## 4 Likelihood Testing for Spatial Variation of Seismicity  
To perform the maximum likelihood estimation of the parameter \(\sigma\) (both for the fixed and adaptive smoothing approach) and to assess the performance of the model, we avoid considering the Poisson distribution of seismic events because this assumption is rarely satisfied by the seismic catalogs [20; 21]. Since we are interested only in the spatial distribution of the events, and with Equation (1), we model the normalized spatial distribution of events, we defined the log-likelihood (\(LL\)) of the observations with:  
\[LL(X|M)=\sum_{i=1}^{N}log(f_{i}) \tag{3}\]  
where \(X\) is the set of the \(N\) observations (i.e., the epicenters of the events in the seismic catalog), \(M\) is the spatial model, \(log\) is the natural logarithm, and \(f_{i}\) is the seismic rate of the spatial cell where the \(i\)-th event is located. We note that this formulation differs from the spatial \(LL\) defined by [22] and has been commonly used in many seismic experiments [23], since the Poisson hypothesis has been abandoned in our study. The \(LL\) of Equation (3) may be ratified as the classical \(LL\) of a bivariate probability density function (represented by the model, \(M\)) in case we assume the independence between the observations in the set \(X\). Additionally, in the case of nonindependent observations, the \(LL\) can be still used for scoring the models (some authors, in this case, called the function \"pseudo-likelihood\", [24].  
To perform a pseudoprospective evaluation of the models first, we calculated the log-likelihood values by dividing the earthquake catalog into two parts: (1) the learning catalog, which contains the events recorded between 1980 and 2009 and is used to construct trial smoothed seismicity models, and (2) the testing catalog, which covers the last ten years of catalogs (2010-2019). The same \(LL\) of Equation (3) is also used to evaluate the performance of the models.  
We applied the fixed and adaptive smoothing methods with and without our correction to include aftershocks and foreshocks for a total number of four different models. First, we used the learning catalog to compute the optimal smoothing parameters from the maximum-likelihood estimations (MLE), which strongly vary with smoothing distance (fixed smoothing) and neighbor number (adaptive smoothing). In the case of fixed smoothing, we used a vector of possible sigma (from 5 km to 200 km, with a spacing of 5 km), while for the adaptive smoothing, a set of possible neighbor numbers) are considered from 1 to 20, with a spacing of 1. The first part of the learning catalog (1980-1999) with a period of twenty years is utilized to build various smoothed seismicity models with different sigma and NN values. Finally, the nearest neighbor numbers and the correlation distances are calculated through maximum-likelihood optimization for the four smoothed seismicity models using the last ten years of the learning catalog (2000-2009). The results of these estimations are summarized in Table 2 and Figure 3.  
We underline that these obtained MLE values are suitable only in the case of a global catalog: regional estimation of these parameters can lead to different MLE values (e.g., smaller sigma and larger NN).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **MLE** \\ \hline Fixed & Sigma = 135 \\ \hline Adaptive & NN = 1 \\ \hline Corrected Fixed & Sigma = 115 \\ \hline Corrected Adaptive & NN = 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: MLE of the parameters.