Spatial Variability of Peak Strong Ground Motions: Implications for ShakeMap Interpolations

An important and commonly asked question regarding ShakeMap is: What is the uncertainty in the ground motion depicted at any location? This is a complex question, requiring consideration of the nominal or dominant frequency content of each considered parameter, the earthquake size (weak versus strong motions), and critically, the distance to the nearest observation(s). We have analyzed the spatial variability of the peak ground motion parameters used in ShakeMap for both weak and strong motions, for earthquakes in California. In our database, we have 289 small events (M<5.6) and 15 large events (M>5.6), and consistent with ShakeMap we analyze peak ground velocity (PGV, also used for instrumental intensity), peak acceleration (PGA), and spectral acceleration at 3 periods (SA, 0.3, 1.0, and 3.0 sec). For each parameter we make two basic comparisons. First, we plot the variance of the amplitude ratios of adjacent stations versus inter-station distance. This allows us to compare our expanded data set with previous studies of spatial variability for arrays and aftershock deployments. Second, we determine the variance associated with ShakeMap by comparing the amplitude at each station with the value that would be mapped in the absence of that station. That is, we drop each station one at a time (i.e., cross-validation, a form of bootstrapping), regenerate the maps, and then compare the interpolated value at the dropped station location with the observed value. In the first comparison, for M>5.6 events, at 5 km separation of stations, 95 percent of the peak ground velocity (PGV) values are within a factor of 3.0 of each other, whereas this factor is 4.2 for weak motion (e.g., Field and Hough, 1997). Compared to previous studies with weak motions, we find the variability of peak strong motions for larger events is less. We suggest this can be attributed to larger events having more coherent source radiation and thus shifting peak values to longer periods, and possible nonlinearity reducing amplification differences at adjacent stations with varying site conditions. We then compared this variability to the variability of the ShakeMap interpolated values as given in the second comparison, using the Northridge earthquake data set. The accuracy of the ShakeMap interpolations is better than suggested by the observed strong motion variability. Now the ratio for the inferred PGV (compared to the observations) values is 1.7 at 5 km, with an average ratio of 1.4 over the range from 0.4 to 5 km. This result suggests that the ShakeMaps PGV rarely predict more than 1.7 times the real value at 5 km from a seismic station, and predict at a 68-percent confidence a value within a ratio of 1.3 from the real value (which is also close to the mean basic error of the ShakeMaps, due to the grid interpolation). Considering a factor-of-two difference in PGV corresponds approximately to a one-unit change in instrumental intensity, for damaging earthquakes, we can infer that the values of intensity should only be infrequently off by a factor more than one intensity unit when a station is within 5 km.