## 2 Optimal Design of the TID
### Optimization Problem
Notably, the estimated parameters are obtained by fitting the Clough-Penzien spectrum \(S(\omega)\) to the averaged power spectral density (PSD) for each group of ground motions. It is seen that the fitting result matches well with the target, indicating the Clough-Penzien spectrum captures well the frequency content of different ground motions. Unlike far-fault ground motions, near-fault ground motions contain more long-period or low-frequency components. Moreover, near-fault ground motions with pulse subset exhibit this feature more obviously. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No.** & **Earthquake** & **Year** & **M** & **Station** & **Distance (km)** & **Site Class** & **Vs\_30** & **Component** \\ \hline
1 & & & & & & & & \\  & & & & & & & & \\
2 & & & & & & & & \\  & & & & & & & & \\
3 & & & & & & & & \\  & & & & & & & & \\
4 & & & & & & & & \\
5 & & & & & & & & \\  & & & & & & & & \\
6 & & & & & & & & \\  & & & & & & & & \\
7 & & & & & & & & \\
8 & & & & & & & & \\
9 & & & & & & & & \\
10 & & & & & & & & \\
11 & & & & & & & & \\
12 & & & & & & & & \\  & & & & & & & \\  & & & & & & & & \\
13 & & & & & & & & \\  & & & & & & & \\
14 & & & & & & & & \\  & & & & & & & \\  & & & & & & & \\  & & & & & & & \\
15 & & & & & & & & \\  & & & & & & & \\  & & & & & & & \\  & & & & & & & \\
16 & & & & & & & & \\  & & & & & & & \\
17 & & & & & & & & \\
18 & & & & & & & \\
19 & & & & & & & \\
20 & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Information of the near-fault ground motions without pulse subset. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No. ** & **Earthquake** & **Year** & **M** & **Station** & **Distance (km)** & **Site Class** & **Vs\_30** & **Component** \\ \hline
1 & & & & & & & & \\  & & & & & & & & \\
2 & & & & & & & \\  & & & & & & & \\
3 & & & & & & & \\  & & & & & & & \\
4 & & & & & & & \\  & & & & & & & \\  & & & & & & & \\
5 & & & & & & & \\  & & & & & & & \\
6 & & & & & & & \\
7 & & & & & & & \\  & & & & & & & \\
8 & & & & & & & \\
9 & & & & & & & \\  & & & & & & \\
10 & & & & & & & \\
11 & & & & & & & \\
12 & & & & & & \\  & & & & & & & \\  & & & & & & \\  & & & & & & \\
13 & & & & & & & \\
14 & & & & & & & \\  & & & & & & \\  & & & & & & \\  & & & & & & \\  & & & & & & \\
15 & & & & & & \\  & & & & & & \\  & & & & & & \\  & & & & & & \\
16 & & & & & & \\  & & & & & & \\  & & & & & & \\  & & & & & & \\
17 & & & & & & & \\
18 & & & & & & \\
19 The power spectrum parameters of the three types of ground motions are as follows: \(\omega_{g}=13.22\), \(\xi_{g}=0.75\), \(\omega_{f}=0.83\), \(\xi_{f}=0.97\) for the far-fault ground motions; \(\omega_{g}=9.04\), \(\xi_{g}=0.98\), \(\omega_{f}=1.38\), \(\xi_{f}=0.89\) for the near-fault ground motions without \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**No.** & **Earthquake** & **Year** & **M** & **Station** &
\begin{tabular}{c} **Distance** \\ **(km)** \\ \end{tabular} & **Site Class** & **Vs\_30(m/s)** & **Component** \\ \hline
1 & Imperial Valley & 1979 & 6.5 &
\begin{tabular}{c} El Centro \\ Array \#6 \\ \end{tabular} & 1.75 & D & 203 & IMPVALL/H-E06140 \\
2 & Imperial Valley & 1979 & 6.5 &
\begin{tabular}{c} El Centro \\ Array \#7 \\ \end{tabular} & 2.1 & D & 211 & IMPVALL/H-E07140 \\
3 & Loma Prieta & 1989 & 6.9 &
\begin{tabular}{c} Saratoga- \\ Aloha \\ \end{tabular} & 8.05 & C & 371 & LOMAP/STG000 \\
4 & Erzican & 1992 & 6.7 & Erzincan & 2.2 & D & 275 & ERZIKAN/ERZ-EW \\
5 & Cape & 1992 & 7.0 & Petrolia & 4.1 & C & 713 & CAPEMEND/PET000 \\
6 & Landers & 1992 & 7.3 & Lucerne & 2.95 & C & 685 & LANDERS/LCN260 \\
7 & Northridge & 1994 & 6.7 &
\begin{tabular}{c} Rinaldi \\ Receiving Sta \\ \end{tabular} & 3.25 & D & 282 & NORTHR/RRS228 \\
8 & Northridge & 1994 & 6.7 &
\begin{tabular}{c} Sylmar-Olive \\ View \\ \end{tabular} & 3.5 & C & 441 & NORTHR/SYL090 \\
9 & Chi-Chi & 1999 & 7.6 & TCU065 & 3.65 & D & 306 & CHCHCH/TCU065-E \\
10 & Chi-Chi & 1999 & 7.6 & TCU102 & 4.6 & C & 714 & CHCHCH/TCU102-E \\ \hline \hline \end{tabular}
\end{table}