# Incorporating Foreshocks in an Epidemic-like Description of Seismic Occurrence in Italy
## 3 The Models
### The Top-Down Loading ETAS Model--The ETAFS Model  
By construction, the previously defined models assume bottom-up triggering as the hypothesis for foreshocks. In other words, a foreshock is considered a normal earthquake that triggers an offspring with a magnitude greater than itself. However, it is important to also consider the possibility of top-down loading as an explanation for foreshocks. The Epidemic Type Aftershocks and Foreshocks Sequence ETAFS model is a generalization of the ETAS model that incorporates the mechanism of loading by aseismic slip to account for top-down triggering. In the ETAFS model, aftershocks are triggered with the same probability rate as the ETAS model, but each earthquake can also be preceded by foreshock activity. Mathematically, this can be expressed as:  
\[\Lambda_{ETAFS}(m,x,y,t)=\Lambda_{ETAS}+\sum_{k}10^{-bm}Q_{f}(d_{ik},t_{k}-t,m_ {k}) \tag{5}\]  
where \(d_{ik}\) is the distance between the event \(i\) and \(k\) and the sum extends over all events with magnitudes \(m_{k}\), occurred at time \(t_{k}\), in the position \((x_{k},y_{k})\) triggered according to the ETAS probability (Equation (2)), whereas \(Q_{f}(d_{ik},t_{k}-t,m_{k})\) is the rate of foreshocks potentially occurring at times \(t<t_{k}\). There is no specific constraint on the functional form of \(Q_{f}\). In order to reduce the number of model parameters we assume [21] that the spatio-temporal organization of the event in the cluster is similar to the aftershock one, setting  
\[Q_{f}(d_{ik},t_{k}-t,m_{k})=K_{f}10^{\gamma_{f}(m_{k}-m_{0})}\frac{(p_{f}-1)c _{f}^{p_{f}-1}}{(t_{k}-t+c_{f})^{p}}\frac{q_{f}-1}{\pi}(\delta(m_{k}))^{q_{f} -1}\Big{(}d_{ik}^{2}+\delta(m_{k})\Big{)}^{-q_{f}} \tag{6}\]  
where \(\delta(m_{k})=d_{f}10^{\gamma_{f}(m_{k}-m_{0})}\), \(\gamma_{f}=\gamma\), \(d_{f}=d\), and \(q_{f}=q\). As in the ETAS model, here we extract the number of events belonging to \(k\)-th cluster from a Poissonian distribution with average \(K_{f}10^{\alpha_{f}(m_{k}-m_{0})}\). Moreover we implement the inverse Omori Law with the same \(p\) as in the aftershock occurrence and \(c_{f}=c\). In principle, different functional forms could be adopted to achieve similar results. Finally, we apply the same aftershock removal procedure as done for the ETAS model in order to take into account the hiding of events caused by the overlapping of coda waves. In mathematical terms  
\[\Lambda_{ETAFS}^{inc}(m,x,y,t)=\Lambda_{ETAFS}(m,x,y,t)\times\prod_{j:t_{j }<t}\Phi(m|m_{j}^{th}(t-t_{j}),\sigma) \tag{7}\]For the simulations of the ETAFS model, we use exactly the same parameters as the ETASI model to generate a complete ETAS catalog. Starting from each simulated ETAS earthquake, we then use the kernel \(Q_{f}\) to generate foreshocks. The magnitude of each foreshock is extracted from the Gutenberg-Richter law, but with the constraint that the magnitude of each event belonging to the cluster must be smaller than the final magnitude \(m_{k}\). We finally apply the filtering procedure by means of the function \(\Phi\), according to Equation (7), to take into account incompleteness. The apparent problem with the ETAFS model is that the spatio-temporal organization of the cluster's events depends directly on the characteristics of the incoming event, the mainshock. In practice, it seems that this model violates the temporal causality principle since the occurrence probability at a certain time depends on the future. However, from the point of view of the point process-like model, the ETAFS remains well defined if one considers each single ETAS event as a cluster of events. In particular, in a formulation practically similar to the ETAS model, the ETAFS model can be viewed as a point process, where each single point has an internal structure represented by an earthquake and its anticipating foreshocks. With this approach, Equation (2) gives the occurrence probability of the last event of the cluster, and all events that belong to the same cluster are deterministically correlated with each other.