# Pseudo-3D Receiver Deghosting of Seismic Streamer Data Based on \(l_{1}\) Norm Sparse Inversion
## 2 Methods
### Pseudo-3D Deghosting Based on Sparse Inversion  
To highlight the data coherence, we introduce the compressed sensing-based sparse inversion methods into the deghosting method. In the pseudo-3D deghosting framework, the linear system of Equation (6) can be rewritten as  
\[\mathbf{L}m=f_{t}^{H}BlockDiag_{\omega}(\mathbf{G}\otimes\mathbf{I})f_{t}m=d \tag{12}\]  
where \(m\) represents the vector of the deghosting result \(\mathbf{M}_{\text{pse-3D}}\) in the radon domain, \(d\) represents the vector of the input dataset \(\mathbf{D}_{\text{pse-3D}}\) in the time-space domain, \(\mathbf{A}\) represents the identity matrix, \(\otimes\) represents the Kronecker product of the matrix, \(f\) and \(f_{t}^{H}\) represent the Fourier transform pair in the time direction, and \(\mathbf{L}\) represents the combination of the ghost delay operator and the Fourier transform operator. Figure 4: Sparsity comparison of the synthetic data. (**a**) Input data. (**b**) Sparse reconstruction in the time domain. (**c**) Sparse reconstruction in the curvelet domain. Figure 3: Demonstration of sparse representation. In actual seismic acquisition, field noise cannot be avoided. Assuming that the input datasets \(d\) are a collection of the noise \(n\) and useful signals \(s=\mathbf{L}m\), then:  
\[d=\mathbf{L}m+n \tag{13}\]  
Based on the compressed sensing theory, the seismic signals are over-complete and satisfy the requirement of sparse representation. Therefore, the useful signal can be reconstructed by the sampling matrix \(\mathbf{A}\):  
\[d=\mathbf{L}\mathbf{A}x_{0}+n \tag{14}\]  
where \(x_{0}\) represents the coefficients in the transformed domain. In general, \(\mathbf{A}\) can be constructed by a sparse transform (wavelet transform [30], curvelet transform [29], etc.). Without considering the noise, the main purpose of this problem is to solve the sparse solution of \(\mathbf{L}\mathbf{A}x=d\), where \(\mathbf{L}\mathbf{A}\) represents a matrix of the size \(m\times n\) and \(d\) represents a vector of the size \(n\times 1\). When \(m\ll n\), this problem is under-determined, which can be summarized as a basic pursuit problem (BP). To solve this BP problem [31], we propose the objective function:  
\[\min_{x}\left\|x\right\|_{1}\ \ \ \text{s.t.}\ \mathbf{L}\mathbf{A}x=d \tag{15}\]  
Considering the existence of noise, the BP problem can be transformed into a BPDN problem (basis pursuit denoising) [32] to avoid the noise interference during inversion:  
\[\min_{x}\left\|x\right\|_{1}\ \ \ \text{s.t.}\ \left\|d-\mathbf{L}\mathbf{A}x \right\|_{2}\leq\sigma \tag{16}\]  
where \(\sigma\) represents the \(l_{2}\) norm residual of the objective function, which is generally set as 5-10% of the \(l_{2}\) norm of \(d\). The proposed method uses the \(l_{2}\) norm to approximate the reconstructed data to the original data while the \(l_{1}\) norm constrains the noise. In this way, the noise \(n\) in the data can be suppressed. To solve the above problem, we adopt the spectral gradient projection method (SPGL1) [33]  
\[\left\{ \begin{array}{ll}\min_{x}\left\|x\right\|_{1}\ \ \ \text{s.t.}\ \ \left\|d- \mathbf{L}\mathbf{A}x\right\|_{2}\leq\sigma\\ \hat{m}=\mathbf{A}\hat{x}\end{array} \right. \tag{17}\]  
where \(\hat{m}\) and \(\hat{x}\) represent the inversion results in the data domain and the sparse domain, respectively, \(\sigma\) represents the noise level, which controls the sparsity, and SLGL1 is a solver for the \(l_{1}\)-regularized least-squares inversion problem, which is suitable for large-scale datasets. In order to improve the inversion accuracy, we adopt a sparse transform to achieve the sparse representation of the curved events with multi-scale and multi-directional characteristics. It can be seen from Section 2.2 that sparse representation can increase the sparseness of the signal. Considering representation accuracy and computational efficiency, we combine a 2D curvelet transform and a 1D wavelet transform as the sampling matrix.