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.