# A Novel Wavelet Selection Method for Seismic Signal Intelligent Processing
## 2 Materials and Methods
### Wavelet Selection Method  
Wavelet packet transform provides good tools for seismic signal analysis. However, the use of wavelet packet transform involves the selection of wavelets. Some commonly used wavelets and their characteristics are listed in Table 1. Therefore, we propose a novel wavelet selection method that considers the stability of wavelet transform or wavelet base transform. This method is suitable for wavelet selection in the intelligent processing of seismic data. Figure 5 shows the specific implementation process of our method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Wavelet**} & **Represent-** & **Orthog-** & **Biortho-** & **Support** & **Symmetry** & **Global** \\  & **entation** & **onality** & **gonality** & **Length** & **Symmetry** & **Moment** \\ \hline Haar & Haar & yes & yes & \(1\) & symmetry & \(1\) \\ Daubechies & db \(N\) & yes & yes & \(2N-1\) & approximate & \(N\) \\ Symlets & sym \(N\) & yes & yes & \(2N-1\) & approximate & \(N\) \\ Coiflets & coif \(N\) & yes & yes & \(6N-1\) & approximate & \(2N\) \\  & & & & reconstruction: & & \\ BiorSplines & bior \(Nr.Nd\) & no & yes & \(2N+1\) & asymmetry & \(Nr-1\) \\  & & & & decomposition: & & \\  & & & & \(2 Nd+1\) & & \\  & & & & reconstruction: & & \\ ReverseBior & rbio \(Nr.Nd\) & no & yes & \(2 Nr+1\) & asymmetry & \(Nr-1\) \\  & & & & decomposition: & & \\  & & & & \(2 Nd+1\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some commonly used wavelets and their characteristics. Figure 4: The difference between wavelet transform and wavelet packet transform. A means the wavelet coefficient of the low-frequency component in the signal. D means the wavelet coefficient of the high-frequency component in the signal. In recent years, intelligent processing methods driven by seismic data have been widely used. Such methods are often based on a sizeable seismic dataset. Intelligent processing techniques, such as deep learning, no longer require us to extract features accurately. In contrast, deep neural networks require that our input data are stable and can reliably represent that type of signal. Therefore, our method is to calculate the decomposition stability of each wavelet on the dataset and select the wavelet with the highest decomposition stability as the best wavelet. Suppose the dataset contains \(l\) seismic signals, \(X_{i}\) is a signal in this dataset, \(i=1,2,\ldots,l\). For a wavelet, its decomposition stability is calculated according to the following steps. 1. Based on this wavelet performing \(n\) levels wavelet packet decomposition of signal \(X_{i}\), \(2^{n}\) wavelet packet coefficients \(\left\{a_{i}^{k}\middle|k\in[1,2^{n}],i\in[1,I]\right\}\) are obtained. 2. Each wavelet packet coefficient is reconstructed separately to obtain \(2^{n}\) reconstructed signals \(\left\{R_{i}^{k}\middle|k\in[1,2^{n}],i\in[1,I]\right\}\), and the correlation coefficients and variance contribution rates are calculated for each of the reconstructed signals and the original seismic signals, respectively. The formula for calculating the correlation coefficient \[co_{i}(k)=\frac{\sum\limits_{s=1}^{S}{(X_{i}(s)-\overline{X_{i}})(R_{i}^{k}(s) -\overline{R_{i}^{k}})}}{\sqrt{\sum\limits_{s=1}^{S}{(X_{i}(s)-\overline{X_{i} })^{2}\sum\limits_{s=1}^{S}{(R_{i}^{k}(s)-\overline{R_{i}^{k}})}^{2}}}},\] (7)  
Figure 5: The calculation approach of the stability \(w\) of wavelet packet transform. For a wavelet, perform wavelet packet decomposition on each signal in the dataset, and then use each wavelet packet coefficient to reconstruct the signal. Correlation coefficients \(co\) and variance contribution rates \(vcr\) were calculated between the reconstructed and original signals. The \(co\) and \(vcr\) can calculate the relevance between a wavelet and a signal. Therefore, the stability obtained from the mean and variance calculations of the variance can be used for wavelet selection, and the wavelet with the highest stability is the most appropriate.