.5, compiled by Jackson and White [1989] and expanded by Collettini and Sibson [2001]. This histogram shows a significant maximum at dip angles y = 45 ± 2.5, which is 15 lower than the fault dip of y 60 predicted for newly formed normal faults. 7. Summary and Conclusions [37] We have simulated the stress field evolution during repeated normal fault earthquake cycles and compared the numerical predictions to constraints obtained from the Figure 8. Comparison of the changes in the stress tensor components in experiment 1 (grey shaded zones) and experiment 2 (black lines) along the contact surface during the first 500 years after the sixth seismic step. (a) Changes in s xx, (b) changes in s yy, and (c) changes in s xy. 12 of 15

13 Table 4. Compilation of Maximum and Average Fault Parallel Shear Stress Drops of Some Recent Earthquakes Earthquake 1959, M7.3 Hebgen Lake (normal faulting) 1969, M5.8 Aleutian Arc (normal faulting) 1972, M6.0 Aleutian Arc (normal faulting) 1983, M6.9 Borah Peak (normal faulting) 1990, M7.5 Mariana (normal faulting) 1992, M6.9 Landers (strike slip) 1994, M6.7 Northridge (reverse slip) Maximum Drop in t f (MPa) Reference Average Drop in t f (MPa) Reference / / 11 to 18 Barrientos et al. [1987] / / 5 to 10 Mori [1983] / / 4 to 18 Mori [1983] / / 2.2 Barrientos et al. [1987] 150 Yoshida et al. [1992] 3.2 Yoshida et al. [1992] 16 to 18 Bouchon [1997] 30 Day et al. [1998] 11 Fletcher and McGarr [2006] 60 Fletcher and McGarr [2006] 25 Bouchon [1997] 30 Day et al. [1998] 17 Fletcher and McGarr [2006] 40 Fletcher and McGarr [2006] geological record of veins crosscutting the high pressurelow temperature metamorphic rocks of the Styra Ochi Unit in south Evia, Greece [Nüchter and Stöckhert, 2007, 2008]. The models were analyzed to estimate the contribution of postseismic stress relaxation processes active close to the lower tip of the fault to the stress field recovery in the upper crust during the interseismic period. The numerical results are as follows. [38] 1. The geometry of coseismic stress redistribution depends on the position relative to the fault: In the upper crust, a differential stress drop results from an increase in the minimum principal stress s 3. Coseismic loading in the deep hanging wall and in the footwall mainly results from an increase in s 1 and a decrease in s 3, respectively. Coseismic stress redistribution is accompanied by a deflection of the principal stresses due to the principally oblique deformation of the crust related to fault slip. [39] 2. Coseismic loading in the deep footwall is due to a drop in s 3, and the magnitude of this drop is consistent with the record of the Evia veins. The results for the associated average fault slips and average fault parallel shear stress drops are within a range observed during natural earthquakes. [40] 3. During the postseismic period, all stress tensor components recover toward their pre earthquake magnitude. This causes a gradual restoration of the magnitude and the orientation of the principal stresses, and a related progressive increase in fault parallel shear stress. The recovery of the stress field geometry remains incomplete in all experiments. [41] 4. The distribution of coseismic loading in the middle and lower crust causes localization of postseismic creep in a downward broadening shear zone below the fault. [42] 5. The contribution of far field tectonic extension to the stress field recovery is significant only for s xx, while stress transfer by localized postseismic creep into the upper crust contributes to the recovery of all stress tensor components. In this context, the results of the numerical models imply that localized postseismic creep close to the lower fault tip, i.e., at the location best constrained by the record of the Evia veins, focusses stress transfer to the upper crust close to the fault. [43] Based on the present numerical results, we conclude that complex and nonsteady state stress field geometries prevail in the crust in seismically active regions. The existence of uniform stress field geometries near active faults seems unlikely at any time during the seismic cycle. The coseismic perturbation of the stress field is controlled by the geometry of the fault, the distribution of slip, and the elastic parameters of the surrounding crust. [44] The results of the present study imply that the understanding of the processes active during postseismic stress relaxation at and directly below the fault tip are essential for the estimation of the reloading rates of the upper crust close to the fault. In this context, the consistency between field studies and the numerical results highlights the importance of investigations of exhumed metamorphic rocks, which provide insight into processes and conditions active at depth during the seismic cycle, which are not accessible by geophysical techniques. [45] Acknowledgments. The present study is funded by Deutsche Forschungsgemeinschaft (DFG) contributions to J.A.N. and PGSF Geological Hazards and Society contract C05X0804 contributions to S.E. This work benefited from discussions with Bernhard Stöckhert, Claudia Trepmann, Richard Sibson, and Rafael Benites. We acknowledge use of the software package R [R Development Core Team, 2009], with which the stresses and displacement fields were calculated and plotted. Constructive comments by four unknown reviewers and Kelin Wang are gratefully acknowledged. References Anderson, E. M. (1905), The dynamics of faulting, Trans. Edinburgh Geol. Soc., 8, Anderson,E.M.(1951),The Dynamics of Faulting, 2nd ed., 206 pp., Oliver and Boyd, Edinburgh, U. K. Atkinson, B. K., and P. G. Meredith (1987), Experimental fracture mechanics data, in Fracture Mechanics of Rock, edited by B. K. Atkinson, pp , Academic, London. Barrientos, S. E., R. S. Stein, and S. N. Ward (1987), Comparison of the 1959 Hebgen Lake, Montana, and the 1983 Borah Peak, Idaho, earthquakes from geodetic observations, Bull. Seismol. Soc. Am., 77, Birtel, S., and B. Stöckhert (2008), Quartz veins record earthquake related brittle failure and short term ductile flow in the deep crust, Tectonophysics, 457, Bouchon, M. (1997), The state of stress on some faults of the San Andreas system as inferred from near field strong motion data, J. Geophys. Res., 102, 11,731 11,744. Brace, W. F., and D. L. Kohlstedt (1980), Limits on lithospheric stress imposed by laboratory experiments, J. Geophys. Res., 85, of 15