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.

翻译：本文提出了一种频域方法，用于从三维空间中一个和稀疏观测点测量的多频近场数据恢复移动点源的轨迹。假设移动点源的辐射周期支撑在实轴上且先验已知。与逆静态源问题不同，需研究可观测与不可观测测量位置。这些概念在远场区域中的类似提法首次出现在作者前一篇论文中（即将发表于SIAM J. Imaging Sciences, 2023）。本文分析了直线运动和圆周运动下的可观测与不可观测测量位置。在近场情形下，我们验证了对于一类可容许的轨道函数，能够以可观测位置为中心、包含轨迹的最小环形区域可被成像。利用稀疏可观测位置的数据，可重构轨迹的Θ-凸域。通过合成数据进行了密集的三维数值测试，验证了新算法的有效性与可行性。