# Identification of Topographic Seismic Site Periods in Sloping Terrains
## 3 Results and Discussion
### Identification of Periods or Frequencies of Interest in a Sloped Terrain
Thus, if \(ARi\) defines the amplification ratio at point \(i\) on the slope of the terrain and \(ARa\) and \(ARb\) are the amplification ratios at points A and B in the horizontal free field (Figure 2), it is possible to calculate a dimensionless ratio between them, i.e., \(\frac{ARi}{(ARa\times ARb)}\). This calculation allows for an indirect evaluation of the behavior of the response resulting from the vibration of the central block. Based on this analysis, as depicted in Figure 5, even after removing the potential effect of the lateral blocks, the amplification peak highlighted in red in Figure 4 persists and maintains the same period for any point located on the slope of the terrain. Therefore, it is inferred that this response is primarily generated by the discontinuity caused by the slope of the terrain. Considering the above, the response of points X, Y, and Z is analyzed for all the cases derived from the set of variables, totaling 144 combinations. In all cases, the period \(Tt\) associated with the previously mentioned amplification peak is identified. Therefore, in Figure 6 the identified periods are plotted as a function of the \(H_{T}/Vs\) ratio, aiming to find a correlation while maintaining the general form of Equation (1). A fitted line with an R\({}^{2}\)-value of 99% is obtained, which allows for the determination of the \(T_{t}\) period, referred to as the topographic site period for practical purposes, using the following expression:  
\[T_{t}=\frac{3H_{T}}{V_{\mathrm{s}}} \tag{2}\]  
The results obtained highlight the impact of topographic features on seismic response, especially in urban environments where variable terrain can pose particular challenges for structural design. The identification of a characteristic period resulting from terrain slope is an analytical element that needs to be evaluated when performing a seismic classification of terrains near or on steep slopes, particularly in cases involving Chilean regulations, where the incorporation of the measured ground vibration period as a seismic classification parameter alongside propagation velocity is currently under discussion. Figure 4: Schematic representation of the spectral response of points on sloping terrain generated by the interaction of the three blocks comprising the domain. The use of 2D finite element models in this study was primarily aimed at isolating and understanding the fundamental effects of topography on seismic response. These models offer a straightforward approach to identifying primary amplification mechanisms due to terrain features. While this simplicity is a strength in terms of clarity and computational efficiency, it is clear that there is an inherent limitation in evaluating complex topographic conditions, for which it is recommended that future research extend to three-dimensional cases under different soil conditions. However, recognizing this limitation, it is important to note that 2D models, under more controlled conditions, contribute significantly to exploring topographic effects. Figure 5: Visualization of the amplification peak associated with topography generated by subtracting the amplification in the horizontal free field. Figure 6: Recorded periods \(T_{t}\) in 144 cases and the fitted trend line for their determination as a function of \(H_{T}\) and \(V\)s.