## 2 Materials and Methods  
### Improving the NERA Algorithm To Incorporate Biot and Gassman Models  
Description of wave propagation in porous fluid-saturated soil began with the studies by [PERSON] [29] and [PERSON] [26; 27]. [PERSON]'s linear theory of poroelastic wave propagation is valid under the assumptions about a bound and isotropic porous medium, the pore-scale much smaller in comparison with the wavelength and small deformations that ensure linear elastic material behavior. [PERSON]'s equations of motion for an isotropic fluid-saturated porous medium are given by [30]:  
\[\frac{\partial\tau_{ij}}{\partial x_{j}}=\rho\ddot{d}_{i}+\rho_{f}\ddot{w}_{i} \tag{1}\]  
\[\frac{\partial p_{f}}{\partial x_{i}}+\rho_{f}\ddot{d}_{i}=-\gamma\dot{w}_{i} \tag{2}\]  
\[w_{i}=\phi\left(d^{f}-d_{i}\right) \tag{3}\]  
\[\gamma=\frac{\eta}{k(\omega)} \tag{4}\]  
where \(d\) is the displacement of the soil matrix, \(d^{i}\) is the displacement of the fluid particles, \(w\) is the displacement of the fluid particles relative to the matrix, \(\rho\) is the total density of the porous medium, \(\rho_{f}\) is the density of the fluid, \(\rho_{f}\) is the pore pressure, \(\phi\) is the porosity, \(d^{f}\) is the displacement of the liquid phase, \(\gamma\) is the viscodynamic operator, \(\eta\) is the dynamic viscosity coefficient of the pore fluid, \(k(\omega)\)--dynamic permeability (frequency-dependent), and \(\tau_{ij}\) are the stress tensor elements. The dot denotes the time derivative. To relate the elastic parameters of a porous medium, [PERSON]'s equations are commonly used [28]. [PERSON]'s theory is based on the assumption that the relative motion of fluid and matrix has a negligible effect on the seismic waves propagation in fluid-saturated soils. [PERSON]'s equations are essentially the lower frequency limit of [PERSON]'s more general equations of motion for poroelastic materials. The boundary of the low-frequency range is defined as follows [31]:  
\[f<0.1\ \frac{\eta\phi}{2\pi k\rho_{f}} \tag{5}\]where \(k\) is the permeability of the rock. In this frequency range, [PERSON]'s theory is in accordance with the conclusions from [PERSON]'s theory. In different studies, the value of the cutoff frequency varies. According to [32], the cutoff frequency is within the range of 1-20 kHz. In [33], it is argued that [PERSON]'s equations give the most accurate results at frequencies lower than 100 Hz. It follows from this that [PERSON]'s theory is well applicable to seismology. According to [PERSON]'s theory, since the fluid and solid matrix particles move together, the density of the porous medium can be averaged as follows:  
\[\rho=\phi\rho_{f}+(1-\phi)\rho_{s} \tag{6}\]  
where \(\rho_{s}\) is the density of the solid grains. Thus, for modeling seismic waves propagation through porous water-saturated soil layers within a seismology frequency band (lower than 100 Hz), the main assumptions for the elementary volume of porous soil are as follows: first, at low frequencies during the passage of the wave, the pore pressure gradients have time to compensate; second, there is no relative motion of the fluid and rock matrix, and, third, there is no movement of the pore fluid beyond the elementary soil volume. Squeezing out fluid from an elementary volume entails an increase in pore pressure in neighboring elements. This process can have an avalanche nature and, under certain conditions, lead to a liquefaction effect. Liquefaction is not considered in the present study.