# Incorporating Foreshocks in an Epidemic-like Description of Seismic Occurrence in Italy
## 2 General Definitions and Declustering  
_Aftershock, Foreshocks and Declustering Techniques_  
There are several ways to decluster a numerical earthquake catalog, resulting in different definitions of aftershocks, foreshocks, and mainshocks. In this paper, we consider the Baiesi-Paczusky-Zaliapin-Ben-Zion (BPZB) declustering method proposed in [32; 33; 34; 35], which relies on a metric to quantify the correlation between events. The nearest neighbor of an event is defined by the metric \(\eta_{ij}=t_{ij}r_{ij}^{d}|10^{-b(m_{i}-m_{c})}\), where \(t_{ij}\) is the time difference between events \(i\) and \(j\), \(r_{ij}\) is their epicentral distance, \(d\) is the fractal dimensionality of epicenters, \(b\) is the exponent of the Gutenberg-Richter (GR) law, \(m_{i}\) is the magnitude of the first event, and \(m_{c}\) is the completeness magnitude. Two earthquakes form a cluster if their distance \(\eta_{ij}\) is below a certain threshold \(\eta_{c}\). To determine \(\eta_{c}\), we define the magnitude-normalized time and space components as follows  
\[\tau_{ij}=t_{ij}\times 10^{-bm_{i}/2}\hskip 28.452756 pts_{ij}=r_{ij}\times 10^{-bm_{ i}/2} \tag{1}\]  
and for any pair of events we plot \(log(s_{ij})\) vs. \(log(\tau_{ij})\) in Figure 1. We use \(d=1.6\) for the fractal dimensionality, and \(b=0.95\) obtained from the magnitude distribution (Figure 2). Similar results are obtained for similar choices of \(b\) and \(d\). In Figure 1, two distinct populations can be observed: the foreshock/aftershock clustering, consisting of events with \(s_{ij}\tau_{ij}<\eta_{c}\), and the stationary Poisson seismicity, consisting of events \(i\) with \(s_{ij}\tau_{ij}>\eta_{c}\) for any \(j\). The red line in Figure 1 represents the clear boundary between these two populations, which is achieved with \(\eta_{c}=10^{6}\). For each event \(i\) in the Poisson population, we identify a seismic sequence that includes event \(i\) and all \(k\) clustered events with \(\eta_{ki}<\eta_{c}\). The largest event in the sequence is defined as the mainshock, while events occurringbefore the mainshock are classified as foreshocks and those occurring after are classified as aftershocks.  
To avoid obscuring the observation of foreshocks, we apply an additional filter proposed by [19] to the declustering procedure. Specifically, if an event \(i\) is identified as a mainshock after the BPZB procedure, but it occurs close enough in time and space to a previous large earthquake (\(m\geq 5\)), then it is considered an aftershock of the earlier event rather than a background event. More precisely, if the probability that the \(i\)-event is triggered by the \(m\geq 5\) earthquake is higher than the background probability \(\mu\), the entire cluster is excluded from the study.  
After identifying the complete cluster of seismic events, we group all mainshocks into magnitude classes \(m\in[m_{M},m_{M}+1)\), and for each aftershock and foreshock, we define a temporal distance \(\Delta t_{M}\) and an epicentral distance \(\Delta r_{M}\) from the corresponding mainshock. We then define \(n_{A}(T,R,m_{L},m_{M})\) and \(n_{F}(T,R,m_{L},m_{M})\) as the total number of aftershocks (and foreshocks) with \(\Delta t_{M}<T\), \(\Delta r_{M}<R\), and magnitude \(m\in[m_{L},m_{L}+0.5)\) linked to a mainshock with magnitude \(m\in[m_{M},m_{M}+1)\).  
In this study, we use \(T=10\) days, \(m_{M}=4,5\), \(m_{L}=2,2.5,3,3.5\), and \(R=L(m_{M})=0.01\times 10^{05m_{M}}\).  
Figure 1: Bimodal distribution of time and space components of the nearest-neighbor for the observed seismicity in Italy. Solid red line corresponds to \(log_{10}(s)+log_{10}(\tau)=6\). The fractal dimensionality is fixed at \(d=1.6\).