## 2 Theoretical Supports
### Extended Probabilistic Seismic Hazard Analysis (EPSHA)  
The PSHA is a method of seismic hazard analysis which aggregates all the possible sources in one platform. Nevertheless, this aggregation may cause sometimes difficulties in understanding which earthquake is most likely to cause _IM_ > [1]. Therefore, the process of removing the terms of summations in the Equation (10) is known by the phenomenon of EPSHA or deaggregation [43; 44]. The deaggregation can be defined as the mean annual rate of exceedance for particular ground motion intensity as a function of magnitude or as a function of source-to-site distance or as a function of both by the following equations:  
\[\lambda\ (IM>x,M\ =\ m)\approx P\Big{(}M_{i}=m_{j}\Big{)}\sum_{j=1}^{N_{M}} \sum_{k=1}^{N_{B}}\lambda\ (M_{i}>m_{min})P(IM>x|m_{j}\kappa_{k})P(R_{i}=r_{k}) \tag{12}\]  
or:  
\[\lambda\ (IM>x,\ R\ =\ r)\approx P(R_{i}=r_{k})\sum_{j=1}^{N_{M}}\sum_{k=1}^{N _{B}}\lambda\ (M_{i}>m_{min})P(IM>x|m_{j}\kappa_{k})P\Big{(}M_{i}=m_{j}\Big{)} \tag{13}\]  
or:  
\[\lambda\ (IM>x,\ M\ =\ m,\ R\ =\ r)\approx P(R_{i}=r_{k})P(M_{i} \tag{14}\] \[=m_{j})\sum_{j=1}^{N_{M}}\sum_{k=1}^{N_{B}}\lambda\ (M_{i}>m_{min})P(IM>x|m_{j}\kappa_{k})\]  
Therefore, for the contribution of each GMPE to its corresponding ground motion parameters, the deaggregation can be mathematically written by as follows:  
\[f(M\ =\ m,\ R\ =\ r,\ GMPE|IM>x)=\frac{\lambda\ (IM>x,\ M\ =\ m,\ R\ =\ r,\ GMPE)}{ \lambda\ (IM>x)} \tag{15}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**GMPE (Reference)** & **Magnitude** & **Distance** & **Fault Mechanism** & **Other Variables** \\ \hline [PERSON] and [PERSON] [12] & \(M_{w}\) & \(R\) & \(S,N,R\) & - \\ [PERSON] and [PERSON] [13] & \(M\) & \(R_{RUP}\) & \(RO,S/N\) & \(V_{S},HW\) \\ [PERSON] [14] & \(M_{w}\) & \(R\) & - & - \\ [PERSON] [15] & \(M_{w}\) & \(R_{JB}\) & \(T,N,S\) & - \\ [PERSON] [18] & \(M_{w}\) & \(R_{SEIS}\) & - & - \\ [PERSON] [19] & \(M_{w}\) & \(R_{SEIS}\) & \(R,T,RO,TO\) & \\ Campbell and Bozorgnia [20] & \(M_{w}\) & \(R_{SEIS}\) & \(S,R,T\) & \\ Campbell and Bozorgnia [21] & \(M_{w}\) & \(R_{JB},R_{RUP}\) & \(S,\) N/NO, \(R/RO\) & \(V_{S},30,\) ZrORz,Z\({}_{1.0}\) \\ Cornell et al. [22] & \(M_{L}\) & \(R_{HTYO}\) & - & - \\ [PERSON] [23] & \(M\) & \(R_{HTYO}\) & - & - \\ [PERSON] [24] & \(M\) & \(R_{HTYO}\) & - & - \\ [PERSON] [25] & \(M\) & \(R_{HYPO}\) & - & - \\ Joyner and Boore [26] & \(M\) & \(R_{JB}\) & \\ Joyner and Boore [27] & \(M\) & \(R_{JB}\) & \\ [PERSON] [28] & \(M\) & \(R_{HTYO}\) & - & \\ Orphal and Lahoud [29] & \(M\) & \(R_{HTYO}\) & - & \\ TERACorporation [42] & \(M_{w}\) & \(R_{RUP}\) & - & \\ Zare and Sabzali [31] & \(M_{w}\) & \(R_{HTYO}\) & \(R,R/S,S\) & \(V_{S},30\) \\ [PERSON] [32] & \(M\) & \(R_{HTYO}\) & - & \\ Podili and Raghukanth [33] & \(M\) & \(R_{RUP}\) & - & - \\ Bajaj and Anbazhagan [34] & \(M_{w}\) & \(R_{HYPO}\) & - & \\ \hline \hline \end{tabular}  
* \(R_{JB}\) = Shortest distance from the recording site to the surface projection of the rupture, \(R_{RUP}\) = Shortest distance from the recording site to the rupture, \(V_{S},30\) = Share wave velocity of the top 30 m of the subsurface profile, \(R_{SEIS}\) = Closest distance to seism genic rupture, \(Z_{\text{TOR}}\) = Depth to the top of the rupture, \(Z_{1.0}\) = Depth to the 1 km/s shear-wave velocity horizontal, S = Strike-slip, N = Normal, NO = Normal-oblique, R = reverse, RO = Reverse-oblique, T = Thrust, TO = Thrust-oblique, HW = Hanging wall.  
\end{table}
Table 1: Variation of GMPE parameters.  
where \(\lambda\) is the return period and by using the given IM (PGA), one can get a return period indirectly from the hazard curve.