## 3 Empirical Scaling Equations vs. Real Records  
In this study, we generate new empirical equations for predicting \(PSA\) values that can be applied to deep soil sites atop deep geological strata. We utilize the same approach as [PERSON] et al. [17; 18; 19], and the GMPE for the horizontal direction is written in the following mathematical form:  
\[ \begin{split}& log[PSA(T)]=c_{1}(T)+c_{2}(T)\cdot M+c_{3}(T) \cdot log(\sqrt{R^{2}+R_{0}(T)^{2}})+\\ & c_{4}(T)\cdot S_{L1}+c_{5}(T)\cdot S_{L2}+c_{6}(T)\cdot S_{G1} +c_{7}(T)\cdot S_{G2}+\sigma_{log}(T)\cdot P\end{split}  \tag{1}\]  
where \(PSA(T)\) stands for horizontal pseudo-absolute acceleration spectra (in [g]) for vibration periods \(T\), \(M\) for earthquake magnitude, and \(R\) for either hypocentral or epicentral distance in kilometers (we develop equations for each type of distance). \(S_{L1}\) and \(S_{L2}\) are categorical variables for local soil (shallow geology) conditions, and \(S_{G1}\) and \(S_{G2}\) are categorical variables for deep geological conditions. Table 2 shows the values of these variables. According to [23; 24], the local soil is divided into three types: \"rock\" soil sites, stiff soil sites, and deep soil sites. According to [15], deep geology is separated into three categories:  
Figure 2: Analyzed area and the regional strong-motion data. Blue circles and red asterisks represent the recording sites and earthquake epicenters for the accelerograms in the EQINFOS [28] and ISESD [29; 30] databases, respectively. Green dashed lines connect the epicenters to the recording stations. Violet circles and orange asterisks indicate the locations of the recording sites and earthquake epicenters for the accelerograms provided by Seismological Survey of Serbia [31]. geological rock, deep geological sediments, and intermediate (or complex) geological site surroundings. Furthermore, the data were assumed to have a log-normal distribution. In Equation (1), \(\sigma_{log}\) is the standard deviation of the common logarithm of \(PSA\), where \(\varepsilon\) is 0 is the median estimate and 1 for the median + 1 \(\sigma_{log}\) estimate. A map of the regional strong-motion data available to the authors is shown in Figure 2. The recording sites and earthquake epicenters for the accelerograms in the EQINFOS [28] and ISESD [29; 30] databases are represented by blue circles and red asterisks, respectively. The locations of the recording sites and earthquake epicenters for the strong-motion accelerograms provided by Seismological Survey of Serbia [31] are indicated by violet circles and orange asterisks, respectively. It should be noted that the north-western Balkans is one of the few regions on the planet where data on deep geological site conditions are accessible for many recording stations. Moreover, ground motion predictions based on scaling equations that simultaneously consider local soil conditions (up to depths of 100 m and more) and deep geology have shown to be in excellent agreement with recorded ground motions and observed intensities in the region [32; 33]. The strong-motion database used to create predictive equations contains 436 horizontal components of strong-motion accelerograms recorded in the north-western Balkans from 112 earthquakes with magnitudes between 3 and 6.8. The majority (418) occurred between 1976 and 1987, with the remainder (18) occurring in 2010. The majority of contributing earthquakes were shallow with focal depths of less than 10 km. The majority of accelerograms were recorded at epicentral distances of less than 100 km, while the fault distance data for many of the accelerograms were either unavailable or unreliable. As a result, we used either epicentral or hypocentral distance for empirical prediction equations. More information on this database can be found in [PERSON] et al. [17]. In two phases, we ran multiple linear regression analyses. The relevant MATLAB(r) scripts for deriving the scaling coefficients were prepared in MATLAB(r) version 8.5, release 2015a.