Stability for the inverse source problem in a two-layered medium separated by rough interface

In this paper, we investigate an inverse source problem for the two-dimensional Helmholtz equation in a two-layered medium. The interface between two media is assumed to be nonlocal and rough, while the compactly supported unknown source is buried in the lower-half medium. For the forward problem, we prove the radiating behaviour of the wave field based on the Angular Spectrum Representation and the asymptotics of Hankel functions. For the inverse problem, using multi-frequency interface measurements, which are limited-aperture, we show an increasing stability estimate which consists of two parts: one part is a Hölder stability estimate, the other part is a logarithmic stability estimate. The latter decreases as the upper bound of the frequency increases. In the derivation of the stability, we require the source function to have an $ H^3 $ regularity to control the high frequency tail of its Fourier transform. To recover the source numerically, we propose a recursive Kaczmarz-Landweber iteration scheme with incomplete data. Numerical examples are presented to justify the theoretical stability estimate and validity of the scheme.