# Sparse Parabolic Radon Transform with Nonconvex Mixed Regularization for Multiple Attenuation
## 1 Introduction
\(L_{1}\) regularization [19] is a convex optimization problem. This means that \(L_{1}\)-norm regularization is easy to solve. [PERSON] et al. [20] proved that sometimes the solution of the \(L_{0}\) regularization is equivalent to that of the \(L_{1}\) regularization for the sparsity problem. \(L_{2}\) regularization can avoid the overfitting of the model, but its solutions do not have the sparse property. Moreover, some numerical experiments showed that sparse signals are recovered from fewer linear measurements by \(L_{q}\) (\(0<q<1\)) regularization [21; 22; 23; 24]. \(L_{1/2}\) regularization can be taken as a representative of the \(L_{q}\) (\(0<q<1\)) regularization, and it has been proved that the solution of \(L_{1/2}\) regularization is sparser and more stable than the  
Figure 1: (**a**) CMP gather after NMO; (**b**) Radon domain data after parabolic Radon transform. The red arrows indicate the smearing. solution of \(L_{1}\) regularization [25]. However, the \(L_{1/2}\) regularization leads to a nonconvex, nonsmooth and non-Lipschitz optimization problem that is difficult to solve quickly and efficiently. Some scholars proposed corresponding solving methods. The reweighted iteration algorithm [25], the iterative reweighted least squares method [26], the iterative half thresholding algorithm [27] and the generalized iterated shrinkage algorithm [28; 29] are efficient methods for solving the \(L_{1/2}\) regularization. Generally, sparse regularization constrains the sparsity of the whole seismic wavefield, which means a unique regularization parameter for different wavefields. However, the different wavefields have different amplitudes, and the same regularization parameter setting easily causes signal damage or redundant residual information. Sparse regularization for multi-tasking exploits differences in features of the data to demix the two distinct components for constraining the sparsity. For the multiple attenuation case in the Radon domain, primary and multiple reflections are distributed in different zones of the Radon domain and have different amplitudes. The unique regularization parameter may limit the performance of multiple attenuation. In this paper, we propose a sparse parabolic Radon transform in the frequency domain with the nonconvex \(L_{q_{1}}-L_{q_{2}}(0<q_{1},q_{2}<1)\) mixed regularization (SPRTL\({}_{q_{1}}-L_{q_{2}}\)) that constrains the sparsity of primary and multiple reflections, respectively. In addition, we use the alternating direction method of multipliers algorithm (ADMM) [30; 31; 32] to approximately solve the multi-task regularization problem. Furthermore, conditions for convergence [33; 34; 35; 36] are indicated. The rationality and effectiveness of the proposed method in multiple attenuation are verified by synthetic and real data. The method in this paper is compared with least squares parabolic Radon transform (LSPRT) and sparse parabolic Radon transform based on \(L_{1}\) regularization (SPRTL\({}_{1}\)) for multiple attenuation. This method improves the precision and focusing ability of Radon transform, obtains a better result of multiple suppression, reduces the loss of primary reflections, and improves the interpretability of seismic data.