# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 2 Methods
### Geometric Mode Decomposition  
Taking inspiration from VMD [18], [PERSON] et al. proposed the geometric mode decomposition (GMD) [21]. GMD considers two-dimensional seismic data as a combination of band-limited modes with line or parabolic geometric features. By iteratively calculating the weighted center of the energy spectrum, GMD can extract modes with different geometric characteristics, such as slopes or curvatures. GMD contains two variants: GMD-F, which performs decomposition in the frequency domain, and GMD-R, which performs decomposition in the Radon domain. In this study, our focus lies on GMD-R as it allows for the suppression of multiples by exploiting the separability of primaries and multiples in the Radon domain. Consequently, the optimization problem is formulated as follows:  
\[\min_{\mathbf{u}_{k},\beta_{k}}\!\!\!\!\  
filter. The new Radon parameters \(\beta_{k}\) are determined using Equation (21). The iteration will continue computing until the convergence standard is satisfied and the convergent modes \(\hat{\mathbf{R}}_{k}^{\star}\) are finally obtained [21]. By applying the inverse RT to these modes, the corresponding decomposed modes in the time-space domain are obtained. Note that the summation of the decomposed modes is equivalent to the input Radon model \(\hat{\mathbf{m}}(\beta)\).  
```
Input : \(\hat{\mathbf{d}},\hat{\mathbf{m}}(\beta),\gamma,\epsilon,K_{m},N_{iter}\) Output : \(\hat{a}_{k}(\beta)\). Initialize \(\beta_{k}^{1}=(\tau_{k}^{1},p_{k}^{1}),\ n=0,\ \hat{\mathbf{m}}(\beta)\) is the Radon model of input data \(\hat{\mathbf{d}},\hat{\mathbf{R}}_{k}^{\star}\) denotes the decomposed modes in the Radon domain, and \(N_{iter}\) is the maximum iteration times.
```
1:for\(n=0:N_{iter}\)do
2:for\(k=1:K_{m}\)do
3:update \(\hat{\mathbf{R}}_{k}\): \[\hat{\mathbf{R}}_{k}^{n+1}(\beta)=\frac{\hat{\mathbf{m}}(\beta)-\sum\limits_ {i\
eq k}\hat{\mathbf{R}}_{i}(\beta)}{1+2\gamma(\beta-\beta_{k})^{2}},\] (20)
4:endfor
5:for\(k=1:K_{m}\)do
6:update \(\beta_{k}\): \[\beta_{k}^{n+1}=\frac{\int\beta_{k}\beta\left|\hat{\mathbf{R}}_{k}(\beta) \right|^{2}\mathrm{d}\beta}{\int\beta_{k}\left|\hat{\mathbf{R}}_{k}(\beta) \right|^{2}\mathrm{d}\beta},\] (21)
7:endfor
8:If\(\sum\limits_{k}\left(\left\|\hat{\mathbf{R}}_{k}^{n+1}-\hat{\mathbf{R}}_{k}^{n} \right\|_{2}\right)<tol\)Exit
9:endfor ```  
**Algorithm 2** The ADMM algorithm for the GMD-R problem  
After NMO correction, multiples and primaries exhibit different curvatures and are distributed in the different areas in the Radon domain, which has a certain separability and distinct geometric features. GMD-R uses the Wiener filter to make the primary and multiple modes converge to their center Radon coefficients gradually, and can decompose them from the original Radon model, thus finally achieving the decomposition of the primary and multiple modes. Starting from this perspective, we developed a novel GMD-based processing flow that can automatically achieve the separation of primaries and multiples. Here, we will provide a detailed explanation of how to apply the GMD-R algorithm for multiple suppression. The procedure of the proposed framework can be divided into three steps:  
1. Apply NMO correction to the input CMP gather data \(\mathbf{d}\) and then calculate the Radon model \(\mathbf{m}\) using the LSRT;
2. By setting the number of decomposition modes to 2, perform GMD-R to the Radon model \(\mathbf{m}\) following Algorithm 2 and obtain the separated primary and multiple modes in the Radon domain;
3. Perform inverse RT to the separated primary modes and the demultiple result is finally obtained. The main advantage of the GMD-based demultiple method over the Radon filter method is that it does not require too many parameters to be set artificially. In the GMD procedure, the decomposition of modes is achieved following the algorithm in Algorithm 2, which is an automatic calculation process. From this point, we can separate primary modes and multiple modes because they have different curvatures.