# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 2 Methods  
### Sparse Radon Transform with the Elastic Half-Norm Constraint  
In seismic exploration, the 2D forward RT [6; 39] and its inverse are given by  
\[ \begin{array}{l}m(\tau,q)=\sum\limits_{\begin{subarray}{c}x=x_{\min}\\ q=\max_{\max}\\ q=\min_{\max}\end{subarray}}^{x=x_{\max}}d(t=\tau+qx^{Ns},x),\\ d(t,x)\approx\sum\limits_{q=q_{\min}}^{x_{\max}}m(\tau=t-qx^{Ns},q),\end{array}  \tag{1}\]  
where \(d(t,x)\) denotes the input seismic data and \(m(\tau,q)\) represents the estimated Radon model. The variables \(\tau,\ q,\ Ns\) in Equation (1) represent the intercept time, slowness, and integration stack path [39; 40]. We can perform the transform in the frequency domain, and the above equation can be represented by [6]  
\[ \begin{array}{l}M(\omega,q)=\sum\limits_{\begin{subarray}{c}x=x_{\min}\\ q=q_{\min}\end{subarray}}^{x=x_{\max}}D(\omega,x)\mathrm{e}^{i\omega qx^{Ns}},\\ D(\omega,x)=\sum\limits_{q=q_{\min}}^{x}M(\omega,q)\mathrm{e}^{-i\omega qx^{Ns }},\end{array}  \tag{2}\]  
where \(D(\omega,x)\) and \(M(\omega,q)\) represents the frequency components of \(d(t,x)\), and \(m(\tau,q)\), respectively. To simplify the derivation, Equation (2) can be represented by  
\[ \begin{array}{l}\textbf{M=L}^{\textbf{H}}\textbf{D},\\ \textbf{D=LM},\end{array}  \tag{3}\]in which \(\mathbf{L}\) denotes the operator of the adjoint RT and \(\mathbf{L^{H}}\) is the conjugate transpose operator. The specific expression of \(\mathbf{L}\) is expressed as [31]  
\[\mathbf{L}=\left[ \begin{array}{ccccc}e^{-i\omega q_{1}x_{1}^{N_{ \text{s}}}}&e^{-i\omega q_{2}x_{1}^{N_{\text{s}}}}&\cdots&e^{-i\omega q_{nq-1}x_ {1}^{N_{\text{s}}}}&e^{-i\omega q_{nq}x_{1}^{N_{\text{s}}}}\\ e^{-i\omega q_{1}x_{2}^{N_{\text{s}}}}&e^{-i\omega q_{2}x_{2}^{N_{\text{s}}}}& \cdots&e^{-i\omega q_{nq-1}x_{2}^{N_{\text{s}}}}&e^{-i\omega q_{nq}x_{2}^{N_{ \text{s}}}}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ e^{-i\omega q_{1}x_{nx-1}^{N_{\text{s}}}}&e^{-i\omega q_{2}x_{nx-1}^{N_{ \text{s}}}}&\cdots&e^{-i\omega q_{nq-1}x_{nx-1}^{N_{\text{s}}}}&e^{-i\omega q _{nq}x_{nx-1}^{N_{\text{s}}}}\\ e^{-i\omega q_{1}x_{nx}^{N_{\text{s}}}}&e^{-i\omega q_{2}x_{nx}^{N_{\text{s}} }}&\cdots&e^{-i\omega q_{nq-1}x_{nx}^{N_{\text{s}}}}&e^{-i\omega q_{nq}x_{nx}^ {N_{\text{s}}}}\\ \end{array} \right]. \tag{4}\]  
The number of rows and columns in the matrix \(\mathbf{L}\) is \(nx\) and \(nq\), with \(nx\) being the trace number of input data and \(nq\) being the sampling number of slowness in the Radon domain, respectively. Using the 1D forward Fourier transform \(f[\cdot]\) and the inverse transform \(f^{-1}[\cdot]\), Equation (3) can be written as [41]  
\[\mathbf{d}=f^{-1}(\mathbf{L}f(\mathbf{m})). \tag{5}\]  
However, the RT encounters challenges in cases when the input data have limited offsets and samplings [25]. These limitations lead to issues such as low resolution, aliasing, and overlapping events in Radon space. The low-resolution Radon model poses difficulties in accurately separating primaries from the input data. To address the problem of low resolution, the RT is always treated as a sparse inverse problem, and the sparse Radon model can be obtained by solving the equation as follows [6; 33]  
\[\underset{\mathbf{m}}{\operatorname{argmin}}\|d-f^{-1}(\mathbf{L}f(\mathbf{m }))\|_{2}^{2}+\lambda U(\mathbf{m}), \tag{6}\]  
where \(U(\mathbf{m})\) denotes the constraint term, which plays a role in ensuring the solution is stable and unique. In general, \(U(\mathbf{m})\) has different types, such as \(l_{0}\), \(l_{1/2}\) and \(l_{1}\) norm. The \(l_{0}\) norm provides the sparsest metric, but its solution is computationally challenging due to the nondeterministic polynomial time complexity. The \(l_{1}\) norm is relatively easier to solve, but it does not yield a sparse enough solution [33]. In recent years, the \(l_{1/2}\) norm regularization has received increasing attention and has been applied to solving inverse problems in various categories, as the \(l_{1/2}\) norm has a stronger sparsity and robustness than the \(l_{1}\) norm.