Data and Methods

We use three-component 100Hz velocity records from the High Resolution borehole Seismic Network (HRSN) to determine the seismic moment rate functions at each station for a Mw2.1 event in the repeating cluster being targeted by the NSF SAFOD experiment. Figure 2.2 shows the location of the target events with respect to the HRSN, and Figure 2.2b shows the relative locations of the Mw2.1 repeating earthquakes, and a nearby Mw0.68 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, giving centroid locations within about 2m of each other. The smaller eGf is located about 10m away and is even within the very small radius inferred for a stress drop of 240MPa after Nadeau and Johnson (2004).

In Figure 2.3, the vertical component waveforms at station VCAB are compared for the Mw2.1 target 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 an ideal case for the empirical Green's function method.

Figure 2.2: Map showing the locations of HRSN stations (inverted triangles), the SAFOD repeating target events (concentric circles), the locations of the 1966 and 2006 mainshock epicentres (stars), and background seismicity (gray dots). The inset shows a cross-sectional view of the relative locations of the repeating events (larger circles), and the Mw0.68 empirical Green's function event (small circle). The size of the circles shows the respective areas of a 240MPa event. For comparison the large gray circle shows the inferred area for a 9MPa event.
\begin{figure}\epsfig{file=dreger07_1_1.eps, width=8cm}\end{figure}

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. Hough and Dreger (1994) and references therein give an overview of the method. The basic concept 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 2.3).

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 2.2) were omitted in all cases due to noisy channels. The remaining stations provide excellent azimuthal coverage of the repeating sequence (Figure 2.2). The moment rate functions may then be inverted for the spatial distribution of fault slip (Dreger, 1994).

In our application we used a 31 by 31 fault with dimensions of 150 x 150 $m^2$, with a corresponding subfault size of 4.8 x 4.8 $m^2$. The fault is assumed to have a strike of 137 and dip of 90. The size of the subfault was chosen to produce a temporally smooth kinematic process with respect to the sample rate of the data. A slip positivity constraint, 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.

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 to 2.3 km/s (8-100%), and rise times from 0.004 to 0.052 sec to find optimal values and to assess the resolution of the parameters.

Figure 2.3: a) Vertical component waveform for the Mw2.1 event recorded at station VCAB. b) Vertical component waveform for the Mw0.68 eGf event recorded at station VCAB. c) The moment rate function obtained by deconvolving (b) from (a).
\begin{figure}\epsfig{file=dreger07_1_2.eps, width=8cm}\end{figure}

Figure 2.4: A) Slip distribution obtained by inverting moment rate functions from 9 HRSN stations. The hypocenter (white square) is in the center of the assumed rupture plane. The white circle shows the area of a 10MPa event. B) Stress drop obtained by applying the method of Ripperger and Mai (2004)
\epsfig{file=dreger07_1_3.eps, width=16cm}\end{center}\end{figure*}


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