# Stochastic Generator of Earthquakes for Mainland France
## 3 Application to Mainland France
### Temporal Description of the Instrumental Catalogue  
#### 3.2.1 Frequency-Magnitude Distribution  
The frequency-magnitude distribution (FMD) describes the temporal occurrence of earthquakes. It consists of computing the annual number of earthquakes as a function of the magnitude. The classical way to compute FMD is to apply the GR law [38] on observed data:  
\[log_{10}(N(\geq M))=a+b\times(M-M_{min})\,, \tag{10}\]  
where \(M\) is the earthquake's magnitude and \(N(\geq M)\) is the annual number of earthquakes with a magnitude greater than or equal to the \(M\) observed in the seismic catalogue. In this study, \(M\) stands for the moment magnitude (\(M_{w}\)), and it is limited by a minimal magnitude \(M_{min}\) (which, here, is equal to 2).  
The seismic catalogue is composed of two types of earthquakes: main shocks and their correlated seismicity (fore shocks or aftershocks). Since they are characterized by different spatial and temporal behaviors, seismologists usually analyze them separately. A declustering algorithm is classically used for that purpose, e.g., [39]. It applies spatio-temporal windows around earthquakes in a catalogue in order to find clusters. In each cluster, the earthquake with the maximal magnitude is considered as the main shock, and the others are considered as correlated events. Thus, it is possible to segregate independent events from dependent ones by conserving only main shocks and alone earthquakes (clusters with only one earthquake). Then, the population of main shocks can be studied separately through a Poisson distribution in time, space and magnitude. In this study, the G85 declustering algorithm [40] is chosen. The Proportion-Magnitude Distribution (PMD) of main shocks obtained by applying this algorithm on French instrumental seismicity is represented by blue empty dots Figure 7.  
Figure 7: Proportion–Magnitude and Frequency–Magnitude Distributions (PMD and FMD) of main shocks for the whole of mainland France. The PMD is obtained by applying the G85 declustering algorithm [40] on data. A Regression of the GR law (Equation (10)) on data has been applied from \(M2\) to \(M5\) and has been extrapolated with the use of the truncated GR law (Equation (11)).  
Orange empty dots in Figure 7 show the annual number of main shocks observed in the exhaustive and declustered catalogue as a function of their magnitude. Applying Equation (10) to these data requires a straight line distribution. Such a distribution is observed between magnitudes 2 and 5 (Figure 7). Thus, we apply Equation (10) between \(M2\) and \(M5\) and obtain \(a=4.41\) and \(b=-1.12\).  
From \(M5.1\), the annual numbers of main shocks fall and do not respect the GR law (Equation (10)). This discrepancy for large magnitudes is classically attributed to a non-completeness of data for these magnitudes. Assuming that, an extrapolation of the GR law that is estimated to be between magnitudes 2 and 5 is carried out for \(M\geq 5.1\).  
As the total energy released by earthquakes is finite, some deviation from the GR straight line is required for largest magnitudes in order to avoid infinite distribution. A truncation is generally applied on the GR law [41]:  
\[\forall M\in[M_{min}M_{max}],N(\geq M)=10^{a}\times\frac{e^{-\beta(M-M_{min}-e ^{-\beta(M_{max}-M_{min})}}}{1-e^{-\beta(M_{max}-M_{min})}}\,, \tag{11}\]  
with \(\beta=b\times log(10)\). Applying Equation (11) with \(a\) and \(b\) estimated before (Equation (10)) gives the FMD of main shocks modeled from the exhaustive and declustered catalogue. This FMD is visible in Figure 7 and is characterized by a mean slope (\(b\) value) that is equal to 1.12 and an asymptotic fall of frequencies for the maximal magnitude \(M_{max}7.3\).  
The mean return periods (inversely equal to annual frequencies) of main shocks are summarized for six magnitude steps in Table 3. These return periods describe the average waiting time between main shocks of magnitudes greater than or equal to \(M\). As expected, the mean return periods increase as the magnitude grows.