# Sparse Parabolic Radon Transform with Nonconvex Mixed Regularization for Multiple Attenuation
## 1 Introduction  
Multiple reflections are usually regarded as coherent noise in reflection seismic data especially in marine exploration. The existence of multiple reflections brings difficulty to seismic data processing [1]. It makes the structural illusion appear in the migrated-stacked section, which can affect the accuracy of subsequent data interpretation [2]. The Radon transform is a common method for multiple attenuation because of its high efficiency and due that it is easily implemented [3; 4]. Due to the kinematic differences between primary and multiple reflections, most multiples are considered as coherent noise with low velocity [5]. Therefore, the primary reflections are upturned or flat, and multiple reflections are parabolic after normal movement correction with certain velocities in the CMP gather, which provides a theoretical basis for multiple attenuation using Radon transform. In 1986, [PERSON] [6] used the parabolic Radon transform to suppress multiples. The parabolic Radon transform makes their seismic events of different curvatures projected in different regions of the Radon domain, so as to separate primary and multiple reflections. However, finite seismic acquisition aperture of seismic data causes energy diffusion in the Radon domain, and it causes the smearing shown by the red arrows in Figure 1b. The energy diffusion limits the effect of multiple attenuation. To improve the above problem, the conception of optimization inversion is introduced into Radon transform [7; 8]. When solving the inverse problems of Radon transform, the solution of inversion Radon transform is not unique. That is to say, there will be multiple sets of different solutions in the Radon domain, and these solutions do not describe the model function well. [PERSON] [6] proposed the least squares inversion method, and the original data are reconstructed by optimizing the data in the Radon domain. This method reduces the smearing in the radon domain, but there are still some artefacts and data outside theoffset that are not restricted. After Radon inverse transform, the data within a restricted offset still consist of the original data, but they are different when the data are outside the restricted range. In addition, the solution of the Radon domain is not optimal. Therefore, to solve this problem, it is hoped that the solution of the Radon domain will be sparser [2]. In 1985, the Radon transform was considered as a sparse inversion problem by [PERSON] et al. [9]. However, a lot of computation is disadvantageous. [PERSON] et al. [10] adopted the method of [PERSON] to implement the sparse inversion Radon transform, and it is in the frequency domain. A high-resolution Radon transform was proposed by [PERSON] to distinguish primary and multiple reflections, and it tackled the aliasing and resolution issues of the Radon transform [11]. [PERSON] et al. [12] discussed fast implementations for the Radon transform in the time and frequency domains. With this sparse constraint, the seismic events have better localization characteristics in the Radon domain and the separation effect of primaries and multiples is guaranteed to a certain extent. [PERSON] [13] proposed iterative 2D model shrinkage to solve the sparse inverse problem of RT and obtained the analogously sparse Radon model. [PERSON] et al. [14] established a mathematical model to combine \(L_{2}\)-norm and the adaptive multiple subtraction method on \(L_{1}\)-norm by weighted combination. This method can suppress the energy of multiples relatively better than non-combination. In order to further improve the separation effect, researchers use the regularization methods of L-norm as the sparsity constraint condition that fully reflects the basic characteristics of the seismic data [15; 16; 17]. We can obtain the high-resolution Radon transform, and it is constrained sparsely, such as in \(L_{0}\) regularization, but it is generally not used because it is hard to solve. [PERSON] [18] proposed \(L_{1}\) regularization, and it provides an alternative.