3.3 Stacking method to obtain an Empirical Correction Spectrum (ECS)
We use an adaptation of the stacking method developed and used by Shearer et al. (2006a) and AS2007 to invert the event spectra for a source model, source parameters and an empirical Correction Spectrum (ECS). Shearer et al. (2006a) stacked the event spectra into small Mw bins, and then inverted for a single ECS common to all events included in the stack and the best fitting stress drop common to all stacked M ranges, assuming the Brune (1970) source model. They also used the same approach to calculate an ECS for each event based on the 200 nearest neighbors, which involves variable spatial averaging due to the variability of the seismicity distribution; AS2007 used the latter approach. Trugman & Shearer (2017) also inverted for a common ECS, and allowed for a Mw dependence of stress drop. Shearer et al. (2019) showed that this Mw dependence may not be resolvable with many data sets, finding strong trade-offs among the scaling factor, spectral fall-off rate, and reference stress drops.
Here, we follow a modified stacking approach proposed by Chen & Abercrombie (2020), known as SNSS (Stacking-No-Self-Similarity assumption) to fit the stacked event spectra and solve for the ECS and mean stress drop. The SNSS approach does not include any assumption about stress drop scaling with magnitude, but inverts for the best fitting ECS common to all bins, while allowing the stress drop in each magnitude bin to vary independently. Chen & Abercrombie (2020) developed a series of synthetic experiments to validate the SNSS approach, and found that it performed better than the original stacking approach. They were unable to test approaches that simultaneously solve for scaling factors, because synthetic experiments indicated that their data set was too limited, with too much inter-event variability to resolve a scaling factor.
We calculate stacked spectra for each calibrated magnitude bin from Mw0.9 to Mw4.0 in increments of 0.3 M units of calibrated magnitudes. Chen & Abercrombie (2020) found that the SNSS approach can recover the true input stress drop when the corner frequency of the lowest magnitude bin is within 80% of the upper limit of the frequency range of the data. This implies that for the upper frequency limit of 60 Hz in this study, the lowest magnitude bin should have a corner frequency of 48 Hz or lower for unbiased stress drop estimation. Assuming the average stress drop about 6 MPa determined by AS2007, the estimated corner frequency of Mw=0.9 (the smallest magnitude bin that is well recorded) would be 75 Hz, Mw=1.2 would be 53 Hz, and Mw=1.5 would be 38 Hz; thus the corner frequency of the Mw0.9 bin, is too high to constrain in the inversion. Based on these estimations, we combine the SNSS and the fixed-stress drop approach in Baltay et al. (2010) to develop a hybrid-adaptive approach that enables us to include the large number of earthquakes in the Mw0.9 bin, but constrain the inversion with larger Mw events. We first apply the SNSS approach to magnitude bins with Mw≥1.5 to obtain the best-fitting reference stress drops for the Mw1.5 bin. Then we fit the stacked event spectra in the Mw0.9 bin fixing the stress drop to the value we obtain for the Mw1.5 bin, to calculate an ECS following Baltay et al., (2010). We refer to this modified approach as the Self-Adaptive SNSS method (but still abbreviate it as SNSS in this study for simplification). This hybrid approach has the advantage of obtaining unbiased stress drop values that are specific to the dataset, instead of an assumed global average value as in Baltay et al. (2010), while also including solutions for small earthquakes and an estimation of ECS from the magnitude bin with most abundant earthquakes.
To calculate actual source parameters, each individual event spectrum is corrected using the common ECS determined from the SNSS inversion, and then fit using the selected source model and assumed constants. We perform a comparison of the original method of Chen & Abercrombie (2020) with our new hybrid SNSS method using a spatially compact dataset with 220 earthquakes, and find generally consistent results. The new method leads to lower magnitude scaling (Figure 2), as a consequence of the different ECS in the two methods, most likely representing the well-known increasing uncertainties and trade-offs as the corner frequency approach the limits of the frequency range of the recorded signal (e.g., Abercrombie, 2015; Ruhl et al., 2017). but there is no solid evidence of how the change compares to the magnitude scaling due to frequency band limitation, or if lower magnitude scaling means more accurate stress drop estimation.