## 2 Theoretical Basis
### Effect of Water Column on Incident Plane P-Waves  
The ratio of the response of the bottom soil layer with the water column above it to that without a water column [21; 36] is as follows:  
\[|F(f,\alpha)|=\frac{1}{\sqrt{1+\alpha^{2}\tan\left(\frac{\pi}{2}\frac{f}{f^{ \prime 1}_{1}}\right)}} \tag{36}\]  
\[f^{\prime 1}_{n}=\frac{c_{w}}{4H}n \tag{37}\]  
\[\alpha=\frac{\rho_{w}C_{w}}{\rho_{ss}C_{ss}} \tag{38}\]where \(F(f,\alpha)\) is the ratio of the response of the bottom soil layer with the water column above it to that without the water column (reduction spectrum) as a function of frequency \(f\) and impedance ratio between water and bottom soil \(\alpha\), \(f_{h}^{res}\) are the resonant frequencies of P-waves in the water column, \(n\) is an odd number, 1,3,5,..., \(c_{w}\) is the P-wave velocity in the water layer, \(H\) is the thickness of the water column, \(f_{1}^{res}\) is the fundamental resonant frequency, \(\rho_{w}\) is the density of water, \(\rho_{ss}\) is the density of seafloor soil, \(C_{ss}\) is the P-wave velocity in seafloor soil (assumed as uniform half space).  
Figure 4 shows the reduction spectra \(\left|F(f,\alpha)\right|\) of the vertical component of bottom movements due to the water column of different thickness \(H\) above the site and different impedance ratios between seawater and seafloor soil \(\alpha\). In order to simulate this reduction, we suggest sequentially applying a series of band-stop filters to the accelerogram corresponding to the bottom:  
\[\begin{array}{c}acc\_filt_{i}=But\big{(}acc\_filt_{i-1}\big{|}\big{[}\,f_{ i}^{res}-\beta;\,f_{i}^{res}+\beta\big{]}\,\big{|}\,\sigma\big{)},\\ for\,i=1,\ldots,\,m;\,f_{1}^{res}\geq\,F_{lb},\,f_{m}^{res}\leq\,F_{hb}\end{array} \tag{39}\]  
where \(But\) is the operator denoting the application of a digital Butterworth filter of order \(\sigma\) and band-stop frequency range \(\big{[}\,f_{i}^{res}-\beta;\,f_{i}^{res}+\beta\big{]}\), \(acc\_filt_{0}\) is the initial accelerogram on the surface of seafloor soil deposits, \(acc\_filt_{i}\) is the filtered accelerogram of iteration \(i\), \(F_{lb}\) is the lower representative frequency bound of the initial accelerogram (usually the natural frequency of the accelerometer), \(F_{hb}\) is the upper representative frequency bound of initial accelerogram (usually Nyquist frequency).  
Figure 4: Reduction spectra for vertical component of bottom movements due to the water column of different thicknesses (H) above the site: (**a**) – H = 20 m, (**b**) – H = 80 m, (**c**) – H = 140 m, (**d**) – H = 200 m (alpha—impedance ratio between seawater and seafloor soil).  
Varying the parameters of band-stop frequency range \(\beta\) and the order of the filters \(\sigma\) it is possible to control the depth and width of the spectral peaks corresponding to the resonant frequencies of the P-wave in the water column. Figure 5 shows the correspondence of the target theoretical reduction spectrum due to the water column and frequency response of a complex digital filter designed to simulate it. The following values were used for calculation: \(c_{w}\) = 1500 m/s, \(H\) = 51 m, \(\alpha\) = 0.5, \(F_{lb}\) = 1 Hz, \(F_{lb}\) = 50 Hz, \(\sigma\) = 2, \(\beta\) = 1.5 Hz.