5.2 Frequency range and Magnitude range of reliable parameters
Previous work has shown that the limited frequency range of the
earthquake spectra available for modeling can significantly bias the
resulting estimates of corner frequency. For example, Shearer et al.
(2019) showed that the frequency bandwidth of regional Southern
California Seismic Network data (~2-20 Hz) is inadequate
to distinguish between different source models and scaling, and
Abercrombie (2015) showed how decreasing frequency range biased the
results in empirical Green’s function analysis. Here our relatively wide
frequency range (2-60 Hz), larger than almost all previous spectral
decomposition studies provide increased resolution.
To further explore the influence of the frequency band on corner
frequency estimations, and guide our interpretation of our new results,
we repeat our analysis limiting the frequency range to first, 2-20 Hz
and second, 2-40Hz. Figure 7 (a, b) shows that corner frequencies
calculated with a narrower frequency band are much more scattered than
those with a wider frequency band, especially when the corner frequency
exceeds the upper limits. In addition, higher corner frequencies are
systematically underestimated when using a lower maximum data frequency
(Figure 7c). Previous work using an empirical Green’s function approach
by Abercrombie (2015), Abercrombie et al. (2017) and Ruhl et al. (2017)
found that systematic low bias in corner frequency estimates starts at a
half or two thirds of the maximum frequency of the data. Chen &
Abercrombie (2020) found similar results using spectral decomposition,
although their synthetic tests suggested that sometimes corner
frequencies of 80% the maximum frequency could be resolved using the
SNSS approach. In Figure S5 we show that our new hybrid SNSS approach
somewhat mitigates the problem compared to the original SNSS approach in
Chen & Abercrombie (2020); in the hybrid approach the smallest
magnitude bin have stress drop derived from the largest magnitude bins
that is used to calculate the empirical correction spectra. We still
observe increased variability at smaller magnitudes, probably largely
reflecting increased uncertainties, but we see less systematic bias.
AS2007 used surface stations and a narrower frequency range in their
analysis, the narrower frequency range being a direct consequence of the
near-surface attenuation and higher noise in the surface recordings. It
is likely that the lower frequency bandwidth is causing the higher
standard deviation and the lower median values reported by AS2007,
compared to the current analysis (Figure 5 and 7). Figure 7d shows a
constant shift of approximately a factor of 1.2 when compared to our
preferred, full bandwidth results. This translates to approximately 1.7
times (1.23) difference in stress drop, consistent
with the difference of the median stress drop values in these two
studies; we obtain ~10MPa for the borehole dataset,
while AS2007 report ~6MPa using similar constants in
equation 5. In Figure 7c we see that the AS2007 corner frequencies are
lower than those from our analysis using the same (2-20 Hz) frequency
range. This suggests that the more attenuated surface data may tend to
underestimate the stress drop during the ECS calculation, although the
random uncertainties are also large from such a limited range (e.g.,
Shearer et al., 2019).
In all analyses shown in Figure 7, some events with corner frequencies
near the limits of the data can be very high, or low. This is possibly
due to a combination of the effects of limited bandwidth with complexity
in the earthquake sources themselves, that is ignored in the simple
spectra fitting. Abercrombie (2021) demonstrates how this can bias the
results, especially for particularly complex events, and Yoshimitsu et
al. (2019) also note how the inappropriateness of the simple Brune
source model individual earthquakes greatly increases the uncertainties
of stress drop estimates.
Given the above discussion, Mw1.5 is the lowest magnitude for which we
should be able to resolve corner frequency without bias using our
approach, given the maximum signal frequency of 60 Hz and a reference
stress drop of 10 MPa. In practice, we find that the median stress drop
of the M>1.1 earthquakes (median\(\operatorname{}\left(\text{Δσ}\right)=1.24)\ \)is not
significantly different from that of the M>1.5 events
(median \(\operatorname{}\left(\text{Δσ}\right)=1.32)\), given the
standard deviation of the whole data set (0.43). We also compare the
spatiotemporal variations of stress drop with different magnitude
cutoffs in Figure S10, and find that patterns with M>1.1 is
similar to higher magnitude cutoffs, therefore, we include earthquakes
with M>1.1 in our interpretations.