Mechanics, slip behavior, and seismic potential of corrugated dip-slip faults

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jb008642, 2012 Mechanics, slip behavior, and seismic potential of corrugated dip-slip faults Scott T. Marshall 1 and Anna C. Morris 2 Received 30 June 2011; revised 29 November 2011; accepted 11 January 2012; published 8 March [1] To better understand the mechanics and seismic potential of nonplanar fault surfaces, we present results from a suite of numerical models of faults with sinusoidal corrugations in the downdip direction. Systematic variations in corrugation wavelength, amplitude, and loading angle are introduced to determine the effects on slip behavior and seismic moment release. We find that corrugated faults, in general, slip less than planar faults. Changes in slip behavior are nearly scale-independent and are dominantly controlled by the amplitude/wavelength of corrugations. Model results suggest that obliquely loaded corrugated faults accumulate less strike slip than a planar fault with the same tip line dimensions and average orientation. This result implies that slip direction is not a reliable indicator of regional stress direction and may at least partially explain repeated, nearly pure dip-slip coseismic events at oblique plate boundaries. Though the scalar seismic moment release is always less for corrugated fault surfaces due to a greater reduction in slip compared to increased surface area, for geologically reasonable corrugation geometries, changes in total scalar moment release are not significantly different than planar faults. Techniques that utilize highly simplified fault geometries may therefore accurately reproduce scalar moment release but will nonetheless incorrectly predict coseismic slip magnitudes and distributions, as well as regional stress orientations. Citation: Marshall, S. T., and A. C. Morris (2012), Mechanics, slip behavior, and seismic potential of corrugated dip-slip faults, J. Geophys. Res., 117,, doi: /2011jb Introduction [2] Although often parameterized as planar dislocations, fault surfaces are nonplanar and have deviations from planarity at all scales [e.g., Candela et al., 2009; Power and Tullis, 1991; Renard et al., 2006; Sagy and Brodsky, 2009; Sagy et al., 2007]. While some wavy surface traces of dipslip faults may be, in part, due to interference with surface topography, kilometer-scale representations of natural fault systems commonly suggest considerable geometric complexity [e.g., Candela et al., 2011; Plesch et al., 2007]. At the outcrop scale, faults commonly exhibit geometrically complex damage zones [e.g., Kim et al., 2004] often with significant variations in host rock rheology [e.g., Dor et al., 2006; Fialko, 2004; Finzi et al., 2009; Hearn and Fialko, 2009; Sagy and Brodsky, 2009]. Despite these complexities, faults tend to localize the vast majority of deformation in a relatively narrow zone of slip along a nonplanar surface that is commonly striated [e.g., Resor and Meer, 2009; Sagy and Brodsky, 2009]. Formation of nonplanar fault surface geometry is poorly understood, but has been suggested to be 1 Department of Geology, Appalachian State University, Boone, North Carolina, USA. 2 Department of Physics and Astronomy, Appalachian State University, Boone, North Carolina, USA. Copyright 2012 by the American Geophysical Union /12/2011JB related to the linkage of preexisting cracks during fault growth [Martel and Pollard, 1989; Myers and Aydin, 2004] and/or fracturing in the near tip region during fault growth [Cowie and Scholz, 1992], possibly in combination with granular flow in fault damage zones during continued slip [Sagy and Brodsky, 2009]. [3] Geometrical analyses of exposed fault surfaces consistently demonstrate that fault surfaces have an anisotropic nonplanar geometry with the least roughness occurring parallel to the slip vector and the most roughness occurring perpendicular to the slip vector [Candela et al., 2011; Power and Tullis, 1991; Power et al., 1987; Renard et al., 2006; Resor and Meer, 2009; Sagy et al., 2007]. This anisotropy has been attributed to wear resulting from slip [Power et al., 1987; Scholz, 2002] and Sagy et al. [2007] suggest that slipparallel smoothing may increase monotonically with cumulative slip. Regardless of the details of the formation mechanisms for fault geometry, the geometrical details of a fault surface clearly exert a primary control on the kinematics [e.g., Bilham and King, 1989; Saucier et al., 1992], slip behavior [e.g., Marshall et al., 2008; Wesnousky, 2008], and seismic potential [e.g., Acharya, 1997; Aydin and Du, 1995; King and Nábĕlek, 1985; Wesnousky, 2006]. [4] Several works have described the mechanics of twodimensional fault surfaces with sinusoidal roughness in the slip direction [e.g., Chester and Chester, 2000; Sagy and Brodsky, 2009; Saucier et al., 1992]. These models 1of19

2 Figure 1. Corrugated fault surfaces in this study are defined using a sinusoidal function of the y coordinate. A single wavelength is defined along the direction of strike. This produces a corrugated fault surface similar to but in simplified form (i.e., no small-scale roughness) of many exhumed natural fault surfaces. demonstrate that slip-parallel undulations produce a complex near fault stress field where so-called residual stresses [Saucier et al., 1992] may accumulate over several seismic cycles and lead to the formation of damage zones [Chester and Chester, 2000] and/or secondary structures [Saucier et al., 1992]. While these models demonstrate that fault surface geometry plays a key role in the slip behavior and the near fault stress field, these existing models do not address the observation that the slip parallel direction is often more than an order of magnitude smoother than the slip perpendicular direction [Resor and Meer, 2009] and that coseismic slip vectors are often subparallel to the most smooth direction on a fault surface [Carena and Suppe, 2002; Resor and Meer, 2009; Sagy et al., 2007]. [5] Slip-perpendicular variations in geometry are often termed slickenlines or slickensides when of approximately centimeter scale or less and corrugations when of centimeter scale or greater. Resor and Meer [2009] suggest that corrugations may act as slip barriers and thus may directly impact the mechanics of a fault surface. This hypothesis is supported by seismological data that indicate that the mean direction of slip on a corrugated fault is nearly identical to the orientation of the fault corrugation [e.g., Carena and Suppe, 2002] and that greater slip and smoother surfaces exist in the slip-parallel direction [Power and Tullis, 1991; Power et al., 1987; Renard et al., 2006; Resor and Meer, 2009; Sagy and Brodsky, 2009; Sagy et al., 2007]. Thus, corrugations may directly alter and control slip. [6] Deviations from planarity on fault surfaces also affect seismic behavior. Fault geometry exerts a primary control on slip distributions in earthquakes [Acharya, 1997; Aydin and Du, 1995; King and Nábĕlek, 1985; Maerten et al., 2005] and may be linked to patterns of earthquake recurrence [Aki, 1984] and termination of ruptures [Nielsen and Knopoff, 1998; Wesnousky, 2006]. Empirical studies have shown that earthquakes and aftershocks also occur more frequently around steps and geometrical irregularities such as lateral steps along strike-slip faults [e.g., Bakun et al., 1980] and slip-perpendicular fault corrugations along dip-slip faults [e.g., Carena and Suppe, 2002]. For example, the 1989 Loma Prieta and 1992 Landers earthquakes in California both occurred in contractional bends along a strike-slip fault [Aydin and Du, 1995; Du and Aydin, 1995; Schwartz et al., 1990], and the 1966 Parkfield earthquake was associated with a right step in the San Andreas fault [Aki, 1968]. Earthquakes often nucleate at bends [Carena and Suppe, 2002; King and Nábĕlek, 1985; Segall and Pollard, 1980; Sibson, 1986], and fault geometry can control earthquake propagation and size [Carena and Suppe, 2002; Nielsen and Knopoff, 1998; Wesnousky, 2006, 2008]. Areas of highenergy release are often located at bends in a fault [Schwartz et al., 1990], and Acharya [1997] proposes that because of the greater stored stress, greater slip during an earthquake may occur around such bends. Both occurrences can lead to greater coseismic slip, and thus larger seismic moment release. [7] Despite the vast literature on fault geometry, the mechanics of corrugated fault surfaces remain largely unknown. In this study, we investigate the effects of fault surface corrugation on slip behavior and seismic moment release. To accomplish this, we define numerous hypothetical three-dimensional fault surfaces with sinusoidal surface corrugations of varying wavelengths and amplitudes. We then use an established numerical method to calculate the three-dimensional slip distribution and seismic moment release on all modeled fault surfaces given a range of horizontal principal stress orientations. 2. Modeled Fault Geometry [8] Because no analytical solution exists for calculating three-dimensional slip distributions on arbitrarily wavy, nonvertical, finite, dip-slip faults, we utilize Poly3D [Thomas, 1993], a well-established polygonal dislocation computer program based on the Boundary Element Method (BEM). For the purposes here, a BEM approach is advantageous over a Finite Element Method because the BEM only requires discretization of the fault surface and not the volume surrounding the fault. To ensure a consistent and uniform fault surface mesh, we have created and utilized a meshing algorithm that uses simple mathematical functions to define the z coordinates of a regular grid of nodes. The resulting point cloud of nodes is then tessellated into a triangulated surface. We chose not to use existing Delaunay triangulation schemes because results at high amplitudes and short wavelengths were unreliable and our mesh generator ensures a consistent tessellation pattern for all surfaces. [9] To achieve a three-dimensional corrugated fault surface of arbitrary strike and dip, we first create a point cloud that defines a two-dimensional square horizontal surface. The z coordinates are then added based on a sinusoidal function. Once the horizontal corrugated surface has been defined, coordinate transformations are applied to rotate the surface to the desired strike and dip. To create the initially planar surface, we define the fault in the horizontal plane so that after rotation, the x axis will point downdip and the y 2of19

3 10 km in the strike and downdip directions and have an average strike of west and dip to the north at 45. Figure 2. Several discretized versions of a smooth analytical sinusoid (gray curve). Because faults in this study are discretized into a series of triangulated points, a perfectly smooth wave (gray curve) is not possible, and aliasing effects must be avoided. The minimum wavelength that can be correctly reproduced (Nyquist) requires a sampling rate of two points per wavelength (black curve). Sampling rates that are less than two per wavelength can result in incorrect wavelengths (red dotted line). Amplitude aliasing occurs when points are not located at peaks and troughs (blue dashed line). To avoid aliasing, all wavelengths tested must have data points at peaks and troughs (i.e., every half wavelength) and only wavelengths greater than twice grid spacing are tested. axis along strike. We then define a 10 km 10 km uniform grid of nodes at 200 m spacing in the x-y plane. These nodes are tessellated into a fault mesh containing 5000 triangular elements. The z coordinate (i.e., the fault normal direction) is then defined based on a single function of the y coordinate given by 2py z ðþ y ¼ A cos ð1þ l strike Thus, (1) creates a point cloud that defines a surface (Figure 1) that is a simplification of the corrugated geometry observed by Resor and Meer [2009]. We use this same process for all modeled faults so that the number of elements and connectivity is the same for each fault regardless of wavelength or amplitude. [10] Although (1) produces a fault surface that is, on average, horizontal, coordinate transformations are used to create fault surfaces that strike west (azimuth 270) and dip north at 45. The final result is shown in Figure 1, with the coordinate axes shown in the orientation prior to the coordinate transformation. [11] The tip to tip length (measured in a straight line from tip to tip, not along the fault surface) is held constant, therefore faults of different wavelengths and amplitudes will have differing fault areas and thus differing element shapes. Because very irregular element shape can lead to numerical artifacts in the BEM matrix inversion [Thomas, 1993], no modeled surface utilizes elements of unusually large aspect ratios. Additionally, because the triangulation pattern is the same for every fault, comparing results on various faults is straightforward. For example, when viewed perpendicular to the average fault orientation, all fault meshes appear identical. This allows for easy identification of numerical artifacts due to the mesh and facilitates direct comparison between all modeled fault surfaces. Although results presented here are normalized, all modeled faults have a tip to tip length of 3. Aliasing and Geometric Parameters [12] When using gridded data to reproduce sinusoidal functions, care must be taken to avoid aliasing; otherwise, unwanted errors in both wavelength and amplitude may be produced (Figure 2). To avoid aliasing effects, we set some simple rules to guide our choices of meshed surfaces. The Nyquist wavelength, defined as twice the grid spacing, is the minimum nonaliased wavelength that is reproducible. For our x-y spacing of 200 m, the smallest wavelength we can generate without aliasing is 400 m. In addition, to prevent amplitude decimation, data points must occur exactly at the peaks and troughs of each wavelength (Figure 2). Given these considerations and assuming that we take the first node at a peak (i.e., x = 0 for the cosine function), any whole number multiple of the minimum wavelength (or equivalently, any even number multiple of the grid spacing) can be produced without incurring any aliasing. We note that 400 and 800 m wavelength faults are not meshed as smooth sinusoids, but are zigzag in shape. We expect that geometric effects on these two shortest-wavelength fault surfaces may be exaggerated or highly simplified; however we see no spurious results on these short-wavelength faults, so mesh refinement was not necessary. [13] Another potential geometrical issue that has to be taken into account is the symmetry of fault surfaces. To ensure that comparisons between fault surfaces of differing wavelengths are meaningful, all surfaces must have identical average orientations. The meshing algorithm therefore creates faults that are symmetrical about the x and y axes so that all fault surfaces have the same average strike (azimuth of 270) and dip (45 ) for every combination of amplitude and wavelength. Furthermore, meshing symmetrical surfaces allows the opportunity to test wavelengths that are larger Figure 3. Examples of several well-constrained, kilometerscale, wavy fault trace geometries of reverse faults in southern California from the Southern California Community Fault Model [Plesch et al., 2007]. The faults shown are (a) Sierra Madre, (b) San Cayetano, (c) Santa Susana, and (d) Pleito. All fault traces are shown in map view. 3of19

4 Table 1. Fault Trace Wavelength and Amplitude Measurements Based on Data in the Southern California Earthquake Center s Community Fault Model [Plesch et al., 2007] a Fault Name and Section Fault Length b (km) l (km) Amplitude (km) Amplitude/l l/length Amplitude/Length Sierra Madre I Sierra Madre II Sierra Madre III Sierra Madre IV San Cayetano I San Cayetano II Santa Susana I Santa Susana II Pleito I a Note that some of the fault traces shown here have wavelength/amplitude ratios greater than 10 1, which extends beyond the amplitude to wavelength ratio range suggested based on small-scale roughness measurements [Power and Tullis, 1991]. b Length is tip to tip. than fault length, which is geologically reasonable because naturally occurring faults often contain bends that do not span full wavelengths (e.g., Figure 3d). [14] Given all of the aforementioned considerations, we choose to test fault wavelength/length ratios of 0.04, 0.08, 0.12, 0.16, 0.20, 0.40, 0.80, 1.00, 1.48, 2.48, and For each wavelength, the fault amplitude/length ratio is varied from 0 to 0.05 in increments of This set of geometries is utilized for loading angles of 0, 15, 30, and 45 for a total of 528 models. [15] Natural fault surfaces commonly have amplitude to wavelength ratios of 10 4 to 10 2 on the submeter scale [Power and Tullis, 1991]. To determine if the suggested range of amplitude to wavelength ratios is reasonable at the kilometer scale, a series of basic measurements were made along the surface traces of several wavy reverse fault traces using data in the Southern California Earthquake Center s Community Fault Model (Figure 3 and Table 1). This admittedly simple exercise demonstrates that faults can have kilometer-scale amplitude/wavelength ratios > 10 1 suggesting some scale dependence of fault roughness. Therefore, while there is at least partial agreement with the submeter-scale amplitude to wavelength ratio data, to understand the mechanics of the most nonplanar faults, we test amplitude/wavelength ratios on the order of 10 3 to We choose to model amplitude/wavelength ratios beyond what is typical for fault surfaces to fully understand the effects of geometry for both typical and extreme examples of fault surface corrugation; however, to focus on more typical geometries, we limit most of our discussion to amplitude/ wavelength ratios of < Numerical Method and Model Boundary Conditions [16] To determine the distribution of slip on arbitrarily corrugated fault surfaces we utilize the numerical BEM computer program Poly3D [Thomas, 1993]. If slip is applied to all elements a priori, Poly3D analytically solves for the displacements, stresses, and strains anywhere in a homogeneous and isotropic half- or whole space. If the distribution of slip is unknown (as is for the problem of interest here), the user specifies traction boundary conditions on each element along with a remote driving stress or strain tensor and Poly3D numerically solves the system of linear equations [i.e., Comninou and Dunders, 1975; Jeyakumaran et al., 1992] to determine the slip vector for each element. The advantage of Poly3D is that it can be used to calculate mechanically valid nonuniform slip distributions on arbitrarily shaped three-dimensional fault surfaces, while simultaneously accounting for mechanical interactions between all modeled fault elements. Thus, Poly3D is well suited for the task of determining the slip behavior of various threedimensional wavy fault surfaces. Poly3D has also been used to better constrain the three dimensional fracturing process [e.g., Kaven and Martel, 2007], fault-perturbed stress fields [e.g., Kattenhorn and Pollard, 1999; Marshall et al., 2010], Figure 4. Schematic sketch showing the various tested loading angles measured clockwise relative to north. The upper tip of the modeled fault is shown by a red wavy line. All modeled faults strike west and dip to the north at 45. 4of19

5 Figure 5. Plots showing the effects of corrugation wavelength on average net slip for the four different principal stress orientations tested. Note that in Figures 6, 7, 8, 11, and 12, l is wavelength of corrugation, L is the fault length, and A is the amplitude. Refer to Figure 4 for definition of loading angle. In general, net slip decreases with wavelength. As loading angle increases, average net slip is more sensitive to wavelength. This suggests that nonplanar fault geometry has a larger effect on obliquely loaded dip-slip faults. and mechanical interaction [e.g., Crider and Pollard, 1998; Willemse et al., 1996], as well as the distribution of slip in complex natural fault systems [e.g., Dair and Cooke, 2009; Griffith and Cooke, 2005; Marshall et al., 2008; Marshall et al., 2009]. [17] In this work, fault surfaces are embedded in an infinite linear elastic whole space, simulating deeply buried faults. This avoids the complications of dipping faults with complex geometry mechanically interacting with the free surface of a half-space and effectively isolates the effects of corrugation geometry on fault slip. Because we choose to model 45 dipping reverse faults in a whole space, the symmetry of our setup implies that the general results presented here should also be applicable to corrugated normal faults. To calculate the slip distribution, all modeled fault elements are given a shear stress free boundary condition, i.e., all resolved shear stresses are released as slip, and no slip is allowed normal to the fault, i.e., no opening or closing. Thus, the resultant slip is governed only by resolved shear stresses. The models are therefore best suited to represent the behavior of frictionally weak faults [e.g., Cooke and Marshall, 2006; Dair and Cooke, 2009; Saucier et al., 1992]. In section 10, we hypothesize as to the potential behavior of frictional corrugated faults. Also, because we do not permit fault surfaces to open or close, we cannot address the potential for or the effects of fault dilation around wavy fault surfaces [e.g., Griffith et al., 2010; Ritz and Pollard, 2011, 2012; Sibson, 1974]. Similar assumptions of zero fault opening have been made by several other studies of wavy faults [e.g., Chester and Chester, 2000; Sagy and Brodsky, 2009; Saucier et al., 1992] and despite observations of preferential polishing along some corrugated exhumed fault surfaces [e.g., Jackson and McKenzie, 1999] observations of slickenlines on both sides of corrugations are common [e.g., Resor and Meer, 2009]. This suggests that while tensile tractions may be resolved onto portions of fault surfaces during fault slip [Griffith et al., 2010; Ritz and Pollard, 2011, 2012] these tensile tractions may not typically overcome lithostatic stresses at seismogenic depths. [18] To drive fault slip, a uniform, uniaxial, horizontal, and compressional driving stress of 1 MPa is applied to the model. Loading is initially defined to be north-south, or orthogonal to the average fault trace orientation. The driving stress tensor in an east (x), north (y), up (z) coordinate system with compression positive is therefore given by s ¼ Because nearly all earthquakes involve some component of oblique slip (implying some degree of oblique loading), we also test a range of loading angles to determine if fault 5of19

6 Figure 6. Plots showing the effects of corrugation amplitude on average net slip for the four different principal stress orientations tested. In general net slip decreases with increasing amplitude, although the effects are greatest for obliquely loaded faults. Normalized wavelengths of less than 0.5 contain complete or multiple corrugations and so these faults have the greatest reductions in slip. As also observed in Figure 5, corrugations have the largest net slip reducing effects on obliquely loaded faults. geometry effects are loading angle dependent. All subsequent loading angles (measured in degrees relative to north/ normal to the fault trace) are calculated by transforming the initial stress tensor [Jaeger et al., 2007, equation 2.30] clockwise about the vertical axis (Figure 4). [19] Oblique loading angles should produce slip that is not completely parallel to corrugation troughs which must lead to deformation of the fault surface. This has been a limiting factor in modeling studies of fault surfaces with small-scale roughness [e.g., Dieterich and Smith, 2009; Dunham et al., 2011a, 2011b; Griffith et al., 2010]; however, since our modeled surfaces have only a single wavelength (i.e., no small-scale roughness) and that wavelength is much larger than the slips we compute, the deformation of our fault geometry due to a single slip event is minimal. For example, because the shortest-wavelength corrugation presented here is 400 m and the computed slips are typically < 0.1 m, the deformation of the fault geometry is negligible. 5. Results [20] Using the nonuniform and spatially variable slip values computed by Poly3D, we calculate weighted average values of net slip, slip vector rake, and total scalar seismic moment release. The average slip values are weighted by element area to ensure that changes in fault area between different fault surface representations are properly accounted for. With the exception of the slip vector rake, all modelcalculated parameters presented here are normalized. Amplitude is normalized by wavelength (or fault length), wavelength by fault length, and calculations of scalar seismic moment, fault area, and average net slip are normalized by the value on the reference planar fault. Thus, a normalized average slip of 0.75 indicates that a fault that has 75% of the average slip compared to a planar fault. We choose to normalize in this manner because normalized parameters emphasize only the amount of deviation from the behavior of the reference planar fault surface, and thus highlight the effects of a given geometrical parameter. [21] All of the results presented here assume a shear modulus of 30 GPa and a Poisson s ratio of 0.25; however, we have explored models with different elastic moduli. Given our normalization process, the results here are completely independent of the shear modulus and nearly independent of the Poisson s ratio chosen. Reasonable variations in Poisson s ratio (0.15) produce slightly less variation in normalized slip values (i.e., less decrease in slip with waviness) with changes of typically < 1% from what is presented here. [22] Although we drive motion on the modeled faults by a simple remote uniaxial stress tensor, crustal stress states most commonly have nonzero intermediate stresses [e.g., Sibson, 1974]. We choose the uniaxial driving stress because it most emphasizes geometrical effects and it does not 6of19

7 Figure 7. Effects of corrugation amplitude/wavelength ratio on average net slip for four different loading angles tested. The consistent behavior of all models suggests that the key geometrical parameter controlling the net slip potential of a fault surface is the amplitude/wavelength ratio. require assumptions of tectonic environment. For example, Sibson [1974] shows that the ratios of the principal stress components systematically varies in different tectonic environments. We have run simulations with s 1 = 1 MPa and s 2 = s 3 = 0.5 MPa (R = 2 in Sibson s [1974] nomenclature) and the general patterns are all similar to those presented here, albeit less dramatic. Nonuniaxial stress states tend to cause smaller changes in slip vector rake with increasing loading angle compared to the uniaxial stress state tested here; however for equivalent rake changes on the planar fault triaxial stress states produce nearly identical results to those presented here. While we have attempted to generalize results as much as is possible, we caution the reader that the results presented here are not universal and are applicable to our specific set of chosen parameters and modeling assumptions. 6. Average Slip Behavior [23] For a loading angle of 0 (see Figure 4), average net slip is always less than the planar fault, with short wavelengths and large amplitudes producing the least slip of the nonplanar surfaces (Figure 5). For any given amplitude, average net slip decreases with wavelength (Figure 5), while for any given wavelength, average net slip decreases with increased amplitude (Figure 6). Most of the significant variations in net slip are seen for wavelengths that are less than half of the fault length (i.e., l / L < 0.5), while wavelengths that are greater than fault length converge toward the net slip values of the planar fault (Figure 6). This result implies that long-wavelength (or gentle) fault bends do not significantly affect the net slip potential of a fault. [24] Like the 0 loading angle results, the modeled nonplanar faults always produce lower average net slip values than the planar faults at any loading angle. As the modeled faults become increasingly obliquely loaded, the effect of the corrugation geometry is greater, especially on fault surfaces that have normalized wavelengths of < 0.5 (Figure 6). Fault surfaces with normalized wavelengths > 0.5 are best described as gently curved or bent because the wavelength is sufficiently long (relative to the fault length) to preclude repeated or even complete single corrugations. Changes in average net slip for oblique loading angles are similar to those for 0, albeit with a slightly smaller minimum average slip values of 0.58 and 0.46, respectively. The same convergence toward planar fault behavior is observed in the obliquely loaded models (Figures 5 and 6). [25] The average net slip results suggest that both wavelength and amplitude exert a primary control on the net slip potential for a corrugated fault surface. Model results suggest that the key geometrical parameter controlling the slip behavior of corrugated faults is the amplitude/wavelength ratio (or the inverse, wavelength/amplitude). For example, when all net slip results from various wavelengths and amplitudes are plotted against the amplitude/wavelength ratio, the models demonstrate dominantly consistent behavior (Figure 7). Remaining differences in the various average net slip results curves show the other key parameter: 7of19

8 Figure 8. Plots of average slip vector rake for the four principal stress directions tested. Values of rake less than 90 indicate a component of left lateral slip, values greater than 90 indicate a component of right lateral slip, and 90 (highlighted with a solid black horizontal line) indicates pure reverse slip. Because all modeled faults are symmetrical and therefore have the same average orientation regardless of wavelength or amplitude, faults with a loading angle of 0 have an average rake of 90 for all wavelength and amplitude combinations (see Figure 10). Note that for obliquely loaded faults the average rakes tend toward dip slip as corrugations increase in size (i.e., larger amplitude/wavelength ratios). Corrugated faults with an amplitude/wavelength ratio of 0.19 (shown by a vertical red dotted line) are predicted to produce nearly pure dip slip regardless of loading angle. For highly corrugated dip-slip faults (amplitude/wavelength ratio > 0.19), the average sense of slip is reversed compared to the planar reference fault loaded at the same angle (see Figure 10). number of corrugations (or l / L). As noted in Figure 6, corrugation wavelengths that are less than one half of the fault length affect the slip potential of a fault most because these surfaces contain more than one complete corrugation. A similar result was noted by Dieterich and Smith [2009] where they found that rough faults to not follow linear length slip scaling. [26] Because the models are discretized into a finite number of elements which do not exactly replicate sinusoids and Poly3D utilizes an approximate numerical solution, some degree of mismatch between models is expected. In general, the consistency of the results is better than expected, especially for the short-wavelength fault surfaces where a smooth sinusoidal geometry is not maintained. For this reason, mesh refinement was not explored. These results suggest that the key corrugation parameter that controls slip behavior is the amplitude to wavelength ratio, although the wavelength/fault length of corrugation also somewhat affects slip behavior. Thus, all subsequent parameters are discussed relative to the amplitude to wavelength ratio. 7. Average Slip Vector Rake [27] Thus far, all presented results have focused on the magnitude of the average net slip; however, both seismologic and geologic evidence suggests that corrugated surfaces affect the direction of fault slip [e.g., Carena and Suppe, 2002; Resor and Meer, 2009]. To better understand the potential effects of fault surface corrugation on slip direction, we calculate the average slip vector rake of each modeled fault surface using the area-weighted average dipslip and strike-slip values (Figure 8). [28] Faults with a loading angle of 0 all produce an average rake of approximately 90, or pure reverse slip, due to symmetry with respect to the loading direction (Figure 8). Therefore, any symmetrically corrugated dip-slip fault that is loaded nearly orthogonally to the fault trace will, on average, 8of19

9 Figure 9a. Slip maps showing the spatial distribution of slip on a modeled fault loaded at 0 with a normalized wavelength of 0.4 and amplitude of 0.04 (i.e., amplitude/wavelength = 0.1). The plots are viewed in map view using an orthographic projection to prevent geometric distortion of the fault shape due to perspective view. The upper tip lines are at the bottom of each fault surface (i.e., the faults all dip to the north). All slip shown here is normalized by the maximum net slip on a planar fault that is loaded at the same angle as the surfaces shown here. While the bull s-eye pattern of net slip magnitude is strikingly similar to planar fault (in an infinite elastic whole space), the dip-slip and strike-slip distributions are more complex. Dip slip is concentrated near the middle of the fault surface along portions that strike orthogonal to the loading direction. Opposing limbs in each corrugation slip with opposite senses of slip and accumulate mostly strike slip that is up to 50% of the planar net slip. This results in opposing limbs of a corrugation having a marked change in slip vector rake. Because the lateral slips are nearly equal and opposite, the average net slip vector is approximately 90, suggesting pure dip slip on average. produce nearly pure dip slip regardless of the wavelength or amplitude of corrugations. While the average behavior is somewhat instructive, corrugated reverse faults loaded orthogonal to the average fault surface strike are not limited to pure dip slip. In this case, each limb of the corrugation will slip with an equal and opposite amount of strike slip (Figures 9a and 9b). In the end, the average rake indicates approximately pure dip slip because the lateral slip on corrugation limbs cancels out due to the fault geometry being symmetrical with respect to the loading direction. [29] Obliquely loaded fault surfaces produce average rakes that vary depending on the amplitude/wavelength of fault surface corrugation (Figure 8). For example, as loading angle increases to 45, the rake on the planar reference fault decreases, as is expected because the obliquely loaded planar faults should have a component of left lateral slip along with reverse slip (see Figure 4). For all loading angles, as the fault corrugations increase from zero to an amplitude/wavelength ratio of approximately 0.19, the rake increases, indicating a relative decrease in average left lateral strike-slip rates (Figure 8). A surprising result is that once corrugations exceed approximately 0.19 in amplitude/wavelength ratio, the fault rake continues to increase indicating an average net slip vector that has an opposite sense of strike slip compared to a planar fault of the same orientation under the same stress state (Figure 8). This reversal in the average sense of strike slip can be explained by inspecting the slip distribution on a corrugated fault surface with an amplitude/ 9of19

10 Figure 9b. Slip maps for the same fault shown in Figure 9a but loaded at 45. When loading angle is increased, dip slip becomes localized along corrugation limbs where the fault surface strikes nearly orthogonal to the loading direction. Opposing limbs accumulate only minor amounts of dip slip due to their orientation relative to the principal stress direction. Conversely, strike slip is at a minimum in corrugation limbs where the dip slip is a maximum, while the opposing limb accumulates a significant amount of strike slip. This results in one limb accumulating dominantly dip slip (the nearly white portions of the rake plot) and the opposing limb being dominantly left lateral strike slip. The combined effects produce a net slip distribution where net slip is maximized in regions that are optimally oriented for dip slip. wavelength ratio > 0.19 (Figure 10). For corrugated faults with an amplitude/wavelength ratio greater than 0.19, left lateral slip is reduced on the left-slipping corrugation limb and right lateral slip is increased on the right-slipping limb. Furthermore, the dip slip on the right-slipping limb is increased due to the local fault surface orientation being well oriented for dip slip. 8. Three-Dimensional Slip Distributions [30] While trends in average net slip may elucidate general fault behavior, finite length nonplanar faults accumulate complex spatial distributions of slip [e.g., Dieterich and Smith, 2009; Griffith et al., 2010; Kaven et al., 2011; Marshall et al., 2008; Ritz and Pollard, 2012]. Loading angles that are nearly orthogonal (0 and 15 ) to the average fault strike produce slip distributions that strongly resemble the planar fault results. For these small loading angles, a single, continuous, bull s eye pattern is observed with maximum net slip near the fault center (Figure 9a). The maximum slip occurring in the middle (with respect to depth) of the fault is a consequence of the fault being embedded in an elastic whole space, whereby the effects of the stress-free surface of the Earth are not accounted for. If the modeled faults were embedded in a half-space, the resultant slip distribution would be similar, but the maximum slip bull s-eye pattern would be shifted toward the Earth s surface, with the amount of shift being larger for shallower faults. As loading angle increases to 30 and beyond, slip begins to decrease at an increased rate (Figure 7), and a single fault surface begins to accumulate a distribution of slip similar to mechanically interacting echelon arrays of fault segments [e.g., Bürgmann et al., 1994; Crider and Pollard, 1998; Segall and Pollard, 1980; Willemse et al., 1996]. Elements with similar orientations mechanically interact most efficiently, effectively forming groups of individual west or east facing segments. Therefore, while any modeled fault is a single connected surface, 10 of 19

11 Figure 10. Slip maps (same view and normalization as in Figures 9a and 9b) of corrugated nonplanar fault with a normalized wavelength of 0.2 and amplitude of 0.05 (amplitude/wavelength = 0.25), loaded at 45. In this case the right-slipping limbs have a maximum normalized strike-slip rate of 0.43 (relative to the planar fault s maximum net slip) while the left-slipping limbs have a maximum strike-slip rate of This occurs because the opposing limbs, when loaded obliquely, receive different magnitudes of resolved shear tractions, inducing different amounts of slip on each limb. Furthermore, when the amplitude/wavelength ratio is sufficiently high, portions of both limbs will receive resolved right lateral shear stress, which acts to slow the strike-slip rate on the left-slipping limbs. The net result is an average slip vector that has an opposite sense of slip compared to a planar fault of the same average orientation loaded at the same angle. Because opposing limbs of corrugations are not equally well oriented for slip, the net slip distribution on obliquely loaded corrugated fault surfaces resembles a segmented echelon array of interacting faults (see also Figure 9b). the differences in resolved stresses due to changes in local fault orientation lead to east and west facing portions effectively behaving as separate mechanically interacting fault segments. 9. Seismic Potential [31] While understanding the patterns of slip on faults may elucidate the general fault mechanics, knowledge of slip alone is insufficient to predict the seismic potential of a fault. Therefore, to better understand the seismic potential of corrugated fault surfaces and to connect the model results to what can be measured during or after an earthquake, we calculate the model-predicted total scalar seismic moment release, M 0, for each modeled surface. Because the modeled surfaces are nonplanar and the slip vector is variable (but always in the plane of the element), we first calculate the moment density tensor for each element using the standard formulation [Aki and Richards, 2002, equation 3.22]. We then integrate over the element area to determine the moment tensor for each element. The moment tensors of each element are summed and the total scalar moment release for each fault is calculated from the total moment tensor using the standard formulation [Dahlen and Tromp, 1998, equation 5.91]. This process assures that the variations in slip direction and fault surface orientation are properly accounted for. We note that all of our modeled faults produce double couple total moment tensors (i.e., the trace of the total moment tensor = 0). This may not be the case for models that allow for fault opening. We also 11 of 19

12 Figure 11. Plots of total scalar seismic moment release for the four principal stress directions tested. In all cases tested, the total scalar moment release for corrugated faults is less than the reference planar fault. Therefore, all other factors equal, a corrugated fault will have less seismic potential than a planar fault. quantify the potential earthquake size using the moment magnitude scale [Hanks and Kanamori, 1979]: M W ¼ 2 3 logm 0 6:03 where M W is the moment magnitude. Note that this equation differs from Hanks and Kanamori [1979] only in units (i.e., we use Newton meters, not dyne centimeters and /3 log (10 7 ) = 181/ ). Because seismic moment release increases linearly with both fault area and average net slip on a planar fault (i.e., for a planar fault M 0 = mda), changes in both net slip and fault surface area will exert influence on the potential seismic moment release on the nonplanar faults modeled here. All calculations of scalar seismic moment are normalized by the scalar moment release of the planar surface loaded at the same angle. Therefore, because results are normalized and shear modulus is constant in all models, the seismic moment results presented here are independent of m. [32] For corrugated faults, the total scalar moment release is always less than the reference planar fault (Figure 11). This is in spite of increases in fault surface area with increasing corrugation size (Figure 12). Because most natural faults have amplitude/wavelengths of 10 4 to 10 2 [e.g., Power and Tullis, 1991], differences in seismic moment on planar compared to corrugated fault representations are not likely to be significant. For the specific case of a fault with amplitude/wavelength corrugations equal to 0.15 (see ð2þ Table 1 for examples of faults with similar geometries) and loaded with a uniaxial driving stress at 45, the normalized value of seismic moment is 0.58, implying a moment magnitude decrease of 0.16 (i.e., 2/3 log (0.58) = 0.16). Therefore, any differences in the scalar moment release of planar versus corrugated fault surfaces are likely to be smaller than Figure 12. Plots showing the relationship between fault surface area and corrugation amplitude/wavelength. Faults with larger amplitude/wavelength ratios have greater fault area compared to a planar surface with the same tip to tip dimensions. 12 of 19

13 Figure 13. Schematic sketch showing magnified views of the middle section of opposing corrugation limbs relative to a principal stress direction oriented 45 from the average fault strike (i.e., large black arrows). (a) Map view of the upper tip lines of a north dipping fault with progressively larger amplitude corrugations. The box shows locations of the midpoints of opposing corrugation limbs where each limb is rotated with respect to the average fault strike. Figures 13b 13e show how the principal stress direction is resolved into fault surface tractions as corrugation amplitude increases, causing the opposing limbs to rotate 0 60 from the average fault strike. (b) For a planar fault, all sections of the fault receive the same resolved stresses. In this case, the resolved stresses induce a combination of dip slip and left lateral strike slip. (c) For a corrugated fault, the limbs rotate with increasing amplitude/wavelength causing asymmetry with respect to the principal stress direction. The left limb is oriented for mostly left slip with some minor amount of dip slip. The right limb is oriented for mostly dip slip with some lesser left slip. (d) For corrugation limbs rotated 45 with respect to the average strike, the left limb receives no resolved stress because the principal stress is parallel to the limb. The right limb slips in pure dip slip. (e) Once the corrugation causes the limbs to be rotated at an angle greater than the principal stress direction both limbs are loaded with the opposite sense of shear compared to the planar fault. In this example, both opposing limbs slip with right slip. Thus, the changing orientation of corrugation limbs relative to the principal stress direction is the key factor controlling the changes in rake presented in Figures 9a, 9b, and 10. typical error bars in moment magnitude calculations, and thus insignificant. 10. Discussion Resolved Stresses on a Corrugated Surface and Implications for Slip Reversals [33] In general, model results suggest that obliquely loaded corrugated faults tend to have average slip vectors consisting of dominantly dip slip; however, some modeled surfaces with high amplitude/wavelength ratios exhibit an average sense of strike slip that is opposite from the planar models (e.g., Figures 8 and 10). To better understand why this seemingly contradictory result is mechanically feasible, one must consider how the driving stress tensor is resolved onto differently oriented portions of faults of varying corrugation amplitude/wavelength. As the amplitude/wavelength ratio increases, the magnitude and direction of resolved stresses on opposing corrugation limbs changes significantly (Figure 13). This effect is most prevalent at higher loading angles, where the upper tip lines of east facing limbs are oriented nearly orthogonal to and west facing limbs being nearly parallel to the driving stress at smaller (and more geologically common) amplitude/wavelength ratios. Once the corrugation amplitude/wavelength becomes large enough to rotate the limbs beyond perpendicular and parallel relative to the corrugation limbs (i.e., Figure 13e), the sense of strike-slip inducing shear stress on the opposing limbs will be the same, potentially resulting in a reversal in average strike slip. [34] While the schematic sketch shown in Figure 13 is useful for understanding the basic two-dimensional geometrical effects, the actual resolved stresses on the modeled surfaces are more complex due to the fault dip. To explore the variations in preslip resolved stresses on the modeled surface shown in Figure 10, we resolve the driving stress tensor onto each fault element by multiplying the stress tensor by each element s unit normal vector. The resolved normal and shear tractions are then calculated using vector projection relative to the along strike, downdip, and element normal directions (Figure 14). [35] When comparing Figures 10 and 14, the causes of the resultant distribution of slip (Figure 10) become clear. The maximum dip slip on this surface does not occur near the middle of the east facing corrugation limbs because the maximum shear stress in the downdip direction is not located near the middle of the limb (Figure 14). This is simply a consequence of the corrugation limbs being rotated to 13 of 19

14 an angle that is not optimally oriented for dip slip. Models predict that this surface (amplitude/wavelength = 0.25) should have left slip on west facing limbs and right slip on east facing limbs (Figure 10); however, the resolved stresses show that both limbs should have a net right lateral shear stress (Figure 14). So, why do the west facing limbs slip opposite of what the resolved stresses suggest? Mechanical interactions must be accounted for. The net shear stresses show that the east facing limbs will accumulate the most slip. Therefore, the east facing limbs will act to effectively pull on the west facing limbs resulting in a net left slip on west facing limbs. This highlights the need to include the effects of mechanical interaction into models of nonplanar faults, even when only a single fault is considered. Simple kinematic models of slip are likely to misrepresent fault slip along nonplanar surfaces because the distribution and direction of slip clearly cannot be known a priori Implications for Frictional Corrugated Faults [36] Because the modeled fault surfaces are given a shear stress free boundary condition (i.e., frictionless), resolved normal tractions have no effect on fault slip. If the modeled Figure of 19

15 surfaces were frictional, the resolved normal stresses on corrugated fault surfaces could significantly alter the distributions of slip presented here. For example, Griffith et al. [2010] show that two-dimensional frictional wavy faults slip considerably less than a frictionless surface; however, if thermal weakening (i.e., formation of melt/pseudotachylite), fault opening, and rate weakening are accounted for, the final distribution slip is not considerably different from the frictionless case. Furthermore, a recent study of twodimensional frictional wavy faults [i.e., Ritz and Pollard, 2012] shows similar slip distributions and similar mechanical interactions, in many cases, to those presented here. Therefore, the frictionless results presented here, should be generally useful for understanding the basic effects of fault surface corrugation on some frictional interfaces. [37] Another issue that arises with wavy faults is the fact that only shear stresses are released during slip. Normal stresses must be eventually released, either by viscous relaxation [e.g., Nielsen and Knopoff, 1998], plastic deformation [e.g., Dunham et al., 2011a, 2011b], secondary fracturing [e.g., Cruikshank et al., 1991; Marshall et al., 2010], or other types of off-fault damage [e.g., Campagna and Aydin, 1991]. Nielsen and Knopoff [1998] show that restraining bends accumulate large amounts of normal stress, and therefore often act as barriers to rupture (at least for the case when viscous relaxation is slow compared to the earthquake repeat time). For the results presented here, our faults only produce a single slip event with slip that is much less than the corrugation wavelength, and any normal stresses that accumulate do not affect the final distribution of slip or strength of the interface. For example, on the obliquely loaded faults presented here, east facing limbs receive the most resolved shear stress but also the most resolved compressive normal stress (Figure 14). On a frictional surface, the effect would be reduced slip potential on east facing limbs due to the increased clamping (resolved compressive stresses). Because the obliquely loaded corrugated fault surfaces modeled here accumulate the most slip on east facing limbs where normal stresses are high, we expect that a frictional fault surface with a similar geometry and loading may accumulate significantly less slip with increasing corrugation amplitude/wavelength [Griffith et al., 2010; Ritz and Pollard, 2012]. [38] To address the potential zones of failure on the fault surface shown in Figure 10, we calculate the Coulomb stresses [i.e., King et al., 1994] on the modeled surface for two coefficients of friction. Faults with friction values of 0.3 or less are most likely to fail in two locations: along east facing corrugation limb edges where our models predict the most slip, and along the middle of the west facing limbs. Therefore, for low-friction faults (characteristic of mature fault zones), or faults with thermal weakening and/or rate weakening friction [i.e., Griffith et al., 2010] we expect similar slip distribution patterns, albeit with less net slip, to what is presented here. For faults with higher friction values, the changes in resolved normal stresses on corrugated fault surfaces are likely to prevent failure along limbs where our frictionless models predict the most slip to occur (Figure 14). This could result, for some extreme cases, in wavy faults rupturing as echelon segments with only west facing limbs ever failing. Such a result may explain why faults of very large amplitude/wavelength ratios (i.e., > > 10 1 ) are not found in nature. Therefore, models incorporating friction and/or relaxation capable rheologies may produce, in some cases, significantly different distributions of slip and potentially different trends in slip vector rake and seismic moment release compared to the results presented here. A detailed three-dimensional study of frictional corrugated surfaces is beyond the scope of this study, but is clearly needed to address these issues Distribution of Slip and Pseudosegmentation [39] When analyzing the spatial distribution of slip (e.g., Figures 9a, 9b, and 10), we gain further insight into the mechanics of corrugated fault surfaces. Although increasing the amplitude/wavelength of corrugations increases area, it also results in long sections of nearly linear left and east facing limbs that are rotated with respect to the average fault orientation (Figure 13). At short wavelengths and high amplitudes, west facing limbs (east facing limbs) effectively form a pseudo-right-stepping array (left-stepping array) of hard-linked echelon fault segments [cf. Crider and Pollard, 1998; Willemse et al., 1996]. Given the oblique loading angles tested here, east facing corrugation limbs consistently accumulate the most slip (Figures 9b and 10) and exhibit similar slip behavior to arrays of right-stepping mechanically Figure 14. Resolved stresses on the modeled fault shown in Figure 10 before slip. (a) Shear tractions in the downdip direction (0.47 maximum) indicate that the east facing corrugation limbs will accumulate most of the dip slip. Note that the maximum resolved downdip tractions are not in the middle of the corrugation limbs. This explains why the dip slip on this fault surface is not at a maximum along the middle of corrugation limbs (see Figure 10). (b) Shear tractions in the along strike direction indicate that both corrugation limbs have both left (positive values; 0.14 maximum) and right (negative values; 0.30 minimum) lateral resolved shear tractions. On both limbs, more elements have resolved right lateral shear tractions and the right lateral shear tractions are generally largest. Despite this, mechanical interaction with the fast-slipping east facing limbs creates a net left lateral slip on west facing corrugation limbs (see Figure 10). (c) Net shear tractions on the modeled surface indicate that the east facing corrugation limbs should accumulate the most slip (in frictionless models) and therefore cause the strongest mechanical interactions with the west facing corrugation limbs. (d) Resolved normal tractions indicate that the most fault clamping is occurring where the maximum shear tractions are resolved and the most slip is occurring. To demonstrate the most likely zones of failure on a frictional surface, coulomb stress (shear traction minus (coefficient of friction times normal traction)) is calculated for a coefficient of friction of (e) 0.3 and (f) 0.6. Zones of failure for the fault surface with a coefficient of friction of 0.3 indicate failure zones that dominantly correspond to the zones of maximum slip presented in Figure 10. For faults with high friction values (0.6), the increased normal tractions on the right-facing limbs overwhelm the high shear tractions resulting in a low probability of failure on the east facing limbs, a result that would produce significantly different distributions of slip than those shown in Figure of 19

16 Figure 15. Net slip vector maps (same view and normalization as in Figures 9a, 9b, and 10) of an increasingly corrugated nonplanar fault with a normalized wavelength of 0.2 and loaded at 45. The colored fault surface shows the magnitude of normalized net slip while the vectors show the direction and relative magnitude of net slip. The slip vectors represent the sum of displacements on both the hanging walls and footwalls of the faults and are mapped as hanging wall motions. (a) Slip on a planar fault surface showing oblique slip. Note that on a planar fault surface there are no predicted variations in slip vector rake (see Figures 9a and 9b). (b) Slip on a fault surface with an amplitude/wavelength ratio of (c) Slip on a fault surface with an amplitude/wavelength ratio of (d) Slip on a fault surface with an amplitude/wavelength ratio of Note that as amplitude/wavelength increases, the slip vectors become parallel to the corrugations. interacting echelon arrays [Bürgmann et al., 1994; Crider and Pollard, 1998; Segall and Pollard, 1980; Willemse et al., 1996], even though the modeled faults are continuous surfaces. For example, if all east facing limbs were laid side by side to form a single connected surface, an approximately elliptical distribution of slip similar to that of a planar fault is formed (Figure 9b). Likewise, all west facing limbs create a similar distribution, albeit with less overall slip. Because interaction is strongest between echelon segments that are closely spaced [Crider and Pollard, 1998; Segall and Pollard, 1980; Willemse et al., 1996], larger amplitude/ wavelengths ratios are more likely to mimic echelon arrays, particularly for high loading angles. Conversely, fault surfaces with smaller amplitude/wavelength ratios do not produce limbs that are sufficiently rotated with respect to the average fault orientation, so the pseudoechelon effect is negligible. Thus, while corrugated reverse faults that are loaded at 0 produce similar distributions of slip compared to their planar counterparts (albeit with less overall slip than the planar fault), the same surface, when loaded obliquely, will produce significantly altered slip distributions relative to a similarly oriented planar fault. 16 of 19