# Horizontal PGA Estimates for Varying Deep Geological Conditions--A Case Study of Banja Luka
## 3 GMPEs for Horizontal PGA and Varying Deep Geology
Therefore, we believe that while accounting for other limitations of the provided database, we can use our GMPEs in seismic hazard calculations with the seismic source zones identified by using the \(M_{W}\)-based seismicity data. Every one of the accelerograms in our database was recorded in the northwestern Balkans. The majority (418) were recorded between 1976 and 1987 [67; 71; 72], with the remaining (18) in 2010. Maps that show the locations of recording stations and the epicenters of contributing earthquakes can be found in our previous papers [7; 58; 59]. MATLAB(r) (version 8.5) scripts were used for calculating the GMPE scaling coefficients \(c_{1},c_{2},c_{3},c_{4},c_{5},c_{6},c_{7}\), and \(\sigma_{log}\) by multiple linear regression analyses. For maximizing the R\({}^{2}\) statistics, \(R_{0}\) was iteratively adjusted. Because around two thirds of the data came from distances shorter than around 30 km, we performed a supplementary analysis that only included data collected at epicentral distances of less than 30 km. This was conducted with the intention of determining whether more reliable GMPEs might be obtained for shorter source-to-site distances. Finally, because the regression was also performed for pseudo-spectral accelerations across different vibration periods, the scaling coefficients were smoothed using the MATLAB(r) function \"smooth\" and the weighted linear least squares and a second-degree polynomial model. The finalized GMPE is as follows:  
\[ \begin{split}\log[\text{PGA}]=-1.2957+0.3946\cdot M-1.3818\cdot \log(\sqrt{R^{2}+19.5^{2}})+0.1772\cdot S_{L1}\\ -0.0953\cdot S_{L2}-0.1469\cdot S_{G1}-0.1059\cdot S_{G2}+0.2691 \cdot P,\end{split}  \tag{3}\]  
The GMPE based on data from epicentral distances shorter than 30 km is as follows:  
\[ \begin{split}\log[\text{PGA}]=3.1868+0.3919\cdot M-3.8728\cdot \log(\sqrt{R^{2}+40.0^{2}})+0.1276\cdot S_{L1}-0.0953\\ \cdot S_{L2}-0.2229\cdot S_{G1}-0.0899\cdot S_{G2}+0.2675\cdot P.\end{split}  \tag{4}\]  
By comparing Equations (3) and (4), we can observe that using only data recorded at distances less than 30 km results in a slightly smaller standard deviation. Figure 2 displays the attenuation of PGA as a function of distance, as calculated with Equations (3) and (4). The coefficients pertaining to \(S_{L}\) and \(S_{G}\) can be used to compute the variations in values calculated by the same equation under different site conditions (refer to Table 3). For example, the ratio between PGA at the geological rock (\(s=2\)) and intermediate deep geology sites (\(s=1\)) is calculated as \(1/10^{\circ}\), while the ratio between the geological rock (\(s=2\)) and deep geological sediments (\(s=0\)) is calculated as \(1/10^{\circ}\). If we use Equation (3), the PGA estimates at the geological rock will be \(1/10^{-0.1469}=1.40\) times larger than at intermediate sites and \(1/10^{-0.1059}=1.28\) times bigger than at geological sediments. The fact that short-period waves travel faster through harder and more compact geological formations (like basalts and granites) than through sediments may explain the amplification at geological rock sites. If the deep geology conditions remain constant, we can calculate ratios between the PGA values at stiff soil (\(s_{L}=1\)) and \"rock\" soil (\(s_{L}=0\)) by calculating \(10^{\circ 4}\), while the PGA ratios between deep soil (\(s_{L}=2\)) and \"rock\" soil (\(s_{L}=0\)) is calculated as \(10^{\circ 5}\). If Equation (3) is used, the PGA value for stiff soil sites is \(10^{0.1772}=1.50\) times larger than for \"rock\" soil sites. Horizontal PGA for deep soil sites will be \(10^{-0.0953}=0.80\) times the PGA for \"rock\" soil, according to the same equation. Put another way, horizontal short-period seismic waves at the surface will be less amplified (or even de-amplified) at deep soil sites than at \"rock\" soil sites. This implies that the energy dissipation of short-period waves in deep soils may outweigh a local soil amplification generated by the impedance difference between the deep soil and the harder rocks beneath.