4.3 Temporal attenuation correction
The 2004 M6 earthquake caused temporal changes in both the velocity and attenuation structure at Parkfield (Kelly et al., 2013; Brenguier et al. 2008; Sheng et al., 2021). To investigate any temporal variation in stress drop, we need to ensure that we are not just misinterpreting temporal variation in attenuation and other path and site effects. Hence, following AS2007 and Abercrombie et al. (2020), we divide the dataset into distinct time periods and calculate separate inversions for each. To balance the number of earthquakes needed for stable inversion, and the temporal dependence of the observed changes, we divide our dataset into three time periods (Figure S6): (1) before 2004 M6 earthquake; (2) one year following the M6 earthquake (Sep. 2004 - Sep. 2005), the approximate duration of significant attenuation and velocity changes; (3) after September 2005. We then repeat the first four different strategies used in our analysis of spatial variation. We refer to them using the same numbering, with the additional note of being temporally corrected. The first time period, before the M6 2004 earthquake has the fewest earthquakes, and so the ECS and resulting source parameters for all spatial bins in this time interval are consequently the least well resolved (Figure S7). We do not make any temporal corrections for changes in velocity because they are too small (~0.25%, Brenguier et al., 2008; Sheng et al., 2021) to have any significant affect within our resolution and uncertainties.
There are insufficient earthquakes to divide them both along strike and into different temporal ranges for Strategies 5 and 6. We take advantage of the coherent ECSs for spatial bins of the same depth range, and adopt a hybrid-approach to combine temporal corrections and the finer-scale spatial binning for Strategies 5 and 6. First we calculate the difference between the ECS in different time periods in each depth bin, then apply these relative temporal differences to the respective ECS for the spatial bins calculated for the entire time period (Section 4.2.3) to obtain “pseudo-” temporal ECSs for each grid during each time period. Finally, we solve for the source parameters for each earthquake using the corresponding ECS based on occurrence time and location. We refer to these, our preferred, parameters as Strategy 6, with temporal correction.
Figure 5 shows the minimal effect of the temporal attenuation correction on the distribution of stress drops obtained using the different strategies. We suspect this is partly related to the increased uncertainty in the inversions, due to the smaller numbers of events in each one, offsets the increased number of free parameters. The inversions with and without temporal binning have very similar median stress drops, and reveal similar behavior of stress drop with respect to depth and magnitude (Table 1 and Figure S7).
Figure S8 shows the ECS changes for each depth bin for the three time periods, with higher amplitudes of ECS indicating lower attenuation (see supplementary text S1). We quantify the relative change in attenuation with time following the approach mentioned above. From Pre-2004 to 2004-2005, the overall ECS amplitude decreases slightly for 1-4 km (t* increase of 0.05ms) and more significantly for 4-5 km, 5-8 km and 8-15 km (t* increases of 1.75ms, 0.84ms and 0.75ms, respectively). From 2004-2005 to Post-2005, we see the reverse behavior, with the overall ECS amplitude increasing and t* decreasing by 0.18ms (1-4 km), 0.30ms (4-5 km), 1.20ms (5-8 km) and 0.32ms (8-15 km). The t* variations suggest increased attenuation immediately following the 2004 M6 earthquake, and gradual recovery over long term, which is consistent with previous studies of attenuation changes (e.g., Kelly et al., 2013). Figure S9 shows individual event corner frequencies before and after temporal correction has an average ratio of 1.011 with standard deviation of 0.34, suggesting the influence of temporal correction is relatively small.