## 4 Numerical Results  
In this section, a numerical test for illustrating the potentialities of the technique just proposed is addressed. Here, we consider a soil deposit consisting of 5 strata. The PGD method was used to calculate the parametric solution for the displacement field \(\bar{\xi}(\omega,\xi_{1},\xi_{2},\xi_{3},\zeta_{4},\zeta_{5})\), as detailed in Appendix A. The problem coordinates are defined in their respective domains: \(z\in\Omega_{z}=(0,7.5)\) m, \(\omega\in 2\pi(0,25)s^{-1}\), and \(\zeta_{i}\in(0.01,0.5)\), \(i=1,\ldots,5\). The different domains were discretized by considering 100 nodes for the spatial coordinate, 1023 nodes required for Fast Fourier Transform (FFT), and 100 nodes for the damping coordinates. Even if 100 nodes for discretizing the parametric domain seem too much, as the calculation of functions depending on the parameters does not imply the solution of any linear system, it is preferable to consider a rich-enough discretization to be sure of representing accurately the parametric solution. Different methodologies were taken into account in the numerical simulations: (i) the Equivalent Linear Response Analysis (EERA); (ii) the Extended Harmonic Modal Hybrid Approach (XHMH) proposed before; and (iii) the nonlinear procedure used in DEEPSOIL software. In the numerical simulations, it was considered 5 layers and a half-space, with the properties listed in Table 1. The Equivalent Linear analysis was carried out using the appropriate material curves given by Reference [23], and, for DEEPSOIL analysis, it was considered the modified hyperbolic model MKZ proposed by Reference [20], as  
\[\tau=\frac{G\gamma}{1+\beta\left(\frac{\gamma}{\gamma_{r}}\right)^{s}}, \tag{29}\]  
with \(\beta\) and \(s\) the parameters that adjust the shape of the backbone curve. The different approaches considered an initial estimation of damping ratio \(\zeta^{0}\) (assuming a linear regime, which are small strains); it was considered for XHMH analysis \(c=1\), and,for DEEPSOIL analysis, \(\beta=0.9\), \(s=0.85\), and \(\sigma_{ref}=0.18\) for each stratum, respectively. In soil analysis, input motion is defined from the response spectrum, or its corresponding time history at bedrock surface (outcrop), known as rock outcropping motion. From the outcropping motion, the objective is to predict the bedrock motion covered by the soil deposit. Thus, the bedrock half-space can be substituted with boundary condition  
\[\tau_{o}^{*}=c_{s}^{*}\rho_{b}A\dot{U}_{s}-c_{s}^{*}A\rho_{b}\dot{U}_{o}^{*}, \tag{30}\]  
where \(\dot{U}_{s}\) the rock outcropping velocity (assumed measurable), and \(\dot{U}_{o}^{*}\) the velocity at the base of the soil column [17; 24], which coincides with the soil-bedrock half-space interface and \(\tau_{o}^{*}\) the shear stress at that position. The solution is obtained assuming the acceleration time history in the outcropping is known; in our simulation, this is the one shown in Figure 1 (recorded at rock site). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Stratum** & **Material** & **Thickness (m)** & **Density (kg/m\({}^{3}\))** & **Gmax (MPa)** & \(\bar{\xi}^{0}\)**(\%)** & \(\gamma_{r}\)**(\%)** & \(\sigma_{v}^{{}^{\prime}}\) **(kPa)** \\ \hline
1 & Silt & 1.5 & 1940 & 15.65 & 0.5 & 0.048 & 14.3 \\ \hline
2 & Silt & 2 & 1940 & 22.79 & 0.5 & 0.048 & 47.6 \\ \hline
3 & Sandy Silt & 2 & 1940 & 26.86 & 0.5 & 0.048 & 85.6 \\ \hline
4 & Silty Sand & 1.3 & 1940 & 31.26 & 0.5 & 0.048 & 117 \\ \hline
5 & Clay & 0.7 & 1940 & 46.49 & 0.5 & 0.048 & 136 \\ \hline Half-space & & & 2200 & 333.6 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Soil properties, with Gmax the initial shear modulus, \(\zeta^{0}\)(%) the damping ratio at small strains, \(\gamma_{r}\)(%) the reference strain, and \(\sigma_{v}^{{}^{\prime}}\) the effective vertical stress. Figure 2 depicts the time evolution of the displacement at the surface, when using different approaches. The truncated solution agrees perfectly with the fully-resolved solution, by considering only two modes.