## 4 Identification and Representation of Multi-Pulse Near-Fault Earthquake Motion
### Representation of Dominant Pulses  
The strong ground motion may contain single or multiple pulse-like components featured by the long period and large velocity amplitude. For example, the recorded horizontal seismic signals at station TCU075 during the Chi-Chi event shown in Figure 4a, are representative of the single one  
Figure 3: The procedure of identifying the horizontal direction \(\theta_{max}\), and the corresponding seismic signal \(\mathbf{S}_{\theta}\). pulse waveform. Its closest distance-to-fault was 3.4 km. The closest distance-to-fault for seismic station NSY, TCU060, and TCU128 were 9.1 km, 8.1 km, and 9.1 km, respectively. The multiple pulse waveform can be easily spotted in the recorded horizontal seismic signals as shown in Figure 4b-d. Thus, the dominant pulse-like components indeed exist in the recorded seismic signals. Representation of the dominant multi-pulse waveform embedded in the signal \(S_{\theta}\) was carried out by applying the CWT to the residual signal \(S_{\theta,i=\theta}=S_{\theta}\) in an iterative manner. The purpose of CWT was to extract dominant pulses, which was different from stage 1 aimed at identifying \(\theta_{max}\). Before extracting the individual dominant pulses, the optimum mother wavelet was determined first by the adaptive mother wavelet selection procedure, considering the specific feature of the signal \(S_{\theta,i}\). The largest wavelet coefficient \(C_{i,max}(l_{0,i},s_{0,i})\) in the time-frequency plane \(C_{i}(l,s)\) produced by the CWT, based on the optimum mother wavelet was used to extract the individual wavelet waveform using Equation (6), where \(p_{i}\) is the \(i_{th}\) individual pulse corresponding to \(C_{i,max}(l_{0,i},s_{0,i})\), and \(y_{base;i}\) is the base value of the mother wavelet adaptively selected by the mother wavelet selection procedure. Due to the varied profile of signal \(S_{\theta,i}\), the actually adopted mother wavelets were different from each other within the iteration process. The residual seismic signal after the \(i\)th iteration of extraction was updated by \(S_{\theta,i}=S_{\theta}-\sum_{k=1}^{k=i}p_{k}\), and it was used as the new target signal for the next iteration. \[p_{i}=y_{base;i}C_{i,max}(l_{0,i},s_{0,i})/\sqrt{s_{0,i}} \tag{6}\]  
A criterion for terminating the extraction iteration reasonably was required. For this purpose, two types of criteria with different threshold values were verified with the help of some common pulse-like near-fault seismic signals. One type of criterion was designed as the peak ground velocity (PGV) ratio between \(p_{i}\) and \(S_{\theta,i}\). The other type of criterion was designed as the energy ratio between \(p_{i}\) and \(S_{\theta,i}\). The energy signals \(p_{i}\) and \(S_{\theta,i}\) was defined by \(E_{p_{i}}=\int p_{i}(t)^{2}dt\) and \(E_{S_{\theta,i}}=\int\ S_{\theta,i}(t)^{2}dt\), respectively. The energy ratio with a threshold 0.4 was found to behave best compared with the criteria for other threshold values or other types. Figure 4: The typical single and multiple pulse waveforms: (**a**) TCU075, (**b**) NSY, (**c**) TCU060, and (**d**) TCU128. The resulting multi-pulse waveform \(\mathbf{P}_{\theta}\) was obtained by superimposing the extracted individual pulse \(\mathrm{p}_{i}\) by Equation (7), where \(\mathrm{p}_{i}\) is the \(i\)th pulse waveform corresponding to the wavelet coefficient \(\mathrm{C}_{i,max}(1_{0,i},s_{0,i})\), and \(N\) is the total number of dominant pulses embedded in the signal \(\mathrm{S}_{\theta}\). This approach can adaptively extract the varied number of dominant pulse components according to the diverse feature of the target signal \(\mathrm{S}_{\theta}\). The individual pulse waveform \(\mathrm{p}_{i}\)\((i=1,2,\cdots N)\), with variant frequency \(\mathrm{f}_{i}\), was located discretely along the time axis, which demonstrates the non-stationary feature of the near-fault earthquake motion.