# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 5 Conclusions  
In this paper, to address the issue that the Radon filter method for multiple suppression requires human setting of filtering parameters, a novel framework that can automatically separate multiples and primaries using GMD-R in the sparse Radon domain is proposed. First, GMD-R is introduced and applied to separate multiple and primary modes in the Radon domain. The GMD-R leverages the principle of Wiener filtering to iteratively decompose the data into modes with varying curvature and intercept, thus achieving the separation of primary modes and multiple modes according to their curvature differences. Based on the GMD-R algorithm, we develop a new processing flow for suppressing multiples. Then, we introduce the EH norm as a constraint and propose a novel sparse Radon transform. The EH norm adds an additional scaled \(l_{2}\) norm to the \(l_{1/2}\) norm, which is added to overcome the numerical oscillation problem. With the help of the EH norm, the new proposed sparse RT has a higher sparsity and stable solution. Finally, by combining the sparse Radon model, we improve the GMD-based multiple removal framework. Two  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test Data** & **Method** & \
\begin{tabular}{c} **Computation** \\ **Time (s)** \\ \end{tabular} & **Method** &
\begin{tabular}{c} **Computation** \\ **Time (s)** \\ \end{tabular} \\ \hline Synthetic data 2 & LSRT & 1.08 & GMD-R & 2.17 \\ EH-SRT & 3.56 & GMD-R & 1.32 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computation time comparison of the LSRT and EH-SRT, and performing GMD on the two Radon models.  
synthetic data tests and a field data test are employed to illustrate the effectiveness of the proposed framework. The results indicate that the GMD-based framework can achieve the suppression of multiples with a high computational efficiency and can be applied in the actual production. Moreover, it reduces the dependence on manual setting parameters and simplifies the processing flow for multiple elimination.  
Conceptualization, A.M. and J.S.; methodology, A.M., J.S. and Y.S.; software, A.M.; validation, J.S.; formal analysis, A.M. and J.S.; investigation, A.M. and C.H.; resources, A.M.; data curation, A.M. and C.H.; writing--original draft preparation, A.M.; writing--review and editing, A.M., J.S. and Y.S.; visualization, A.M.; supervision, J.S.; project administration, A.M. and J.S.; funding acquisition, J.S. All authors have read and agreed to the published version of the manuscript.  
This research was supported in part by the Key Laboratory of Deep Oil and Gas, China University of Petroleum (East China); in part by the General Program of National Natural Science Foundation of China under Grant 41674125; and in part by the Major scientific and technological project of CNPC under Grant no. ZD2019-183-003.  
Not applicable.  
The data used in this study are available by contacting the corresponding author.  
We are grateful to all of the reviewers for their constructive comments on this paper. We also thank the open-source Madagascar for providing us with excellent tools. We extend our gratitude to [PERSON] from China Oilfield Services Limited (COSL) and [PERSON] from Train oil field branch of CNPC for their advices of seismic data processing.  
The authors declare no conflict of interest.