# Identification of Topographic Seismic Site Periods in Sloping Terrains
## 3 Results and Discussion  
### Identification of Periods or Frequencies of Interest in a Sloped Terrain  
For horizontally homogeneous terrain, the periods at which amplification peaks associated with different vibration modes are generated are estimated by using the well-known one-dimensional approximation:  
\[T_{n}=\frac{4H_{T}}{(n+1)V_{s}} \tag{1}\]  
where \(H_{T}\) is the total thickness of the soil measured from the bedrock, and \(n\) is the vibration mode. Therefore, if the domain shown in Figure 1 is simplified and decomposed into three blocks, specifically as two lateral rectangular blocks and one central trapezoidal block, as shown in Figure 2, for points in free-field conditions in the lateral zones with horizontal surfaces, such as points A and B, it is possible to identify the vibration periods using Equation (1), provided these points are sufficiently far from the sloping area of the terrain and also that a constant \(Vs\) condition is maintained throughout the domain. However, since the dynamic response cannot be decoupled, the central block, which represents the sloping area, is influenced by the vibration of the surrounding blocks, and vice versa, both in terms of the amplitude and frequency content. While this coupled response and interaction among all blocks is present, there is an influence zone beyond which any analyzed point will exhibit spectral behavior analogous to the response in free-field conditions generated on a horizontal terrain. This influence zone is delimited by the distances \(X_{a}\) and \(X_{b}\) in Figure 2.  
Focusing on the identification of the periods in which amplification ratio peaks occur, the periods were recorded in the different conducted models, with their representative response shown in Figure 3 for a reference case. Notably, this response was observed in the different modeled cases. In Figure 3, the presence of two main amplification peaks can be observed for different points located on the slope of the terrain, specifically the central trapezoidal block, whose associated period coincides with the primary mode of vibration of the lateral blocks, calculated using Equation (1) with \(H_{T}=H_{a}\) and \(H_{T}=H_{b}\). Additionally, other peaks are recorded, some associated with values close to the period of the second mode of the vibration of the lateral blocks, although they are not the only peaks present, such as the case of the period identified as \(T_{j}\) in Figure 3. It is worth emphasizing that the periods identified in the studied models were practically independent of the inclination angle \(\beta\) or the location of the point on the slope. Schematically, the observed response in the modeled cases is presented in Figure 4, which illustrates that the spectral behavior of a point that is located on the slope of the terrain is predominantly a composition of the dynamic response of the three blocks together, where the self-vibration of the lateral blocks, in particular, induces the highest amplification peaks. However, a consistent pattern was also observed; the presence of a peak that is not associated with the vibration of the lateral soil blocks, as shown in Figure 3, and highlighted in red in Figure 4. This peak may be related to the discontinuity caused by the inclination of the terrain. While this study does not specifically focus on the magnitude of the amplification ratio peaks, a trend of alternating maximum values is observed based on the position of the analysis point, as shown in Figure 4. Specifically, for points near the base, the maximum occurs at the period associated with the first mode of the vibration of the lower block. Conversely, for points approaching the crest, the maximum amplification corresponds to the period of the tallest soil column. Considering that the response at points X, Y, and Z in Figure 1 is a composition with significant influence from the lateral blocks, for the purpose of analysis, a simplified approach is used to subtract the free-field response at points A and B from the response of the points on the slope.