## 2 Methodology
### Extraction Approach of the Depth-Domain Generalized Seismic Wavelet  
The depth-domain seismic data can be considered as the convolution of depth-domain reflectivity (reflection coefficients) and a series of depth-domain wavelets, expressed as follows [23]:  
\[s(h)=\int_{-\infty}^{\infty}w(h-l,l)r(l)dl, \tag{19}\]  
where \(s\), \(w\), and \(r\) represent the depth-domain seismic trace, depth-domain wavelets, and depth-domain reflectivity series, respectively. Equation (19) can be rewritten as follows:  
\[\mathbf{s}=\mathbf{W}\times\mathbf{r}+\mathbf{n}, \tag{20}\]  
where \(\mathbf{s}\), \(\mathbf{W}\), \(\mathbf{r}\), and \(\mathbf{n}\) denote the seismic trace vector, the wavelet kernel matrix, the reflectivity vector, and the noise vector in the depth domain, respectively. Equation (19) indicates that the depth-domain seismic data are the summation of a series of depth-domain wavelets. Thus, the seismic data could be disassembled into the depth-domain generalized seismic wavelets series by selecting a reasonable method. OMP is a suitable method to decompose the seismic data into a batch of wavelets based on the pre-constructed overcomplete wavelet basis. Generally, the alternative wavelet basis is called the 'wavelet atom', and their collection--as a set--is called the 'dictionary' [41]. The OMP converges fast because it orthogonalizes all the atoms chosen at each iteration [37]. According to Equation (18), the overcomplete dictionary for depth-domain seismic signals, \(D\), is constructed by the following:  
\[D=\left\{g_{\gamma}(h)\right\}=\left\{g_{\gamma=(u,k_{0},l_{0})}(h)\right\}_{ \gamma\in\Gamma^{\prime}}, \tag{21}\]  
where \(g_{\gamma}\) represents the depth-domain generalized seismic wavelet. The implementation of OMP is iterative. The optimal depth-domain generalized seismic wavelet \(g_{\gamma_{n}}\) is extracted from the overcomplete dictionary at each iteration adaptively, where \(n\) represents the \(n\)-th iteration. Before conducting OMP on the seismic trace \(s(h)\), one should initialize the residuals, \(R^{(0)}s=s\), and the index set, \(\Lambda_{0}=\varnothing\), which is the set of optimal wavelet atoms. Then, the optimal wavelet, \(g_{\gamma_{n}}(h)\), at the \(n\)-th iteration can be obtained from the following equation:  
\[g_{\gamma_{n}}(h)=\arg\max_{g_{\gamma_{n}}\in D}\frac{\left|\left\langle R^{(n )}s,g_{\gamma_{n}}\right\rangle\right|}{\left\|g_{\gamma_{n}}\right\|}, \tag{22}\]  
where \(\left\langle\cdot,\cdot\right\rangle\) denotes the inner product and \(\left\|g_{\gamma_{n}}\right\|=\sqrt{\left\langle g_{\gamma_{n^{\prime}}},g_{ \gamma_{n}}\right\rangle}\) normalizes the wavelet \(g_{\gamma_{n}}\). The optimal wavelet atom is added to the index set by \(\Lambda_{n}=\left[\Lambda_{n-1},g_{\gamma_{n}}\right]\). Then, the amplitude coefficient, \(a_{n}\), is solved by the following:  
\[a_{n}=\arg\min\|s-\Lambda_{n}a_{n}\|, \tag{23}\]  
Figure 5: PCCs between the depth-domain generalized seismic wavelet and the depth-domain attenuated wavelet shown in Figure 4.  
The solution to Equation (23) is as follows:  
\[a_{n}=\left(\Lambda_{n}^{T}\Lambda_{n}\right)^{-1}\Lambda_{n}^{T}s, \tag{24}\]  
where \(\Lambda_{n}^{T}\) denotes the transpose of \(\Lambda_{n}\). Finally, the residual is updated through the following equation:  
\[R^{(n)}s=s-\Lambda_{n}a_{n}. \tag{25}\]  
Equations (22)-(25) represent one complete iteration of the depth-domain generalized seismic wavelet overcomplete dictionary-based OMP (OMP-G\({}_{\text{d}}\)) decomposition of the depth-domain seismic data. The algorithm kills when the pre-set threshold is larger than the residual error or \(n\) exceeds a pre-set maximum iteration number. The seismic trace after \(N\) iterations is written as follows:  
\[s(h)=\sum_{n=0}^{N-1}a_{n}g_{\gamma_{n}}(h)+R^{(N)}s, \tag{26}\]  
where the minimum residual, \(R^{(N)}s\), is regarded as the data noise.