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We examine the partition of long-term geologic slip on the Hayward fault into inter- seismic creep .... southwest fracture zone located about 1 km southwest of the.
Bulletin of the Seismological Society of America, Vol. 102, No. 3, pp. 961–979, June 2012, doi: 10.1785/0120110200

Probabilistic Estimates of Surface Coseismic Slip and Afterslip for Hayward Fault Earthquakes by Brad T. Aagaard, James J. Lienkaemper, and David P. Schwartz We examine the partition of long-term geologic slip on the Hayward fault into interseismic creep, coseismic slip, and afterslip. Using Monte Carlo simulations, we compute expected coseismic slip and afterslip at three alinement array sites for Hayward fault earthquakes with nominal moment magnitudes ranging from about 6.5 to 7.1. We consider how interseismic creep might affect the coseismic slip distribution as well as the variability in locations of large and small slip patches and the magnitude of an earthquake for a given rupture area. We calibrate the estimates to be consistent with the ratio of interseismic creep rate at the alinement array sites to the geologic slip rate for the Hayward fault. We find that the coseismic slip at the surface is expected to comprise only a small fraction of the long-term geologic slip. The median values of coseismic slip are less than 0.2 m in nearly all cases as a result of the influence of interseismic creep and afterslip. However, afterslip makes a substantial contribution to the long-term geologic slip and may be responsible for up to 0.5–1.5 m (median plus one standard deviation [S.D.]) of additional slip following an earthquake rupture. Thus, utility and transportation infrastructure could be severely impacted by afterslip in the hours and days following a large earthquake on the Hayward fault that generated little coseismic slip. Inherent spatial variability in earthquake slip combined with the uncertainty in how interseismic creep affects coseismic slip results in large uncertainties in these slip estimates.

Introduction Surface rupture is an engineering concern for man-made structures that cross active faults. Several major utility and transportation systems cross the Hayward fault, which runs along the eastern edge of the San Francisco Bay. The Working Group on California Earthquake Probabilities (WGCEP, 2008) estimated the mean 30-year probability (2007–2036) for a magnitude 6.7 or larger event on the Hayward–Rodgers Creek system is 31%. Another recent study using simulations of repeating single-segment ruptures with a viscoelastic rheology and a Coulomb failure criterion suggested that the probability may exceed 50% (Pollitz and Schwartz, 2008). For a fault that is locked during the interseismic period, we expect coseismic slip from large earthquakes over many earthquake cycles to match the long-term geologic slip. In this case surface slip only occurs in the large earthquakes, so estimating the expected surface slip at a site simply involves estimating the surface slip in these large events. The Hayward fault behaves differently from many other faults by accommodating part of its long-term slip via interseismic creep. Numerous man-made features along the fault have been deformed or offset by interseismic creep at the surface. Several studies have identified creeping and locked regions of the fault based on various combinations of micro-

seismicity, alinement arrays, GPS surveys, and interferometric synthetic aperture radar (Bürgmann et al., 2000; Simpson et al., 2001; Wyss, 2001; Waldhauser and Ellsworth, 2002; Schmidt et al., 2005; Funning et al., 2007; Lienkaemper et al., 2012). The sizes and shapes of the creeping and locked regions vary considerably among the models. Those controlled primarily by seismicity (Wyss, 2001; Waldhauser and Ellsworth, 2002) have smaller locked patches than those controlled primarily by geodetic measurements (Bürgmann et al., 2000; Simpson et al., 2001; Funning et al., 2007; Lienkaemper et al., 2012). All four of these models constrained by geodetic measurements support a locked region beneath Oakland and a creeping region beneath Berkeley. The Funning et al. (2007) geodetic model provided the most recently available demarcation and quantitative description of the locked and creeping regions. These creeping regions and their spatial distribution within the seismogenic zone likely influence the decomposition of the long-term geologic slip into interseismic creep, coseismic slip, and afterslip. In this case, estimating the surface slip involves developing models for how the creeping regions affect the coseismic slip distribution and how much afterslip occurs at a site. In our approach we estimate the nominal coseismic 961

962 slip based on a uniform long-term geologic slip rate for the Hayward fault and the imaging of the locked and creeping regions. It is consistent with earthquake ruptures releasing stress that accumulates in the locked regions and, perhaps, extending into creeping regions where the interseismic creep rate is slower than the long-term geologic slip rate. We focus on computing the surface slip at three sites along the Hayward fault (Fig. 1), where alinement arrays provide estimates of the interseismic creep rate. The creep behavior around the three sites differs considerably as imaged in the geodetic models (Bürgmann et al., 2000; Simpson et al., 2001; Funning et al., 2007). Alinement array site HTEM sits near the southern edge of the large, deep creeping region beneath Berkeley, California. Alinement array site H73A lies within a shallow, slowly creeping region above the large, locked patch beneath Oakland, California. Alinement array site HAPP sits within the moderately deep, fast creeping region near Fremont, California. None of the geodetic models are able to constrain whether the fault is locked or creeping at its north end where it extends under the San Francisco Bay. This uncertainty in behavior has little impact on our estimates of surface slip at the three alinement array sites that lie south of this region. The extent to which slip occurs in creeping regions during earthquake ruptures of the Hayward fault remains an open question. The limited description of the moletrack associated with the surface rupture from the 1868 Hayward fault earthquake (Lawson, 1908) does not indicate over what

Figure 1. Location of alinement array sites (triangles) along the Hayward fault (thick solid line). The fault trace corresponds to the 90-km rupture length used in the modeling. See Figure 3 for the other rupture lengths, which are site specific. The color version of this figure is available only in the electronic edition.

B. T. Aagaard, J. J. Lienkaemper, and D. P. Schwartz

time span it formed, so it could have been produced during the rupture as coseismic slip, after the rupture as afterslip from postseismic deformation, or both. Lawson (1908) also includes a description of afterslip in Haywards (now the city of Hayward, California) during the first two weeks following the earthquake as measured by the overlap in slats of a fence where it crossed the fault. Yu and Segall (1996) suggested that the 1868 earthquake had a moment magnitude of 7.0 based on an analysis of triangulated data collected between 1853 and 1860 and 1876 and 1891. The long time span of the data means that it includes coseismic slip and afterslip, as well as other postseismic deformation. Bakun (1999) and Boatwright and Bundock (2008) conclude that the shaking intensities are more consistent with a magnitude 6.8 event, which is supported by the ground-motion modeling of Aagaard, Graves, Rodgers, et al. (2010). Thus, the coseismic slip may have yielded a magnitude 6.8 earthquake, whereas the coseismic slip plus afterslip may have corresponded to a magnitude 7.0 event. Observations from other faults with interseismic creep and/or afterslip provide modest additional guidance. Continuous GPS recordings indicate that coseismic slip in the 2004 Mw 6.0 Parkfield, California, earthquake was confined to the southwest fracture zone located about 1 km southwest of the main trace of the San Andreas fault (Murray and Langbein, 2006), whereas afterslip occurred exclusively on the main surface trace (Lienkaemper et al., 2006). Furthermore, geodetic models of coseismic slip and afterslip on the fault surface suggest that afterslip started almost immediately after the earthquake rupture, was roughly complementary in spatial distribution to the coseismic slip, and led to a total seismic moment (coseismic slip plus afterslip) that was two to three times larger than the seismic moment for the coseismic slip (Johanson et al., 2006; Johnson et al., 2006; Murray and Langbein, 2006). At the southern end of the rupture, a creepmeter recorded strain associated with the coseismic slip, but surface slip in the form of en echelon cracks did not appear until at least four days after the earthquake (Bilham, 2005). A lack of real-time measurements precluded direct observation of coseismic slip in the 1966 Mw 6.0 Parkfield and 1987 M w 6.6 Superstition Hills, California, earthquakes. The first surface slip measurements were made about 10 hours after the 1966 Parkfield earthquake (Oakeshott et al., 1966) and about 7 hours after the 1987 Superstition Hills earthquake (Sharp et al., 1989). Williams and Magistrale (1989) extrapolated their creep measurements to 1 minute after the earthquake and concluded that coseismic slip was only 0.05– 0.23 m and grew to as high as 0.485 m one day after the earthquake and 0.71 m two months later. Bilham (1989) observed similar behavior on three creepmeters. From these observations and interpretations of the distribution of interseismic creep on the Hayward fault and observations of postseismic deformation (afterslip) from historical earthquakes, it is apparent that (1) coseismic slip at the surface on creeping faults such as the Hayward fault is likely to be smaller than that for faults without interseismic creep,

Probabilistic Estimates of Surface Coseismic Slip and Afterslip for Hayward Fault Earthquakes and (2) the amount of surface slip will likely increase with time as a result of postseismic deformation in the form of afterslip. In other words, we expect the contribution of coseismic slip to the long-term geologic slip to be smaller for faults with interseismic creep compared with faults without creep (due to the effects of creep on both coseismic slip and afterslip). This view of Hayward fault behavior represents a departure from analyses that neglect partitioning of slip on the fault into coseismic slip and afterslip. Kinematic rupture models used in ground-motion simulations require a complete description of the spatial and temporal evolution of slip over the entire earthquake rupture. Determining slip at any point on the rupture simply involves extracting the coseismic slip at the desired location from the final slip distribution for the kinematic rupture model. One of the advantages of determining slip at a location with this approach is that models of processes affecting the coseismic slip distribution, such as interseismic creep, can be included. The effect of interseismic creep on coseismic slip is an area of active research in earthquake source modeling. Recent work has examined the connection between creep and coseismic rupture using simple parameterizations. For example, Chen and Lapusta (2009) simulated small, repeating earthquakes that are thought to occur on locked patches surrounded by steadily creeping regions. Kaneko et al. (2010) characterized the likelihood that two locked patches separated by a small creeping region would rupture together by varying the size of the creeping region. Our relatively limited knowledge of how creep affects coseismic slip, and vice versa, suggests that simple models remain a viable approach in accounting for how interseismic creep affects coseismic slip in large earthquakes. The Working Group on California Earthquake Probabilities (WGCEP, 2003) assumed that portions of the creeping regions may rupture with the locked regions in large Hayward fault earthquakes, resulting in rupture areas that might be about 67% larger than an event with the same magnitude on a completely locked fault (WGCEP, 2003). In developing the earthquake rupture probabilities the WGCEP did not consider the spatial variability of slip, so it did not need to develop a detailed model for how interseismic creep affects the spatial distribution of coseismic slip. Aagaard, Graves, Schwartz, et al. (2010) employed two models to account for how interseismic creep affects the coseismic slip distribution in a study of ground motions for Hayward fault scenario earthquakes. In the slip-predictable approach, interseismic creep reduces the coseismic slip in accordance with the rate of interseismic creep and the time since the most recent large earthquake. While this model makes intuitive sense, it often overestimates the amount of slip in earthquake ruptures (Murray and Langbein, 2006). The second model and the one preferred by Aagaard, Graves, Schwartz, et al. (2010) reduces the slip in creeping areas by imposing a vertical gradient in the slip distribution in regions where interseismic creep occurs. This results in a progressive decrease in the amount of coseismic slip as the rupture

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extends from the deep, locked portions of the fault into the shallow creeping regions. Aagaard, Graves, Schwartz, et al. (2010) chose a vertical slip gradient of −0:12 m=km for consistency with the effective rupture area approach used by the WGCEP (2003), and also considered the end-member cases of no coseismic slip occurring in creeping regions and interseismic creep having no effect on the coseismic slip distribution. The kinematic rupture models that account for the effects of interseismic creep on coseismic rupture generated by Aagaard, Graves, Schwartz, et al. (2010) do not explicitly include observational constraints on coseismic slip at the surface, such as those from paleoseismic trenches and field measurements of surface rupture. Additionally, the kinematic rupture models involve only a few realizations that span a range of the viable models. In the present study we refine the method for generating the slip distribution in the kinematic rupture models in order to explicitly include observational constraints on coseismic slip associated with surface rupture and generate a sufficient number of models using a Monte Carlo approach to sample the range of viable models. We apply this methodology to create probabilistic estimates of the coseismic slip and afterslip at alinement array sites HTEM, H73A, and HAPP (McFarland et al., 2009) along the Hayward fault (Fig. 1).

Methodology We want to estimate the coseismic slip at sites along the Hayward fault for expected earthquake ruptures. Although the size of the next large Hayward fault earthquake is not known, events in the moment magnitude range of 6.5 to 7.1 are expected and of primary interest because they will likely generate coseismic slip and/or afterslip at the surface. In constructing slip models from which we extract the slip at the three alinement array sites, we include variability in the magnitude for a given rupture area, the vertical gradient in slip within the creeping regions, and random short length-scale variations in the magnitude of the slip. This variability in the parameters yields a small range of moment magnitudes for rupture of a given area, so it is convenient to construct families of slip models based on a given rupture length. We build upon the procedure used to construct slip models for the ground-motion modeling of Hayward scenario earthquakes (Aagaard, Graves, Schwartz, et al., 2010). Figure 2 outlines the procedure for computing the coseismic slip distribution. We merge a uniform nominal slip distribution with a short length-scale stochastic slip distribution in the wavenumber domain. We then account for a reduction in coseismic slip due to creep by reducing the slip amplitude using a vertical gradient in slip within the creeping regions (as we will discuss in more detail later in this section). We also taper the slip along the lateral edges and bottom of the rupture. Finally, we scale the slip to match the desired moment magnitude.

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B. T. Aagaard, J. J. Lienkaemper, and D. P. Schwartz

Figure 2. Diagram of the procedure used to compute the coseismic slip at the alinement array sites. We generate a deterministic nominal distribution of slip based upon the rupture length and width and imaging of creeping regions. This is merged in the wavenumber domain, with a stochastic slip distribution that follows a von Karman distribution for the power spectra of the magnitude of slip in the wavenumber domain, to create a coseismic slip distribution with realistic spatial heterogeneity. We extract the coseismic slip at the site of interest from this coseismic slip distribution that spans the entire rupture area. The color version of this figure is available only in the electronic edition. In constructing the kinematic slip model, we begin by setting the extent of the rupture over the fault surface. We discretize the fault using a uniform 1-km resolution grid (a 500-m resolution grid gives almost identical results for estimates of the slip at the alinement array sites). In our kinematic rupture approach, the results are independent of fault roughness, so we

use a vertical, planar fault for simplicity; the slip could be mapped onto a nonplanar fault surface without affecting the results. For our nominal moment magnitudes of 6.5, 6.7, 6.9, and 7.1, we consider rupture lengths of 35 km, 48 km, 65 km, and 90 km, respectively. Rather than select different endpoints for each realization of an earthquake with a given rupture

Probabilistic Estimates of Surface Coseismic Slip and Afterslip for Hayward Fault Earthquakes length, we choose fixed endpoints for each rupture length and each site (Table 1; Fig. 3). This simplifies the task of tapering the slip along the buried edges of the faults. The alinement array sites are not centered within the rupture, because we consider the finite extent of the Hayward fault. We follow Aagaard, Graves, Schwartz, et al. (2010) and set the nominal rupture width to 13 km, which corresponds to the maximum depth of 95% of the microseismicity along the Hayward fault (WGCEP, 2003; WGCEP, 2008). We decrease the rupture width to 10 km along the southernmost portion of the Hayward fault where the seismicity merges with that on the Calaveras fault (distances greater than 73.5 km south of Point Pinole, California), consistent with its shallower maximum depth of seismicity. We set the uniform slip value based on the Hanks and Bakun (2008) magnitude–area relation and a shear modulus of 30.0 GPa (other magnitude–area relations, such as Wells and Coppersmith, 1994 and Somerville et al., 1999 give very similar magnitudes for these rupture areas). In using the magnitude–area relation we account for the tapering of slip in the creeping regions extending the slip over a larger area for a given magnitude compared with a locked fault. We compute an effective area with uniform slip that yields the same seismic moment as the rupture area with reduced slip in areas associated with a linear taper along the buried edges of rupture and the vertical gradient in slip in creeping regions. This effective area approach was also used by the WGCEP (2003). For the stochastic distribution, we incorporate random variability at length-scales shorter than one-half of the rupture length. The power spectra for the magnitude of slip in the wavenumber domain follows a von Karman distribution, Pk 

as ad ; 1  k2 H1

where H is the Hurst exponent, ks and kd are the along-strike and down-dip wavenumbers, and as and ad are the alongstrike and down-dip correlation lengths. We use the median value of 0.75 from Mai and Beroza (2002) for the Hurst exponent and their simple scaling of correlation lengths with magnitude, 1 log as  −2:5  M w ; 2

(2)

(3)

and 1 log ad  −1:5  M w : (4) 3 In Figure 2 the slip variance parameter corresponds to the variance of the normal distribution of the amplitude in the initial distribution before modification to fit the von Karman distribution. In the creeping regions delineated by the Funning et al. (2007) model, we decrease the slip toward the ground surface with a uniform vertical gradient. We consider a wide range of values for this vertical gradient as described in the Monte Carlo Sampling section. On the lateral edges we apply a linear taper over 5 km, whereas on the bottom edge we apply a linear taper over 3 km. Our estimates of slip at the alinement array sites are insensitive to the widths of these tapering regions; we simply use the same tapering widths as those used in the ground-motion modeling of scenario events on the Hayward fault by Aagaard, Graves, Schwartz, et al. (2010). We achieve the desired moment magnitude by scaling the slip after tapering. Having set the rupture dimensions and uniform slip based on the Hanks and Bakun (2008) magnitude–area relation, the scaling factor is close to 1.

(1)

k2  a2s k2s  a2d k2d ;

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Calibration The rupture models used in the Hayward ground-motion modeling were calibrated against the next generation attenuation (NGA) ground-motion prediction models

Table 1 Endpoints for Each Rupture Length North Rupture Endpoint Site

HTEM

H73A

HAPP

Rupture Length

35 48 65 90 35 48 65 90 35 48 65 90

km km km km km km km km km km km km

Nominal M w

6.5 6.7 6.9 7.1 6.5 6.7 6.9 7.1 6.5 6.7 6.9 7.1

Dist. Point Pinole

−10.0 −10.0 −10.0 −10.0 17.0 17.0 6.0 −10.0 36.0 23.0 6.0 −10.0

km km km km km km km km km km km km

South Rupture Endpoint

Longitude

Latitude

−122.4345° −122.4345° −122.4345° −122.4345° −122.2555° −122.2555° −122.3276° −122.4345° −122.1333° −122.2159° −122.3276° −122.4345°

38.0746° 38.0746° 38.0746° 38.0746° 37.8769° 37.8769° 37.9580° 38.0746° 37.7354° 37.8327° 37.9580° 38.0746°

Dist. Point Pinole

25.0 38.0 55.0 80.0 52.0 65.0 71.0 80.0 71.0 71.0 71.0 80.0

km km km km km km km km km km km km

Longitude

Latitude

−122.2043° −122.1222° −122.0060° −121.8329° −122.0272° −121.9436° −121.9064° −121.8329° −121.9064° −121.9064° −121.9064° −121.8329°

37.8172° 37.7196° 37.5970° 37.4175° 37.6183° 37.5216° 37.4761° 37.4175° 37.4761° 37.4761° 37.4761° 37.4175°

The location of the endpoints is given both in terms of distance south from Point Pinole along the Hayward fault and longitude and latitude in the WGS84 horizontal datum.

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B. T. Aagaard, J. J. Lienkaemper, and D. P. Schwartz

Figure 3. Examples of slip distributions without short length-scale stochastic variability for the four rupture lengths using site HTEM (top two rows) and slip distributions with short length-scale stochastic variability for the four rupture lengths using sites H73A (middle two rows) and HAPP (bottom two rows). In the Monte Carlo simulations, all of the slip distributions include short length-scale stochastic variability. The vertical gradients in creeping regions for the scenarios shown for four rupture lengths are −0:1 mm=km (35 km), −0:05 m=km (48 km), −0:4 m=km (65 km), and −0:025 m=km (90 km). See Figure 6 for a diagram of how the vertical gradient in slip affects the slip distribution. The vertical gradients are most easily seen in the slip distributions without stochastic variability. The filled circle indicates the location of the alinement array site in each scenario. The endpoints for each rupture length are given in Table 1. The color version of this figure is available only in the electronic edition. (Aagaard, Graves, Schwartz, et al., 2010). Matching spectral accelerations at long periods between the ground-motion synthetics and empirical regressions from the NGA models places some constraints on the amplitude of the slip. It does not explicitly constrain the average or maximum surface slip. Because we want accurate estimates of surface slip in this study, we calibrate the slip models based on observations of average and maximum surface slip. Most faults do not contain significant creeping regions, so we perform the calibration using slip models without any vertical gradients in the slip distribution. That is, we perform the calibration using the coseismic slip models described in the previous section neglecting the effects of creep. Wells and Coppersmith (1994) include an empirical regression for average surface slip as a function of moment magnitude based on observations of surface rupture. Surface slip data available at the time of their study were limited, with relatively sparse observations along the rupture traces. As

shown in Figure 4, the Wells and Coppersmith (1994) regression for average surface slip predicts values about 30% less than the average slip over the entire rupture surface computed using magnitude–area relations. For large earthquakes one expects, on average, the slip at depth to match slip at the surface (which may occur on a single surface or be distributed across a broader zone) for consistency with the longterm accumulation of offset across the fault. While one can develop several possible explanations for the apparent discrepancy between surface slip and average slip for the entire rupture observed by Wells and Coppersmith, see for example King and Wesnousky (2007), technological improvements have facilitated collection of much more complete surface slip data for many earthquake ruptures since the Wells and Coppersmith (1994) study. Wesnousky (2008) examined surface rupture data from 37 large earthquakes, 8 of which have occurred since 1994. As a consistency check, we combine his regressions for

Probabilistic Estimates of Surface Coseismic Slip and Afterslip for Hayward Fault Earthquakes

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Figure 4. Comparison of regressions for average surface slip and average slip over the entire rupture area. W & C refers to Wells and Coppersmith (1994), W refers to Wesnousky (2008), and H & B refers to Hanks and Bakun (2008). The average surface slip from Wesnousky (2008) is consistent with the average slip over the entire rupture from the Hanks and Bakun (2008) and Wells and Coppersmith (1994) magnitude–area relations. The color version of this figure is available only in the electronic edition. magnitude as a function of rupture length and for coseismic slip as a function of rupture length. This yields a linear relationship between moment magnitude and surface slip rather than the linear relationship between the magnitude and the logarithm of surface slip, which we expect based on magnitude–area relationships. Nevertheless, the combination of the two regressions implies that the average surface

slip does not systematically differ from the slip averaged over the entire rupture area (Fig. 4). This is consistent (within 20%) with the relationship between magnitude and surface slip implied by the Hanks and Bakun (2008) magnitude–area relationship for magnitudes greater than about 6.7. As shown in Figure 5, the average surface slips extracted from our slip models that ignore interseismic creep closely match the

Figure 5. Calibrated estimates of surface slip neglecting the effects of interseismic creep on coseismic slip. The plots include 2500 realizations of the slip distribution for each of the four rupture lengths for site HTEM. The rupture models are consistent with the Wesnousky (2008) regressions for average surface slip and ratio of average surface slip to maximum surface slip. The color version of this figure is available only in the electronic edition.

968 combined empirical regression for average surface slip from Wesnousky (2008). In addition to a regression for average surface slip as a function of magnitude, Wells and Coppersmith (1994) also developed a regression for maximum surface slip as a function of magnitude. In their analysis of the average and maximum surface slip, Wells and Coppersmith computed the ratio of the average to maximum surface slip for 57 earthquakes. The ratio of average to maximum slip ranges from about 0.2 to 0.8 with no apparent trend with magnitude. On the other hand, the ratio of average to maximum surface slip, based on their regressions for the average surface slip and maximum surface slip as a function of magnitude, gives a ratio of 0.63–0.85 for earthquakes with magnitudes in the range of 6.0–7.0. Wesnousky (2008) constructed a regression of the ratio of average to maximum surface slip and found the ratio to be 0:44  0:14. Thus, the data from both Wells and Coppersmith (1994) and Wesnousky (2008) appear to favor a ratio of average-to-maximum surface slip ranging from 0.2 to 0.8 with an average of about 0.5; the Wells and Coppersmith (1994) regression for maximum surface slip as a function of magnitude likely suffers from incomplete surface rupture information much like the regression for average surface slip. We favor the ratio of average to maximum surface slip of 0.44 from Wesnousky (2008), because the data set contains earthquakes with more observations per event. The parameters of the stochastic distribution control the ratio of the average to maximum surface slip. In constructing the stochastic distribution, we use a normal distribution for the slip magnitude where the power spectra follows the von Karman distribution. The variance of the normal distribution of the slip in the stochastic distribution (i.e., the slip variance parameter) influences the ratio of the average to maximum slip values. A variance of 1.37 times the average slip for this parameter yields a mean ratio of average to maximum surface slip across the four rupture lengths of 0.44 with a standard deviation (S.D.) of 0.07 with values ranging from about 0.25 to 0.65. These ratios closely match both the regression obtained by Wesnousky (2008) and the range of values found by Wells and Coppersmith (1994). The smaller S.D. in the ratio of average to maximum slip in the synthetic slip models compared with the observations may be associated with Wesnousky’s observations containing strike-slip events across a wide range of tectonic environments. Estimating the variability in the ratio from the observations requires employing the ergodic assumption (mean value of a process parameter over time is equal to the mean over an ensemble of realizations of the process), because the data do not contain repeated ruptures over the same section of a fault. For our synthetic slip models, we do not need to assume an ergodic process and can directly compute the variability at a site over many ruptures of the fault. Because the synthetic models are specific to the Hayward fault, we would expect less variability compared with observations spanning a large number of strike-slip faults.

B. T. Aagaard, J. J. Lienkaemper, and D. P. Schwartz

Monte Carlo Sampling In order to estimate the expected slip at the three alinement array sites and the associated uncertainty, we use Monte Carlo sampling to construct probability density functions for the slip at the sites. For each of the four rupture lengths we consider the aleatory variability in the earthquake magnitude for a given rupture area and the spatial heterogeneity of the slip and the epistemic uncertainty of how interseismic creep reduces coseismic slip through variability of the vertical gradient of slip in creeping regions. We do not consider variability in the nominal rupture width; however, slip heterogeneity provides small variations in rupture width through variability in the depth extent of rupture along strike. In modeling ground motions for Hayward fault scenario earthquakes, Aagaard, Graves, Schwartz, et al. (2010) included only one intermediate case for the vertical gradient in slip in creeping regions and the two end-member cases for how interseismic creep affects coseismic slip. In this study we examine the range of behavior with finer sampling. We consider 10 discrete values for the vertical gradients in slip, where the gradient varies by a factor of 2 between values (Fig. 6; Table 2). To facilitate selection of the gradients in

Figure 6.

Diagram of vertical slip gradients used to reduce slip in creeping regions. Larger gradients reduce the extent of rupture into the creeping regions. An infinite gradient prevents any coseismic slip from occurring in creeping regions and a gradient of zero ignores the effect of interseismic creep on coseismic slip. The color version of this figure is available only in the electronic edition.

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Table 2 Effective Rupture Areas for Each Rupture Length and Vertical Gradient in Slip Slip Gradient (m=km) and Bin Number† Site HTEM

H73A

HAPP

Rupture Length* 35 48 65 90 35 48 65 90 35 48 65 90

km km km km km km km km km km km km

Nominal Mw

0 bin −4

−0.0125 bin −3

−0.025 bin −2

−0.05 bin −1

−0.1 bin 0

−0.2 bin +1

−0.4 bin +2

−0.8 bin +3

−1.6 bin +4

−∞ bin +5

6.5 6.7 6.9 7.1 6.5 6.7 6.9 7.1 6.5 6.7 6.9 7.1

455 624 845 1150 455 624 845 1149 455 624 845 1149

440 611 831 1130 444 615 828 1133 440 611 828 1133

424 597 816 1120 432 605 811 1116 425 597 811 1116

391 566 786 1080 409 584 773 1082 393 568 773 1082

332 503 715 1000 359 538 683 1004 319 502 683 1004

293 447 601 824 311 482 533 824 259 417 533 824

274 412 536 692 283 446 465 692 233 372 465 692

265 394 507 636 270 429 434 636 221 352 434 636

260 386 493 611 263 420 419 611 216 342 419 611

256 378 480 588 257 412 404 588 210 332 404 588

Columns 4–13 give the effective rupture area (km2 ) for each combination of alinement array site, rupture length, and vertical gradient in slip in creeping regions. *Spatial variations in creep result in slightly different rupture areas for a given rupture length across the sites. † The bin number corresponds to the integer value used in the Monte Carlo sampling.

our Monte Carlo sampling, we associate integer values from −4 to 5 with the 10 bins (Fig. 7). The end-member cases (interseismic creep has no affect on the coseismic slip and interseismic creep prevents any coseismic slip in creeping regions) are much less likely than the intermediate cases of interseismic creep reducing coseismic slip to some degree in creeping regions, so greater weights are given to the intermediate cases by selecting vertical gradients from a normal distribution. We center a normal distribution on 0.5 (the boundary between the bins 0 and

Figure 7. Sampling of vertical gradients using a normal distribution centered at 0.5 with bins ranging from −4 to 5 corresponding to gradients ranging from 0 to infinity. The color version of this figure is available only in the electronic edition.

1 corresponding to the slip gradients of −0:20 m=km and −0:10 m=km) with an S.D. of 1.8, while truncating the distribution at values of −4 and 5 (Fig. 7). This choice was driven by calibration of the ratio of coseismic slip to the long-term geologic slip rate as discussed at the end of the Surface Afterslip section. We do not examine the effects of varying the S.D. in the normal distribution. Gradients approaching infinity imply larger coefficients of friction on the fault surface in order to sustain the larger stress concentrations associated with sharp spatial boundaries between regions dominated by aseismic creep and regions dominated by coseismic slip. Figure 3 illustrates slip distributions without any short length-scale stochastic variability for various slip gradients and rupture lengths using site HTEM and slip distributions with short length-scale stochastic variability for various slip gradients and rupture lengths using sites H73A and HAPP. In the Monte Carlo simulations, we generate slip distributions with short length-scale stochastic variability for all of the scenarios for each site and rupture length. A sample size of 200 provides sufficient resolution to capture the range of gradients as shown in Figure 8. Observations of rupture areas for large earthquakes indicate that there is variability in the earthquake magnitude for a given rupture area. The magnitude–area regression developed by Hanks and Bakun (2008) includes this aleatory uncertainty in the regression. This allows us to compute the magnitude for a given rupture area and then sample a normal distribution with the S.D. reported by Hanks and Bakun (2008) to include the variability. We use a sample size of 50 to capture this variability in the magnitude for a given rupture area. Figure 8 displays the distribution in moment magnitude of the earthquakes in the Monte Carlo simulations for site HTEM and includes the epistemic uncertainty in the vertical

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Figure 8. Distributions of vertical gradient in slip and the earthquakes magnitude in the Monte Carlo sampling for the four rupture lengths for alinement array site HTEM. The color version of this figure is available only in the electronic edition. gradient in slip in creeping regions and the aleatory uncertainty in the magnitude for a given rupture area. We associate the short length-scale stochastic variations in the slip distribution with the inherent complexity of the earthquake rupture process and geologic structure. As a result, we simply use a different random seed when generating 50 realizations of the stochastic portion of the slip distribution to include variability in the location of large and small slip patches. The Monte Carlo sampling provides 200 realizations for the vertical gradient in slip and 50 realizations each for the magnitude–area regression and the short length-scale stochastic variations in slip. This leads to 500,000 realizations for each of the four rupture lengths for each site and 2 million total realizations of slip distributions from which we extract the coseismic slip for a site. Surface Afterslip Surface afterslip has been observed following several earthquakes but it is not common. New technologies, such as subpixel correlation of optical imagery and terrestrial laser scanning will enhance our abilities to detect afterslip, and it could be more common than previously thought. For example, following the 2009 M w 6.3 L’Aquila, Italy, earthquake Wilkinson et al. (2010) used repeated terrestrial laser scans to document 3.0 cm of afterslip, which was comparable to the coseismic slip. Three events, the 1976 M w 7.5 Guatemala (Bucknam et al., 1978), the 1987 M w 6.6 Superstition Hills (Sharp et al., 1989), and the 2004 M w 6.0 Parkfield (Lienkaemper et al., 2006) earthquakes, have sufficiently detailed data to adequately constrain the time history of the afterslip (Fig. 9). Furthermore, the afterslip for the 2004 Parkfield earthquake suggests that it has made a significant contribution to the long-term geologic slip over the most recent earthquake cycle (Murray and Langbein, 2006). Constraining the progression of afterslip requires measurements within hours (ideally) or a day or so of the event along with subsequent

measurements months to years later. Empirical studies of afterslip creep data have firmly established that afterslip accumulates as a logarithmic progression in time (Smith and Wyss, 1968; Boatwright et al., 1989; Savage and Langbein, 2008). To analyze the three key afterslip data sets for a relationship between afterslip rates and magnitude, we apply computational tools developed by Boatwright et al. (1989) and Budding et al. (1989) in their analyses of the 1987 Superstition Hills earthquake. In the simple power-law function adopted by Boatwright et al. (1989), Dt  Dtotal =1  T=tC ;

(5)

the accumulated surface afterslip, D, approaches a final value, Dtotal , asymptotically in time depending on only two power-law parameters, the exponent, C, and the time constant, T. We show the range of values for the power-law parameters in Figure 10 for the three data sets. The power-law exponent, C, varies distinctly with magnitude. The time constant, T, tends to be poorly determined, but in most cases it is about one year or more. For simplicity we assume a time constant of one year to characterize the range of the powerlaw exponent and its variation with magnitude. The powerlaw exponent varies inversely proportional to the earthquake magnitude for consistency with increasing coseismic slip with magnitude and theoretical formulations for afterslip and empirical results. This suggests surface coseismic slip is proportional to the slip duration and slip available at depth (Marone et al., 1991). A few high outliers for the Superstition Hills data are from a section of the fault lying within the Imperial Valley, an unusual tectonic environment (Elders et al., 1972) characterized by high heat flow and a few kilometers of weak sediment cover (Kohler and Fuis, 1986). We include this Superstition Hills data because they are limited to just a few outliers in a large data set, and they do not skew the results significantly. However, the entire data set for the 1979

Probabilistic Estimates of Surface Coseismic Slip and Afterslip for Hayward Fault Earthquakes

(a)

(b)

(c)

(d)

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Figure 9. Surface afterslip along four historic surface ruptures for (a) the 1979 Mw 6.5 Imperial Valley, (b) the 1987 Mw 6.6 Superstition Hills, (c) the 2004 Mw 6.0 Parkfield, and (d) the 1976 Mw 7.5 Guatemala earthquake. The plots indicate the total amount of accumulated slip along the fault. For the Guatemala earthquake the stars indicate the sparse locations of repeated measurements. The color version of this figure is available only in the electronic edition. Imperial Valley earthquake reflects this unusual tectonic environment that we judge would make a poor analog for the Hayward fault. The San Francisco Bay region has much lower crustal heat flow in much more consolidated rock. Consequently, we exclude the data from the 1979 Imperial Valley earthquake from our analysis. The ranges of power-law exponents are expressed as a function of moment magnitude in Figure 11 based on linear regression of median and  one S.D. ranges of the data shown in Figure 10. The three equations shown in Figure 11 are used to estimate the ranges of afterslip variation expected for the four rupture lengths (magnitudes) of scenario earthquakes on the Hayward fault. This range of power-law exponents is consistent with the observations of afterslip at the Temescal creepmeter following the 2007 magnitude 4.2 Oakland, California, earthquake (Lienkaemper et al., 2012). Although surface creep as measured in creepmeters often accelerates after these small events on the Hayward fault, the initiation of slip is usually delayed several days to a week, suggesting the physical process is more appropriately classified as along-strike propagation of creep (Lienkaemper et al., 2012), making direct comparison with afterslip from large earthquakes difficult.

In our analysis of slip on the Hayward fault, we explicitly include the effects of both coseismic slip and afterslip. As a result, we modify the power-law expression for afterslip from Boatwright et al. (1989) given in equation (5) to match the coseismic slip value at 1 s (comparable to the rise time of the coseismic slip), D1 s  Dcoseismic , and end at the total slip value, D∞  Dtotal . This requires adding a constant term to equation (5), and we have Dt  A  B

and

1 ; 1  T=tC

(6)

A

1 D − aDcoseismic ; 1 − a total

(7)

B

−a D − Dcoseismic ; 1 − a total

(8)



T a 1 1s

C ;

(9)

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Figure 10.

Constraints on afterslip power-law time history parameters from observations of afterslip for (a) the 2004 Mw 6.0 Parkfield earthquake (alinement arrays), (b) the 1987 M w 6.6 Superstition Hills earthquake (surveyed quadrilaterals and repeated observations of offset features and nails across surface rupture), and (c) the 1976 Mw 7.5 Guatemala earthquake (repeated observations of offset cultural features and survey markers). Each point displays the parameters calculated for the time series at each observation location and the solid lines give the median values and the dashed lines delineate  one S.D..

where Dt is the total slip (coseismic slip plus afterslip) as a function of time, Dcoseismic is the coseismic slip, Dtotal is the final value of total slip, T is the afterslip time constant

(which we normalize by 1 s), and C is the afterslip power-law exponent. The uncertainty in the total slip depends on the uncertainty in the coseismic slip, final total slip, and power-law exponent. Although the final total slip is not independent of the coseismic slip, we assume independent random variables in calculating the uncertainty in the total slip,

Figure 12.

Figure 11.

Linear regression for the afterslip power-law exponent as a function of moment magnitude using the data from the 2004 Mw 6.0 Parkfield, 1987 Mw 6.6 Superstition Hills, and 1976 Mw 7.5 Guatemala earthquakes. The color version of this figure is available only in the electronic edition.

Diagram illustrating how we extract the maximum slip in a 2 km or 3 km region around each site from the distribution of coseismic slip on a 1 km grid (open circles). We assume that interseismic creep tends to relieve stress concentrations associated with gradients in the coseismic slip distribution, so that the amount of afterslip at a site is the difference between the maximum coseismic slip in the region (shown by the filled circles in this case) and the coseismic slip at the site, Dafterslip  Dmax − Dsite . The color version of this figure is available only in the electronic edition.

Probabilistic Estimates of Surface Coseismic Slip and Afterslip for Hayward Fault Earthquakes  δD 

∂D

2

δD ∂Dcoseismic coseismic 2 1=2 2   ∂D ∂D δC  δDcoseismic  ; ∂Dcoseismic ∂C

(10) where ∂D −a a 1   ; ∂Dcoseismic 1 − a 1 − a 1  T=tC

(11)

∂D 1 a 1 −  ; ∂Dtotal 1 − a 1 − a 1  T=tC

(12)

∂D 1  D − Dcoseismic a ln1  T=1 s ∂C 1 − a2 total  −1  Dtotal − Dcoseismic a ln1  T=1 s 1 − a2  1 ; (13) − B ln1  T=t 1  T=tC and δX denotes uncertainty in value X. Assuming independent random variables simplifies the expression for the uncertainties in our results but will tend to slightly overestimate the uncertainties. In order to apply this afterslip model we need to determine a value for the total slip (coseismic slip plus afterslip). In developing a procedure for determining the amount of afterslip, we keep in mind that the long-term geologic slip is the sum of the interseismic creep, coseismic slip, and afterslip over many earthquake cycles. Following a large earthquake on the Hayward fault, afterslip would be manifested as creep at a rate that is faster than the interseismic rate. We associate the creep processes with reduction of shear stress concentrations, so we expect creep to be related to gradients in the coseismic slip distribution (assuming interseismic creep eliminated the significant stress concentrations prior to the earthquake). This is analogous to viscoelastic relaxation in a bulk material reducing deviatoric stress. Thus, we use the spatial variation in the coseismic slip to develop a model for the amount of afterslip at a site. As afterslip reduces stress concentrations to remove gradients in the slip field, the slip will trend toward the maximum value in the region of afterslip. Figure 12 shows a schematic of our method for determining the afterslip at a site. We set the afterslip at each site to be the difference between the maximum coseismic slip within a semicircular region and the coseismic slip at the site. As we consider larger semicircular regions in determining the afterslip, the afterslip will tend toward the difference between the maximum slip over the entire rupture and the expected coseismic slip at the site. On the other hand, as the size of the semicircular region shrinks to zero, the afterslip approaches zero. Interseismic creep contributes to the long-term slip rate, so

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we expect the coseismic slip plus afterslip to be less than the coseismic slip in the locked regions. A semicircular region smaller than the width of the creeping region produces afterslip estimates within this reasonable range. We calibrate the afterslip model using the ratio of the interseismic creep rate to the geologic slip rate. Based on a geologic slip rate of 9:0 mm=yr for the Hayward fault (WGCEP, 2003, 2008) and interseismic creep rates of 3:9 mm=yr, 3:3 mm=yr, and 6:4 mm=yr at the HTEM, H73A, and HAPP alinement arrays (McFarland et al., 2009)1, the corresponding accumulating slip deficits are 5:1 mm=yr, 5:7 mm=yr, and 2:6 mm=yr, or 57%, 63%, and 29% of the geologic slip rate. This means we expect the total slip (coseismic slip plus afterslip), on average, to be about 30%–60% of the value for a locked fault at the three sites. Slip gradients in the creeping regions of between −0:10 m=km and −0:20 m=km yield total slip values in this desired range using radii of 2–3 km in the afterslip calculation, as shown in Table 3. Consequently, we compute the afterslip based on semicircular regions with radii of 2–3 km and center the probability density function for selecting the slip gradient between −0:10 m=km and −0:20 m=km as stated in the Monte Carlo Sampling section. For semicircular regions with larger radii to be preferred for computing the afterslip, the geologic slip rate for the Hayward fault would need to be larger and/or the interseismic creep rate at the sites would need to be much lower.

Results and Discussion In this study we develop simple models for how creep reduces the coseismic slip at the three alinement array sites and how spatial variability in coseismic slip affects afterslip. We find that the effects of creep and afterslip significantly reduce the amount of coseismic slip expected in large Hayward fault earthquakes compared with a locked fault without interseismic creep and afterslip. Figure 13 and Table 4 summarize the coseismic slip and afterslip from the Monte Carlo simulations. The probability density functions are skewed due to the constraint of nonnegative slip values. As a result, median values, which correspond to the fiftieth percentile of the distribution, are more useful than mean values. Whereas in each scenario the total slip at each site is the sum of the coseismic slip and afterslip, the median total slip values are not simply the sum of the median coseismic slip and median afterslip values, because the coseismic slip and afterslip distributions are not independent. The amount of afterslip depends on the coseismic slip at a site and the maximum coseismic slip value in the surrounding region, which are spatially correlated in equations (1–4). 1

The interseismic creep rate of 3:9 mm=yr at HTEM excludes a transient surge associated with a slow slip event in 2007; the interseismic creep rate is 4:1 mm=yr if the surge in creep is included (Lienkaemper et al., 2012).

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Table 3 Median Coseismic and Total Slip (Coseismic Slip + Afterslip) for Three Vertical Gradients of Slip in Creeping Regions

Slip Gradient −0:20 m=km Rupture Length (km) HTEM 35 48 65 90 H73A 35 48 65 90 HAPP 35 48 65 90

Slip Gradient 0 m=km (locked)

Slip Gradient −0:10 m=km

Nominal Mw

Coseismic (m)

Coseismic + Afterslip (2 km) (m)

Coseismic + Afterslip (3 km) (m)

Coseismic (m)

Coseismic + Afterslip (2 km) (m)

Coseismic + Afterslip (3 km) (m)

Coseismic (m)

6.5 6.7 6.9 7.1

0.0 0.0 0.0 0.0

0.2 0.4 0.5 0.9

0.4 0.6 0.8 1.2

0.0 0.1 0.3 0.7

0.5 0.7 1.0 1.6

0.7 0.9 1.3 1.9

0.7 0.9 1.2 1.6

6.5 6.7 6.9 7.1

0.0 0.1 0.1 0.4

0.5 0.8 0.7 1.2

0.7 1.0 0.9 1.4

0.3 0.4 0.6 1.0

0.8 1.0 1.2 1.8

0.9 1.2 1.4 2.1

0.7 0.9 1.2 1.6

6.5 6.7 6.9 7.1

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.4

0.0 0.1 0.2 0.7

0.0 0.0 0.2 0.7

0.3 0.5 0.8 1.4

0.5 0.7 1.0 1.7

0.7 0.9 1.2 1.6

Afterslip is computed using regions with radii of 2 km and 3 km for two vertical gradients of slip in creeping regions. Gradients between −0:20 m=km and −0:10 m=km yield total slip values consistent with slip deficit associated with the ratio of interseismic creep rate to the geologic slip rate. Based on interseismic creep rates of 3:9 mm=yr (HTEM), 3:3 mm=yr (H73A), and 6:4 mm=yr (HAPP) and a geologic slip rate of 9:0 mm=yr for the Hayward fault, we expect median total slip values to be about 57% (HTEM), 63% (H73A), and 29% (HAPP) of the coseismic slip values for the locked (0 m=km slip gradient) case.

Figure 13.

Coseismic and afterslip estimates at the three alinement array sites for 2 million realizations of the slip models. The afterslip analysis uses regions of 2 km and 3 km radius around each site. The color version of this figure is available only in the electronic edition.

Probabilistic Estimates of Surface Coseismic Slip and Afterslip for Hayward Fault Earthquakes

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Table 4 Median and Median Plus One Standard Deviation (σ) Estimates of Coseismic Slip, Afterslip, and Total Slip (Coseismic + Afterslip) Coseismic Slip

Site HTEM

H73A

HAPP

Afterslip (2 km)

Afterslip (3 km)

Coseismic + Afterslip (2 km)

Coseismic + Afterslip (3 km)

Rupture Length (km)

Nominal Mw

Median (m)

Median σ (m)

Median (m)

Median σ (m)

Median (m)

Median σ (m)

Median (m)

Median σ (m)

Median (m)

Median σ (m)

35 48 65 90 35 48 65 90 35 48 65 90

6.5 6.7 6.9 7.1 6.5 6.7 6.9 7.1 6.5 6.7 6.9 7.1

0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.5 0.0 0.0 0.0 0.1

0.3 0.4 0.7 1.2 0.5 0.6 0.9 1.6 0.3 0.4 0.7 1.1

0.2 0.3 0.4 0.5 0.3 0.4 0.4 0.5 0.0 0.0 0.1 0.3

0.5 0.6 0.8 1.1 0.6 0.8 0.8 1.1 0.3 0.3 0.5 0.8

0.4 0.5 0.6 0.8 0.6 0.7 0.7 0.8 0.1 0.2 0.2 0.5

0.7 0.9 1.2 1.5 0.9 1.2 1.2 1.5 0.5 0.6 0.8 1.2

0.4 0.5 0.7 1.1 0.6 0.8 0.8 1.3 0.1 0.0 0.2 0.8

0.8 1.1 1.6 2.4 1.1 1.5 1.8 2.6 0.5 0.7 1.2 2.1

0.6 0.7 1.0 1.4 0.9 1.1 1.2 1.7 0.3 0.3 0.5 1.1

1.1 1.4 1.9 2.8 1.3 1.8 2.1 2.9 0.8 0.9 1.5 2.5

Estimates of coseismic slip, afterslip, and total slip (coseismic + afterslip) at the three alinement array sites for four rupture lengths. Expected coseismic slip values are relatively small due to the effects of interseismic creep, whereas median values for afterslip range from 0.0–1.5 m. Ruptures that do not reach the surface at the site result in zero values for the coseismic slip. Afterslip estimates are given for two models of afterslip with different sized zones influencing the amount of afterslip; a zone 2–3 km in radius is most consistent with the accumulating slip deficit at each alinement array site. The median total slip is not the sum of the median coseismic and median afterslip values because afterslip is not independent of the coseismic slip.

Coseismic Slip At all three sites minimal coseismic slip is expected with median values of 0.1 m or less for all four rupture lengths (which encompass earthquakes ranging from magnitude 6.5 to 7.1), except the 90 km rupture length for site H73A, which has a median coseismic slip of 0.5 m. A median value of 0.0 means that in at least 50% of the Monte Carlo realizations coseismic slip did not extend all of the way to the surface and reach the site. Coseismic slip exceeding 0.6 m is unlikely for the two shorter rupture lengths, and a value exceeding 1.6 m is unlikely for the two longer rupture lengths. Alinement array sites HTEM and HAPP lie within two of the regions with deeper interseismic creep imaged by geodetic models (Bürgmann et al., 2000; Simpson et al., 2001; Funning et al., 2007; Lienkaemper et al., 2012). As a result, with our model of how interseismic creep affects coseismic slip, the coseismic slip values at these sites tend to be less than those at many other locations along the Hayward fault where the interseismic creep is limited to shallower depths, such as alinement array site H73A. Paleoseismic studies of the Hayward fault, for example, Lienkaemper et al. (1999), Lienkaemper et al. (2002), and Lienkaemper and Williams (2007), do not provide estimates of the slip per event to which we can compare our coseismic slip estimates. Nevertheless, the scarp colluvium features and fissure fills used by Lienkaemper et al. (2002) and Lienkaemper and Williams (2007) to identify 12 earthquakes at Tyson’s Lagoon in the past 1900 years appear to be consistent with formation by coseismic slip. Cracks and fissures generated by rapid afterslip following the 1987 Mw 6.6 Superstition Hills earthquake (Sharp et al., 1989) exhibit

similar traits, so rapid afterslip may have contributed to the geomorphic features observed at Tyson’s Lagoon. Ordinary large creep events on the Hayward fault with slip in the range of 0.5–2.0 cm would not develop such features, because erosion would rapidly degrade these smaller offsets before they could accumulate and develop features large and distinct enough to be detected in a trench wall. We believe that our model of surface coseismic slip and afterslip are consistent with the paleoseismic observations. Although our median coseismic slip is less than 0.1 m, the coseismic slip values for one S.D. above the median value fall in the range of 0.2– 1.1 m at site HAPP. Coseismic slip in this range combined with early, very rapid afterslip could generate the scarp colluvium and fissure fills observed by Lienkaemper et al. (2002) and Lienkaemper and Williams (2007). The observed variability in surface slip and the uncertainty in how interseismic creep affects coseismic slip dominate the uncertainty in the expected coseismic and total slip values. As discussed earlier, the ratio of average surface slip to maximum surface slip is independent of magnitude (Wesnousky, 2008), so the larger average slip in larger magnitude events leads to greater maximum values, which in turn leads to greater variability of slip at a fixed point. As a result, the variability in the coseismic slip at each site increases with magnitude. Our results suggest that for sites along the surface trace of the Hayward fault where we observe interseismic creep, we should expect coseismic slip to have a minimal contribution to the long-term geologic slip. Instead, most of the slip will occur in the form of interseismic creep and afterslip immediately following large earthquakes.

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Afterslip Afterslip immediately following an earthquake rupture involves creep much faster than the interseismic rate. The temporal evolution of afterslip follows a power-law function with an exponent that depends on the magnitude of the earthquake. As a result, the afterslip develops very quickly and can have similar impacts as coseismic slip, with the added complexity that the slip continues for months to years, albeit with a decreasing rate. Figure 14 shows the median total slip (coseismic slip plus afterslip) as a function of time for each rupture length for site HTEM (the temporal evolution at the other two sites is very similar). We use the mean magnitude (which is almost identical to the nominal magnitude) in computing the powerlaw exponent and include the uncertainty in the coseismic and final total slip as described earlier. The curves for the total slip illustrate the logarithmic growth in time of the afterslip. Each of the rupture lengths uses a value of 365 days for the relatively poorly constrained time constant parameter. This corresponds to afterslip progressing at a rate of about 10% in the first minute, 25% in the first hour, 35% in the first 6 hours, 40% in the first 24 hours, 70% in the first 30 days, 85% in the first 6 months, and a little more than 90% in the first year. A time constant of 1000 days increases the duration of the afterslip, but more than 80% of the slip still occurs in the first year. Figure 14 also includes curves showing the evolution of afterslip in three scenarios for each rupture length. These curves illustrate the variety of behavior captured in the Monte Carlo sampling relative to the curve for the median value. Some cases show progression that varies very little relative to the median curves. That is, the coseismic slip and total slip are both scaled by about the same amount relative to the median. In other cases, the coseismic slip value may be smaller than the median while the total slip is larger and vice versa. The variability in behavior reaffirms the need to acquire measurements of coseismic slip and afterslip as soon as possible following a large event for accurate estimates of the total slip expected to develop in the months to years following it.

Afterslip behavior can vary significantly from location to location along the length of the rupture. The 1986 Mw 6.6 Superstition Hills earthquake provides a good example of this variability with dozens of measurements along the strike of the fault (Fig. 11). Most of the Superstition Hills afterslip observations fall between those of the 1976 M w 7.5 Guatemala earthquake and those of the 2004 Mw 6.0 Parkfield earthquake, with the power-law exponent, C, ranging from 0.1 to 0.2, except for a southern section of the rupture within the high heat flow region of the Imperial Valley. Consequently, the local variation in the proportion of coseismic slip to total slip (coseismic slip plus afterslip) for the Superstition Hills earthquake ranged from about 2%–18% (95%-confidence, assuming 1-s rise time), and we expect similar afterslip for comparable magnitude earthquakes on the Hayward fault. Quantifying the Influence of Creep In characterizing the effect of interseismic creep on the coseismic slip distribution, we consider the entire spectrum of possibilities, from the creeping regions exhibiting no influence on the coseismic slip (a vertical gradient of zero in the creeping regions) to the interseismic creep preventing any coseismic slip in creeping regions (a vertical gradient of infinity in the creeping regions). We give greater weight in the Monte Carlo sampling to the intermediate cases in which interseismic creep has a moderate influence on the coseismic slip, with the greatest weight given to the cases that yield total slip estimates consistent with the geologic slip rate and the interseismic creep rate measured with the alinement arrays. Reducing the uncertainty related to how interseismic creep affects coseismic slip would generally require more observations of large (Mw 6.0–7.0) earthquakes on faults with interseismic surface creep combined with detailed coseismic slip and afterslip measurements, or models incorporating a more complete physical description of how interseismic creep affects rupture behavior and afterslip. In this study the expected coseismic slip is smaller than the value obtained from analyses ignoring the potential effects of interseismic creep on coseismic slip. For moderate

Figure 14. Median and median  one S.D. (thick solid and short dashed lines) for total slip (coseismic slip plus afterslip) as a function of time from earthquake rupture at the HTEM alinement array site for each of the four rupture lengths using a 3 km zone for computing the afterslip. The amount of afterslip can vary considerably due to the spatial variation of coseismic slip as illustrated by the thin, long dashed lines that show the total slip evolution for a few illustrative cases. The color version of this figure is available only in the electronic edition.

Probabilistic Estimates of Surface Coseismic Slip and Afterslip for Hayward Fault Earthquakes influences of interseismic creep on coseismic slip (corresponding to vertical gradients in slip of −0:10 m=km to −0:20 m=km), the expected (median) coseismic slip at the sites ranges from about 0.0 m for the 35 km rupture length (Mw 6.5) to about 0.5 m for the 90 km rupture length (Mw 7.1). This corresponds to more than a 60% reduction in the expected coseismic slip relative to the case where the effects of interseismic creep on coseismic slip are ignored (vertical gradient in slip of 0 m/km). The total slip (coseismic slip plus afterslip) was computed using the 3-km zone for afterslip at the sites, and these slip gradients span a large range. The gradient of −0:20 m=km gives total slip values of about half as large as those for a completely locked fault (vertical gradient of 0), while the gradient of −0:10 m=km gives values that are roughly the same as those for a completely locked fault. The larger total slip values associated with smaller gradients would imply reduction in the interseismic creep rate following the afterslip in order to be consistent with the geologic slip rate for the Hayward fault. Such a scenario is inconsistent with the observations that the creep rates do not appear to have decreased following the 1868 earthquake (Lienkaemper and Galehouse, 1997; Schmidt and Bürgmann, 2008). The uncertainty in the Funning et al. (2007) geodetic model of creeping regions was excluded from the Monte Carlo sampling. However, the wide range of vertical gradients in the sampling captures the general effects of varying the boundaries of the creeping regions. This is especially true for the depth extent of interseismic creep, which is relatively poorly constrained in the geodetic model. Relative to a nominal vertical gradient in slip in creeping regions and the boundaries of the creeping region, the effects of deeper or shallower interseismic creep are equivalent to smaller or larger vertical gradients in the slip distribution within the creeping regions. Thus, our sampling of a wide range of vertical gradients in the slip distribution also captures, to some degree, the uncertainty associated with the location and size of creeping regions.

Slip on Multiple Fault Strands Observations of coseismic slip and afterslip at Parkfield indicate that afterslip can occur on a different fault strand than the coseismic slip. In the 2004 Parkfield earthquake, afterslip but no coseismic slip occurred on the main trace of the San Andreas, whereas coseismic slip but no afterslip occurred on the surface trace known as the southwest fracture zone (Murray and Langbein, 2006). Our analysis provides estimates of the amount of deformation that can be expected at three sites but does not distribute the slip onto different strands of the fault. Careful measurements across all Hayward fault strands following an earthquake rupture are critical to assessing where coseismic slip and afterslip are occurring. Such measurements can also be used to identify the afterslip progression relative to the median and  one S.D. curve in Figure 14.

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Lateral versus Oblique Slip In estimating the coseismic slip and afterslip we assumed purely right-lateral slip. Observations suggest that afterslìp tends to have the same orientation as the coseismic slip (Sharp et al., 1982). The 1964 elevation profile of the Claremont water tunnel (Blanchard and Laverty, 1966), located about 1 km north of the HTEM alinement array, shows little if any change from its original 1929 grade. This suggests little or no (< 10%) vertical interseismic creep has occurred in the vicinity of site HTEM. The only geologic measurement of vertical slip rate for the Hayward fault, 0:5–0:6  0:1 mm=yr (Lienkaemper and Borchardt, 1996) compared with 9  2 mm=yr of right-lateral slip, also indicates that the ratio of vertical to horizontal slip rate is less than about 10%. Therefore, we believe 10% represents a reasonable upper limit on the amount of vertical component of slip to expect at the three sites and other locations along the Hayward fault. Because we expect afterslip to follow the same slip direction as the coseismic slip, measurement of the three components of fault offset following a large Hayward fault earthquake will provide the best estimate of the orientation of the afterslip. Earthquake Response Although the magnitude of the next large Hayward earthquake is not known, it is expected to be in the range of Mw 6.5 to 7.1. Our estimates of coseismic slip and the amount and duration of afterslip (Table 4; Figs. 13, 14) provide guidance for planning postearthquake response efforts for earthquakes in this magnitude range. Following such a large earthquake event, the earthquake magnitude, amount of coseismic slip, and rate of afterslip will provide a clearer estimate of the total slip expected to develop in the hours, days, months, and years following the earthquake. Small or negligible values of coseismic slip could lead to overly optimistic estimates of the effect of a large earthquake on utility and transportation infrastructure immediately following such an earthquake, because the infrastructure could be significantly affected by up to 0.5–1.5 m (median plus one S.D.) of afterslip in the following hours and days. Due the large uncertainty in the coseismic and afterslip estimates, real-time geodetic and rapid postearthquake response measurements of coseismic and early afterslip are critical for accurate estimates of the total slip that will develop.

Conclusions We further develop models by Aagaard, Graves, Schwartz, et al. (2010) of how creep affects coseismic slip on the Hayward fault through incorporation of additional constraints on surface slip and extending them to provide estimates of afterslip. These models suggest a complex interaction between creep, coseismic slip, and afterslip. In contrast to a locked fault without interseismic creep in which the long-term geologic slip is composed almost entirely of coseismic

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slip, for the Hayward fault the models suggest that the long-term geologic slip will be dominated by contributions from interseismic creep and afterslip with a minimal contribution from coseismic slip. The large uncertainties in our coseismic slip and afterslip estimates illustrate the need for further development of models that can use laboratory experiments and additional observations to more tightly constrain the relationships among interseismic creep, coseismic slip, and afterslip. Similarly, these models also need higher resolution images of the location and rate of interseismic creep in order to construct more precise estimates of coseismic slip and afterslip for large earthquakes on the Hayward fault.

Data and Resources The Funning et al. (2007) creep model was obtained directly from the authors. Many of the figures were generated using Generic Mapping Tools (Wessel and Smith, 1998).

Acknowledgments We thank Thomas Hanks and Jessica Murray-Moraleda for careful reviews of the paper and Tracy Johnson, Tom Horton, Steve Thompson, Keith Kelson, and Dan O’Connell for feedback throughout the course of this study. This work was funded by Bay Area Rapid Transit and the Earthquake Hazards Program of the U.S. Geological Survey.

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Manuscript received 14 July 2011