# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 1 Introduction
As a result, the separation of primary and multiple modes can be achieved using GMD-R. Based on GMD-R, we have developed a framework for automatically separating primary and multiple modes in the Radon domain. Compared to the Radon filter method, the developed GMD-based multiple suppression method significantly reduces the need for human intervention, as GMD-R automatically performs computational decomposition based on the predefined number of modes. However, the result of GMD-R is dependent on the resolution and sparsity of the Radon model, which is affected by the limited sampling and finite aperture in offset [25]. To address the challenges associated with achieving a high-resolution Radon model, RT is regarded as an inversion problem, and various inversion methods are employed to solve this problem [6]. In 1995, [PERSON] et al. [26] presented a new approach to achieve sparse Radon transform, implementing it in the frequency domain (FSRT) through a Bayesian framework. While this algorithm proved to be highly efficient, its limitation lies in the fact that RT in the frequency domain assigns equal weight to all events, constraining its ability to enhance temporal resolution. In response to this constraint, the time domain sparse RT (TSRT) was introduced [27], offering a higher level of sparsity in the time domain and thereby achieving enhanced resolution. However, it is worth noting that TSRT comes with a substantial computational cost [28]. Combining the advantages of TSRT and FSRT, a novel RT was developed by [PERSON] et al. [6]. The approach performs RT and its inverse in the frequency domain, formulates a new equation with the \(l_{1}\) norm constraint, and solves it iteratively using the re-weighted least-squares algorithm. However, the large dimension of the inverse matrix in the time domain leads to a high computational cost. To overcome this disadvantage, an iterative 2D model shrinkage algorithm was introduced by [PERSON] to accelerate the RT [29]. [PERSON] et al. employed an iterative algorithm that combinesthe Radon transform with a \(l_{1}\) norm regularization term for denoising and spares the representation of seismic data [30]. A new sparse RT was proposed by [PERSON] et al. to attenuate the outlier effects in the Radon domain [31]. [PERSON] and [PERSON] developed a new RT along the offset direction and modified the basic Radon function to alleviate the ill-posed problem [32]. Furthermore, [PERSON] et al. developed a novel sparse RT with the \(l_{1-2}\) norm constraint (SRTL\({}_{1-2}\)) and extended the method to three dimensions for multiple attenuation [33; 34]. The \(l_{p}\) norm is added to promote sparsity of RT [35; 36]. However, the method is computationally complex and requires a large number of iterative operations. To enhance the sparsity of the Radon model while concurrently reducing computational complexity, this paper introduces the elastic half (EH) norm [37; 38] as a constraint on the RT and proposes a sparse Radon transform. The EH norm contains a \(l_{1/2}\) norm and a scaled \(l_{2}\) norm, which is added to overcome the numerical oscillation problem of the \(l_{1/2}\) norm. To solve the optimization problem of the developed sparse RT, an efficient alternating multiplier iteration algorithm is employed. Then, we improve the GMD-based demultiple framework by applying GMD-R to the sparse Radon model of the proposed RT. The high-sparsity Radon model can not only simplify the separation of primary and multiple modes but also improves the computational efficiency of GMD-R. In addition, we conduct an analysis of the impact of the number of mode decompositions and provide recommendations for selecting an appropriate number. Finally, synthetic and field dataset tests are given to verify the favorable performance of the proposed GMD-based approach in multiple elimination.