## 2 Empirical Frequency-Dependent Scaling Equations for Vertical Ground Motion
The solid lines reflect median empirical predictions based on the data recorded at all epicentral distances, whereas the dashed lines reflect predictions based on distances less than 30 km. To calculate the exact differences between the attenuation curves for different site conditions, we can apply the coefficients corresponding to the categorical variables \(S_{L}\) and \(S_{G}\). If the coefficients from Table 2 are utilized, the spectral amplitude for \(T\) = 0.05 s will be \(1/10^{-0.021}\) = 1.05 times larger at deep geological rock (\(S=2\)) than at deep geological sediments (\(S=0\)), as seen in the top left-hand plot in Figure 2. This is most likely because short-period waves travel through more compact rocks more easily. The opposite occurs for larger periods; the spectral amplitude for \(T\) = 0.50 s will be \(1/10^{0.147}\) = 1.40 times larger at deep geological sediments than at rock, as seen in the bottom right-hand plot in the same figure. The exact differences between the \(PSA\) attenuation curves for different site conditions can be also calculated by using the coefficients \(c_{4}\), \(c_{5}\), \(c_{6}\), or \(c_{7}\). For example, the spectral amplitude for \(T\) = 0.05 s will be \(1/10^{0.052}\) = 1.13 times larger at deep soil sites (\(S_{L}=2\)) than at rock soil sites (\(S_{L}=0\)), while the amplitude for \(T\) = 0.30 s will be \(1/10^{0.088}\) = 1.23 times larger. The coefficient \(c_{5}\) will have positive values for all vibration periods up to 0.5 s. This indicates that deep soil sites produce amplification of vertical seismic waves for all vibration periods smaller or equal to 0.5. This is somewhat surprising because our investigation of the horizontal spectra revealed that short-period waves at the deep soil sites are de-amplified [18]. Coefficients \(c_{5}\) and \(c_{7}\) from Tables 2 and 3 can be used together to examine the combined influence of deep soils and deep geological sediments. When the coefficients from Table 2 are taken into account, the \(PSA\) (0.05 s) for deep soil above deep geological sediments will be \(10^{0.052}\)\(10^{-0.021}\) = 1.07 of the value for rock soil atop the geological rock. The amplification for \(PSA\) (0.30 s) in deep soil above deep geological strata is \(10^{0.088}\)\(10^{0.083}\) = 1.48. On the other hand, coefficients \(c_{5}\) and \(c_{7}\) show that \(PSA\) amplitudes in deep soils atop deep geological sediments will be de-amplified up to 26% for vibration periods 0.1 s (\(10^{0.090}\)\(10^{-0.181}\) = 0.81), 0.15 s (\(10^{0.107}\)\(10^{-0.178}\) = 0.85), 1.5 s (\(10^{-0.175}\)\(10^{-0.142}\) = 0.93), and 2.0 s (\(10^{-0.278}\)\(10^{-0.147}\) = 0.74). Figure 3: Median empirical estimates of vertical \(PSA\) at deep soil (\(S_{L}=2\)), rock soil (\(S_{L}=0\)), and stiff soil (\(S_{L}=1\)) sites, as well as median \(\pm\) 1 \(\sigma_{log}\) and median \(\pm\) 2 \(\sigma_{log}\) (shaded area) estimates for the deep soil sites. Figure 4 compares the spectra of 15 vertical accelerograms recorded at 10 different deep soil sites to the empirical predictions obtained by Equation (1) and the scaling coefficients shown in Table 3. As can be observed, the actual and anticipated spectra have a high degree of agreement.