1 Introduction
The stress drop is a measure of the average stress changes during an
earthquake. It is one of the most important source parameters that is
directly related to strong ground motion and fundamental problems
earthquake physics. It is superficially easy to compute stress drop from
spectral analysis based on corner frequency, but the high variability,
both within individual studies (e.g., Abercrombie et al., 2017) and
between different studies (e.g., Pennington et al., 2021), indicates
that stress drop measurements are often subject to large uncertainties
(e.g., Abercrombie 2021). Published measurements span 3 orders of
magnitude, and until real variability can be distinguished from the
large uncertainties, the potential for using stress drop measurements to
probe the physics of the rupture process, and to assist in the
prediction of future ground motions is severely limited (e.g.,
Hardebeck, 2020; Molkenthin et al., 2017).
Earthquake stress drop can help understand whether earthquakes with
different magnitudes are associated with the same physical processes
(self-similarity) (Aki, 1967). Stress drop has been found to be
independent of magnitude by multiple studies, and compilations of
studies, covering different magnitude ranges, from small and moderate
(Allmann & Shearer, 2007; Goebel et al., 2015; Imanishi & Ellsworth,
2006; Uchide et al., 2014) to large ones (Allmann & Shearer, 2009). In
contrast, other studies have reported non-self-similarity, typically
over smaller magnitude ranges (Imanishi & Uchide, 2017; Mayeda et al.,
2005; Oye et al., 2005). Spatial and temporal variations of stress drop
have the potential to reveal heterogeneities and changes in stress
distribution within fault zones (e.g., Allmann & Shearer, 2007; Chaves
et al., 2020; Chen & Shearer, 2013; Moyer et al., 2018; Ruhl et al.,
2017; Shearer et al., 2006a; Uchide et al., 2014)). However, different
studies of the same region have resolved different spatial and temporal
distributions; for example, the spatial pattern in Japan from (Oth,
2013) appears different from that by Uchide et al., (2014) and Yoshida
et al., (2017), who used more localized attenuation corrections. Some
studies have also observed an increase in stress drop with depth that
may reflect increasing stress on the faults, but such dependence may be,
at least partially, an artifact of changes in attenuation and velocity
with depth (Allmann & Shearer, 2007; Sumy et al., 2017; Abercrombie et
al., 2020). Abercrombie (2014) analyzed three repeating clusters of
small earthquakes on the San Andreas Fault at Parkfield, and found the
temporal changes of only one cluster (largest magnitudes) agreed with
the results of Allmann & Shearer (2007) for the same earthquakes.
The variation in stress drop observed between different earthquakes
could be due to real variation in the rupture processes, but the
discrepancy among different studies indicates that random and systematic
uncertainties are significantly distorting results; the differences
between studies are often significantly larger than the calculated
uncertainties (e.g., Huang et al., 2017; Pennington et al., 2021). The
most likely causes of these problems are the simplifying assumptions
required (concerning source geometry and attenuation structure), and the
inherent ambiguities in separating source and path effects using
seismograms with limited frequency range (e.g., Abercrombie, 2021).
Shearer et al. (2019) and Pennington et al. (2021) both demonstrated
that relative variability is more reliable than absolute values, and
that consistency between different approaches provides confidence in the
results.
To understand better the real uncertainties in stress drop estimates,
and improve the quality of the measurements, it is important to perform
detailed analyses of the effects caused by frequency bandwidth limits of
the data, spatiotemporal variations of attenuation, and possible rupture
velocity changes, among other things. As the problems became clearer,
many studies have focused on reducing the scatter and improving the
resolution of stress drop measurements, and quantifying more realistic
uncertainty measurements using high-quality datasets and careful data
processing. Baltay et al. (2011) showed that results using stacked
Empirical Green’s Functions (EGF) to correct for path and site effects
had much lower standard deviation than previous studies; Kwiatek et al.,
(2014) f ound the stress drop scattering from EGF analysis is
significantly reduced from fixed attenuation correction. Chen &
Abercrombie (2020) used synthetic tests to develop an improved stacking
approach and retrieved stress drop measurements with low standard
deviation and reliable spatiotemporal patterns for an induced sequence
in Oklahoma. Shearer et al. (2019) compared small earthquake corner
frequencies estimated using a spectral ratio method (local EGF varying
by earthquake) and spectral decomposition method (a single global EGF
for all the earthquakes), and found the spectral ratio corner
frequencies are slightly larger than those from the spectral
decomposition method. They also showed that the frequency range of the
Southern California regional seismic network data is insufficient to
resolve source scaling and absolute values of stress drop, and their
variation, independently.
To investigate in more detail the relative effects of limited frequency
range and assumptions about attenuation structure on stress drop
measurements, we need an exceptionally well-recorded data set. The
Parkfield segment of the San Andreas Fault in California has been
densely instrumented for decades as a consequence of the Parkfield
Earthquake Prediction Experiment (Bakun & Lindh, 1985), and the San
Andreas Fault Observatory at Depth (SAFOD, Zoback et al., 2011). This
instrumentation includes a borehole seismic network (Malin et al.,
1989), that is able to record higher frequency signals than the
surrounding surface networks. Allmann & Shearer (2007) performed a
spectral-decomposition based study of earthquake stress drop in the
region using the surface recordings. The occurrence of the 2004 M6
earthquake enabled them to look for temporal as well as spatial
variation in stress drop. However, many of the earthquakes included in
the analysis were relatively small (M<2) compared to the
frequency range of the data available. Here we use the higher-frequency
borehole recordings to perform a similar analysis, and can investigate
directly the effects of using the limited frequency range of the surface
data. We are also able to include data for a further decade following
the 2004 M6 earthquake than the earlier study.
The large body of previous work in the region provides us with context
in which to interpret our results. The seismicity is well-located (e.g.,
Waldhauser et al., 2004), and includes sequences of near-identical,
repeating sequences (Nadeau & McEvilly, 1999, 2004). Analysis of
variability in the moment and timing of these repeating earthquakes has
revealed intriguing spatial and temporal variation in the stressing rate
(e.g., Chen et al., 2010; Lengliné & Marsan, 2009; Nadeau & Johnson,
1998). Measurements of aseismic slip in the region have also revealed
the gradual transition from creeping north-west of Middle Mountain to
fully locked to the south-east of the HRSN, as well as temporal
variations related to the 2004 M6 earthquake, and other events (e.g.,
Murray & Langbein, 2006). There have also been multiple studies of the
coseismic and postseismic slip, and stress changes associated with the
2004 earthquake (e.g., Dreger et al., 2005; Jiang et al., 2021). The
velocity structure (e.g., Thurber et al., 2004) and attenuation
structure (e.g., Abercrombie et al., 2000; Bennington et al., 2008) are
well known, and changes in both following the 2004 M6 earthquake have
been observed (e.g., Brenguier et al., 2008; Kelly et al., 2013; Sheng
et al., 2021). Allmann & Shearer (2007), subsequently referred to as
AS2007, found spatial variations in stress drop that did not obviously
correlate with other parameters, and remained relatively stable,
unaffected by the 2004 M6 earthquake. They also observed temporal
changes in both stress drop and attenuation following the M6 earthquake,
with different regions of decrease, and increase in each, but the
temporal changes in stress drop were small compared to the spatial
variation. Detailed studies of individual specific repeating sequences
using the borehole recordings reported a decrease in stress drop
followed by recovery for some sequences, but not others (e.g.,
Abercrombie, 2014, 2021; Kim et al., 2016), and also increases in
attenuation and recovery on a similar time scale (Kelly et al., 2013).
No large-scale analysis of the borehole recordings for earthquake source
parameters has been performed to date.
We use borehole recordings and the spectral decomposition approach
(Shearer et al., 2006a; AS2007; Chen & Abercrombie, 2020) to estimate
stress drop for over four thousand earthquakes M0-4, from 2001-2016, in
the San Andreas Fault zone at Parkfield. First, we investigate the
effects of using different inversion methods, and allowing for spatial,
depth, and temporal variation in attenuation and rupture velocity. We
then compare the results obtained with different frequency ranges to
quantify the effects of the limited bandwidth that is typically
available for studies using surface recordings. Finally, we interpret
our preferred results of spatial and temporal stress drop variation in
the context of existing observations of the structure, and distribution
of seismic and aseismic slip on the fault.