# Identification of Topographic Seismic Site Periods in Sloping Terrains
## 2 Material and Methods  
### Terrain Characteristics Used  
In this study, stratigraphic conditions and lateral heterogeneities were not considered to compare the spectral response using a uniform shear wave velocity, _Vs_, for the entire model domain. This simplification was intentional to isolate and focus on the topographic effects on seismic response, allowing for a clearer analysis of how terrain features influence spectral amplification. The uniformity of _Vs_ provides a controlled environment to assess the pure impact of topography, free from the additional complexities introduced by stratigraphic variability and lateral heterogeneities.  
By standardizing the shear wave velocity, it becomes easier to compare the findings with previous studies that adopted similar modeling assumptions. This approach aims to make the results more relevant to understanding topographic effects while also providing a basis for future studies to potentially incorporate more complex variables for further validation.  
The parametric analysis focused on the inclination angle and stiffness of the terrain as principal variables, also allowing for a comparison of the potential variation of the period recorded on the sloping terrains, with respect to the period calculated in a one-dimensional soil column, where, for homogeneous and slope-free terrains, it is known that the period is equal to \(T_{0}=4H/Vs\). Thus, the terrain was modeled with linear elastic behavior characterized mainly by the soil stiffness variation of the shear wave velocity, _Vs_.  
A sensitivity analysis of the initial stress conditions on the principal analysis variables, amplification ratio and period, was performed. From this, no significant influence of the variables associated with the initial stress state was observed, for which geostatic conditions with \(\mathrm{k_{0}=0.4}\), \(\mathrm{\gamma=18\leavevmode\
obreak\ kN/m^{3}}\), and \(\mathrm{\
u=0.3}\) were defined for all models.  
The range of values for \(\beta\) and \(H\) were defined based on criteria from various seismic design codes [23; 24], which converge for angles below 20\({}^{\circ}\) and at heights below 30 m. Therefore, for the present study, the finite element parametric analysis was performed using \(\beta\) angles of 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), and 50\({}^{\circ}\); \(H_{a}\) values of 30 m, 50 m, and 100 m; and \(H\) values of 30 m, 60 m, 90 m, and 120 m. In this range, the potential influence of topography on the seismic response may not necessarily be significantly different from what is expected on a horizontal terrain.  
The terrains were characterized with _Vs_ values of 150 m/s, 350 m/s, and 500 m/s to cover a broad range of soil stiffness. These values were chosen to represent typical soil conditions encountered in engineering practice, ranging from softer, more deformable soils to stiffer, less deformable soils. By selecting this range of _Vs_ values, the study aims to reflect the variability in soil conditions that can significantly influence seismic response. This approach aligns with the general guidelines found in standard seismic design codes, which often account for such variations in soil stiffness. Considering the combination of parameters, a total of 144 cases were modeled.