A Sparsity-Promoting Electromagnetic Inversion Method Regularized by $\ell_1 / \ell_2$-Norm of the Model Gradient

Abstract

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.