## 5 Numerical Example  
In this section, we present a numerical example of the identification and representation of multi-pulse near-fault seismic signals by the proposed method. The horizontal velocity time-histories at station TCU101, TCU131, CHY029, and TCU070 during the Chi-Chi event (1999-09-20 17:47:16 UTC, Mw = 7.6) are plotted in Figure 5. The closest distance-to-fault for these four sets of seismic records were 1.9 km, 26.2 km, 16.4 km, and 18.4 km, respectively. A few prominent pulse waveforms with large amplitude exist in the velocity traces. The signal \(\mathbf{S}_{EW}\) and \(\mathbf{S}_{NS}\) at station TCU101 reached their peak at the same phase, while this pattern was not so clear for the other three stations. The results from Section 6 indicate that the signal set at station TCU101 and TCU131 qualified as pulse-like earthquake motions, while the other two sets did not. Although some prominent, large amplitude velocity pulses exist in the two orthogonal channels at stations CHY029 and TCU070, the difference of peak phase may result in the offset of pulse energy when the two orthogonal horizontal records were composited at the specific direction in the horizontal plane. Therefore, it is unreliable to identify pulse-like earthquake motions by single, one channel signals recorded at the seismic station. The signal set at station TCU101 was used as the numerical example to demonstrate the detailed analysis procedure of the proposed approach. Prior to identifying \(\mathrm{\theta}_{max}\), the variance values for all candidate mother wavelets in the repository were evaluated by the adaptive mother wavelet selection procedure. Based on the resulting variance shown in Figure 6, the mother wavelet 'gaus3' was the optimum mother wavelet for the signal set at station TCU101, thus it was used as the mother wavelet to project the signal set into the time-frequency planes \(\mathbf{C}_{EW}\) and \(\mathbf{C}_{NS}\). Their resultant plane \(\mathbf{C}_{RST}\) shown in Figure 7, was calculated by the square sum of \(\mathbf{C}_{\text{EW}}\) and \(\mathbf{C}_{\text{NS}}\). In Figure 7, the x-axis is time, the y-axis is the scales (or frequencies), and the color bar illustrates the amplitude of the wavelet coefficients in the time-frequency plane. The column \(\text{l}_{0}\) and row \(\text{s}_{0}\) for the largest wavelet coefficient \(C_{max}(\text{l}_{0},\text{s}_{0})\) in \(\mathbf{C}_{RST}\) were 1558 and 318, respectively. The corresponding wavelet coefficient located at (1558, 318) in \(\mathbf{C}_{\text{NS}}\) and \(\mathbf{C}_{\text{EW}}\) were 447.0 and 746.5, respectively. Thus, the most unfavorable direction was identified by \(\theta_{\text{max}}=\tan^{-1}(746.5/447.0)=59.09^{\circ}\), and the resultant signal \(\text{s}_{\theta_{max}}\) was easily obtained by \(\text{s}_{\theta_{max}}=\text{s}_{NS}\cos(\theta_{max})+\text{s}_{EW}\sin( \theta_{max})\) as shown in Figure 8a. Figure 5: The recorded horizontal velocity time–histories: (**a**) station TCU101, (**b**) station TCU131, (**c**) station CHY029, and (**d**) station TCU070. Figure 6: The variance value for all candidate mother wavelets in the repository by the adaptive wavelet selection procedure. The horizontal direction containing the largest pulse energy for the signals at station TCU101 was 59.09\({}^{\circ}\) from north to east, while the strike angle of the Chi-Chi event was 5\({}^{\circ}\). The largest pulse energy theoretically lay in the direction normal to the strike line [7], at 95\({}^{\circ}\). There was an error \(\sim\)34.35% between the direction from analysis and theory. This error may result from both the unpredicted distribution of the rock medium between the hypocenter and seismic station, and the unevenly distributed slip direction in the rupture plane. In order to identify the dominant pulse components embedded in \(S_{\theta=59.09^{\circ}}\), it was projected by CWT into the time-frequency plane \(\mathrm{C}_{i}(l,s)\) by iteration pattern.