The rupture process of the Parkfield SAFOD target earthquakes obtained from the empirical Greens function waveform inversion method

Ahyi Kim, Douglas Dreger, and Taka'aki Taira


Nadeau et al. (1995) found that the seismicity on the San Andreas fault at Parkfield is highly clustered and individual clusters consist of sequences of near periodically repeating small earthquakes of similar seismic moment. Nadeau and Johnston (1998) studied the moments, slip rate, and recurrence intervals of these repeating earthquakes, and found that the stress drop of small earthquakes is quite high and, remarkably, increases with decreasing seismic moment. For example, for one of the SAFOD target M2.1 (Hickman et al., 2004) earthquakes, the relationship developed by Nadeau and Johnston (1998) yields a stress drop of 100 MPa (correcting for a rigidity of 12 GPa for hypocentral depth of 2.1 km; e.g. Dreger et al., 2007). On the other hand, a spectral corner frequency method yields an average stress drop for the same earthquake of 8.9-22.1 MPa (Imanishi et al. 2004; Imanishi and Ellsworth, 2006). Recently, Dreger et al. (2007) investigated the rupture processes of a sequence of repeating M2.1 SAFOD target earthquakes using the eGf deconvolution method. In that study, they found that peak stress drops between 66.7-93.9 MPa. We applied an eGf waveform inversion method to the $M_w$2.1 repeating earthquakes and compared rupture process models, static stress drop distributions, and waveform fits with those from the eGf deconvolution method.


Inspired by Hartzell et al. (1978), we directly inverted observed seismograms using the linear least-squares method of Hartzell and Heaton (1983) in which the finite-source is discretized with a finite distribution of point sources in both space and time instead of inverting deconvolved moment rate functions. In this method, the point sources are triggered by the passage of a circular rupture front. The Greens function from each subfault to station is defined by the waveform of a small earthquake, or eGf, located near the hypocenter of each subfault. One advantage of this method is that propagation differences over the fault are better represented for each station. The observed seismogram at location x for the main event, U, is expressed by the discrete form of the general representation theorem:

\begin{displaymath}U(x,t)=\sum_{k=1}^{K}\mu _{k}A_{k}u_{k}\cdot eGf(x,x_{k},t+\delta t_{k})\end{displaymath}

where ${x_{k}}$ is the location of subfault. ${A_{k}}$,${\mu _{k}}$, and ${u_{k}}$ are subfault area, rigidity, and slip for the kth subfault of a total of K subfaults. The phase delay term includes both the delay due to rupture propagation and the travel time difference between the eGf hypocenter and the subfault. We assume that each eGf has the same mechanism as the main event, an acceptable approximation for small earthquakes. The variation of mechanism in the main shock can also potentially be accounted for if eGfs are well distributed over the fault plane. For our San Andreas fault applications, the focal mechanisms of events vary little over a wide range of magnitudes (Thurber et al., 2006). To stabilize the inversion, we employed a slip positivity constraint using the non-negative, least-squares routine of Lawson and Hanson (1974), as well as spatial smoothing. 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,

\begin{displaymath}VR = \left [ 1-\frac{\sum_{i}^{}\left ( d_{i}-s_{i} \right )^{2}}{\sum_{i}^{}d_{i}^{2}}\ \right ]\times 100\end{displaymath}

where ${d_{i}}$ and ${s_{i}}$ represent the data and synthetic time series, respectively. The subscript i is an index over station, component, and time. The size of the subfault was chosen to produce a temporally smooth kinematic process with respect to the sample rate of the data.

Computational setup

We used three-component velocity waveforms recorded at five of the Berkeley Seismological Laboratory's High Resolution borehole Seismic Network (HRSN) stations to examine the rupture processes of the same 5 repeating earthquakes studied by Dreger et al. (2007). We also performed eGf deconvolution inversions using the 5 stations, to allow comparison with our waveform inversion results. Locations of repeating earthquakes and stations are shown in Figure 2.22. For the inversion, we used the same fault parametrization as Dreger et al. (2007), a ${150 \times 150 m^{2}}$ fault decomposed into 31 by 31 square subfaults of ${4.8 \times 4.8 m^{2}}$. The strike, dip, and rake are 137, 90, and 180, respectively. The records from HRSN are sampled at 250 sps, which gives an effective bandwidth of 100 Hz. The subfault size is consistent with approximately a quarter of the wavelength of S waves assuming a velocity of 2.3 km/s at 2.1 km depth. We used the same $M_w$0.68 event (eGf 1) as Dreger et al. (2007) for the Greens function. eGf 1 is located about 10 m away from the centroid of the target events (Dreger et al., 2007). Since the eGf is a real earthquake, the waveforms contain a rise time which can bias the slip model to be more compact than it actually is. To cancel this effect, we convolved an assumed eGf rise time of 0.008 seconds with the data before performing the inversion. (Dreger et al. (2007)).

Figure 2.22: Map showing the HRSN station locations (blue triangles are the stations used in this study) and SAFOD repeating target events (star). The inset shows the cross-sectional view of the relative locations of the five studied M2.1 repeating events (larger colored circles), and $M_w$0.68 eGf 1 (red dot) and $M_w$0.64 eGf 6 (blue dot). The size of the circles shows the respective areas of a 100 MPa event. For comparison, the large gray circle shows the inferred area for a 10 MPa event.
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Inversion results and interpretations

As an example of the results, slip model and stress drop distribution are shown for EVT4 in Figure 2.23. Typically, the slip distribution is circular with a diameter of about 50 m, and the average slip of the main asperity is 3.3 -4.0 cm. The peak slip amplitudes were found to be 10-13 cm. Using our spatially variable slip distribution model, we computed the coseismic stress change on the fault plane using the method of Ripperger and Mai (2007) (Figure 2.23). The computed static stress drop distribution shows that the small patch has a peak stress drop of 63.9-89.4 MPa, which is consistent with the values reported by Dreger et al. (2007). However, the average stress drop of 5 MPa is consistent with the typical range of between 1-10 MPa. The rupture process of small earthquakes is complicated, just as in large earthquakes, in terms of spatially variable slip. Also, the SAFOD target events appear to occur on a localized fault patch of high strength capable of earthquakes with high stress drop The very high peak stress drop that is obtained implies that small-dimension, high-strength asperities exist on the San Andreas Fault.

Figure 2.23: Left: Slip model obtained for EVT4. Right: Stress drop distribution for EVT4. For both figures, the number indicates the peak/average slip for each event.
\begin{figure}\centering\epsfig{, width=8cm}\setlength{\abovecaptionskip}{0pt}\setlength{\belowcaptionskip}{0pt}\end{figure}


We thank Bob Nadeau for providing the repeating earthquake catalog.


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