We investigate the rupture process of a sequence of repeating Mw 2.1 earthquakes on the San Andreas Fault in Parkfield spanning the occurrence of the September 28, 2004 mainshock by inverting seismic moment rate functions obtained from empirical Green's function deconvolution. The results show that these events have extremely concentrated slip patches with radii on the order of 10–20 m, with peak slip between 8.4 and 11.4 cm. The rupture speed and rise time are consistent with values of larger earthquakes. The spatial distribution of stress drop for the events shows low average values 2.5–5.6 MPa and very large peak values of 66.7–93.9 MPa. The results show that strong asperities can exist at small scales on an otherwise weak fault, and helps reconcile differences between traditional spectra-based and tectonic loading methods for determining the stress drop of small repeating earthquakes.
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 On the Parkfield segment of the San Andreas Fault repeating earthquake seismicity is observed with highly similar waveforms (correlation coefficients exceeding 0.98) suggesting that the events occur on the same patch of fault repeatedly [Nadeau et al., 1995]. Surrounding these repeating clusters are areas inferred to creep. A shallow cluster of such repeating events is the drilling target of the NSF EarthScope San Andreas Fault Observatory at Depth (SAFOD) experiment [Hickman et al., 2004]. Imanishi et al.  studied waveforms and spectra for one of the SAFOD target events using data obtained from the SAFOD Pilot Hole, and determined a Mw 2.1 and a depth of 2.1 km. From their corner frequency measurements they find a static stress drop of 8.9 MPa, which is consistent with commonly found values, and a mechanism of frictional sliding on a weak fault as proposed by Lachenbruch and Sass .
 On the other hand, Nadeau and Johnson  proposed an asperity-loading model in which an interseismic loading rate of 2.3 ± 0.2 cm/yr [Murray and Langbein, 2006] and the observed average recurrence interval of the repeating earthquake sequence of 2.89 ± 0.32 yrs [Nadeau et al., 2004] were used to infer that the slip in each event was on the order of 6.6 cm. Using an independent estimate of the scalar seismic moment they then determined the rupture area (radial dimension of 14.3 m) and a stress drop of 240 MPa. If a rigidity of 12 GPa, more appropriate for the shallow depth of the event is used, a stress drop of 100 MPa is obtained with their method. The difference between 8.9 and 100 MPa bears directly on the nature of the faulting mechanics, whether frictional sliding or rock fracture processes are operating in these events. In this study we reconcile the difference in the reported stress drop estimates using a finite source inverse method to determine the rupture area, slip distribution, the spatially variable stress drop and rupture velocity of the small repeating earthquakes.
 We use three-component velocity records from the Berkeley Seismological Laboratory's High Resolution borehole Seismic Network (HRSN) to determine the seismic moment rate functions at each station for 5 events in the sequence of repeating earthquakes (Table 1), one prior to the September 28, 2004, Mw6 mainshock, and four afterward. Figure 1 shows the location of the target events with respect to the HRSN, and Figure 1(inset) shows the relative locations of the Mw2.1 repeating earthquakes and a nearby Mw 0.68 event used as an empirical Green's function (eGf). The relative event locations are based on sub-sample precision waveform cross-correlation measurements and the double-difference relocation method [Waldhauser, 2001]. The 5 Mw2.1 repeating events have centroid locations within about 2 m of each other. The smaller eGf is located about 10 m away and is even within the very small radius inferred for a stress drop of 240 MPa after Nadeau et al. .
Relative locations are based cross-correlation measurements, and Mws are from relative spectral scaling.
 The HRSN is sampled at 250 sps giving an effective bandwith of 100 Hz, and the instruments are located in boreholes at depths between 63 and 298 m, providing an exceptional signal to noise ratio (SNR) in the recordings. In Figure 2 the vertical component waveforms at station VCAB are compared for the Mw2.1 target (EVT4) and the Mw0.68 eGf illustrating an extremely high degree of waveform similarity. This level of waveform similarity is observed for all three components at all stations, and attests to the nearly collocated nature of the events as well as the similarity in their respective focal mechanisms. Together with the exceptional SNR these events represent near ideal case for the empirical Green's function method.
 To obtain the seismic moment rate functions at each station we employed a commonly used spectral domain deconvolution approach in which the complex spectrum of the eGf is divided out of the complex spectrum of the main event. We applied a 1% water level to avoid problems associated with the division of relatively small values in the eGf spectrum. Hough and Dreger [1994, and references therein] give an overview of the method. The basic concept of the method is that if the smaller eGf event is collocated and has the same radiation pattern as the larger event, then the common instrument response, propagation, attenuation, and site effects are removed by the deconvolution process, resulting in the unfettered source spectrum. The inverse Fourier transform yields the pulse-like seismic moment rate function (Figure 2), which is the result of the convolution of two source terms, one dependent on the slip rate and the other the on rupture velocity.
 We performed deconvolutions separately on each of the three components at 8 or 9 of the HRSN stations depending on availability and SNR. Stations EADB and GHIB (Figure 1) were omitted in all cases because of noisy channels. The remaining stations provide excellent azimuthal coverage of the repeating sequence (Figure 1). In order to further reduce noise we stacked the 3-component moment rate functions at each station. The noise level was very low in each case, and this last step was not really necessary but it was included to remain consistent with previous work [e.g., Dreger, 1994].
 Moment rate functions obtained in this manner may be inverted for the spatial distribution of fault slip, and this kinematic source inversion approach has been applied to events ranging from small (MW ∼ 3) to great (MW8) earthquakes [e.g., Mori, 1993; Hough and Dreger, 1994; Dreger, 1994; Antolik et al., 1996] and is therefore well suited for investigating the scaling characteristics of earthquakes. The basic kinematic rupture model is one in which the source nucleates at a point on the fault surface and the spread of slip over the fault plane is governed by a constant rupture velocity. At each point on the fault the slip occurs over a finite rise time governed by a prescribed slip velocity function, assumed here to have the form of a boxcar with a constant duration. Both the duration of the slip velocity function and the rupture velocity are varied to find an optimal set of values.
 In our application we used a 31 by 31 fault with dimensions of 150 × 150 m2, with a corresponding subfault size of 4.8 × 4.8 m2. The fault is assumed to have a strike of 137 and dip of 90 following A. Kim and D. S. Dreger (Rupture process of the 2004 Parkfield Earthquake from near-fault seismic waveform and geodetic records, submitted to Journal of Geophysical Research, 2007, hereinafter referred to as Kim and Dreger, submitted manuscript, 2007). The size of the subfault was chosen to produce a temporally smooth kinematic process with respect to the sample rate of the data, and this subfault dimension is consistent with the 5m wavelength for S waves considering Vs = 2.3 km/s at 2.1 km depth (Kim and Dreger, submitted manuscript, 2007) and the 100 Hz bandwidth.. A slip positivity constraint using the non-negative least squares method of Lawson and Hanson  and a smoothing operator minimizing the spatial derivative of slip were applied. The weight of the smoothing constraint was determined by trial and error by finding the smallest value that produced a smoothed model with close to the maximum fit to the data measured by the variance reduction,
where d and s are the data and synthetic time histories and i is an index over station, component, and time. In addition, we apply the constraint that moment rate functions integrate to the scalar seismic moment obtained independently from Imanishi et al.  for the Mw2.1 2003 event (EVT1, Table 1). The Mws for the other events were obtained from low-frequency spectral ratios with respect to EVT1.
 In each inversion it is assumed that the rupture velocity is constant and that the boxcar slip velocity function has a constant rise time. Using a grid search, we tested rupture velocities of 0.2–2.3 km/s (8–100%) and rise times from 0.004 to 0.052 s to find optimal values and to assess the resolution of the parameters.
 In Figure 3a we show the slip model for EVT4. This model has a rise time of 0.008 s and a rupture velocity of 1.8 km/s (78% of the local shear wave velocity), the median from models within 2% of the peak fit. The allowable range in the kinematic parameters given this level of fit is 1.2–2.3 km/s (52–100% of the shear wave velocity) in rupture velocity and 0.004–0.012 s for rise time. Within this population of solutions there is a trade off in the rise time and rupture velocity where long rise times are associated with fast rupture velocities (or short rupture times) and vice versa. This is expected because of the simple form of the moment rate functions, which are fit very well by the model (Figure 4). It is notable that the obtained rise time is more consistent with a slip pulse rather than crack-like rupture [e.g., Heaton, 1990] and the slip velocity inferred from the ratio of slip to rise time (Table 2) is consistent with values obtained for larger events. For all of the events there is a dominant asperity in which the slip is found to be extremely concentrated, roughly circular with a diameter of about 40 m. The slip in the model varies spatially but is relatively smooth because of the simple form of the obtained moment rate functions (Figure 4) and the applied smoothing constraint. For EVT4 the slip is peaked at the center with a value of 8.6 cm, which is of similar magnitude to the 6.6 cm inferred by Nadeau and Johnson . The average of the dominant asperity is 3.3 cm.
Simple average/average considering only subfaults with slip and positive stress change (stress drop)/stress drop from the close-to-circular primary asperity assuming the circular fault relationship (see Figure 3).
 Because the slip distribution is non-uniform, we use the method of Ripperger and Mai , shown to be consistent with static or dynamic elastic dislocation models, to determine the coseismic stress change (stress drop). This method maps the spatially variable slip on the fault to the spatially variable stress change, or stress drop. Results applied to the SAFOD target event are shown in Figure 3b. In regions of high slip the stress change is positive, indicating a stress drop during rupture. The method also determines the degree of stress increase (negative stress change) on the region surrounding the rupture. The model for EVT4 has a peak stress drop of 80 MPa and averages ranging from 3.7 to 19.7 MPa depending on how the average is calculated (Table 2). Although the peak stress drop does decrease with increased subfault size, we found that it is in the range of 65.8–80MPa for models with the original to 2× coarser discretizations and 38.9–48.2 MPa for 4× and 8× coarser discretizations. The 4× and 8× coarser models do not fit the moment rate data as well.
5. Discussion and Conclusions
 The very high stress drop we obtain for much of the rupture area of the SAFOD repeaters (Figure 3b) is at odds with more traditional spectrally based estimates [e.g., Imanishi et al., 2004]. However, the stress drop averaged from Figure 3b over areas with positive stress drop is only 11.6 MPa, which is close to Imanishi et al.'s  result. On the other hand, the spatially variable high stress drop we obtain is required to fit the shape of the moment rate functions, and the peak is closer to the estimate obtained using the method of Nadeau and Johnson . Thus, the finite source results reconcile these disparate estimates of stress drop, illustrating that the two methods are apparently sensitive to different aspects of the rupture.
 Assuming an average density of 2000 kg/m3, hydrostatic pore pressure and a coefficient of friction of 0.4 give a maximum frictional strength of only 7.8 MPa at the depth of the events. On the other hand, it has been proposed that small dimension asperities with strength approaching that of intact rock can concentrate substantial stress levels [Nadeau and Johnson, 1998; Sammis et al., 1999; Johnson and Nadeau, 2002], and the large peak values we obtain (Figure 3b) are consistent with shear strength from fresh rock fracture experiments [Ohnaka, 2003]. High stress drop repeating earthquakes may represent those relatively isolated, small-scale contact points where large stress concentrations can develop and be released on a fairly regular basis. The much larger fault areas of bigger earthquakes may be frictionally weak but studded with sparsely distributed high strength asperities producing relatively low average stress drops during large earthquake rupture.
 The repeating nature of the events requires a healing mechanism that recovers the high strength in a very short time. Vidale et al.  correlated stress drop and rupture area inferred from extrapolated estimates of corner frequencies for repeating Calaveras fault events with recurrence interval and found that scalar moment increased and source duration decreased with increased recurrence interval consistent with progressive fault healing. We find no such systematic changes in the events we studied. Following the 2004 mainshock, the repeat time of this sequence shortened to a minimum of 69 days because of the elevated post-mainshock loading of the asperity. Fracture and subsequent sliding at the obtained high stress allows for the possibility of rapid thermally controlled mechanisms for the short healing time [Nadeau and Johnson, 1998; Rice, 2006]. Subsequent rapid cooling of melts and gels formed in thermal weakening mechanisms [Rice, 2006] could provide the rapid restrengthening required by the microrepeating earthquake data.
 Lastly, in the first 60 days following the 2004 mainshock, considerable postseismic deformation was observed. Langbein et al.  modelled the observations finding between 5 and 15 cm of postseismic slip in the region of the SAFOD repeating events (inferred from Langbein et al. [2006, Figure 8]). Summing the finite source slip models for EVT2 and EVT3 yields peak and average values of 18.4 and 6.5 cm, respectively, remarkably similar to both the amount of postseismic slip in the Langbein et al.  model and the 13.2 cm of slip from the Nadeau and Johnson  asperity loading model. Please refer also to the auxiliary material for this article.
 Supported by the National Science Foundation (EAR-0510108 & EAR-0537641). We thank an anonymous reviewer and Martin Mai for helpful comments. Contribution 07-15 of the Berkeley Seismological Laboratory.