# Incorporating Foreshocks in an Epidemic-like Description of Seismic Occurrence in Italy
## 6 Conclusions  
In this study, we analyzed the Italian seismic catalog (ISIDE) and compared different types of ETAS models. The first is the standard ETAS model with the only constraint that \(\alpha=b\), as proposed in [19]. The second and third variants, ETASI [30] and ETAFS [21], explicitly account for the incompleteness of the experimental catalog and a higher probability of foreshock occurrence, respectively. The ETAFS model combines the concept of bottom-up triggering for the aftershock side and top-down loading for the occurrence of foreshocks. By simulating the Italian seismicity with the ETAS model, we show that the fixed-\(\alpha\) ETAS model accurately reproduces the number of aftershocks in the ISIDE catalog when the magnitude difference between mainshock and aftershocks is relatively small. However, an excess of aftershocks is observed in the ETAS model compared to the ISIDE catalog when this difference increases. We attribute this effect to the short-term aftershock incompleteness, which becomes much more important as the magnitude difference between mainshock and aftershocks increases. To address this issue, we demonstrate that introducing the incompleteness ingredient in the ETAS model (ETASI model) makes it possible to better describe the occurrence of aftershocks by removing events that were not recorded by seismic stations. The key observation, however, is that both the ETAS model and the ETASI model fail to capture the number of foreshocks present in the ISIDE catalog. Our study clearly shows an excess of foreshocks in the ISIDE catalog for mainshocks with magnitude \(m<6\) compared to what is predicted by the ETAS and ETASI models. In contrast, the ETAFS model, which introduces a preparatory phase accompanied by foreshocks in a point process description, accurately describes the seismicity reported by the ISIDE catalog for both foreshocks and aftershocks. These patterns are consistent with those found for the Southern California seismic catalog in ref. [21], suggesting that they are a stable feature of seismic occurrences. The situation for the three mainshocks with \(m>6\) in the ISIDE catalog is different. Indeed, two of them, the Amatrice and the Norcia earthquakes, do not present foreshocks, whereas the L'Aquila earthquake presents a number of foreshocks significantly larger than those predicted by the ETAFS model. We finally remark that the ETAFS model assumes that the magnitude of the mainshock is encoded in the spatial organization of the foreshocks. The spatial kernel of \(Q_{f}\) is of the order of the area fractured by the mainshock. Therefore, while the prediction of a mainshock with a bottom-up triggering ETAS model is purely random, in the case of a top-down loading model, it would be possible to use the organization of foreshock epicenters as forecasting information. It is important to note that we did not assume a direct dependency between the magnitude of foreshocks and the magnitude of the mainshock, since the introduction of such dependence is still being studied [51] and is beyond the scope of this article. [PERSON] and [PERSON] contributed to the research, numerical results and writing of the manuscript. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The complete seismic catalog is available at [[http://terremoti.ingv.it/en/iside](http://terremoti.ingv.it/en/iside)]([http://terremoti.ingv.it/en/iside](http://terremoti.ingv.it/en/iside)), accessed on 30 April 2021. \begin{table}
\begin{tabular}{c c c} \hline
**Name** & **Mw** & **Date** \\ \hline L’Aquila & 6.0 & 6 April 2009 \\ \hline Amatrice & 6.0 & 24 August 2016 \\ \hline Norcia & 6.5 & 30 October 2016 \\ \hline \end{tabular}
\end{table} 
Table 3: The Mw6 mainshock registered in the ISIDE seismic catalogue. Acknowledgments. [PERSON] acknowledges support by MEXT Project for Seismology TowArd Research innovation with Data of Earthquake (STAR-E Project), Grant Number: JPJ010217.