## 5 Results  
The final smoothed seismicity models are constructed using the entire learning catalog and the optimized correlation distances, previously obtained and given in Table 2. The models represent the bidimensional probability density function (PDF) of the seismicity (the sum of all the rates is 1). The corrected fixed smoothed seismicity model is calculated with a smoothing distance of 115 km, and it is 135 km in the case of the uncorrected model. Both adaptive smoothed seismicity models are determined using the nearest neighbor number equal to 1. These fixed and adaptive smoothed seismicity rate models are illustrated in Figure 4a,b (not corrected, hereafter fixed and adaptive) and Figure 4c,d (corrected, hereafter Fixed,corrected and Adaptivecorrected). To check if our corrected models perform better than those uncorrected smoothed seismicity models, we tested the \(\text{Fixed}_{\text{corrected}}\) and \(\text{Adaptive}_{\text{corrected}}\) models against the two standard fixed and adaptive smoothed seismicity models. Therefore, we performed a global pseudoprospective test, computing the \(LL\) (Equation (2)) of the four models using the ten-year testing catalog (2010-2019). Here, we outline that our testing catalog is entirely independent of the developed models. We preferred to endorse a similar computation procedure adopted in the real global prospective tests of the Collaboratory for the Study of Earthquake Predictability, CSEP, [23] and the global experiments [25; 26]. We evaluated the performance of the models using two different magnitude thresholds, M\({}_{\text{w}}\) 5.5+ and M\({}_{\text{w}}\)6.5+, to check the robustness of our models' forecasting locations and rates for future earthquakes. The results of these comparisons are presented in Tables 3 and 4 for the four developed models. Figure 3: MLE for the parameters sigma and NN for the Fixed (**a**), Adaptive (**b**), \(\text{Fixed}_{\text{corrected}}\) (**c**), and \(\text{Adaptive}_{\text{corrected}}\) (**d**) models. Blue curves represent the log-likelihood functions, red dots the maximum of these functions. For a correct interpretation of the models' \(LL\), we recall that large \(LL\) values (i.e., the ones nearest to zero) indicate relatively good performances of the models, and small \(LL\) values (i.e., the ones further from zero) indicate relative bad performances of the models. In general, our results show that the adaptive smoothed seismicity models (Adaptive and Adaptive\({}_{\text{corrected}}\)) produce larger \(LL\) values and reveal better forecasting performance with respect to those from the fixed smoothed ones (Fixed and Fixed\({}_{\text{corrected}}\)). The \(LL\) values are also consistent with the results of the adaptive smoothed seismicity models. \begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Log-Likelihood (\(LL\))** \\ \hline Corrected Adaptive & \(-2850\) \\ \hline Adaptive & \(-2857\) \\ \hline Corrected Fixed & \(-2931\) \\ \hline Fixed & \(-2949\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Log-likelihood (\(LL\)) values for the smoothed seismicity models for testing catalog from magnitude M\({}_{w}\) 6.5 (300 events). Figure 4: Spatially smoothed seismicity models using (**a**) 135 km smoothing distance from the fixed and (**b**) the nearest neighbor number NN = \(1\) from the adaptive smoothing seismicity approaches; spatially smoothed seismicity-corrected models using (**c**) 115 km smoothing distance from the fixed and (**d**) the nearest neighbor number NN = 1 from the adaptive smoothing seismicity approaches, employing the epicenters of the earthquakes for M\({}_{w}\geq\) 5.5 in the global CMT catalog (normalized seismicity rates, i.e., PDF, are in log10 scale). values are \(-29\),632 and \(-29\),639 for the corrected and uncorrected adaptive smoothed models, while they are \(-31\),297 and \(-31\),198 in the case of the fixed corrected and corrected smoothed seismicity models, respectively (Table 3).