## 6 Discussion  
The comparison of the four global models, fixed and adaptive smoothed seismicity with and without our correction for the inclusion of aftershocks and foreshocks, clearly shows better performance of the models that use the correction. This positive result indicates that using all events of a seismic sequence instead of only the mainshock increases the forecasting capabilities of the smoothing seismicity models. Another very interesting result is the better performance of the adaptive approach concerning the fixed approach,  
Figure 5: The difference between the normalized seismicity rates (linear scale) of the adaptive and fixed corrected models (\(\mathrm{Adaptive_{corrected}}-\mathrm{Fixed_{corrected}}\)) in some zones in the world: Chile (**a**), Indonesia (**b**), and Mexico (**c**).  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Models**} & **Magnitude for the** & **Log-Likelihood Difference,** \\  & **Comparison** & **\(\Delta LL\)** \\ \hline Corrected Adaptive vs. Adaptive & 5.5+ & 7 \\ \hline Corrected Adaptive vs. Adaptive & 6.5+ & 7 \\ \hline Corrected Fixed vs. Fixed & 5.5+ & 99 \\ \hline Corrected Fixed vs. Fixed & 6.5+ & 18 \\ \hline Adaptive vs. Fixed & 5.5+ & 1658 \\ \hline Adaptive vs. Fixed & 6.5+ & 92 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Log-likelihood differences (\(\Delta LL\)) between the models.  
here demonstrated for a global catalog and two different magnitude thresholds, M\({}_{\rm w}\) 5.5+ and 6.5+. Looking at the normalized seismicity rates in Figure 4a-d, it is possible to note the larger smoothing for the fixed models compared to the adaptive models in the zones where the seismicity is higher. The difference between the adaptive and fixed smoothing approaches is evidenced in Figure 5: the large smoothing for the fixed model leads to lower rates with respect to the adaptive model in the zones where the earthquake rate for the testing catalog is higher (pink and red colors in Figure 5). On the contrary, the rates of the fixed model are higher with respect to the adaptive model in the areas adjacent to the more seismic active zones (blue colors in Figure 5). Zones far from the main seismic regions (e.g., intraplate zones with very few earthquakes) have a very small difference between the fixed and adaptive seismicity rate models (light pink color in Figure 5).  
The significantly better performances obtained by the adaptive smoothed approach (Table 5) confirm at a global scale the regional results obtained by [4] for California and [31] for Italy. Our method is more straightforward than that of [10], because it does not require a sophisticated stochastic declustering procedure [12]. Still, it only needs to identify the events in a seismic sequence, in this work made with the classical [16] declustering algorithm. Despite its simplicity, our method gives encouraging good results. A possible future work could be a comparison between our approach and the [10] approach.  
Our method is based on the assumption of stationarity of the seismicity (usually accepted in long-term modeling); however, working in smaller time and spatial scales, some regions may exhibit different spatiotemporal variations, useful to forecast stronger seismic events [32; 33]. Abandoning the stationarity assumption, smaller earthquakes can also be used to try to determine the current state of the seismic cycle [34] and then identify possible temporal variations in the long-term seismic rates.