# Probabilistic Seismic Hazard Assessment of the Southwestern Region of Saudi Arabia
## 2 Methodology
### Estimation of Seismic Hazard by Using a Logic Tree
Consequently, this can lead to different hazard results, influencing building design codes and risk mitigation strategies in the assessed region. In the current investigation, the seismic hazard estimates were calculated using the well-known probability theorem as reported in [28]. The exceedance rates for a given quantity of ground motion were expressed for the site classes that were considered. The main output of a PSHA is the seismic hazard curve, which describes the annual frequency of exceedance for different values of a selected ground motion parameter. The fundamental concept behind the computations to obtain this curve is relatively straightforward. First, a target ground motion level (y) for a ground motion parameter (Y; usually PGA or SA) is chosen. Next, the probability of exceeding this target value is calculated for one of the magnitude-distance scenarios considered in the seismicity model and then multiplied by the probability that this specific scenario would occur, P(Y \(>\) y | M, r, \(\epsilon\)). Finally, this probability is multiplied by the overall rate of earthquakes with magnitudes above the Mmin of interest in the seismic source corresponding to the magnitude-distance scenario. This process is repeated for all possible magnitude-distance scenarios in the seismicity model, and the associated rates of each scenario are summed. When this is carried out for different ground motion levels, a seismic hazard curve is generated. Therefore, the average rate of exceeding a specified ground motion level is represented by Equation (6). Figure 3: Logic tree designed for the present hazard evaluation. References included here refer to the considered NGA-West2 attenuation equations [75; 76; 77]. \[\gamma(\mathrm{Y}>y)=\sum_{\mathrm{i=1}}^{\mathrm{N_{sources}}}\lambda_{\mathrm{i}} \int\limits_{\mathrm{M_{min}}}^{\mathrm{M_{max}}}\int\limits_{\mathrm{r}=0}^{ \infty}\int\limits_{\mathrm{\varepsilon}=-\infty}^{\infty}\mathrm{f_{mi}(M)f_{ \mathrm{ri}}(r)f_{\mathrm{e}}(\varepsilon)p(Y>y|M,r,\varepsilon)\cdot dM\cdot dr \cdot d\varepsilon \tag{6}\]  
where P(Y > y | M, r,\(\varepsilon\)) is derived from the ground motion model, and fmi(M) and fmi(r) represent the probability distributions for magnitude and distance, respectively. \(\varepsilon\) denotes the number of standard deviations in the ground motion model needed to achieve the target ground motion for a specific magnitude and distance, and \(\lambda\)i denotes the earthquake occurrence rate for events with magnitudes equal to or greater than Mmin for source i. Events with magnitudes below Mmin are considered incapable of generating ground motions that could cause structural damage. This theory underpins the computation of seismic hazard estimates through the integration of probabilistic elements including ground motion models, distributions of magnitude and distance and earthquake occurrence rates. It helps mitigate uncertainties in hazard estimation by quantifying the probability of various ground motion levels occurring over time, accounting for the variability in seismic sources and GMPEs. The accessible visual interface of the Windows-based R-CRISIS software [97; 98], a proven and verified tool with the help of ArcGIS, was utilized to execute the evaluation and implement all input data models. In order to conduct this evaluation, calculations were carried out using a grid covering the entire region of interest with a 0.1\({}^{\circ}\)\(\times\) 0.1\({}^{\circ}\) spacing. The present study computes the seismic hazard for two different site classes, corresponding to the soil types NEHRP B/C and C.  
Seismic hazard values were calculated for mean PGA and SA for oscillation periods ranging from 0.01 to 10.0 s (0.01, 0.02, 0.03, 0.05, 0.075, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, 7.5, and 10.0 s). For a 50-year period, these computations were conducted at 10% and 5% exceedance rates, corresponding to 475- and 975-year return periods, respectively.