论文标题
通过稀疏的正则化模型和地震波场建模中的应用来反抗不完整的傅立叶变换
Inverting Incomplete Fourier Transforms by a Sparse Regularization Model and Applications in Seismic Wavefield Modeling
论文作者
论文摘要
我们提出了一个稀疏的正则化模型，用于反转不完整的傅立叶变换，并将其应用于地震波场建模。拟议模型的目标函数采用了紧密的帧系统下的$ \ ell_0 $ norm的莫罗包络，以促进稀疏性。该模型导致了一个非平滑，非凸优化的问题，传统迭代方案效率低下甚至发散。通过利用$ \ ell_0 $规范的特殊结构，我们通过凸优化问题的全球最小化器确定了所提出的非凸优化问题的本地最小化器，这为我们提供了有效和融合的洞察力，可以保证算法可以保证解决它。我们根据$ \ ell_0 $ narm的接近运算符定义的地图的固定点来表征正则化模型的解决方案，并开发了解决该算法的定点迭代算法来解决它。通过将映射与$α$平均的非专用运算符连接，我们证明了由建议的固定点接近算法生成的序列会收敛到所提出模型的局部最小化器。我们的数值示例证实，所提出的模型基于$ \ ell_1 $ -norm的现有模型的表现大大优于现有模型。频域中的地震波场建模需要求解一系列具有较大波数的Helmholtz方程，这是一项计算密集的任务。将提出的稀疏正则化模型应用于地震波场建模只需要几个低频的数据，避免求解具有较大波数的Helmholtz方程。数值结果表明，根据$ \ ell_1 $ norm的现有方法，就恢复的合成地震图的SNR值和视觉质量而言，其执行效果更好。
We propose a sparse regularization model for inversion of incomplete Fourier transforms and apply it to seismic wavefield modeling. The objective function of the proposed model employs the Moreau envelope of the $\ell_0$ norm under a tight framelet system as a regularization to promote sparsity. This model leads to a non-smooth, non-convex optimization problem for which traditional iteration schemes are inefficient or even divergent. By exploiting special structures of the $\ell_0$ norm, we identify a local minimizer of the proposed non-convex optimization problem with a global minimizer of a convex optimization problem, which provides us insights for the development of efficient and convergence guaranteed algorithms to solve it. We characterize the solution of the regularization model in terms of a fixed-point of a map defined by the proximity operator of the $\ell_0$ norm and develop a fixed-point iteration algorithm to solve it. By connecting the map with an $α$-averaged nonexpansive operator, we prove that the sequence generated by the proposed fixed-point proximity algorithm converges to a local minimizer of the proposed model. Our numerical examples confirm that the proposed model outperforms significantly the existing model based on the $\ell_1$-norm. The seismic wavefield modeling in the frequency domain requires solving a series of the Helmholtz equation with large wave numbers, which is a computationally intensive task. Applying the proposed sparse regularization model to the seismic wavefield modeling requires data of only a few low frequencies, avoiding solving the Helmholtz equation with large wave numbers. Numerical results show that the proposed method performs better than the existing method based on the $\ell_1$ norm in terms of the SNR values and visual quality of the restored synthetic seismograms.
