一种由模型梯度 $\ell_1 / \ell_2$ 范数正则化的促进稀疏性的电磁反演方法

中文摘要 AI 辅助

提出了一种由模型梯度的 $\ell_1 / \ell_2$ 范数正则化的非线性电磁（EM）反演方法。$\ell_1 / \ell_2$ 范数定义为 $\ell_1$ 范数与 $\ell_2$ 范数之比，具有尺度不变性，与传统 $\ell_1$ 范数相比，能更准确地刻画解中的稀疏性。针对 $\ell_1 / \ell_2$ 范数的商结构引入的非凸性和非光滑性，将反演问题重新表述，并使用交替方向乘子法（ADMM）求解，得到一个包含五个子步骤的高效优化算法。在第一个子步骤中引入 Gauss-Newton 方法对非线性反演进行线性化。通过一系列使用合成和实验数据集的数值算例验证了所提方法，并与 Tikhonov 和全变分（total variation）等标准正则化方法进行了比较。结果表明，所提方法在重建介电常数分布方面性能更优，尤其是在保持尖锐边缘和抑制伪影方面，且未增加额外计算成本。此外，该方法对测量噪声表现出很强的鲁棒性，并且对正则化权重选择的敏感性降低。

英文摘要

A nonlinear EM inversion method regularized by $\ell_1 / \ell_2$-norm of the model gradient is proposed. The $\ell_1 / \ell_2$-norm, defined as the ratio of $\ell_1$-norm to $\ell_2$-norm, exhibits a scale-invariant property that enables it to more accurately characterize sparsity in the solution compared with the conventional $\ell_1$-norm. To address the nonconvexity and nonsmoothness introduced by the quotient structure of $\ell_1 / \ell_2$-norm, the inversion problem is reformulated and solved using the alternating direction method of multipliers, resulting in an efficient optimization algorithm comprising five sub-steps. The Gauss--Newton method is incorporated into the first sub-step to linearize the nonlinear inversion. The proposed method is validated by a series of numerical examples using both synthetic and experimental datasets, and is compared against standard regularization methods such as Tikhonov and total variation. The results demonstrate that the proposed method yields superior reconstructions of permittivity profiles, particularly in preserving sharp edges and suppressing artifacts, without incurring additional computational cost. Furthermore, the method exhibits strong robustness to measurement noise and reduced sensitivity to the choice of the regularization weight.

发表机构

• King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）

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