## 3 Ground Motion Time Histories
The synthetic ground acceleration time histories have been generated as spectrum-compatible and energy-compatible with the two selected real records, based on the method of [PERSON] et al. (2020), see Equations (2)-(6), and by using the corresponding Matlab subroutines. The method of [PERSON] et al. (2020) is \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**RSN**} & \multirow{2}{*}{**Station**} & **Pulse** & \multicolumn{2}{c}{**5-95\%**} & \multirow{2}{*}{**PGA**} & \multicolumn{2}{c}{**Earthquake**} & \multirow{2}{*}{**M\({}_{\mathbf{w}}\)**} & \multirow{2}{*}{**Mechanism**} & **Rrup** & **Vs30** \\  & & **Period** & **(s)** & **(s)** & & **Name** & & & **(km)** & **(m/s)** \\ \hline \multirow{4}{*}{8161} & “El & & & & & & & & & \\  & Centro & & & & & & & & & \\  & Array & & & & & & & & & \\  & \#12” & & & & & & & & & \\ \hline \multirow{4}{*}{8606} & “Westside & & & & & & & & & & \\  & Elementary & & & & & & & & & \\  & School” & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic features of the two selected records from the PEER Ground Motion Database (2019). Figure 3: (**a**). Acceleration time histories of the two horizontal components of the real record from Table 1 for the station “El Centro Array #12” and five corresponding spectrum-compatible and energy-compatible synthetic acceleration time histories; (**b**) Acceleration time histories of the two horizontal components of the real record from Table 1 for the station “Westside Elementary School” and five corresponding spectrum-compatible and energy-compatible synthetic acceleration time histories. partially based on the same procedure used by [PERSON] and [PERSON] [51], i.e., on the fact that any periodic function can be expanded into a series of sinusoidal waves:  
\[x(t)=\sum_{i=1}^{n}A_{i}\sin(\omega_{i}t+\varphi_{i}) \tag{2}\]  
where \(A_{i}\). is the amplitude and \(\varphi_{i}\) is the phase angle of the i-th contributing sinusoid. By fixing an array of amplitudes and generating different arrays of phase angles, one obtains different motions with the same general appearance but different details. The computer uses a random number generator to produce strings of phase angles with the uniform likelihood in the range between 0 and 2\(\pi\). In order to match the target spectrum, the amplitudes \(A_{i}\) have to be iteratively adjusted:  
\[A_{i,j+1}=A_{i,j}\frac{PSA_{t}}{PSA_{computal}} \tag{3}\]  
where j is the number of the current iteration and \(PSA_{t}\) is the target pseudo-acceleration spectrum. To simulate the transient character of real earthquakes, the steady-state motions are multiplied by an envelope function I(t). The artificial motion z(t) then becomes:  
\[z(t)=I(t)\sum_{i=1}^{n}A_{i}\sin(\omega_{i}t+\varphi_{i}) \tag{4}\]  
Usually, the envelope function is a deterministic function that does not change during the iteration process. In their method, [PERSON] et al. [50] adjust \(I(t)\) according to the differences between the target and simulated energy distributions. For this purpose, they proposed iterative adjustment of \(I(t)\) based on the values of Arias Intensity (AI). The Arias Intensity is the measure of the energy contained in the accelerogram and can be defined as:  
\[AI=\frac{\pi}{2g}\int\limits_{t=0}^{t=t_{d}}a_{g}^{2}(t)dt \tag{5}\]  
where \(t_{d}\) is the total duration of strong ground motion and \(a_{g}\) is the acceleration time history. In order to match the target \(AI\), the envelope \(I(t)\) has to be iteratively adjusted:  
\[I(t)_{j+1}=I(t)_{j}(\frac{a_{g}^{2}(t)_{t}}{a_{g}^{2}(t)_{computed}}) \tag{6}\]  
where j is the number of the current iteration, \(a_{g}\) is the target pseudo-acceleration spectrum, and \(p\) is a factor controlling the convergence speed. Table 2 demonstrates the PGA and AI values of the synthetic accelerograms compatible with the real records for the stations \"El Centro Array #12\" and \"Westside Elementary School\".