# Stochastic Generator of Earthquakes for Mainland France
## 2 Methodology
### Main Shock Generation
#### 2.1.2 Main Shock Generation in Space  
Every main shock simulated in terms of year and magnitude is then localized somewhere in the whole of mainland France.  
A recent study [16] stipulates that the structural inheritance--that is, crust or mantle weakening from past tectonic--can play an important role in deformation localization in intraplate context. According to previous works and new observation, these authors estimate that 55-95% of the stable region's seismicity occurs near such weaknesses. Based on this, we suppose here that such geological objects, that give testimony to hundreds of millions years of seismicity, give more exhaustive information than past seismicity in predicting possible location of future earthquakes. In this paper, for a first approximation, we propose guiding the location of generated main shocks by using a fault probability map.  
However, even in active regions around the world, fault activity could be difficult to address, e.g., [20] (for the Canterbury earthquake in New Zealand). Recent studies in China [21] or earthquakes in France [22] and Australia [23] serve as reminders for how difficult it is to define fault activity in stable regions. Since, in these regions, faults can be inactive over long periods of time [14; 21], we decided to consider all faults, regardless of their presumed importance and activity.  
The map of faults is converted into a density map by computing the average length of fault lines per unit area on a Cartesian grid. After that, the density values are normalized in order to obtain a sum equal to 1, which leads to a probability map. In that sense, each cell of the grid is characterized by a probability of hosting faults. Grid resolution is set by the user.  
Looking at past seismicity, one would notice that earthquakes with the highest magnitudes are concentrated in the most active regions, even in intraplate domains. For example, these regions in France are the Alps, the Pyrenees and the Rhine basin. We decide to apply a magnitude threshold to drive highest magnitudes' earthquakes in areas where they have most chance to happen. To achieve this, we segregate the fault probability map by regionalization. Each region is characterized by a maximal magnitude \(M_{max\,region}\). This magnitude corresponds to that of the largest earthquakes that this region can host.  
Coupling regionalization with probability map leads to a set of regions defined by their own probability map and maximal magnitude \(M_{max\,region}\). A main shock of magnitude \(M\) generated at year \(y\) can only be localized in eligible regions, i.e., regions where \(M\leq M_{max\,region}\). Once the set of eligible regions is defined, the location of this main shock is drawn through the cumulative distribution function of the corresponding probability map (Figure 2). This procedure is a heterogeneous Poisson process whose density depends on the magnitude of the synthetic main shock to be located.  
The Cartesian grid on which fault probability map is computed is used to select a cell, whose resolution depends on the user, where the main shocks' coordinates (the black point in Figure 2) are uniformly drawn. Figure 2 summarizes the localization process. Owing to this approach, spatial distribution of generated main shocks is limited by magnitude, with respect to the fault probability map that characterizes seismicity.  
This step is referred as 2A in Figure 1.