# Saturation Effect on the Coherency Loss of Spatially Varying Vertical Ground Motions
## 3 Numerical Example  
In this section, the information from the study cases and the parametric study conducted herein is introduced first, and then relevant results are analyzed. Three cases (numbered as case-1, case-2, and case-3) were considered in this study: case-1 evaluates the parametric study with varying SD, case-2 examines the varying GWL, and case-3 investigates the varying WLT. Based on these three cases, the effect of SD, GWL, and WLT on the coherency loss between vertical SVEGMs was investigated. Figure 1 shows the diagram of the study cases performed, where \(\rho\) is medium density, \(\mu\) the [PERSON]'s ratio of the soil skeleton, \(G\) is the shear modulus, \(\xi\) is the damping ratio, \(n\) is the porosity, \(P\) is the absolute fluid pressure, \(K\)w is the bulk modulus of water, and \(S\), is the saturation degree. Besides, herein, the bulk modulus of the solid skeleton is \(K_{\rm b}\) = 8.67 \(\times\) 10\({}^{7}\) Pa; the bulk modulus of solid grains is \(K_{\rm s}\) = 3.6 \(\times\) 10\({}^{10}\) Pa. Specific cases involved are summarized in Table 1, and fundamental information of simulated seismic motions is shown in Table 2. Simulation points, \(j\)  
The amplitudes of the transfer functions for all cases are shown in Figure 2. From Figure 2a, it can be found that as the SD increases, the amplitude of the transfer function decreases, and the predominant frequency shifts toward the high frequency end; especially when the SD increases from 98% to 100%, the amplitude decreases sharply. This is consistent with the results of [PERSON] and [PERSON] [27], which verifies the reliability of the results in this paper, to some extent. Moreover, by observing the transfer functions in Figure 2b,c it can be found that: (i) for site \(j\), the vertical motion is amplified until the site is fully saturated; (ii) for site \(k\), when the water table is under the site surface, the vertical motion is amplified; in contrast, when there is a water layer on the surface (the water table is above the surface), the vertical motion is suppressed at a certain frequency point. Therefore, the state of full saturation (i.e., SD = 100% in case-1, the GWL = 30 m for site \(k\) in case-2, or the WLT = 0 m for site \(k\) in case-3) is the critical state of amplification and suppression. In other words, when the site is incompletely saturated, amplification can be observed; when the site is fully saturated, vertical motion would be slightly suppressed; and when the site is fully saturated and under a water layer, the vertical wave would be significantly suppressed. Therefore, the state of saturation is the turning point. The resonant frequency of P-wave,\(f_{n}\), in the water layer can be defined as [13]  
\[f_{n}=n\frac{c_{p}^{*w}}{4d\sin\theta_{p}^{w}},\ \ \ n=1,3,5\ldots \tag{10}\]  
in which \(d\) denotes the depth of water layer, \(c_{p}^{*w}\) the wave speed of the P-wave in the water layer, and \(\theta_{p}^{w}\) the inclined angle of the P-wave. For the numerical example in the present paper, when \(d\) = 20 and 30 m, the corresponding first-order resonant frequencies of the water layer are 21.81 and 14.54 Hz, respectively. Then, from Figure 2c, it can be found that the transfer functions are suppressed near 21.81 and 14.54 Hz, respectively, which verifies the validity of the results of this paper, to some extent again. Figure 3 gives the mean lagged coherency loss and corresponding mean phase angle of the coherence function for the generated vertical in-plane motions, where the corresponding modulus of the spectral ratio between sites \(j\) and \(k\) are also given (here, spectral ratio is defined as the ratio between the transfer functions at sites \(j\) and \(k\), and the peaks and troughs of the amplitude curve of the spectral ratio reflect the difference between the transfer function between sites \(j\) and \(k\)).