# Seismic Risk in Alborz: Insights from Geological Moment Rate Estimation and Fault Activity Analysis
## 3 Results and Discussion  
The Gutenberg-Richter [28] method is a widely used mathematical approach for describing the seismicity of a region. This method involves investigating the seismic state of the region and defining the magnitude-frequency relationship of earthquake events. The constant coefficients in this equation provide insights into the different levels of seismicity in the area. The equation for the Gutenberg-Richter method is represented as:  
\[\text{Log~{}Nc}=\text{a}-\text{bMs} \tag{1}\]  
where Nc represents the cumulative frequency of earthquakes, a and b are constant coefficients (with a denoting the number of earthquakes and b denoting the relative frequency of earthquakes with different magnitudes), and Ms indicates the magnitude of the earthquakes. For this study, earthquakes with a magnitude of Ms \(\geq\) 3 that occurred over 120 years (from 1900 to 2020) in the Alborz mountain range were selected. By plotting the graph of Log N \(-\) Ms, the seismic parameters of the region (a and b) were obtained (Figure 4). According to the seismic formula obtained for the region (Figure 4), the values of the seismic parameters were determined as a = 1.832508913 and b = 0.6918. The earthquake return period formula can be used to calculate the time of occurrence of two similar events. The average time interval between a given event and a more significant event or equivalent is called the return period (T), which is calculated from the following relation [29]:  
\[\text{T}=1/\text{N} \tag{2}\]  
where N is the number of earthquakes of equal or greater magnitude than the one of interest that occurred during the time period of interest. According to Equation (1), the average return period for earthquakes in the Alborz region can be calculated using the following Equation (3):  
\[\text{T}=10^{\circ}(-\text{a}+\text{bM}) \tag{3}\]  
where M is the magnitude of the earthquake. The return period of earthquakes in the Alborz region is calculated and presented in Table 3.  
Figure 4: Graph of Log N \(-\) Ms in order to calculate the seismic formula of the region using the Gutenberg-Richter method. Horizontal ground acceleration rates are typically chosen based on two possible boundaries: 10% and 64% [31]. The limit of 10% represents the maximum credible earthquake (MCE), which is the highest magnitude earthquake that can be expected to occur within the next 50 years, with a probability of 10% or less. The limit of 64% represents the design basis earthquake (DBE), which is the earthquake that can be expected to occur within the next 50 years with a probability of 64%. These parameters can be calculated using the relations proposed by [PERSON] and [PERSON] [32]. The equation used to calculate the probability of exceedance for a given return period (Rp) is:  
\[\text{Rp}=1-\text{exp}(-\text{T}\times 10^{7}(\text{a}-\text{bMs})) \tag{4}\]  
where T is the return period in years, Ms is the earthquake magnitude, and a and b are the coefficients obtained from the Gutenberg-Richter relation. Therefore, the equation used to calculate the DBE is:  
\[0.64=1-\text{exp}(-\text{T}\times 10^{7}(\text{a}-\text{bMs})) \tag{5}\]  
and the equation used to calculate the MCE is:  
\[0.1=1-\text{exp}(-\text{T}\times 10^{7}(\text{a}-\text{bMs})) \tag{6}\]  
For the region, the DBE and MCE were calculated to have a useful life of 10 to 100 years (Table 4).