# Identification of Topographic Seismic Site Periods in Sloping Terrains
## 2 Material and Methods
### Overview of the Numerical Model and Terrain Characteristics Used  
Since the present study is aimed at identifying the periods associated with a variation in the amplification ratio on a sloping terrain, dynamic response modeling was carried out using finite elements in the Plaxis 2D CONNECT Edition V22 [25], based on the geometric configuration shown in Figure 1. To achieve this, amplification ratios were determined by comparing Fourier acceleration spectra at different study points with the spectra recorded in the free field in horizontal zones and at the level of the bedrock, shown in Figure 1.  
A geometric configuration as shown in Figures 1 and 2 was used, whose main variables were the inclination angle, \(\beta\), the shear wave velocity, _Vs_, and the heights of the base and crest of the slope up to the bedrock, \(H_{a}\) and \(H_{b}\), respectively.  
For the finite element model, a mesh composed of 6-node triangular elements with a maximum size of \(\lambda/15\) was used, where \(\lambda\) is the wavelength associated with a maximum frequency of 20 Hz [25]. The free-field boundary was employed for the lateral boundary conditions, which allows the model surface to behave as a free surface, without any restrictions or influence from structures or other elements [25].  
A time-domain, two-dimensional, finite element model was used, with its base resting on bedrock subjected to vertically propagating in-plane shear waves. This represented a conservative analysis from the point of view of topographic amplification compared to analyses performed using SH waves (out of plane) [26; 27; 28; 29].  
Since the study focused on analyzing the identification of fundamental vibration periods, a controlled excitation was applied to the model to decouple the influence of other frequency sources and identify the fundamental vibration modes of the terrain [21; 22; 24]. Following this approach, a sensitivity analysis was conducted on models activated with a single pulse, recording the free vibration response of the terrain until it ceased. The analysis included variations in period (0.1 to 0.5 s), amplitude (0.2 g to 1 g), and signal type, considering both rectangular and sinusoidal pulses. Based on this, no significant  
Figure 1: General description of the finite element model and location of the analysis points.  
Figure 2: Discretization of the sloped terrain into a three-block domain, indicating the expected response in the horizontal free field.  
influence of the signal characteristics was observed in the magnitude and identification of the vibration periods of interest in this study.  
Based on the above, the models were triggered with a single acceleration rectangular pulse characterized by an amplitude of 1 g and a period of 0.1 s, applied at the bottom boundary in the horizontal direction, and the vertical displacements were restricted, which was consistent with the rigid bedrock assumption.  
The ratio of topography dimension to wavelength was identified in the literature as a key parameter affecting the magnitude of topographic effects [5; 14; 18; 30; 31]. However, it was recorded that the \(H/\lambda\) ratio does not show significant changes in the identified periods in terrains characterized as homogeneous [19]. Nevertheless, a wide range of \(H/\lambda\) values from 0.1 to 0.6 were used.  
The wave incidence angle may affect the amplification effects. However, in the present study, aiming to identify the periods in which amplification peaks occur, site response analysis is conducted by evaluating the accelerations, and it is reasonable to consider vertically propagating waves, as acceleration magnitudes are generally not significantly affected by the incident wave angle [18; 27].