## 5 Seismic Vulnerability Assessment of Campi Alto di Norcia
### Reliability of the Seismic Vulnerability Methods  
The reliability of the selected statistical methods for the buildings' typologies of Campi Alto di Norcia was assessed by comparing the real damage data recorded after the 2016 earthquake sequence with theoretical damage scenarios evaluated through the  
Figure 11: (**a**) Distribution of the I\({}_{V}\) through the different methods for 67 SUs; (**b**) distribution of the I\({}_{V}\) through the different methods for 50 SUs in the aggregate. Figure 10: Normalized I\({}_{V}\) distribution in the case study of Campi Alto di Norcia, according to (**a**) the GDNT method, (**b**) the Formisano method and (**c**) the [PERSON] method. above-presented I\({}_{\rm V}\) methods. For the evaluation of the theoretical damage connected to the different approaches, the formulation proposed by [31] was adopted:  
\[\mu_{\rm T}=2.5+3\cdot\tan\hslash\left(\frac{\rm I+6.25\cdot V\ -12.7}{Q} \right)\cdot\mathrm{f}(V,\mathrm{I}) \tag{3}\]  
\[\mathrm{f}(V,\mathrm{I})=\left\{ \begin{array}{ll}\mathrm{e}^{\frac{V}{2} \cdot(1-7)}&\mathrm{I}\leq 7\\ 1&\mathrm{I}>7\end{array} \right. \tag{4}\]  
\[\mathrm{V}=0.56+0.0064\cdot\mathrm{I}_{\rm V} \tag{5}\]  
being V the vulnerability calculated from the I\({}_{\rm V}\) value, I the intensity using the European Macroseismic Scale 1998 (EMS98) using numbers between I (not felt) and XII (complete destruction) for the individual classification of earthquakes, Q the ductility factor and \(\mu_{\rm T}\) the damage value in the EMS98 scale, varying between 1 (negligible to slight damage) and 5 (destruction) according to [29]. Based on post-seismic damage evaluations of irregular brick masonry buildings and on [36], a ductility factor of 2.5 was adopted for masonry buildings with sufficient ductile behaviour. To measure the strength of shaking produced by the 2016 earthquake at Campi Alto di Norcia in October 2016, the magnitude, as well as the accelerograms, of the Campi Station (Table 2) were transformed into a macroseismic intensity; the results were then compared with data collected by the technician, who surveyed the damages just after the seismic events [37]. Finally, an intensity I\({}_{\rm MCS}\) equal to 8.5, evaluated with the MCS scale, was assigned to the case study [38]. Since the damage assessment was performed, as already said in the previous paragraph, through the European Macroseismic Intensity, to correlate the two intensity measures the analytical expression provided by [39] was used:  
\[\mathrm{I}_{\rm EMS98}=0.734+0.814\cdot\mathrm{I}_{\rm MCS}=7.65\approx 8.0 \tag{6}\]  
The relative error was evaluated as the difference between the theoretical and the observed damage in the case study after the seismic events of October 2016, then related to that specific seismic event's intensity (6). The I\({}_{\rm V}\) method reliability was calculated as the mean relative error of the values achieved for the whole building stock analysed. The uncertainties affecting the methods are both epistemic and aleatory, being related to the features characterizing each aggregate, to the classification of buildings into a vulnerability class or construction typology [40]. To overcome these issues and to control with more accuracy the reliability of the I\({}_{\rm V}\) method, the vulnerability of each building was evaluated not as a single value but as a range referred to the whole historical city centre. By this way, the upper and lower bounds of the I\({}_{\rm V}\) were defined for each SU, considering the SD of the vulnerability assessment of Campi Alto di Norcia. This procedure allows obtaining, for each I\({}_{\rm V}\), the possible vulnerability variation by taking into account the different assessments performed in the historic centre for the construction typology analysed. The aim was to evaluate a range of possible results for the seismic vulnerability of each building, whose amplitude is defined by the dispersion of the data with respect to the sample space.