# 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
Equation (10) is a quadratic equation, which can be solved by setting its derivation of variable \(\mathbf{m}\) be zero and the update equation of \(\mathbf{m}\) is given by  
\[\mathbf{m}^{k+1}=f^{-1}\bigg{\{}\Big{[}\mathbf{L}^{T}\mathbf{L}+(2\sigma+\xi) \mathbf{I}\Big{]}^{-1}\Big{[}\mathbf{L}^{T}f(\mathbf{d})+\xi\Big{(}f\Big{(} \mathbf{T}^{k}\Big{)}-f\Big{(}\mathbf{z}^{k}\Big{)}\Big{)}\Big{]}\bigg{\}}, \tag{13}\]  
The optimization problem in Equation (11) follows a typical \(l_{1-2}\) norm formulation, and its solution can be achieved by employing the half-threshold function [37]:  
\[\mathbf{T}^{k+1}=H_{\eta}\Big{(}\mathbf{m}^{k+1}+\mathbf{z}^{k}\Big{)}, \tag{14}\]  
where  
\[H_{\eta}(\mathbf{x})= \begin{cases}g_{\eta}(\mathbf{x}_{i}),&|\mathbf{x}_{i}| >\frac{3\sqrt[3]{2}}{4}\eta^{\frac{2}{3}}\\ 0,&\text{otherwise}\end{cases} , \tag{15}\]  
\[g_{\eta}(\mathbf{x}_{i})=\frac{2\mathbf{x}_{i}}{3}\bigg{(}1+\text{cos}\bigg{(} \frac{2\pi}{3}-\frac{2\psi_{\eta}(\mathbf{x}_{i})}{3}\bigg{)}\bigg{)}, \tag{16}\]  
\[\psi_{\eta}(\mathbf{x}_{i})=\arccos\Bigg{(}\frac{\eta}{8}\bigg{(}\frac{|\mathbf{ x}_{i}|}{3}\bigg{)}^{-3/2}\Bigg{)}, \tag{17}\]  
where \(\eta=\lambda/\xi\). Finally, the optimization problem of Equation (7) can be solved through several iterations and a sparse Radon model \(\mathbf{m}\) can be obtained using Equation (13). It is important to note that only a single pseudoinverse matrix \(\big{[}\mathbf{L}^{T}\mathbf{L}+(2\beta+\xi)\mathbf{I}\big{]}^{-1}\) needs to be calculated in the procedure. And the inverse matrix is irrelevant with input data values, but only with the dimension of input data. Once the matrix is calculated, it can be used to all seismic data that have the same size. The two three-dimensional matrixes \(\mathbf{L}(\omega,x,q)\) and \(\big{[}\mathbf{L}^{T}\mathbf{L}+(2\sigma+\xi)\mathbf{I}\big{]}^{-1}\) do not need repeat calculation in every iteration, therefore it may take a certain amount of memory to store them. In each iteration, the forward Fourier transform of \(\mathbf{T}^{k}\) and \(\mathbf{z}^{k}\) needs to be calculated and the inverse Fourier transform is performed to derive the final sparse Radon model \(\mathbf{m}^{k+1}\). The pseudocode for performing the EH-SRT is summarized in Algorithm 1. ```
Input: \(\mathbf{L}\), \(\mathbf{d}\), \(\lambda\), \(\sigma\), \(\xi\), \(N_{iter}\). Output: \(\mathbf{m}=\mathbf{m}^{k}\). Initialize: \(\mathbf{m}^{0}\) is the estimated Radon model of LSRT, \(\mathbf{T}^{0}=\mathbf{m}^{0}\), \(\mathbf{c}=0\) and \(k=0\)
1:for\(k=0\) to \(N_{iter}\)do
2: update the sparse Radon model \(\mathbf{m}^{k+1}\) with \(\mathbf{m}^{k+1}=f^{-1}\Big{\{}\big{[}\mathbf{L}^{T}\mathbf{L}+(2\sigma+\xi) \mathbf{I}\big{]}^{-1}\Big{[}\mathbf{L}^{T}f(\mathbf{d})+\xi\Big{(}f\Big{(} \mathbf{T}^{k}\Big{)}-f\Big{(}\mathbf{z}^{k}\Big{)}\Big{)}\Big{]}\Big{\}}\)
3: update \(\mathbf{T}^{k+1}\) with Equations (14)-(17)
4: update \(\mathbf{z}^{k+1}\) with \(\mathbf{z}^{k+1}=\mathbf{z}^{k}+\mathbf{m}^{k+1}-\mathbf{T}^{k+1}\)
5:endfor
```  
**Algorithm 1** The pseudocode for the proposed EH-SRT