A Multi-Frequency Iterative Method for Reconstruction of Rough Surfaces Separating Two Penetrable Media

TL;DR

The paper develops a Tikhonov-regularized Newton framework to reconstruct rough interfaces between two dielectric media from scattered fields, leveraging multi-frequency data to mitigate ill-posedness and nonlinearity. By solving a sequence of regularized linearized problems using the Frechet derivative of the integral-operator forward model, the surface profile $s(x)$ is iteratively refined across frequencies $f_{(m)}$, with each frequency initialized by the previous reconstruction. Numerical experiments show that multi-frequency reconstructions yield higher accuracy, better resolution of sharp surface features, and robustness to noise and measurement sparsity compared with single-frequency approaches. The method has practical implications for nondestructive testing and remote sensing where accurate surface profiling under noisy measurements is essential, and it can be extended to layered media and 3D imaging in future work.

Abstract

A numerical scheme that uses multi-frequency Newton iterations to reconstruct a rough surface profile between two dielectric media is proposed. At each frequency sample, the scheme employs Newton iterations to solve the nonlinear inverse scattering problem. At every iteration, the Newton step is computed by solving a linear system that involves the Frechet derivative of the integral operator, which represents the scattered fields, and the difference between these fields and the measurements. This linear system is regularized using the Tikhonov method. The multi-frequency data is accounted for in a recursive manner. More specifically, the profile reconstructed at a given frequency is used as an initial guess for the iterations at the next frequency. The effectiveness of the proposed method is validated through numerical examples, which demonstrate its ability to accurately reconstruct surface profiles even in the presence of measurement noise. The results also show the superiority of the multi-frequency approach over single-frequency reconstructions, particularly in terms of handling surfaces with sharp variations.

Figures (11)

• Figure 1: 2D scattering problem involving a rough surface separating two dielectric media.
• Figure 2: Actual surface profile and the reconstructions at $350\,\mathrm{MHz}$, $800\,\mathrm{MHz}$, and $900\,\mathrm{MHz}$.
• Figure 3: Error in reconstruction computed using \ref{['err_def']} versus frequency.
• Figure 4: Actual surface profile and the reconstructions obtained by the multi- and single-frequency simulations at $600\,\mathrm{MHz}$.
• Figure 5: Error in reconstruction computed using \ref{['err_def']} versus frequency for the simulations with the frequency increment $10\,\mathrm{MHz}$, $20\,\mathrm{MHz}$, $50\,\mathrm{MHz}$, $150\,\mathrm{MHz}$, $300\,\mathrm{MHz}$.