# Horizontal PGA Estimates for Varying Deep Geological Conditions--A Case Study of Banja Luka
## 4 PSHA Analysis for the Case Study Location in Banja Luka
For the calculation of the total hazard, we will employ the law (or formula) of total probability [97], which defines the probability of an event, \(A\), in case that a series of \(n\) mutually exclusive events, \(H_{i}\), occurs, and \(H_{i}\) comprises a full system of hypotheses about the event, \(A\), as follows:  
\[P(A)=\sum_{i=1}^{n}P(A|H_{i})\cdot P(H_{i}). \tag{5}\]  
Based on Equation (5), we can apply the following equation to calculate the average yearly rate of occurrence of seismic events that may cause PGA to exceed the expected _pga_:  
\[N(pga)=\sum_{i\in I}v_{i}\int_{M_{min}}^{M_{max}}\int_{R_{min}}^{R_{max}}f_{ gmpe}(\text{PGA}>pga|M,R)\cdot f_{m_{i}}(M)\cdot f_{r_{i}|m_{i}}(R|M)dMR, \tag{6}\]  
where \(i\) denotes the seismic source zone serial number (from the set \(I\)), and \(\
u\) is the annual rate of events exceeding \(M_{min}\), which is defined as follows:  
\[\
u=e^{dn10-(bh10)M_{min}}. \tag{7}\]  
In Equation (7), \(a\) and \(b\) are the coefficients that characterize the total degree of seismicity in the analyzed area and the ratio between events of smaller and larger sizes, respectively, and are defined as follows:  
\[\log N_{GR}(M)=a-b\cdot M, \tag{8}\]  
where \(N_{GR}\) represents the rate of occurrence of earthquakes of different magnitudes in each zone, \(i\); \(M_{max}\) is the maximum magnitude for each zone, \(i\); \(M_{min}\) is the minimum magnitude (although the SHARE seismic-hazard map for all of Europe was calculated with \(M_{min}=4.5\)[84], we chose to use \(M_{min}=4.0\) for all zones in this study to obtain more conservative PGA estimates for shorter return periods); and \(R_{min}\) and \(R_{max}\) are the minimum and maximum epicentral distances equal to 0 and 300 km, respectively. Furthermore, \(f_{gmpe}\) is the conditional cumulative distribution function, which is based on the GMPE and determines the likelihood that PGA will surpass _pga_ if an event of magnitude, \(M\), occurs at a source-to-site distance, \(R\);\(f_{r\mid m}\) is the probability density function for distance; and \(f_{m}\) is the probability density function for magnitudes, which is defined as follows:  
\[f_{m}(M)=\beta\frac{e^{-\beta(M-M_{min})}}{1-e^{-\beta(M_{max}-M_{min})}},M_{ min}\leq M\leq M_{max}, \tag{9}\]  
where \(\beta=(\text{ln}10)b\). We will now explain the alternative metrics for seismic hazard, shown in Table 4. The so-called \"return period\" is calculated as follows:  
\[Tr=N\Big{(}pga)^{-1}. \tag{10}\]  
Although \(Tr\) is commonly used in the engineering literature as an instinctive measure of hazard levels (a larger \(Tr\) indicates stronger events that occur less often and vice versa), it lacks a distinct physical meaning in the sense that it does not (generally) correspond to a single earthquake of engineering concern. This is because \(N(pga)\) for a single location is calculated by combining and summing data from all analyzed seismic source zones across all magnitudes and distances. When homogeneous Poisson distributions are assumed, the annual probability, or the likelihood of at least one yearly exceedance of the expectation _pga_, may be expressed as follows [97]:  
\[P(pga)=1-e^{-N(pga)}. \tag{11}\]Finally, if a binomial distribution is assumed, the probability that the expectation _pga_ will be exceeded at least once during \(t\) years can be computed as follows [97]:  
Figure 4:0  
A procedure that is called \"seismic hazard disaggregation\" can be used to identify the earthquakes that contribute the most to the computed hazard [98; 99]. It is effectively the inverse of the process described in Equation (6), and it allows us to distinguish independent contributions from distinct magnitude and distance combinations. The four plots of Figure 5 show the PSHA disaggregation for 44\({}^{\circ}\)46.5\({}^{\prime}\)N, 17\({}^{\circ}\)15\({}^{\prime}\) E (refer to the solid blue circles in Figures 1 and 4) and _Tr_ = 95, 475, 975, and 2475 years. These plots show that the most contributing magnitudes increase with the return period, _Tr_.