# Stochastic Generator of Earthquakes for Mainland France
## 3 Application to Mainland France
### Results
#### 3.3.1 Main Shock Generation  
Cumulative Seismic Moment Distribution  
Cumulative seismic moments produced by both synthetic and instrumental catalogues are compared Figure 9. The seismic moment \(M_{0}\) (N.m) released by an earthquake is calculated for each real or synthetic main shock directly from its moment magnitude \(M_{w}\) following [42]:  
\[M_{0}=10^{1.5\,M_{w}+9.1}\,. \tag{12}\]  
According to Table 2, the completeness period associated with instrumental \(M\geq 4\) earthquakes is 1965-2020, which represents 56 years. Comparisons are thus made over this completeness period. Since 100,000 years have been generated, we divided the synthetic seismicity into a set of 1785 sub-catalog of 56 years. Each of these sub-catalog are comparable to the complete instrumental catalogue. The final seismic moment observed in the instrumental catalogue is smaller than the final mean seismic moment generated. This slight difference is due to the fact that the return periods of \(M\geq 4\) main shocks given as an input are equal to 1.14 years (Table 3) and 1.24 years, according to the initial catalogue. Thus, the initial catalogue (red line Figure 9) is composed of 45 \(M\geq 4\) main shocks over 56 years, whereas the sub-catalog of 56 years are on average composed of 49 \(M\geq 4\) main shocks (black line Figure 9). Thus, the generator produces a few more \(M\geq 4\) earthquakes than we observe in the initial catalogue. Figure 8: Probability density function of frequency of main shocks for different magnitudes according to the stochastic FMD of main shocks visible in Figure 7.  
The mean distribution of generated seismic moments is, by definition, smooth, and could differ from the one observed in the instrumental catalogue, but trends are similar. Distributions visible in the synthetic sub-catalogues show variations, as well as the instrumental distribution. The Frequency-Magnitude Distribution (FMD) of generated main shocks can be observed in the 100,000 years of generated seismicity. This FMD of main shocks is compared Figure 10 with the mean one given as an input to the generator (Figure 7). The generator manages to reproduce this FMD of main shocks, especially for magnitudes smaller than 6.5. For higher magnitudes, slight variations are observed between the generator's FMD and the data's FMD. This part needs to be improved in order to avoid these slight variations. The FMD of generated main shocks for the six SHARE regions (Figure 4) are also shown in Figure 10. Table 4 presents the b-values computed by regressing the GR law (Equation (10)) on these FMDs of main shocks, from magnitudes 4 to 5 and between \(M5\) and \(M6\). Aside from the Rhine basin region (n\({}^{\circ}\)5 Table 4), every region shares practically the same slope of the FMD of generated main shocks before magnitude 5 (Figure 10): from 1.09 to 1.15. These slopes are close to the slope of the FMD of main shocks given as an input: 1.12. This is expected, since we only use this FMD as an input. Homogeneity is thus observed in the results. Table 5 lists b-values calculated on the data in each region. Please refer to Appendix B for more details on how these values have been computed. Regardless of what is obtained from the observed (Table 5) or synthetic (b(M4-5), Table 4) seismicities, the b-values of a given region keep the same order with respect to the b-value used as the input. In fact, regions 1, 5 and 6 are characterized by both b-values being greater than or equal to the reference b-value (1.12). In parallel, regions 2, 3 and 4 are characterized by both b-values being lower than the reference b-value. Thus, variations of synthetic b-values around the initial value are consistent with b-values obtained from data. Moreover, for \(M\geq 5.1\), slopes of FMD diverge from one region to another. Between magnitudes 5 and 6, the FMD of stable oceanic crust and the Pyrenees are, respectively, characterized by the highest and the lowest slopes: 1.98 and 0.98 (Table 4). The higher the slope, the lower the seismic activity.