# A Seismic Inversion Method Based on Multi-Scale Super-Asymmetric Cycle-JNet Network
## 2 Multi-Scale Supersymmetric Network (Cycle-JNet) Method
### Wavelet Transform  
Wavelet transform plays an important role in various signal processing applications [29; 30; 31]. Among them, discrete wavelet transform (DWT) can decompose the signal into several wavelets, which is an ideal tool to map the change characteristics of unsteady signals. As shown in Figure 1, this is the basic principle of DWT. Seismic data processed by DWT will show the characteristics of multi-scale and high resolution, which helps in improving the accuracy of machine learning and thus improving the accuracy and resolution of inversion results. DWT is the convolution between the data sequence and the parent wavelet. The mother wavelet \(\psi\) is:  
\[\psi_{a,b}(t)=|a|^{-1/2}\psi\left(\frac{(t-b)}{a}\right),\;a,b\in R,\;a\
eq 0 \tag{1}\]  
where a is the scaling factor and b is the translation factor. The inversion target of seismic inversion body P(t) is the mud content, and its DWT is defined as:  
\[P(a,b)=|a|^{-1/2}\int P(t)\overline{\psi}\left(\frac{(t-b)}{a}\right)dt=\langle P (t),\psi_{a,b}(t)\rangle \tag{2}\]  
where P(a,b) is the wavelet coefficient. Common mother wavelet functions include Mexican-hat wavelet, Morlet wavelet, and so on. This paper focuses on the wavelet decomposition of well vibration curve data. In the curve data, the sand value is 0, and the mud value is 1; the composition of sand and mud changes frequently and complicatedly. The db 1 wavelet in Daubechies wavelet [32] is selected to analyze the characteristics of well seismic data, because it has low complexity and strong mutation detection ability, which can improve the calculation efficiency and accurately identify the frequently changing mud. However, the decomposition of other wavelets in Daubechies wavelet, such as db 2 and db 4, is too complex and too computationally intensive to be suitable for a large number of well vibration data. The permissible conditions for Daubechies wavelets used in this study are:  
\[C_{\psi}=\int\frac{|\hat{\psi}(\omega)|}{|\omega|}d\omega<\infty \tag{3}\]  
In Equation (3), \(\hat{\psi}(\omega)\) is obtained after the Fourier transform of \(\psi(t)\). After the inverse transformation of Equation (2), the following equation is obtained:  
Figure 1: DWT working diagram. \[\mathrm{P}(t)=C_{\psi}^{-1}\iint\frac{\mathrm{P}(a,b)\psi_{a,b}(t)}{a^{2}}\mathrm{ dadb} \tag{4}\]  
The negative frequency of \(\hat{\psi}(\omega)\) has no meaning; therefore, Equation (4) can be written as Equation (5):  
\[\mathrm{P}(t)=2C_{\psi}^{-1}\int_{0}^{\infty}\int_{0}^{\infty}\frac{\mathrm{P} (a,b)\psi_{a,b}(t)}{a^{2}}\mathrm{dadb} \tag{5}\]  
In order to improve calculation speed and reduce data redundancy, the scale and displacement parameters of DWT are amplified. The discrete mother wavelet function can be defined as  
\[\psi_{j,k}(t)=a_{0}^{-j/2}\psi\left(a_{0}^{-j/2}t-kb_{0}\right),\;j,k\in Z \tag{6}\]  
In a J level decomposition, \(j=1+2+\ldots+J\). The dilated and translated basis functions at different resolution levels are described by the scaling function, which is called the father wavelet \(\Phi_{j,k}(t)\) and is given by  
\[\Phi_{j,k}(t)=a_{0}^{-j/2}\Phi\left(a_{0}^{-j/2}t-kb_{0}\right),\;j,k\in Z \tag{7}\]  
where \(a_{0}\) and \(b_{0}\) are the scale factors and \(J\) is the scale level.