Direct Imaging Methods for Reconstructing a Locally Rough Interface from Phaseless Total-Field Data or Phased Far-Field Data

This paper is concerned with the problem of inverse scattering of time -harmonic acoustic plane waves by a two -layered medium with a locally rough interface in two dimensions. A direct imaging method is proposed to reconstruct the locally rough interface from the phaseless total -field data measured on the upper half of the circle with a large radius at a fixed frequency or from the phased far -field data measured on the upper half of the unit circle at a fixed frequency. The presence of the locally rough interface poses challenges in the theoretical analysis of the imaging methods. To address these challenges, a technically involved asymptotic analysis is provided for the relevant oscillatory integrals involved in the imaging methods, based mainly on the techniques and results in our recent work [L. Li, J. Yang, B. Zhang, and H. Zhang, arXiv:2208.00456, 2022] on the uniform far -field asymptotics of the scattered field for acoustic scattering in a two -layered medium. Finally, extensive numerical experiments are conducted to demonstrate the feasibility and robustness of our imaging algorithms.