# Seismic and Geodetic Crustal Moment-Rates Comparison: New Insights on the Seismic Hazard of Egypt
## 7 Estimation of Seismicity Parameters and Seismic Moment-Rates  
The magnitude-frequency recurrence relationship (Equation (10)) by [PERSON] and [PERSON] [18] describes the seismic activity within a certain seismic source. \[\log_{10}N(m)=a-bm \tag{10}\]  
\(N(m)\) is the annual number of earthquakes equal to or greater than a specific magnitude \(m\), and \(a\)- and \(b\)-parameters characterize the seismicity within the source. The \(a\)-value indicates the seismicity level during a certain period, while the \(b\)-value provides indications about the low- to large-magnitude earthquakes ratio. The later parameter typically lies between 0.6 and 1.5 [48; 103; 104]. In order to estimate the seismicity parameters, separate earthquake sub-catalogs were created for each delimited source, considering only crustal earthquakes, i.e., earthquakes with depth less than 35 km. The seismicity recurrence parameters for the defined sources were estimated by applying the linear least-square method, taking into account the completeness period for each magnitude range [105] (Tables 1 and 2). This method, as stated by the author, is the maximum likelihood estimation of the \(a\)- and \(b\)-parameters extended to the case of events grouped in magnitude with each group observed over individual periods. Additionally, the maximum expected magnitude (m\({}_{\text{max}}\)) and its variance have been also determined following the Robson-Whitlock-Cooke methodology [106; 107; 108] (Equations (11) and (12)). It is an optimal approach when seismicity data, as happens to be the case for this region, are scarce or incomplete. Given an ordered set including \(n\) recorded magnitudes, \(m_{1}\leq\dots\leq m_{n-1}\leq m_{\text{max}}^{obs}\), maximum estimated magnitude (\(m_{\text{max}}\)) and its uncertainty (\(\sigma\)) are given by  
\[m_{\text{max}}=m_{\text{max}}^{obs.}+0.5\Big{(}m_{\text{max}}^{obs.}-m_{n-1} \Big{)} \tag{11}\]  
\[\sigma(m_{\text{max}})=0.5\Big{[}3\sigma_{M}^{2}+0.5\Big{(}m_{\text{max}}^{obs. }-m_{n-1}\Big{)}^{2}\Big{]} \tag{12}\]  
\(\sigma_{M}\) being the standard deviation of the maximum observed magnitude. All estimated parameters for each seismic source and their associated uncertainties are reported in Table 2. Depths D50 and D90, the depths above which 50% and 90% of the earthquakes occur, respectively, were also computed. To do that, depths that seem to be artificial (equal to 0, 10, 30, and 33 km) were removed before the computation. Finally, D90 (in Tables 2 and 3) has been proposed as the seismogenic depth for each source. Concerning uncertainty in the \(m_{\rm max}\) value, when computing the model uncertainty given by Equation (12), it was considered conservatively that uncertainties (\(\sigma_{M}\)) are equal to 0.6 previously to the year 1900, 0.5 in the period 1900-1962, 0.3 in the period 1962-2002, and 0.2 since 2002 [109]. Two sources need some additional explanation. In the EG-10 source, the largest magnitude values, corresponding to the 857 A.D. and 1264, Mw 6.2, earthquakes, do not follow the Gutenberg-Richter model (Table 2). Then, these two earthquakes have been considered as characteristic earthquakes. To include them, a constant seismic rate equal to 0.001 earthquakes by year (2 earthquakes every \(\sim\)2000 years) in the magnitude interval \(m^{u}-\Delta m_{c}/2\leq m\leq m^{u}+\Delta m_{c}/2\)[19] has been considered, being in this case \(m^{u}\) equal to Mw 6.2, the biggest recorded magnitudes in the source, and \(\Delta m_{c}\) equal to 0.6, the individual uncertainty for earthquakes before 1900. The second source is the EG-02 zone, initially presenting the same problem as the EG-10 one. In addition, the obtained \(b\)-value presents an odd high value, above 1.5, then providing an unrealistic seismic moment-rate. This is clearly due to the incompleteness of the seismic catalog in this source.