## 4 Results
### Deriving an ESI-07 IPE for Normal Faulting Earthquakes with Global Validity  
To derive an IPE with global validity for normal faulting earthquakes, we built a dataset (Table 1) of 26 global earthquakes that occurred between 1688 and 2020. The selected earthquakes (magnitude range 5.4-7.4) occurred in Italy, Greece, Spain, and Mexico and have an ESI-07 I\({}_{0}\) within the range of IX to XI. The distribution of data, in relation to ESI-07 intensity, shows a Gaussian-type distribution, with about 70% of the data between ESI-07 VI and VIII (Figure 5a). Following the data decimation process, our dataset encompasses a total of 1106 intensity data points (Figure 5b); 48% of the IDPs are located within 15 km from the epicenter and 91% within 40 km.  
In Figure 6a, we introduce the intensity prediction equation developed using the new global dataset, expressed in terms of \(\Delta\Gamma\) versus epicentral distance; the curve and 95% confidence bounds are shown; numerical coefficients are presented in Table 3. The attenuation is steeper in the range between about 10 and 25 km, then gradually decreases.  
In Figure 6b, we draw the curves of the expected ESI-07, considering the I\({}_{0}\) values of XI, X, and IX; solid lines represent the results obtained in this study while dashed lines are the same using the equation of [PERSON] et al. (2020) [10]. This comparison provides insights into the consistency and reliability of our proposed IPE equation. Figure 6b shows that the slope of the IPE derived in this study using epicentral distance is more pronounced than the one developed by [PERSON] et al. (2020) [10], resulting in a faster attenuation. Nevertheless, the two equations are fairly similar since the difference between the two is generally within one degree of intensity (Figure 6c).  
Given the distribution of the input data, we consider the proposed equation reliable up to 40 km, and we do not recommend extrapolation beyond these limits.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **1980 Irpinia–Basilicata, Italy** & **ESI-07 VI** & **ESI-07 VII** & **ESI-07 VIII** & **ESI-07 IX** & **ESI-07 X** \\ \hline \multirow{3}{*}{Nr. Observations} & IDP Serva et al., 2007 [37] & 4 & 20 & 35 & 4 & 2 \\ \cline{2-7}  & IDP Pizza et al., 2023 [38] & 137 & 248 & 89 & 14 & 38 \\ \cline{2-7}  & Isoseisimals & 6017 & 4581 & 1444 & 404 & 188 \\ \hline \multirow{3}{*}{Median distance (km)} & IDP Pizza et al., 2023 [38] & 20 & 17 & 15 & 17 & 13 \\ \cline{2-7}  & Isoseisimals & 63 & 40 & 23 & 15 & 15 \\ \cline{2-7}  & Predicted ([PERSON] et al., 2020 [10], assuming I\({}_{0}=\) X) & 35 & 24 & 17 & 11 & 7 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Number of observations and median epicentral distances for the 1980 Irpinia–Basilicata earthquake.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Log-Linear Model: R = a + b \(\times\) log\({}_{\text{int}}\)(R) + c \(\times\) R, where R is the Distance (km):**} \\ \hline
**Type of Distance (R)** & **a (95\% Conf. Bounds)** & **b (95\%)** & **c (95\%)** & **SSE** & **R\({}^{\text{2}}\)** & **Adj. R\({}^{\text{2}}\)** & **RMSE** \\ \hline D\({}_{\text{Epi}}\) & \(-16.14\,(-31.15,-1.128)\) & \(17.81\,(1.406,34.21)\) & \(-0.183\,(-0.4639,0.09788)\) & 0.9941 & 0.9432 & 0.9053 & 0.5756 \\ \hline D\({}_{\text{RnpX}}\) & \(1.167\,(-0.5689,2.903)\) & \(0.8557\,(-0.8146,2.526)\) & \(0.06127\,(-0.0605,0.1826)\) & \(2.846\,\) & \(0.8374\,\) & \(0.7289\,\) & \(0.974\,\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of the results obtained in terms of “ epicentral distance” (D\({}_{\text{Epi}}\)) and “horizontal distance from the fault” (D\({}_{\text{RnpX}}\)). SSE: Sum of Squared Errors, RMSE: Root Mean Square Error.