# High-Fidelity OC-Seislet Stacking Method for Low-SNR Seismic Data
## 2 Materials and Methods
### High-Order OC-Seislet Stacking  
In the processing of data with complex subsurface structures, the NMO correction in the time-space-scale domain cannot handle cross events or accurately correct for the time shifts of dipping layers at large offsets. In such cases, DMO can compensate for the flaw of NMO processing. According to [PERSON]'s DMO theory, DMO can be executed in cascade with NMO. Phase analysis has significant potential for advanced feature analysis in non-linear data, such as characterizing transient signals [40] and predicting earthquakes [41]. Based on the mathematical relationship between the frequency-wavenumber-scale domain and the time-space-scale domain through the double Fourier transform, phase shift processing in the frequency-wavenumber-scale domain corresponds to time shift in the time-space-scale domain, and completes the DMO for non-zero-offset gathers. This phase shift, known as the DMO operator, can be obtained by solving a special initial value problem of the OC equation. After NMO and the logarithmic stretching transform of the time axis, a double Fourier transform is performed to obtain the transformed data \(u(\Omega,k,h)\). The phase shift conversion relationship for different half-offset distances in the frequency-wavenumber-scale domain is given by [30]  
\[u(\Omega,k,h_{2})\approx u(\Omega,k,h_{1})\frac{F(\frac{2k\cdot h_{2}}{\Omega} )exp[i\Omega\psi(\frac{2k\cdot h_{2}}{\Omega})]}{F(\frac{2k\cdot h_{1}}{\Omega })exp[i\Omega\psi(\frac{2k\cdot h_{1}}{\Omega})]}. \tag{7}\]  
where  
\[F(x)=\sqrt{\frac{1+\sqrt{1+x^{2}}}{2\sqrt{1+x^{2}}}}exp(\frac{1-\sqrt{1+x^{2}} }{2}), \tag{8}\]  
\[\psi(x)=\frac{1}{2}(1-\sqrt{1+x^{2}}+ln(\frac{1+\sqrt{1+x^{2}}}{2})). \tag{9}\]  
Equation (7) represents the trace prediction operator in the frequency-wavenumber-scale domain. The bold black symbols denote vectors, where \(\Omega\) corresponds to the dimensionless frequency of the stretched time coordinate, \(k\) is the wavenumber, and \(h\) represents the half-offset distance. After obtaining the prediction operator, modifications to the prediction and update operations can be made within the framework of the seislet transform. This paper uses the CDF 9/7 biorthogonal basis to replace the prediction operator \(P\) in Equation (2) and the update operator \(U\) in Equation (5). This replacement involves two consecutive processes and a scaling operation. The first-step prediction and update operators are  
\[\left\{ \begin{array}{l}\dot{P}[e]_{k}=[S_{k}^{+(e\to e)}(e_{k} -1)+S_{k}^{-(e\to e)}(e_{k})]*\alpha,\;k>1\\ \dot{P}[e]_{k}=2S_{k}^{-(e\to e)}(e_{k})*\alpha,\;k=1\\ \dot{U}[r]_{k}=[S_{k}^{+(e\to e)}(r_{k-1})+S_{k}^{-(e\to e)}(r_{k})]*\beta,\;k>1 \\ \dot{U}[r]_{k}=2S_{k}^{+(o\to e)}(\dot{r}_{k})*\beta,\;k=1\end{array} \right. \tag{10}\]  
Substituting the above equation into Equations (1) and (3), the first-step scale coefficients are  
\[\left\{ \begin{array}{l}\dot{c}_{k}=\epsilon_{k}-\alpha\beta S_{k}^{-(o\to e )}S_{k}^{-(e\to e)}(e_{k})+\beta S_{k}^{-(o\to e)}(\alpha_{k})-\alpha\beta S_{k }^{+(o\to e)}S_{k-1}^{-(e\to e)}(e_{k-1})-\\ \alpha\beta S_{k}^{-(o\to e)}S_{k}^{+(e\to e)}(e_{k-1})+\beta S_{k}^{+(o\to e)}( e_{k-1})-\alpha\beta S_{k}^{+(o\to e)}S_{k-1}^{+(o\to e)}(e_{k-2}),k>2\\ \dot{c}_{k}=\epsilon_{k}-\alpha\beta S_{k}^{-(o\to e)}S_{k}^{-(o\to e)}( \epsilon_{k})+\beta S_{k}^{-(o\to e)}(\alpha_{k})-\alpha\beta S_{k}^{+(o\to e)}S _{k}^{+(e\to e)}(\epsilon_{k-1})+\\ \beta S_{k}^{+(o\to e)}(\alpha_{k-1})-2\alpha\beta S_{k}^{+(o\to e)}S_{k-1}^{-( e\to e)}(\epsilon_{k-1}),k=2\\ \dot{c}_{k}=\epsilon_{k}-4\alpha\beta S_{k}^{+(o\to e)}S_{k}^{-(e\to e)}( \epsilon_{k})+2\beta S_{k}^{+(o\to e)}(\epsilon_{k}),k=1\end{array} \right. \tag{11}\]The simplification results are as follows:  
\[\left\{