Chiou and Youngs PEER-NGA Empirical Ground Motion Model for ...

faulting earthquakes at long periods is due to more compact dip-slip earthquakes being less
efficient at producing long period motions than long strike-slip earthquakes, then perhaps the
values of c1a and c1b should be similar in this period range. We have made this assumption in
developing the period dependence of c1b.
Step 6: With c1a and c1b fixed, the period variation of the hanging wall scaling factor, c9 was
defined. The smoothed value of c9 decreases to 0 at long periods.
Step 7: The next step was completion of the soil amplification model, fSite. The variation of
φ2 and φ3, with period was re-evaluated holding the previously smoothed parameters of the
model fixed. This assessment resulted in modifications to φ2 that indicated slightly more
nonlinearity. These parameters were then smoothed and φ1, the linear scaling term, was
recomputed and smoothed.
Step 8: The next step was to fix the remaining parameter for main shocks, the dependence
on the depth to top of rupture defined by parameter c7. This parameter was found to decay to
zero as the spectral period increases, consistent with the concept that it represents an increase
in dynamic stress drop or the “stress parameter” with increasing depth of rupture.
Step 9: The remaining parameters in the ground motion model for main shock motions are
the intercept term c1 and the variance terms τ and σ. Because all of the other model
parameters have been fixed, the aftershock data can now be removed from the analysis. The
resulting effect on the intercept term c1 is negligible (less than 0.01 percent) and produces a
slight reduction in the inter-event variability τ.
The estimated values of c1 exhibited noticeable steps at periods of 0.8, 4, and 8 seconds, as
shown in Figure 31. These steps occur at periods where there are large reductions in the
number of usable data, defined by the minimum usable frequency, as shown on Figure 32.
The derivative of c1 with respect to spectral period was computed numerically. These values,
shown on Figure 33, show peaks that usually coincide with a sudden drop in the number of
usable data (Figure 32). These observations suggest that the estimated values of c1 may be
biased by the systematic removal of weaker motion, which would tend to leave larger ground
motion amplitudes in the remaining data. We therefore took the following steps to smooth
parameter c1. We first estimated the bias due to data drop out at 0.8, 1.1, and 1.6 seconds.
The bias is taken as the increase in the c1 estimate when the dataset is replaced by the
reduced set for the next longer spectral period. Secondly, at periods longer than 1.6 second,
we impose a constant slope of -1.4 (the horizontal dotted line Figure 33), matching the trend
in the data. As a result, the large-magnitude spectra do not reach constant displacement
scaling at periods up to 10 seconds, the limit of our model. The resulting values for c1 are
shown by the red curve in Figure 21.
C&Y2006 Page 42