## 3 Adaptive Mother Wavelet Choosing  
Wavelet analysis is used as a method of transforming time-sequential data into data on a time-frequency plane [21]. During the last thirty years, this analysis technique has been under rapid theoretical development and has been used to solve many problems [22]. The fundamental principle of  
Figure 1: The relative frequency distribution of the closest distance-to-fault for all the 221 sets of seismic signals in the Chi-Chi event collected from the virtual data center (VDC). wavelet transform can be found in numerous textbooks. Thus only the near-fault earthquake-related aspects will be discussed in this section. It is beneficial for the interpretation of the fundamental principle of wavelet transform to be compared with the Fourier transform. In Fourier transform, seismic signals are approximated by summarizing a series of infinite sinusoid signals with solo frequencies. This method uses the time averaging technique to decompose signals, thus phase information cannot be reserved by the Fourier transform. While in the wavelet transform, recorded seismic signals are represented by summarizing a group of wavelet waveforms, each being a narrow-band signal located at the specific time point, which is best suited for the representation of the nonlinear and non-stationary seismic pulse. The dominant pulse waveforms of interest can be represented briefly by just a few wavelets, with the elaborately selected mother wavelets, time, and scale parameters. Both the amplitude and phase information can be reserved by the wavelet transform. The wavelet function at time t is defined mathematically by Equation (1), where \(\Phi(\cdot)\) is the mother wavelet, \(s\) is the scale factor, \(l\) is time location parameter. The parameter \(s\) is intended to dilate the mother wavelet, which scales the central frequency of the mother wavelet to match the interested pulse waveform embedded in the signal. Parameter \(l\) is intended to translate the wavelet along the time axis. There are many kinds of mother wavelets used in scientific research. Different mother wavelets result in different time-frequency planes [23]. In order to represent the dominant pulse waveform embedded in the strong ground motion with the best resolution, careful selection of the optimum mother wavelet is always necessary. If the profile of the chosen mother wavelet is close to the interested pulse waveform, then a limited number of wavelets is enough to represent the main profile of the signal, with relatively high resolution. Considering the variant feature of the near-fault earthquake motions, different mother wavelets should be adaptively used to extract each individual dominant pulse embedded in the near-fault earthquake motion, instead of using a single mother wavelet for all seismic signals as in References [1; 20]. \[\Phi_{s,l}(t)=\frac{1}{\sqrt{s}}\Phi\left(\frac{t-l}{s}\right) \tag{1}\]  
Regarding the typical profile of near-fault earthquake motions, the mother wavelets shown in Figure 2 were used in this paper as the candidate mother wavelets. The repository was composed of 15 types of mother wavelet, including the Haar wavelet, the Gaussian wavelet family from orders 1 to 8, the Daubechies wavelet family from orders 2 to 6, and the Morlet wavelet. These mother wavelets were adequate for the majority of pulse-like seismic signals. An adaptive procedure when choosing a mother wavelet is necessary to achieve the best resolution for the time-frequency plane by continuous wavelet transform (CWT). For the specific seismic signal \(\mathsf{f}(t)\), the waveform of extracted dominant pulses should represent the main profile of the source signal to the maximum extent. In order to evaluate the level of similarity for each mother wavelet \(\Phi_{i}(\cdot),i=1,2,\ldots 15\) in the wavelet repository, an iteration process was carried out to extract the dominant pulse \(p_{1}\) embedded in the source signal \(\mathsf{f}(t)\) by CWT, using mother wavelet \(\Phi_{i}(\cdot),i=1,2,\ldots 15\).