# Incorporating Foreshocks in an Epidemic-like Description of Seismic Occurrence in Italy
## 4 Data Catalog, Methods and Null Models  
The purpose of the statistical test that will be carried out in this paper is to evaluate whether the three models defined in the previous section are able to describe the experimental data. In the null-hypothesis test we compare the number of aftershocks and foreshocks for each mainshock. In practice we consider the ISIDE Italian Catalog (from 2005/04/16 to 2021/04/30). We use the BPZB selection procedure to identify aftershocks and foreshocks and we evaluate the quantities \(n_{A}(T,R,m_{L},m_{M})\) and \(n_{F}(T,R,m_{L},m_{M})\) for a wide range of parameter \(m_{L}=[2,2.5,3,3.5]\), \(m_{M}=[4,5,6]\), and for \(T=1,3,10\) days, always considering \(R=L(m_{M})\) km. We compute these quantities for each model we defined before, namely ETAS, ETASI and ETAFS and we consider each of them as null model for the hypothesis test. We choose model parameters such as the models lead to synthetic catalogs of equal duration of the ISIDE catalog and roughly the same number of events with \(M>2.5\). For each model and for a defined set of model parameters, we generate 1000 catalogues that we concatenate and we identify seismic sequences by means of a BPZB procedure. After the identification of the sequences, we divide the whole catalog in \(N_{s}\) subset where each one contains the same number of the earthquakes of instrumental Italian catalogue. Now we compute the quantity \(n^{X}_{A,j}\) and \(n^{X}_{E,j}\), which represents the number of aftershocks and foreshocks, respectively, in the \(j\)-th subset produced by the \(X\) model. The same quantity is computed for the instrumental Italian catalogue. The superscript, \(X\), indicates a specific numerical model with the three possible entries \(X=[ETAS,ETASI,ETAFS]\), according to the corresponding model. Conversely, we no longer use the superscript \(X\) for the number of aftershocks and foreshocks in the instrumental Italian catalogue. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Model** & \(\mathbf{K_{0}}\) & \(\mathbf{\kappa}\) & \(\mathbf{b}\) & \(\mathbf{p}\) & \(\mathbf{c}\) & \(\mathbf{d}\) & \(\gamma\) & \(\mathbf{q}\) & \(\mathbf{\omega}\) & \(\mathbf{\delta_{0}}\) & \(\mathbf{K_{f}}\) & \(\mathbf{\alpha_{f}}\) \\ \hline ETAS & 0.07 & 0.95 & 0.95 & 1.2 & 0.024 & 0.006 & 1.958 & 1.3 & - & - & - & - \\ ETASI & 0.1 & 0.93 & 0.95 & 1.2 & 0.01 & 0.006 & 1.958 & 1.3 & 0.3 & 1 & - & - \\ ETAFS & 0.1 & 0.93 & 0.95 & 1.2 & 0.01 & 0.006 & 1.958 & 1.3 & 0.3 & 1 & 0.05 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the three models presented in this study. We consider the lower magnitude threshold \(m_{0}=2\). We evaluate the difference \(\psi_{j,A}=\frac{n_{A,j}^{X}-n_{A}}{n_{A,j}^{X}+n_{A}}\) and \(\psi_{j,F}=\frac{n_{A,j}^{X}-n_{F}}{n_{A,j}^{X}+n_{F}}\). This allows us to quantify the difference in the number of aftershocks and foreshocks, respectively, between the \(j\)-th realization of the synthetic catalogue of the \(X\) model and the instrumental ISIDE catalogue. In particular we evaluate these quantities for a different sets of parameters \((T,R,m_{L},m_{M})\). and compute two histograms \(H(\psi_{j,A})\) and \(H(\psi_{j,F})\) defined as the fraction of the \(N_{s}\) subsets with a value \(\psi_{A}=\psi\pm d\psi\), or \(\psi_{F}=\psi\pm d\psi\), where \(d\psi=0.005\). If the histogram \(H(\psi)\) is very peaked around \(\psi=0\), the prediction of the model is in good agreement with the experimental catalogue, i.e., the number of aftershocks (foreshocks) in the model is similar the instrumental one. The two limits \(\psi=-1\) and \(\psi=1\) represents the worst cases. In particular, with \(\psi=-1\), the synthetic catalogue predicts a number of events equal to 0. The opposite behaviour is observed with \(\psi=1\), which represents the limit for infinite events predicted in the simulated catalogue.