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.