## 5 Seismic Vulnerability Assessment of Campi Alto di Norcia
### Reliability of the Seismic Vulnerability Methods
Resorting to Equation (3), it was then possible to achieve a reliable and credible range for the expected damage. The accuracy of each statistical method was then evaluated as the minimum relative error achieved considering the theoretical damage scenario defined with the reduced I\({}_{\rm V}\) and with the increased I\({}_{\rm V}\) compared to the real/observed one in the case study. A range of the initial level of expected damage was then established to perform the parametric analysis of the seismic vulnerability of the considered construction typology, evaluating the influence of the I\({}_{\rm V}\) methods in the seismic response of the buildings. The reliability of the I\({}_{\rm V}\) method was assessed, considering two typologies of relative error:  
* at the individual building level (\(\overline{Err}_{mean,d}\)) considering the difference of the theoretical damage (\(\overline{\mu_{T}}\)), evaluated with the direct measure of the Iv of the single building, with the real damage observed (\(\mu_{R}\));at a global level (\(\overline{Err}_{mean,r}\)), considering the difference of the theoretical damage scenarios defined with the reduced I\({}_{V}\) (\(\mu_{T}^{-}\)) and with the increased I\({}_{V}\) (\(\mu_{T}^{+}\)), with the real damage observed (\(\mu_{R}\)). The comparison of the results achieved with the \(\overline{Err}_{mean,r}\) and \(\overline{Err}_{mean,d}\) provide import information about the possibility of evaluating the I\({}_{V}\) of each building within a range, rather than for a single value. The next expressions (7) and (8) try to clarify the procedure adopted for each building:  
\[\text{I}_{\text{v}}=\left\{ \begin{array}{l}\text{I}_{\text{v}}^{-}=\overline{ \text{I}_{\text{v}}}-SD\rightarrow\mu_{T}^{-}\text{ (}lower bound\text{)}\rightarrow Err_{-}=\left|\mu_{R}-\mu_{T}^{-}\right|\\ \overline{\text{I}_{\text{v}}}\rightarrow\overline{\mu_{T}}\text{ (} direct measure\text{)}\rightarrow Err=\left|\mu_{R}-\overline{\mu_{T}}\right|\\ \text{I}_{\text{v}}^{+}=\overline{\text{I}_{\text{v}}}+SD\rightarrow\mu_{T}^ {+}\text{ (}upper bound\text{)}\rightarrow Err_{+}=\left|\mu_{R}-\mu_{T}^{+}\right| \end{array} \right. \tag{7}\]  
\[\overline{Err}_{mean,d}=\left(\frac{\Sigma Err}{n}\right) \tag{8}\]  
Being \(Err\) the error for a given building defined as the difference between theoretical and real damage detected in the case study in absolute value, \(n\) the number of buildings, and \(Err_{-}\) and \(Err_{+}\) the errors in a given building as the minimum difference between theoretical damage, considering the upper and lower bounds, and the real one of the case study in absolute value. Figure 12 summarizes the procedure adopted for the reliability evaluation of the selected I\({}_{V}\) methods in the case study considering the vulnerability range for each building. The process was applied to each of the three I\({}_{V}\) methods adopted, comparing the results obtained and assessing the lowest relative error (Table 6). As visible, none of the methods was able to estimate the observed damage of Campi Alto di Norcia with high accuracy, resulting in differences between real and theoretical damage, evaluated for each building, higher than 0.85 for all procedures (Figure 13). Considering a probable theoretical damage range, the relative errors decreased, remaining otherwise higher than 0.58. As already said, results highlight the poor accuracy of the methods if used under very different conditions than those assumed in their calibration processes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **I\({}_{V}\) Method Accuracy (Direct Measure)** & **I\({}_{V}\) Method Accuracy (Assuming a Range)** \\ \hline I\({}_{V}\) Methods & Average relative errors & Variance relative errors & Average relative errors & Variance relative errors \\ GDNT II & 0.87 & 0.67 & 0.67 & 0.37 \\ Formisano & 1.13 & 0.72 & 0.72 & 0.55 \\ Vicente & 0.85 & 0.61 & 0.58 & 0.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Evaluation of the Iv method reliability in the case study analysed.