# MCPSHA: A New Tool for Probabilistic Seismic Hazard Analysis Based on Monte Carlo Simulation
## 2 Method
### Strategy for Generating a Stochastic Earthquake Catalogue  
The seismic zoning map of China establishes the corresponding earthquake occurrence probability model, a tri-class seismic source model (Figure 1), and provides the basic parameters based on geophysical data obtained from each region. China's earthquake activity model is based on a three-class seismic source model, which includes the seismic belt, the background seismicity zone, and the tectonic feature zone [37; 38]. Based on the basic assumptions and seismic activity parameters of the Chinese ground motion parameter zoning map, the following steps can be used to synthesize the seismic event set:  
1. [PERSON] and [PERSON] [39] propose the exponential magnitude probability distribution approach to deal with the possible maximum (\(m_{uz}\)) and minimum (\(m_{0}\)) magnitudes, as expressed in Equation (4). Based on the basic assumptions of Chinese probabilistic seismic hazard analysis, in the seismic belt, the magnitude distribution of earthquakes satisfies the doubly-truncated Gutenberg-Richter recurrence relation Equation (4) expressed below. \[F(m)= \begin{cases}\frac{1-\mathrm{e}^{-\beta(m-m_{0})}}{1-\mathrm{e}^{-\beta( m_{uz}-m_{0})}}&,\;\;m_{0}\leq m\leq m_{uz}\\ 0&,\;\;other\;\;conditions\end{cases} \] (4)  
The value of \(F(m)\) is monotonous in \([0,1]\). Let \(F(m)=r\) (r follows uniform distribution in \([0,1]\)). We can obtain Equation (5) through the inversion transformation of Equation (4). \[m=m_{0}-\ln\Bigl{\{}1-r\times\Bigl{[}1-e^{-\beta(m_{uz}-m_{0})}\Bigr{]}\Bigr{\}} /\beta \tag{5}\]  
Therefore, given a random number r from \([0,1]\), the magnitude of the earthquake \(m\) can be determined (Figure 2) according to Equation (5). 1. The seismic activity in the seismic province follows the Poisson distribution [37]. In \([0,\mathrm{t}]\), the probability of a total of \(n\) earthquakes will be observed, as shown in Equation (6). \[P(n=k)=\frac{\left(v_{0}\times t\right)^{k}}{k!}.e^{-v_{0}\times t}\;\;\;,(k=1,2,3\ldots\ldots)\] (6) where \(k\) is the number of earthquakes; \(v_{0}\) describes the mean activity rate of earthquake occurrence with magnitude equal or greater than \(m_{0}\). We can acquire the cumulative distribution function of Equation (6) as shown in Equation (7):  
Figure 1: Sketch Map of an earthquake activity model in China—a tri-class seismic source model. \[P(1\leq n\leq k)=\sum_{n=1}^{k}\frac{\left(v_{0}\times t\right)^{k}}{k!}.e^{-v_{0}t} \tag{7}\]  
The value of \(P(1\leq n\leq k)\) is continuity and monotonicity, and thus we can get the value of k through the DST. If \(t_{i}(i=1,2,\ldots,k)\) is set to the original time of an earthquake, the time interval of adjacent earthquakes (\(\tau_{i}\)=\(t_{i}-t_{i-1}\)) follows the exponential distribution [40], as shown in Equation (8):  
\[F\tau_{i}=\left\{ \begin{matrix}0&,&\tau_{i}<0\\ 1-e^{-v_{0}\times\tau_{i}}&\tau_{i}\geq 0\end{matrix} \right. \tag{8}\]  
According to the DST, the time interval \(\tau_{i}\) of the earthquake event \(i\) can be obtained. Let \(t_{i}=0\), then, the sequence of earthquake events can be obtained. 3. Determination of the location of the epicenter. The non-uniform spatial distribution of earthquake activity within the seismic belt is manifested by the activity of earthquakes at the potential source (which consists of the background seismicity zone and the tectonic features zone). In practice, the magnitude (\(m\)) is divided into \(N_{m}\) intervals, and \(m_{j}\left(m_{j}-\frac{1}{2}\bigtriangleup m\leq m_{j}\leq m_{j}+\frac{1}{2} \bigtriangleup m\right)\) represent the mid-value of the magnitude interval \(j\). The spatial distribution function (\(f_{hl,mj}\)) implies the occurrence probability of an earthquake generated by an individual potential source \(l\) with a magnitude interval \(mj\).