Imaging a moving point source from multi-frequency data measured at one and sparse observation points (part II): near-field case in 3D

In this paper we propose a frequency-domain method for recovering the trajectory of a moving point source from multi-frequency near-field data measured at one and sparse observation points in three dimensions. The radiating period of the moving point source is supposed to be supported on the real axis and a priori known. In contrast to inverse stationary source problems, one needs to study observable and non-observable measurement positions. The analogue of these concepts in the far-field regime were firstly proposed in the authors' previous paper (to appear in SIAM J. Imaging Sciences, 2023). In this paper the observable and non-observable measurement positions for straight and circular motions are analyzed. In the near-field case, we verify that the smallest annular region that contains the trajectory and centered at an observable position can be imaged for an admissible class of orbit functions. Using the data from sparse observable positions, it is possible to reconstruct the $\Theta$-convex domain of the trajectory. Intensive 3D numerical tests with synthetic data are performed to show effectiveness and feasibility of this new algorithm.