  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 2D. 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] 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.
