## 4 Identification and Representation of Multi-Pulse Near-Fault Earthquake Motion
### Identification of the Most Unfavorable Horizontal Direction  
Specific directions in the horizontal plane may contain larger pulse energies compared to other horizontal directions [1]. The most unfavorable horizontal direction (denoted as \(\theta_{max}\) hereinafter) may not be the seismic station direction. The structural response was controlled by the structure's period ratio and the input seismic signal. However, for one specific structure, whose basic period is a specific value, the seismic signal at the direction of \(\theta_{max}\) was responsible for the most unfavorable seismic response. Therefore, it is important to identify this direction and investigate its engineering features. In this paper, the horizontal direction \((\theta)\) was defined as zero toward the direction of channel 1 at a seismic station, and positive clockwise. Some seismic stations' main axis were set according to the local fault direction. While, some other seismic station's main axis were set perpendicular to NS/EW,  
Figure 2: The repository of the mother wavelets used in the adaptive selection of the optimum mother waveform in the wavelet transform. This paper rotates the seismic station to the NS/EW direction for convenience. Namely, the EW component corresponded to \(\theta=90^{\circ}/270^{\circ}\), and the NS component corresponded to \(\theta=0^{\circ}/180^{\circ}\). To identify the most unfavorable horizontal direction \(\theta_{max}\), the two-source horizontal components \(\mathbf{S}_{EW}\) and \(\mathbf{S}_{NS}\) were projected into their time-frequency planes \(\mathbf{C}_{\text{EW}}\) and \(\mathbf{C}_{\text{NS}}\), respectively. In the time-frequency plane, the x-axis was time and the y-axis was the scale value predefined. The scale value in the y-axis was converted into the instantaneous frequency by Equation (2), where f\({}_{0}\) is the central frequency of the chosen mother wavelet, \(s\) is the scale value in the y-axis, and \(\Delta t\) is the sampling period of the signal. A reasonable period band should be set to cover the whole interested range of the period for the multi-pulse near-fault earthquake motion. In the proposed method, this period band was set from 0.25 s to 15 s, which was adequate to cover all periods of interest. \[f=\text{f}_{0}/(s\cdot\Delta t) \tag{2}\]  
\[C(l,s)=\int_{-\infty}^{+\infty}f(t)\frac{1}{\sqrt{s}}\Phi\left(\frac{t-l}{s} \right)\text{d}t \tag{3}\]  
The wavelet coefficient at location \((l,s)\) in the time-frequency plane is defined by Equation (3), where \(f(t)\) is one of the source signals (\(\mathbf{S}_{EW}\) or \(\mathbf{S}_{NS}\)). The absolute value of the wavelet coefficient \(C(l,s)\) in the time-frequency plane is an indicator of the pulse energy level. The two time-frequency planes obtained above (\(\mathbf{C}_{EW}\) and \(\mathbf{C}_{NS}\)) can be combined into the resultant time-frequency plane \(\mathbf{C}_{RST}\) by the square sum, since the energy coefficient is a kind of scalar value. The wavelet coefficient corresponding to the maximum pulse energy (denoted by \(C_{max}(l_{0},s_{0})\)) was obtained by finding the largest absolute value of the coefficient in the plane \(\mathbf{C}_{RST}\), where \(l_{0}\) and \(s_{0}\) are the time and scale locations for \(C_{max}(l_{0},s_{0})\), respectively. Because the scope of the predefined scale gave a time-frequency plane with low resolution, it was difficult to locate the exact scale location of \(C_{max}(l_{0},s_{0})\). Thus a \"zoom-in\" approach was carried out by refining the scale around \(s_{0}\) to get the more precise maximum wavelet coefficient in \(\mathbf{C}_{RST}\). Finally, \(\theta_{max}\) was calculated using Equation (4), and the corresponding signal \(\mathbf{S}_{\theta}\) at the direction of \(\theta_{max}\) was obtained by Equation (5). Figure 3 summarizes the procedure discussed above.