# Pseudo-3D Receiver Deghosting of Seismic Streamer Data Based on \(l_{1}\) Norm Sparse Inversion
## 2 Methods  
### Pseudo-3D Deghosting Framework with the Data Coherence along the Shot Direction  
In marine seismic exploration, the most conventional acquisition method is the streamer observation system, which is where a power boat drags several streamers for movement. The mainstream processing strategy is streamer-by-streamer, and a single streamer record can be viewed as a dataset of several shot gathers. For conventional multi-shot gathers, the radon domain deghosting algorithm can be expressed as  
\[\mathbf{D}_{i}(x_{j},\omega)=\sum_{i=1}^{np}\mathbf{M}_{i}(p_{ip},\omega)\cdot e ^{-i\omega p_{ip}x_{j}}\cdot(1-e^{-i\omega\Delta t_{ip}}) \tag{1}\]  
where \(\mathbf{D}_{i}(x_{j},\omega)\) represents the \(i\)-th shot of the seismic data in the frequency domain, \(\mathbf{M}_{i}(p_{ip},\omega)\) represents the \(i\)-th shot of the deghosting result in the frequency domain, and \(x_{j}\), \(p_{ip}\), and \(\omega\) represent the offset index of the \(j\)-th trace, the slowness index of the \(ip\)-th trace, and the angular frequency, respectively. \(\triangle t_{ip}\) represents the delay time between the ghost and primary, which is known to be a function of the water velocity \(v\), the receiver depth \(z\), and the incident angle \(\theta\) (Figure 1b). In the radon domain, the slowness \(p\) is calculated by the water velocity \(v\) and the incident angle \(\theta\) as follows:  
\[p=\frac{1}{v}\sin(\theta) \tag{2}\]The delay time \(\triangle t_{ip}\) in the radon domain is calculated by  
\[\triangle t_{ip}=\frac{2z}{v}\cos(\theta)=\frac{2z\sqrt{1-p_{ip}^{2}v^{2}}}{v} \tag{3}\]  
To simplify Equation (1), we redefine the 2D deghosting operator \(\mathbf{G}\) in the radon domain, which can be regarded as a combination of the radon transform operator and the ghosting operator as follows:  
\[\mathbf{G}_{ip,j}=e^{-i\omega p_{ip}x_{j}}\cdot(1-e^{-i\omega\Delta t_{ip}}) \tag{4}\]  
Equation (1) can then be rewritten as a form of matrix multiplication:  
\[\left[ \begin{array}{c}\mathbf{D}_{i}(x_{1},\omega)\\ \mathbf{D}_{i}(x_{2},\omega)\\ \vdots\\ \mathbf{D}_{i}(x_{n-1},\omega)\\ \mathbf{D}_{i}(x_{n},\omega)\\ \end{array} \right]=\left[ \begin{array}{cccc}\mathbf{G}_{1,1}&\mathbf{G}_{1,2}&\cdots&\mathbf{G}_{1,np-1}&\mathbf{G}_{1,np}\\ \mathbf{G}_{2,1}&\mathbf{G}_{2,2}&\cdots&\mathbf{G}_{2,np-1}&\mathbf{G}_{2,np} \\ \vdots&\vdots&\mathbf{G}_{ip,j}&\vdots&\vdots\\ \mathbf{G}_{n-1,1}&\mathbf{G}_{n-1,2}&\cdots&\mathbf{G}_{n-1,np-1}&\mathbf{G} _{n-1,np}\\ \mathbf{G}_{n,1}&\mathbf{G}_{n,2}&\cdots&\mathbf{G}_{n,np-1}&\mathbf{G}_{n,np} \\ \end{array} \right]\left[ \begin{array}{c}\mathbf{M}_{i}(p_{1},\omega)\\ \mathbf{M}_{i}(p_{2},\omega)\\ \vdots\\ \mathbf{M}_{i}(p_{np-1},\omega)\\ \end{array} \right] \tag{5}\]  
The conventional 2D deghosting method indicates a shot-by-shot deghosting manner. Because every shot gather is processed independently, the inversion accuracy of the 2D deghosting algorithm is easily limited by the field noise and accuracy of the 2D operator, which is necessary to introduce a priori information to constrain the inversion. During the horizontal streamer acquisition, the location of the sources and receivers is relatively stable, and so the offset parameters of the different shots are consistent. According to this property, we can rearrange the dataset into the data cube shown in Figure 2. In this data cube, the seismic records are arranged in a source-offset-time manner. Then, we can adopt a Fourier transform to the data cube, and the frequency slice \(\mathbf{D}_{\text{pse-3D}}(\omega_{k})\) can be expressed as:  
\[\mathbf{D}_{\text{pse-3D}}(\omega_{k})=