# High-Fidelity OC-Seislet Stacking Method for Low-SNR Seismic Data
## 2 Materials and Methods
### High-Order OC-Seislet Stacking
\begin{array}{l}\dot{c}_{k}=e_{k}-\alpha\beta e_{k}^{\prime}+\beta\o_{k}^ {\prime}-2\alpha\beta e_{k-1}^{\prime}+\beta\o_{k-1}^{\prime}-\alpha\beta e_{k-2 }^{\prime},k>2\\ \dot{c}_{k}=e_{k}-\alpha\beta e_{k}^{\prime}+\beta\o_{k}^{\prime}-3\alpha\beta e _{k-1}^{\prime}+\beta\o_{k-1}^{\prime},k=2\\ \dot{c}_{k}=e_{k}-4\alpha\beta e_{k}^{\prime}+2\beta\o_{k}^{\prime},k=1\end{array} \right. \tag{12}\]  
The first-step scale coefficients \(\dot{c}\) are used as the new even sequences, and the first-step difference coefficients \(\dot{r}\) (from Equation (1)) are used as the new odd sequences for the second step of prediction and updating:  
\[\left\{ \begin{array}{l}\bar{P}[e]_{k}=[S_{k}^{+(e\to o)}(\dot{c}_{k-1})+S_{k}^ {-(e\to o)}(\dot{c}_{k})]*\gamma,\;k>1\\ \bar{P}[e]_{k}=2S_{k}^{-(e\to o)}(\dot{c}_{k})*\gamma,\;k=1\\ \bar{U}[r]_{k}=[S_{k}^{+(o\to e)}(\dot{r}_{k-1})+S_{k}^{-(o\to e)}(\ddot{r}_{k })]*\delta,\;k>1\\ \bar{U}[r]_{k}=2S_{k}^{+(o\to e)}(\ddot{r}_{k})*\delta,\;k=1\end{array} \right. \tag{13}\]  
The second-step scale coefficients \(\ddot{c}\) obtained are  
\[\left\{ \begin{array}{l}\ddot{c}_{k}=\dot{c}_{k}-\gamma\delta\dot{c}_{k}^{ \prime}+\delta\dot{r}_{k}^{\prime}-2\gamma\delta\dot{c}_{k-1}^{\prime}+\delta \dot{r}_{k-1}^{\prime}-\gamma\delta\dot{c}_{k-2}^{\prime},k>2\\ \dot{c}_{k}=\dot{c}_{k}-\gamma\delta\dot{c}_{k}^{\prime}+\delta\dot{r}_{k}^{ \prime}-3\gamma\delta\dot{c}_{k-1}^{\prime}+\delta\dot{r}_{k-1}^{\prime},k=2\\ \dot{c}_{k}=\dot{c}_{k}-4\gamma\dot{\gamma}\dot{c}_{k}^{\prime}+2\delta\dot{r }_{k}^{\prime},k=1\end{array} \right. \tag{14}\]  
where marked points on superscripts \(\bar{P}\) and \(\bar{P}\) indicate the first and second stages. According to the parameter decomposition algorithm of the CDF 9/7 biorthogonal wavelet transform [42], the coefficients \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are defined as follows:  
\[ \begin{array}{l}\alpha=-1.586134342\\ \beta=-0.052980118\\ \gamma=0.882911076\\ \delta=0.443506852\end{array}  \tag{15}\]  
The first stage can be completed by combining Equations (1), (3), and (10) and repeating the processing by using Equations (1), (3), and (13). Finally, the scaling ratios for the second stage's scale coefficients and approximation sequences are as follows:  
\[c_{k}=\ddot{c}_{k}*K, \tag{16}\]  
\[r_{k}=\ddot{r}_{k}*(1/K), \tag{17}\]  
where \(K=1.230174105\), obtaining the result of a single iteration. The detailed workflow is illustrated in Figure 1, where the marked points in the superscripts indicate the first and second stages and the scaled coefficient \(c\) must be rearranged in position before being incorporated into the next cycle:  
The OC-seislet transform uses multiple partial moveout and iterative computations to achieve DMO and emphasize the frequency-wavenumber information of effective  
Figure 1: Illustration of high-order OC-seislet transform in one iteration. waves. The final zero-order scale coefficient represents the stacking result in the frequency-wavenumber-scale domain. Each iteration involves multiple weightings, which will affect the true amplitude of the seismic signals. Thus, weighting coefficients need normalization. To normalize the coefficients, we set all values of the frequency-wavenumber domain data \(u(\Omega,\mathbf{k},\mathbf{h})\) to one, then perform the CDF 9/7 wavelet transform to obtain the weighting coefficients \(w\). Each frequency stacking sample point has the same coefficient, after dividing the zero-order scale information from the OC-seislet transform by the weighting coefficient, and the normalized OC-seislet stacking coefficients are obtained. Benefiting from the high compression advantage of the high-order OC-seislet transform, the stacking coefficients of the effective reflection and diffraction waves are larger and more prominent. Ordinary soft thresholding [36] can efficiently remove smaller noise coefficients, achieving efficient denoising in the frequency-wavenumber-scale domain.