## 5 Numerical Example
The signal length of \(S_{\theta}\) was 49 s. The y-axis was the transient frequency of the signal converted from the central frequency of the mother wavelet, the scale value, and the sampling frequency. There were two dominant wavelet components embedded in the signal \(S_{\theta}\). Each component comprised only one wavelet waveform, which was the result of translating and dilating the corresponding mother wavelet by the index \(l_{0,i}\) and \(s_{0,i}\) of the wavelet coefficient \(\mathrm{C}_{i,\mathrm{max}}(\mathrm{l}_{0,i},\mathrm{s}_{0,i})\) in the time-frequency plane shown in Figure 9 as dictated by Equation (6). The signal \(P_{\theta}\) was obtained by superimposing the two extracted pulse components: \(\mathrm{p}_{1}\) and \(\mathrm{p}_{2}\) by Equation (7) as shown in Figure 10. These data show that the signal \(P_{\theta}\) captured all the dominant pulses in \(S_{\theta}\). The parameters for these extracted pulse components are listed in Table 1, where \(f_{i}\) is the central frequency of the pulse component calculated by Equation (9). The first pulse waveform \(\mathrm{p}_{1}\) was represented by the 'gaus3' mother wavelet located at 15.58 s with instantaneous frequency 0.1307 Hz (scale = 306), and the value of PGV is 44.9 cm/s. The second pulse waveform \(\mathrm{p}_{2}\) was represented by the 'db6' mother wavelet located at 27.6 s with instantaneous frequency 0.1388 Hz (scale = 524), and the value of PGV was 0.1388 Hz (scale = 524). The value of PGV was 0.1388 Hz (scale = 524). 27.0 cm/s. Due to the the lower frequency and the higher PGV value of the extracted pulse component \(\mathrm{p_{1}}\) compared with that of \(\mathrm{p_{2}}\), its energy was larger than that of \(\mathrm{p_{2}}\). This was in accordance with the larger wavelet coefficient for \(\mathrm{p_{1}}\) compared to \(\mathrm{p_{2}}\). The criterion of early arrival of the pulse energy was required to ensure that the directivity pulse was of primary concern, as predicted by the theoretical seismology. The pulse index \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathrm{p_{i}}\) & **Mother Wavelet** & \(\mathbf{l_{0,i}}\) & \(\mathbf{s_{0,i}}\) & \(\mathbf{C_{i,max}(\mathbf{l_{0,i}}\mathbf{s_{0,i}})}\) & **Time\(\mathbf{l_{\text{s}}}\)** & \(\mathbf{f_{\text{f}}/\text{Hz}}\) & **PGV/cm\(\cdot\)s** \\ \hline \(\mathrm{p_{1}}\) & ’gaus3’ & 1558 & 306 & 871.60 & 15.58 & 0.1307 & 44.9 \\ \(\mathrm{p_{2}}\) & ’db6’ & 2757 & 524 & 549.18 & 27.57 & 0.1388 & 27.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters for the extracted pulse components at station TCU101. Figure 10: The extracted dominant pulse components embedded in \(\mathbf{S_{0}}\) at station TCU101: (**a**) \(\mathrm{p_{1}}\), (**b**) \(\mathrm{p_{2}}\), and (**c**) \(\mathrm{P_{0}}\). Figure 9: The time–frequency plane for the extracted pulse components at station TCU101: (**a**) \(\mathrm{p_{1}}\) with the guas3 mother wavelet, and (**b**) \(\mathrm{p_{2}}\) with db6 mother wavelet. \(\mathbf{S_{0}}\), which was in agreement with the criterion (1):PI \(>0.85\). Figure 8 shows the resultant signal \(\mathbf{S_{0}}\) and the pulse signal \(\mathbf{P_{0}}\) embedded in \(\mathbf{S_{0}}\). The time required by signal \(\mathbf{S_{0}}\) to reach 20% of its CSV was 14.59 s. While the time required by signal \(\mathbf{P_{0}}\) to reach 10% of its CSV was 13.45 s. Thus, the signal \(\mathbf{S_{0}}\) qualified as a pulse early-arriving signal as dictated by criterion (2): \(t_{10\%\text{-pulse}}<t_{20\%\text{,original}}\). The PGV of signal \(\mathbf{P_{0}}\) was 0.497 m/s, which was in agreement with criterion (3): \(\text{PGV}>0.3\) m/s. Finally, the recorded horizontal signals at station TCU101 qualified as multi-pulse near-fault earthquake motions. Figure 11 shows comparison of the extracted signal by both the author's and [PERSON]'s methods [1]. The signal extracted by the author's method captured all the dominant pulses embedded in the signal \(\mathbf{S_{0}}\).