A methodology for the estimation of kappa (k) for large datasets. Example application to rock sites in the NGA East database

This report reviews four of the main approaches (two band limited and two broadband) currently used for estimating the site k0: the acceleration slope (AS) above the corner frequency, the displacement slope (DS) below the corner frequency, the broadband (BB) fit of the spectrum, and the response spectral shape (RESP) template. Using these four methods, estimates of k0 for rock sites in Central Eastern North America (CENA) in the shallow crustal dataset from NGAEast are computed for distances less than 100 km. Using all of the data within 100 km, the mean k0 values are 8 msec for the AS approach and 27 msec for the DS approach. These mean values include negative k estimates for some sites. If the negative k values are removed, then the mean values are 25 msec and 42 msec, respectively. Stacking all spectra together led to mean k0 values of 7 and 29 msec, respectively. Overall, the DS approach yields 2 3 times higher values than the AS, which agrees with previous observations, but the uncertainty of the estimates in each case is large. The AS approach seems consistent for magnitudes down to M3 but not below. There is large within station variability of k that may be related to differences in distance, Q, complexity along the path, or particular source characteristics, such as higher or lower stress drop. The station to station differences may be due to site related factors. Because most sites have been assigned Vs30 2000 m/sec, it is not possible to correlate variations in k0 with rock stiffness. Based on the available profile, the individual spectra are corrected for crustal amplification and only affect results below 15 Hz. Since the AS and DS approaches are applied over different frequency ranges, we find that only the DS results are sensitive to the amplification correction. More detailed knowledge of individual near surface profiles may have effects on AS results, too. Although k is considered to be caused solely by damping in the shallow crust, measurement techniques often cannot separate the effects of damping and amplification, and yield the net effect of both phenomena. The two broadband approaches, BB and RESP, yield similar results. The mean k0_BB is 5+ 0.5 msec across all NEHRP class A sites. The k0_RESP for the two events examined is 5 and 6 msec. From literature, the average value of k0 in CENA is 6 2 msec. This typical value is similar to the broadband estimates of this study and to the mean kAS when all available recordings are used along with all flags. When only recordings with down going FAS slope are selected from the dataset, the mean value of kAS increases by a factor of 2 3. To evaluate the scaling of high frequency ground motion with k, we analyze residuals from ground motion prediction equations (GMPEs) versus k estimates. Using the k values from the AS approach, the average trend of the ln(PSA) residuals for hard rock data do not show the expected strong dependence on k, but when using k values from the DS approach, there is a stronger correlation of the residuals, i.e. a k that is more consistent with the commonly used analytically based scaling. The kDS estimates may better reflect the damping in the shallow crust, while the kAS estimates may reflect a net effect of damping and amplification that has not been decoupled. The kDS estimates are higher than the kAS estimates, so the expected effect on the high frequency ground motion is smaller than that expected for the kAS estimates. An empirical hard rock site factor model is developed that represents the combined Vs k0 site factor relative to a 760 m/sec reference site condition. At low frequencies 10 Hz) the residuals do not show the strong increase in the site factors as seen in the analytical model results. A second hard rock dataset from British Columbia, Canada, is also used. This BC hard rock residuals show an increase in the 15 50 Hz range that is consistent with the analytical k0 scaling for a hard rock k0 of about 0.015 sec. The variability of the PSA residuals is also used to evaluate the k0 scaling for hard rock sites from analytical modeling. The scatter in existing k0 values found in literature is disproportionately large compared to the observed variability in high frequency ground motions. We compared the predicted ground motion variability based on analytical modeling to the observed variability in our residuals. While the hard rock sites are more variable at high frequencies due to the additional k0 variability, this additional variability is much less than the variability predicted by the analytical modeling using the variability from k0 Vs30 correlations. This is consistent with weaker k0 scaling compared to that predicted by the analytical modelling seen in the mean residuals.