Due to the finite acquisition aperture and sampling of seismic data, the Radon transform (RT) suffers from a smearing problem which reduces the resolution of the estimated model. In addition, inverting the RT is typically an ill-posed problem. To address these challenges, a sparse RT mixing the ${\ell }_1$ and ${\ell }_2$ norms of the RT coefficients in the mixed frequency-time domain is developed, and it is denoted as SRTL_1-2. In most conventional sparse RTs, the sparse constraint term often is the ${\ell }_1$ norm of the Radon model. We prove that the sparsity effect of the ${\ell }_{1\text{-}2}$ minimization is better than that of the ${\ell }_1$ norm alone by comparing and analyzing their 2D distribution patterns and threshold functions. The difference of the convex functions algorithm and the alternating direction method of multipliers algorithm are modified by combining the forward and inverse Fourier transforms to solve the corresponding sparse inverse problem in the mixed frequency-time domain. Our method is compared with three RT methods, including a least-squares RT (LSRT), a frequency-domain sparse RT (FSRT), and a time-invariant RT in the mixed frequency-time domain based on an iterative 2D model shrinkage method (SRTIS). Furthermore, we modify the basis function in SRTL_1-2 by including an orthogonal polynomial transform to fit the amplitude-variation-with-offset (AVO) signatures found in seismic data, and we denote this as high-order SRTL_1-2. Compared to the SRTL_1-2, the high-order SRTL_1-2 performs better when processing seismic data with AVO signatures. Synthetic and real data examples indicate that our method has better performance than the LSRT, FSRT, and SRTIS in terms of attenuation of multiples, noise mitigation, and computational efficiency.

中文翻译：

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具有 [math] 最小化的混合频时域中的稀疏 Radon 变换

由于有限的采集孔径和地震数据的采样，氡变换（RT）存在拖尾问题，从而降低了估计模型的分辨率。此外，反演 RT 通常是一个不适定问题。为了应对这些挑战，一种稀疏 RT 混合${\ell }_1$和${\ell }_2$开发了混合频时域中的 RT 系数范数，并表示为 SRTL _1-2。在大多数传统的稀疏 RT 中，稀疏约束项通常是${\ell }_1$氡模型的范数。我们证明了稀疏效应${\ell }_{1\text{-}2}$最小化优于${\ell }_1$通过比较和分析它们的 2D 分布模式和阈值函数来单独进行范数。将凸函数算法与乘法器的交替方向法的区别进行了改进，结合正向和反向傅里叶变换，解决了相应的混合频时域稀疏逆问题。我们的方法与三种 RT 方法进行了比较，包括最小二乘 RT (LSRT)、频域稀疏 RT (FSRT) 和基于迭代 2D 模型收缩方法的混合频时域中的时不变 RT （SRTIS）。此外，我们通过包含正交多项式变换来修改 SRTL _1-2中的基函数，以拟合在地震数据中发现的振幅随偏移量变化 (AVO) 特征，我们将其表示为高阶 SRTL_1-2。与 SRTL _1-2相比，高阶 SRTL _1-2在处理具有 AVO 特征的地震数据时表现更好。合成和实际数据示例表明，我们的方法在倍数衰减、噪声缓解和计算效率方面比 LSRT、FSRT 和 SRTIS 具有更好的性能。