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.