# Incorporating Foreshocks in an Epidemic-like Description of Seismic Occurrence in Italy
## 3 The Models
### The Incomplete ETAS Model: The ETASI Model  
A crucial prerequisite for an accurate estimation of the ETAS parameters is the completeness of the seismic catalog. Unfortunately, due to the strong temporal correlation between earthquakes, there is a significant lack of events, particularly in the immediate aftermath of high-magnitude mainshocks [39; 40; 41; 42; 43]. This is caused by the overlapping of coda waves, which is even more pronounced when focusing on larger earthquakes where aftershocks are more challenging to detect and report in experimental catalogs [30; 44; 45; 46; 47; 48]. To address this issue, we introduce the concept of short-term aftershock incompleteness into the ETAS model, resulting in the ETAS Incomplete (ETASI) model [30]. The core idea is that the probability of detecting an event at time \(t-t_{i}\) after an \(m_{i}\) earthquake depends on the difference between its magnitude \(m\) and the detection threshold \(m_{i}^{th}(t-t_{i})\). Specifically, we adopt the functional form proposed by [39; 40].  
\[m_{i}^{th}(t-t_{i})=m_{i}-w\log(t-t_{i})-\delta_{0} \tag{3}\]  
where \(w\) and \(\delta_{0}\) are two parameters obtained to reach the best agreement with the instrumental catalog. The functional form for \(m_{i}^{th}\) (Equation (3)) is the most compatible with experimental data, and its logarithmic decay can be explained by the behavior of the seismic  
Figure 2: Instrumental magnitude distribution \(P(m)\) for the Italian seismicity. Black circles represents only the background activity whereas red squares the whole catalog.  
waveform envelope \(\mu(t)\) after a mainshock [44; 49]. This envelope is always greater than a minimum value \(\mu_{c}(t)\), which decays logarithmically. [PERSON] et al. [44] have linked the existence of \(\mu_{c}(t)\) to the overlap between aftershock coda waves, and demonstrated that its decay incorporates parameters related to the Omori-Utsu law governing the decay of aftershocks [50]. Therefore, the expected number of aftershocks in the immediate aftermath of a mainshock can be estimated [49]. We implement the expression Equation (3) in the ETAS model by means of the function \(\Phi(m|m_{j}^{th}(t-t_{j}),\sigma)\) which represents the cumulative distribution of the normal distribution. In particular, \(\Phi\) is a decreasing function of \(m\) which is roughly equal 1 when \(m>m_{j}^{th}(t-t_{j})+\sigma\), whereas \(\Phi\simeq 0\) when \(m<\lambda-\sigma\).  
The function \(\Phi\) can be implemented in the ETAS model Equation (2) as  
\[\Lambda(m,x,y,t)=\Lambda(x,y,t)\times\prod_{j:t_{j}<t}\Phi(m|m_{j}^{th}(t-t_{j} ),\sigma) \tag{4}\]  
The implementation of the ETASI model is straightforward, as it involves generating a synthetic ETAS catalog and then applying the \(\Phi\) function to remove a portion of the events. The parameters of the ETASI model include all those of the ETAS, as well as two additional parameters that account for the incompleteness function. However, we do not restrict our analysis to fitting only the last two parameters (\(w\) and \(\delta_{0}\)); rather, we perform a global optimization of all 10 parameters in the model.