# A Novel Wavelet Selection Method for Seismic Signal Intelligent Processing
## 2 Materials and Methods
### Wavelet Packet Transform  
[PERSON] first proposed wavelet transform in 1974. However, it was not until 1986 that the first wavelet basis was inadvertently constructed by [PERSON] [42]. Compared with traditional signal analysis methods, wavelet transform has the ability of a digital  
Figure 3: The amplitude distribution of all used seismic records.  
Figure 2: Records collected by mining-induced microseismic monitoring system in Huangtupo Copper and Zinc Mine.  
microscope, which can effectively perform detailed time-frequency analysis of signals and extract more characteristic information [43; 44].  
The time-frequency analysis procedure using wavelet transform is comparable to the description of the windowed Fourier transform, but the two are essentially different. First, the expression of wavelet transform is as follows  
\[T_{f}^{uav}(a,b)=<f,\psi^{a,b}>=|a|^{-1/2}\int_{-\infty}^{+\infty}f(t)\psi\left( \frac{t-b}{a}\right)dt, \tag{1}\]  
where \(<\), \(>\) represents the inner product in \(L^{2}\), \(\psi^{a,b}(t)=|a|^{-1/2}\psi\left(\frac{t-b}{a}\right)\), \(\psi^{a,b}(t)\) is called wavelet, and \(\psi\) is called mother wavelet.  
The mother wavelet needs to meet the following conditions:  
1 Absolutely integrable and squarely integrable, as \(\psi\in L^{1}(R)\cap L^{2}(R)\);  
2 Offset between positives and negatives, as \(\int_{-\infty}^{+\infty}\psi(x)dx=0\) and \(\hat{\psi}(0)=0\), where \(\hat{\psi}\) is the Fourier transform of \(\psi\);  
3 Meet the allowable conditions: \(\int_{-\infty}^{+\infty}\frac{|\hat{\psi}(w)|^{2}}{w}dw<\infty\).  
The principle of wavelet transform can be understood as a continuous translation of the mother wavelet. It can also enlarge and shrink the mother wavelet and adapt to various frequency bands by changing its scale. When the parameters \(a\) and \(b\) in Formula (1) are continuous values, it is called continuous wavelet transform (CWT). It is often necessary to discretize the expansion factor \(a\) and translation factor \(b\), that is, \(a=a_{0}^{m}\), and \(b=nb_{0}a_{0}^{m}\), where \(m\) and \(n\) are integers, \(a_{0}\) is a constant greater than 1, and \(b_{0}\) is a constant greater than 0. The corresponding discrete wavelet function is expressed as  
\[\psi_{m,n}(t)=|a_{0}^{m}|^{-1/2}\psi\left(\frac{t-nb_{0}a_{0}^{m}}{a_{0}^{m}} \right)=|a_{0}^{m}|^{-1/2}\psi(a_{0}^{-m}t-nb_{0}). \tag{2}\]  
As a result, the discrete wavelet transform (DWT) is  
\[T_{m,n}^{uav}(f)=a_{0}^{-m/2}\int dtf(t)\psi(a_{0}^{-m}t-nb_{0}). \tag{3}\]  
Wavelet transform is used to analyze seismic signals, which can effectively separate high-frequency and low-frequency signals, and further analyze and process the low-frequency signals. In the actual analysis, the high-frequency signal in seismic events is also a crucial component. Therefore, wavelet transform will lose more detailed information. The wavelet packet transform can simultaneously decompose the signal into different frequency bands so that both the high-frequency and low-frequency parts of the signal are analyzed, analyzing the signal more precisely [45]. The wavelet packet transform can also select the frequency band most consistent with the signal characteristics.  
Suppose \(\{H_{n}\}\) and \(\{L_{n}\}\) are orthogonal high and low pass filters corresponding to orthogonal wavelet function \(\alpha(t)\) and orthogonal scaling function \(\beta(t)\), respectively, then  
\[\begin{  
Therefore, the function \(\xi_{n}\) can be obtained by iteration  
\[\begin{split}\xi_{2n}(t)&=\sqrt{2\sum\limits_{k}L_{k} \xi_{n}(2t-k)},\\ \xi_{2n+1}(t)&=\sqrt{2\sum\limits_{k}^{k}H_{k}\xi_{n} (2t-k)}.\end{split} \tag{6}\]  
The function \(\xi_{n}(t)\) obtained by iteration is the wavelet packet determined by \(\beta(t)\). As shown in Figure 4, the difference between wavelet transform and wavelet packet transform lies in the process of high-frequency components of the signal.