## 2 Theoretical Basis  
### Scheme of One-Dimensional Horizontally Layered Soil System and Governing Equations of Motion  
The study implies one-dimensional vertical propagation of P- and SH-waves in horizontally homogeneous soil layers of infinite horizontal extent. The soil is assumed to be a porous saturated medium. In the general case, the pore fluid may consist of a water, gas mixture, and ice (if we consider the case of a site in the permafrost zone). Figure 1 shows schematically the direction of movement of soil particles relative to the direction of waves propagation through the soil layers in the cases of their occurrence on land (with a certain position of the water table) and in the water area. Wave equations for propagating shear waves (SH-wave) and compression waves (P-wave) in a horizontally layered system have a simple form:  
\[\frac{\partial^{2}u_{h}}{\partial z^{2}}=\frac{1}{V_{S}^{2}}\ \frac{\partial^{2}u_{h}}{ \partial t^{2}} \tag{1}\]  
\[\frac{\partial^{2}u_{v}}{\partial z^{2}}=\frac{1}{V_{P}^{2}}\ \frac{\partial^{2}u_{v}}{ \partial t^{2}} \tag{2}\]  
where \(u_{h}\) and \(u_{v}\) are the horizontal and vertical movement, respectively, as a function of time \(t\) and vertical coordinate \(z\), \(V_{S}\) and \(V_{P}\) are the shear and compression waves velocities, respectively. The shear wave velocity correlates with the shear modulus, which in turn depends on the stress and strain as follows:  
\[G=\rho_{b}V_{s}^{2}=\frac{\tau_{h}}{\gamma_{h}} \tag{3}\]  
where \(G\) is the shear modulus, \(\rho_{b}\) is the bulk density, \(\tau_{h}\) is the horizontal stress, \(\gamma_{h}\) is the horizontal strain. Similarly, compression wave velocity correlates with constrained modulus, which in turn depends on stress and strain as follows:  
\[M=\rho_{b}V_{P}^{2}=\frac{\tau_{v}}{\gamma_{v}} \tag{4}\]  
where \(M\) is the constrained modulus, \(\tau_{v}\) is the vertical stress, \(\gamma_{v}\) is the vertical strain. The relationship between \(M\) and \(G\) is expressed as follows:  
\[M=2G\frac{1-\vartheta}{1-2\vartheta} \tag{5}\]  
where \(\vartheta\) is the Poisson's ratio. In the presence of pore water, the constrained modulus \(M^{*}\) differs from that in the absence of underground water \(M\)[12, 24, 27]:  
\[M^{*}=M+\frac{Kf}{\phi} \tag{6}\]  
where \(Kf\) is the bulk modulus of pore water, -\(\phi\) is the porosity of the soil. Figure 1: One-dimensional horizontally layered system of porous saturated soils and its spatial discretization in the case of vertically propagated waves: (**a**) a shear wave (on-land site), (**b**) a compression wave (on-land site), (**c**) a shear wave (site in water area), (**d**) a compression wave (site in water area). Thus, the relationship between constrained modulus for saturated soil and compression wave velocity and then, with stress and strain is as follows:  
\[M^{*}=\rho_{b}{V_{P}^{*2}}=\frac{\tau_{v}^{*}}{\gamma_{v}^{*}} \tag{7}\]  
where \(M^{*}\) is the constrained modulus for saturated soil (i.e., below water table), \(V_{P}^{*}\) is the compression waves velocity for saturated soil, \(\tau_{v}^{*}\) and \(\gamma_{v}^{*}\) are vertical stress and strain for saturated soil, respectively. Taking into account Formulas (3), (4) and (7), expressions (1) and (2) can be rewritten as follows:  
\[\rho_{b}\frac{\partial^{2}u_{h}}{\partial t^{2}}=\rho_{b}V_{P}^{2}\frac{ \partial^{2}u_{h}}{\partial z^{2}}=G\frac{\partial^{2}u_{h}}{\partial z^{2}} \tag{8}\]  
\[\rho_{b}\frac{\partial^{2}u_{v}}{\partial t^{2}}=\rho_{b}V_{P}^{2}\frac{ \partial^{2}u_{v}}{\partial z^{2}}=M\frac{\partial^{2}u_{v}}{\partial z^{2}} \tag{9}\]  
\[\frac{\partial^{2}u_{v}}{\partial z^{2}}=\frac{1}{V_{P}^{2}}\ \frac{\partial^{2}u_{v}}{ \partial t^{2}} \tag{2}\]0  
where \(u_{v}^{*}\) is the vertical movement below the water table.