## 2 Empirical Frequency-Dependent Scaling Equations for Vertical Ground Motion  
We start with the development of new empirical formulae for estimating vertical _PSA_ values that can be used for deep soil sites atop deep geological deposits. The following mathematical form will be used to express the attenuation equations:  
\[ \begin{split} log[PSA_{vert}(T)]=c_{1}(T)+c_{2}(T)\cdot M+c_{3}(T )\cdot log(\sqrt{R^{2}+R_{0}(T)^{2}})+c_{4}(T)\cdot\\ S_{L1}+c_{5}(T)\cdot S_{L2}+c_{6}(T)\cdot S_{G1}+c_{7}(T)\cdot S_{G2}+ \sigma_{log}(T)\cdot P1\end{split}  \tag{1}\]  
where \(PSA_{vert}(T)\) denotes the amplitudes of the vertical pseudo-absolute acceleration spectrum for a 5 percent viscous damping ratio, \(T\) is the vibration period of an undamped single-degree-of-freedom oscillator, \(M\) is the earthquake magnitude, and \(S_{L1}\) and \(S_{L2}\) (the local soil categorical variables) are 0 for the rock soil sites (further denoted as \(S_{L}=0\)), \(S_{L1}=1\) and \(S_{L2}=0\) for stiff soil (\(S_{L}=1\)), and \(S_{L1}=0\) and \(S_{L2}=1\) for deep soil (\(S_{L}=2\)). It should be noted that in this equation, local soil (i.e., shallow geology) is classified by [PERSON] et al. [12; 13], while deep geology is separated into three categories using [PERSON]'s [14] classification: (i) geological rock (further denoted as \(S=2\)) with the corresponding categorical variables \(S_{G1}\) and \(S_{G2}\) equal to 0; (ii) deep geological sediments (\(S=0\)) with \(S_{G1}=0\) and \(S_{G2}=1\); and (iii) intermediate (or complex) geological site surroundings (\(S=1\)) for which \(S_{G1}=1\) and \(S_{G2}=0\). The unknown parameters resulting from the regression analysis are the coefficients \(c_{1}\) through \(c_{6}\). Table 2 displays the coefficients \(c_{1}\) to \(c_{6}\) that can be used to predict the \(PSA_{vert}\) values for various vibration periods. The regression analyses were conducted on the assumption that the studied data had a log-normal distribution. The standard deviation of the logarithm of \(PSA_{vert}(T)\) is denoted by \(\sigma_{log}(T)\), where \(p=0\) for median estimations and \(p=1\) for median plus one standard deviation estimations. Finally, \(R\) stands for hypocentral or epicentral distance, and \(R_{0}(T)\) was iteratively calculated for each \(PSA(T)\) to minimize the standard deviation in the related prediction equation. The employed strong motion database comprises 218 vertical components of accelerograms obtained in the north-western Balkans from 112 earthquakes with magnitudes between 3 and 6.8. The majority (209) occurred between 1976 and 1987, with the remainder (9) occurring between 2010 and 2011. 203 are from the EQINFOS database [32] and 6 are from the ISESD database for the period 1976-1987 [33; 34]. The Seismological Survey of Serbia used an accelerograph network to collect remaining 9 data in 2010 [35]. Because the majority of the accelerograms had a low signal-to-noise ratio for \(T>2\) s [16; 36], most of the records were high-pass filtered with a relatively large corner frequency, and the results for \(T>2\) s are unreliable. Hence, we calculated scaling coefficients up to a 2 s vibration period. [PERSON] et al. [17; 24] provide more information on the used data. The MATLAB(r) function \"regress\" was used to perform two-phase multiple linear regression analyses, the first of which consisted of fitting the Equation (1) model to a dataset that only included 203 vertical acceleration components from the EQINFOS database [32], for which [PERSON] et al. [37] established the soil categorization of the recording locations. Because there were no deep soil sites among the 203 components, we fitted Equation (1) without the coefficient \(c_{5}\). To lower the root mean squared error, the \(R_{0}\) values were  
Figures 2 and 3 show the attenuation with the distance of \(PSA(T)\) for four different vibration periods, \(T\) = 0.05 s, 0.10 s, 0.30 s, and 0.50 s. Figure 2 shows the empirical predictions at deep soil sites (\(S_{L}=2\)) for three different deep geology types.