4.2.4 Summary of effects of spatial attenuation and velocity
corrections in stress drop calculation
Figure 3 compares the results of our inversions to investigate the
effects of spatial binning (with depth and along-strike) and varying
rupture velocity on the separation of source and path effects in the
Parkfield region. The relatively wide frequency range of the borehole
data compared to previous spectral decomposition studies provides
improved resolution of source from the path and site effects. Figure 5
shows the distribution of stress drops obtained using each strategy. Not
surprisingly, the largest effects are on the depth dependence of the
resulting stress drop estimates, but the magnitude dependence is also
affected.
Including a depth-dependent attenuation correction (Strategy 3)
significantly reduces both the magnitude and depth dependence, and the
standard deviation of the resulting stress drops (Figure 5) compared to
Strategy 1. The finer scale lateral binning of Strategy 5 (10 km
along-strike binning) only slightly reduces the dependence and standard
deviation of Strategy 3 (no along-strike binning), suggesting that
along-strike variation in attenuation is less significant that with
depth in the study area.
The comparisons between Strategies 6 and 5, between 4 and 3, as well as
between 2 and 1, show that velocity correction also significantly
reduces the dependence of the calculated stress drops on both magnitude
and depth.
The variation in velocity has more effect at shallow depths, where the
velocity is increasing rapidly with depth, than deeper, where it is
changing more slowly. Conversely, the ECS correction has a larger effect
at greater depths, implying that the difference in path effects
continues to be significant in this depth range.
The ECS attenuation correction and velocity corrections also have
different effects on the magnitude and depth dependence of stress drops.
From Table 1 and Figure 4, magnitude dependence is more significantly
reduced by improved attenuation corrections than depth-dependent
velocity correction, while for depth dependence, the influence of
velocity correction is more significant. AS2007 also found that depth
dependence in stress drop is sensitive to the assumed velocity
structure, and did not resolve any significant increase in stress drop
with depth. They also found no dependence of stress drop on magnitude in
a similar range to this study.
Figure 6 compares the empirical correction spectra (ECS) obtained,
showing greater variability with depth than along strike, consistent
with the effects on stress drop. These spatial-bin specific ECS correct
for site effects common to all events. They also attempt to correct for
the difference between the average regional travel-time dependent path
terms from the original spectral decomposition, and the real, along path
attenuation that depends on the individual source and station locations.
It is hard to interpret the ECS in terms of absolute attenuation, but
their differences do provide quantitative information about the
variability in attenuation for earthquakes in different source
locations. We fit the slopes of ECS spectra between 20 and 60 Hz,
following Anderson & Hough (1984), to quantify the variability of t* in
different depth ranges, and along strike (Figure 6).
As we apply increased correction for spatial variation of velocity and
attenuation, the standard deviations progressively decrease from 0.66
for Strategy 1 to 0.41 for Strategy 6 (Figure 5a), implying that the
variability in stress drop in an earthquake population strongly depends
on data processing, and trades-off with attenuation and velocity
corrections. Meanwhile, we calculate the root mean square (RMS) of the
misfits between ECS and individual source spectra by frequency sample
for Strategy 2, 4 and 6, and find that the median RMS of misfits from
the 3 strategies synchronizes with the drop of stress drop standard
deviations, which suggests that the proposed corrections may lead to
better spectral fitting; however, by introducing the corrections we also
introduce more free parameters that trade-off with the misfits.