## 3 A New Smoothed Seismicity Approach  
Building a spatial grid is the first step to constructing a spatial smoothed seismicity model [3; 4]. In this work, we used a global spatial regular grid, \(0.5^{\circ}\) by \(0.5^{\circ}\). Therefore, we need to compute the contribution to each event in the seismic catalog to the generic \(i\)-th spatial grid; the following equation describes that contribution:  
\[f_{i}=\sum_{j=1}^{N}cK_{ij}A_{i}d_{j} \tag{1}\]  
where \(f_{i}\) represents the normalized total seismic rate for the \(i\)-th spatial grid, \(N\) is the total number of events in the complete (i.e., not declustered) catalog, \(c\) is the normalization factor \(\left(c=\frac{1}{\sum f_{i}}\right)\), \(K_{ij}\) is the kernel function that depends on the distance between the center of the \(i\)-_th_ spatial cell and the epicentre of the \(j\)-th earthquake, \(A_{i}\) is the area of the \(i\)-th spatial cell, and \(d_{j}\) is the correction to take into account the foreshocks and aftershocks contribution to the spatial model.  
The following Gaussian kernel function [3] is used:  
\[K_{ij}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{r_{ij}^{2}}{2\sigma^{2}}} \tag{2}\]  
where \(r_{ij}\) is the distance between the center of the \(i\)-th spatial cell and the epicentre of the \(j\)-th earthquake, and \(\sigma\) is the free parameter of the model that rules the amplitude of the smoothing. However, we note that different kernel functions can also be employed in smoothing the epicenters from the earthquake catalog [4; 18]. The smoothing distance, \(\sigma\), involved in each earthquake may be defined differently in various smoothed seismicity models. For example, the fixed smoothed seismicity models practiced a single smoothing distance for all earthquakes. The adaptive smoothed seismicity models represent unique smoothing distances for each earthquake between an event and its \(n\)th closest neighbors (NN), resulting in spatially varying smoothing distances [4]. The distance becomes smaller in regions of high seismicity than in areas with sparse seismicity. It is one of the crucial parameters in the smoothed seismicity models both for the earthquake rates and the spatial variations of the earthquake activity rates in a region [19]. The correction parameter \(d_{j}\) represents the innovative part of our method. It is defined as following \(d_{j}=\frac{1}{5_{j}}\), where \(S_{j}\) is the number of events in the seismic sequence and contains the \(j\)-th event. For example, if a seismic sequence contains ten events, one mainshock, and nine aftershocks, each event receives a weight, \(=\frac{1}{10}\). Since the sum of all the weights is equal to one, the inclusion of aftershocks does not create a spatial bias in the model [6], and it leads to a better description of the fault that generated the sequence.  
Using this simple correction may help better identify the active fault structures and their features in a region. Removing all the aftershocks and foreshocks [3; 4], giving very high weight to the mainshocks only [10], may lead to an incomplete or biased view of the spatial distribution of future seismicity. Conversely, considering all the events in the sequence with a uniform weight, as in our method, increases the model's forecasting performance.  
We underline that with Equation (1), we build normalized smoothed seismicity models, i.e., the sum of all the rates in the spatial cells are equal to 1. In this work, we do not face the problem of the total number of events and their magnitude frequency distribution, already treated in [9]. In that work, the seismicity rates are corrected by a proposed technique that allows counting all events in the complete seismic catalog by quickly adjusting the magnitude frequency relationships. Our method differs from theirs, since we only deal with the spatial distribution of the seismicity by using an equal weight for all the events of the corresponding seismic sequence and incorporating aftershocks to improve the spatial resolution of the model.