# Saturation Effect on the Coherency Loss of Spatially Varying Vertical Ground Motions
## 3 Numerical Example
As can be seen, the troughs of each lagged coherency loss function are directly related to the peaks of the corresponding spectral ratios, and due to the topography effect, the amplitude of the coherency function exhibits additional loss, which is consistent with previous results [25]. As the SD increases, the lagged coherency loss for vertical seismic motions gradually approaches that for bedrock motions, indicating  
Figure 2: Amplitude of transfer function for varying: (**a**) SD; (**b**) GWL; (**c**) WLT. that the existence of groundwater can make up the extra lagged coherency loss due to the topography effect. The higher the SD, the higher the P-wave velocity [27] in the medium and the stiffness of the site in the vertical direction; thus, the amplitude of the transfer function for sites \(j\) and \(k\) tends to be simultaneously lower, and the spectral ratio may decrease. Note that when the site is fully saturated, both the transfer functions at sites \(j\) and \(k\) are approximately 1 or slightly lower than 1, indicating that the amplification and filtering effect of the site on the vertical motion almost vanish, and thus, at this time, the difference between the seismic waves at sites \(j\) and \(k\) is mild, and the extra coherency loss is very small. In other words, the higher the SD, the smaller the extra lagged coherency loss due to the topography effect. When it comes to water level, the authors analyze relevant results by combining case-2 and case-3 herein. The extra loss of the lagged coherence function decreases with increasing water levels, except for the values corresponding to extremely sharp peaks (corresponding to the P-wave resonant frequency of the water layer) of the spectral ratio curve. The existence of a fully saturated soil layer dramatically decreases the propagation time of the vertical wave inner site (\(c_{p}=2224.7\) m/s in fully saturated soil layer, while \(c_{p}=282.79\) m/s in a soil layer with SD = 95%) and then the time delay between sites \(j\) and \(k\), causing the superposition modes of the wave at sites \(j\) and \(k\) to be similar, and as thickness of fully saturated soil layer increases, the time delay would become lower and lower. Therefore, to a certain extent, the extra coherency loss due to irregular topography can be made up by an underlying fully saturated soil layer, and this phenomenon tends to be more remarkable as water table becomes higher and higher. At the frequencies where the vertical motion is suppressed significantly by the surface water layer, the value of coherency loss is extremely low. This is because around the frequencies where vertical motion is suppressed significantly by the surface water layer, the frequency content of the motions at \(j\) and \(k\) is extremely different. Coherency loss function only evaluates the difference between motions. Therefore, the larger the difference between the frequency content of motions, the lower the coherency loss. As to the phase angle of the coherence function, there is always a remarkable change (e.g., local bold lines shown in Figure 3c) showing up around the frequency point corre  
Figure 3: Coherence functions and relevant spectral ratios for varying: (**a**) site SD; (**b**) GWL; (**c**) WLT. sponding to each peak of the spectral ratio curve, indicating that when the amplitudes of transfer functions at sites \(j\) and \(k\) differ greatly, the phase angle of the coherency function exhibits significant variation. Besides, the curves for the phase angle also gradually shift towards the high frequency end as spectral ratio varies with increasing SD, GWL, and WLT. All of the above mentioned indicate that the phase angle of the coherency function is also directly related to the spectral ratio.