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.