3.1 Spectral decomposition to obtain relative event source spectra
To measure the source parameters, we need to isolate the source contribution from the other effects within the recorded earthquake waveforms. An observed waveform (Figure S1) can be represented as the convolution:
\begin{equation} S\left(t\right)=\text{ET}\left(t\right)*\text{ST}\left(t\right)*\text{PT}\left(t\right)\text{\ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\nonumber \\ \end{equation}
where ET, ST and PT refer to the event term, site (or station) term and path term, respectively, all functions of time (t ). Transforming Equation 1 to the frequency domain, and taking the logarithm converts the equation into a linear system that can be solved iteratively for ET, ST and PT as functions of frequency, following the Spectral Decomposition method developed by Shearer et al., (2006a) using a large number of earthquakes and stations:
\begin{equation} S\left(f\right)=\text{ET}\left(f\right)+\text{ST}\left(f\right)+\text{PT}\left(f\right)+R(f)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\nonumber \\ \end{equation}
where \(R(f)\) is a residual term from the solution of the overdetermined equations. However, spectral decomposition only obtains the relative shape of the source spectra, and an additional correction is required to remove any site effects that are common to all events.
After calculating the event spectra (ET (f ), which can be considered relative source spectra), we follow Shearer et al. (2006a) to calculate the relative seismic moment of each event assuming it is proportional to their low frequency (2-4 Hz) amplitudes. To convert these relative moment estimates to actual moments and moment-magnitudes, we calculate their relationship with the catalog (Waldhauser & Schaff, 2008) local magnitudes. We observe a linear relationship for events with either ML≥1.40 or ML<=0.83, but an excess of events with ML~1 (Figure S2). We therefore exclude these ML~1 earthquakes from the calibration. We calculate the best fitting linear relationship for events with ML≥1.40 and ML<=0.83, and assume that ML = Mw when ML = 3.0 (Shearer et al., 2006a; AS2007) to derive moment estimates for all earthquakes in our dataset. The linear relationship has a slope of 0.92, consistent with previous studies of earthquakes in this magnitude range that typically find values of about 1 (e.g., Abercrombie, 1996; Ben-Zion & Zhu, 2002; Hanks & Boore, 1984), smaller than the 1.5 assumed for larger earthquakes in the original definition of Mw (Kanamori, 1977).
Estimating the actual source spectra from the event terms requires either an assumption of a reference site (e.g., Bindi et al., 2020; Oth et al., 2011 ), or a source model (Shearer et al ., 2006a) to correct for higher frequency attenuation and amplification effects. We follow the approach of Shearer et al. (2006a), calculating empirical correction spectra (ECS) to extract estimates of the absolute source spectra. The basic Spectral Decomposition approach also assumes a simplified attenuation structure in which PT(t) depends only on the travel time. Any spatial variation in attenuation, including a dependence on source depth (as observed by Bennington et al., 2008) is not included, and will be absorbed into the event term and bias the resulting source spectra (e.g., Shearer et al., 2019; Abercrombie et al., 2020). In other words, ET(t) in equation (1) will include both the real source term and a function that includes common effects at all events and all sites, and source region specific attenuation. We investigate the effects of this assumption, and whether it is possible to address its effects by comparing a single ECS for the entire data set with separately calculated ECSs for different spatial source regions. Tomography models have shown strong spatial variations of material properties in the study region (e.g., Bennington et al., 2008; Thurber et al., 2004; Thurber et al., 2006; Zeng & Thurber, 2019; Zhang et al., 2007). We search for the most appropriate strategy that allows us to remove the influence of heterogeneous attenuation on stress drop estimations while maintaining an adequate number of events for stable stacking analysis.