**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 Fixedcorrected and Adaptivecorrected).

To check if our corrected models perform better than those uncorrected smoothed seismicity models, we tested the Fixedcorrected and Adaptivecorrected 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, Mw 5.5+ and Mw6.5+, 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 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 Mw ≥ 5.5 in the global CMT catalog (normalized seismicity rates, i.e., PDF, are in log10 scale).

**Table 3.** Log-likelihood (*LL*) values for the smoothed seismicity models for testing catalog from magnitude Mw 5.5 (3161 events).


**Table 4.** Log-likelihood (*LL*) values for the smoothed seismicity models for testing catalog from magnitude Mw 6.5 (300 events).


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 (Adoptive and Adaptivecorrected) produce larger *LL* values and reveal better forecasting performance with respect to those from the fixed smoothed ones (Fixed and Fixedcorrected). The *LL* 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). The largest *LL* value calculated for the adopted smoothed seismicity models arises from the use of the correction parameter including the foreshocks and aftershocks in the global catalog. So, in general, including smaller earthquakes in the clusters increases the performance of the future M w ≥ 5.5 and M w ≥ 6.5 earthquake forecasting capability in the smoothed seismicity models.

To understand if this increase is rather significant, we interpreted the difference of the *LL* values for two models in terms of the Bayes factor [27], a common interpretation for pseudoprospective experiments [28–30]. According to [27] table, we obtained "very strong evidence" (difference in log-likelihood Δ*LL* > 5) in favor of our proposed method, both for the fixed and adaptive approaches (Table 5). In Figure 5a–c, we also present the different maps calculated between the normalized seismicity rates (linear scale) of the adaptive and fixed corrected models (as Adaptivecorrected − Fixedcorrected), along with the events of the testing catalog, in the same zones of Figure 2: Indonesia (Figure 2a), Mexico (Figure 2b), and Chile (Figure 2c). Colors in light blue to red represent positive differences (i.e., the rate of the adaptive model is higher with respect to the fixed model), deep blue represents negative differences (i.e., the rate of the adaptive model is lower with respect to the fixed model), and blue represents no difference.

**Table 5.** Log-likelihood differences (Δ*LL*) between the models.


**Figure 5.** The difference between the normalized seismicity rates (linear scale) of the adaptive and fixed corrected models (Adaptivecorrected − Fixedcorrected) in some zones in the world: Chile (**a**), Indonesia (**b**), and Mexico (**c**).
