# Incorporating Foreshocks in an Epidemic-like Description of Seismic Occurrence in Italy
## 5 Results
### Foreshock Comparison  
In Figure 5 we plot the histograms \(H(\psi)\) of \(\psi_{j,F}\) for the ETAS and ETAFS catalogues. We will not present results for the ETASI model which are comparable to those of the ETAS model, since the two models only differ in the implementation of short term incompleteness,  
Figure 4: The fraction of numerical subsets \(H(\psi)\) of the ETAFS (ETASI) and ETAS simulated catalogue, with a number of aftershocks \(\frac{n_{M}^{2}(T,R,m_{L},m_{M})-n_{A}(T,R,m_{L},m_{M})}{n_{M}^{2}(T,R,m_{L}, m_{M})+n_{A}(T,R,m_{L},m_{M})}=\psi\pm 0.005\). Results are for different values of the time window \(T\). Moving horizontally, from left to right, \(m_{L}=2,2.5,3,3.5\) whereas vertically, from top to bottom, \(m_{M}=4,5,6\). The orange dashed line indicate the optimal description of the seismicity \(\psi=0\). which is not relevant for foreshocks. As for the aftershocks, we consider different windows of time \(T\), foreshock minimum magnitude \(m_{L}\) and mainshock threshold \(m_{M}=4\), \(5\). We will present results for mainshocks with \(m>6\) separately. This figure shows that the ETAS catalogue presents significant less foreshocks than the instrumental Italian one. Indeed \(H(\psi)\) is in all cases peaked around \(\psi\simeq-0.5\), which indicates that the number of foreshocks in the ETAS catalogue is, on average, about \(1/3\) the number found in the Italian catalogue. Moreover, we note that the probability of producing a numerical catalog that has the same number of foreshocks as the observed catalog is practically zero. Indeed, no ETAS synthetic catalog presents a number of foreshocks equal or larger than the one observed in the instrumental catalog when \(m_{M}=4\) and, at the same time, when \(m_{M}=5\) less than 1% of synthetic catalogs satisfies this condition. This conclusion can be drawn for all the foreshocks magnitude thresholds \(m_{L}\), and for each time window \(T\) considered. This interpretation is supported from the measurement of the quantity \(I_{+}^{ETAS}\) reported in Table 2. Indeed, for all values of \(m_{M}\), \(m_{L}\) and \(T\) the value of \(I_{+}^{ETAS}\) is nearly zero. We conclude that it is very unlikely or even impossible that a synthetic catalog for the ETAS model presents a number of foreshocks equal or larger than the one observed in the Italian seismic catalogue. The situation slightly changes when one considers the ETASI model, with \(I_{+}^{ETASI}\) (Table 2) still indicating that the hypothesis that the ETASI model predicts a number of foreshocks equal to or larger than those actually observed can be rejected with a high confidence. On the contrary, the numerical catalog simulated by means of the ETAFS model captures the seismicity of the foreshocks very well; in fact, all the \(H(\psi)\) histograms exhibit a peak that is very close to \(\psi=0\). We observe this behavior for all minimum magnitude \(m_{L}\) and temporal domains \(T\). It is worth noting that a dramatic change in \(H(\psi)\) is obtained by adding just a small percentage of earthquakes (about 10%) to the ETASI catalog. This small percentage of earthquakes are foreshocks triggered according to the second term in Equation (5), indicating that their addition is strictly necessary to obtain a good agreement with the observed seismicity in the Italian catalogue. This is confirmed quantitatively in Table 2, where all \(I_{+}^{ETAFS}\) values are significantly larger than 0 and smaller than 1, presenting fluctuations around the optimal value (\(I_{+}^{ETAFS}=0.5\)). The ISIDE catalog presents only three \(m_{M}\geq 6\) earthquakes, Table 3, which are here analysed separately. The L'Aquila earthquake was a Mw6 magnitude event that occurred on 6 April 2009 at 01:32 UTC in the Abruzzo region of central Italy. After BPZB declustering, we identified 540 foreshocks without considering any time or magnitude constraints. The Amatrice and Norcia events, on the other hand, were a Mw6 and Mw6.5 magnitude earthquake that occurred on 24 August 2016 and 30 October 2016, respectively.