# 2-D Seismic Response Analysis of a Slope in the Tyrrhenian Area (Italy)
## 3 Site Response Analysis
### Topographic Effects  
It is well known that the topography affects the responses at the free surface by complicated cycles of amplification and de-amplification. In general, the responses are amplified at the upper surface of slopes and attenuated at their base. Among the most important study of topographic amplification, we recall the experience of [PERSON] [45,46] and [PERSON] et al. [47], related to the seismic response of steep natural slopes. In this study, the site response was characterized by maximum acceleration at various locations as follows (Figure 20):  
- \(a_{\text{fit}}\), the maximum free field acceleration in front of the toe;  
- \(a_{\text{fit}}\), the maximum free field acceleration behind the crest see Equation (4);  
- \(a_{\text{max}}\), the maximum crest acceleration. In this study, the topographic and site amplification were treated separately in order to determine the contribution of the different factors; therefore, three measures of amplifications were computed: \(A_{t}\)--Equation (3)--\"topographic amplification,\" the amplification of the free field motion at the crest; \(A_{s}\)--Equation (4)--\"site amplification,\" the amplification due to the natural frequencies of the site; \(A_{a}\)--Equations (5) and (6)--\"apparent amplification,\" the apparent amplification of the motion between the base and the crest. The proposed parameters were obtained as follows:  
\[A_{t}=\frac{a_{\text{max}}-a_{ffc}}{a_{ffc}} \tag{3}\]  
\[A_{s}=\frac{a_{ffc}-a_{fft}}{a_{fft}} \tag{4}\]  
\[A_{a}=\frac{a_{\text{max}}-a_{fft}}{a_{fft}} \tag{5}\]  
\[A_{a}=(1+A_{t})(1+A_{s})-1 \tag{6}\]  
The results of numerical analyses carried out for the input ([PERSON] et al. [24]; [PERSON] et al. [25]; [PERSON] et al. [26]; DISS Aspromonte Est; DISS Gioia Tauro; DISS Messina Strait), and based on the [PERSON] approach, are shown in the Table 9.  
Figure 20: [PERSON]’s [46] model for the topographical amplification evaluation. Results show a topographical amplification coefficient, given by \(1+A_{t}\) greater than 1.20, while the same coefficient given by NTC 2018 for the study area is equal to 1.20. Topographic amplification can be also measured by the ratio between the Fourier spectrum of the seismogram at the crest (10 m away) and the Fourier spectrum of the free field seismogram behind the crest (250 m away). This is the so-called TAF (Topographic-Aggravation Factor) which is defined in the frequency domain, after [PERSON] et al. [48]. Figure 21 shows the Topographic Aggravation Factor (TAF), after the [PERSON] et al. [48] approach, for the [PERSON] et al. [26] input. Results show values of TAF reaching on average about 1.45 in the frequency range f of 1-5 Hz. The topographic amplification can be also evaluated in the time domain following the approach proposed by [PERSON] and [PERSON] [49] in terms of the (2-D)/(I-D) ratio of elastic response spectra obtained at the crest. Figure 22 shows the Topographic Aggravation Factor (TAF) after the [PERSON] and [PERSON] [49] approach for the [PERSON] et al. [26] input. In this case, results show high values of TAF, reaching on average about 1.42 for the period range T of 0.20 to 1.0 s. Obtained values are in agreement with large amplifications obtained for similar studies performed on ridges, slopes and valleys. Figure 21: Topographic Aggravation Factor (TAF), after the [PERSON] et al. [48] approach, obtained as the ratio of the Fourier amplitude spectra at the crest and behind the crest, for the [PERSON] et al. [26] input. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Input** & \(A_{t}\) & \(A_{s}\) & \(A_{a}\) \\ \hline [PERSON] et al. [24] & 0.10 & 4.29 & 4.81 \\ [PERSON] et al. [25] & 0.29 & 2.76 & 3.83 \\ [PERSON] et al. [26] & 0.65 & 0.76 & 1.91 \\ DISS Messina Strait & 0.22 & 1.88 & 2.52 \\ DISS Gioia Tauro & 0.01 & 0.87 & 0.89 \\ DISS Aspromonte Est & 0.09 & 2.16 & 2.43 \\
1693 Catania Earthquake & 0.16 & 1.24 & 1.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Different results for the [PERSON] et al. [47] model.