In this paper, we introduce a frequency-domain approach to extract information on the trajectory of a moving point source. The method hinges on the analysis of multi-frequency near-field data recorded 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 classify observable and non-observable measurement positions. The analogue of these concepts in the far-field regime were firstly proposed in the authors' previous paper (SIAM J. Imag. Sci., 16 (2023): 1535-1571). In this paper we shall derive the observable and non-observable measurement positions for straight and circular motions in $\R^3$. In the near-field case, we verify that the smallest annular region centered at an observable position that contains the trajectory 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，用于提取运动点源轨迹信息。该方法基于对三维空间中单点及稀疏观测点记录的多频近场数据的分析。运动点源的辐射周期假定为支撑于实轴上的已知先验信息。与逆静态源问题不同，需要区分可观测与不可观测的测量位置。远场情形中这些概念的类比最早由作者前文（SIAM J. Imag. Sci., 16 (2023): 1535-1571）提出。本文将在$\R^3$中推导直线运动与圆周运动的可观测与不可观测测量位置。对于近场情形，我们验证了：对于容许类轨道函数，可对以可观测位置为中心包含轨迹的最小环形区域进行成像。利用稀疏可观测位置的数据，可重建轨迹的$\Theta$-凸区域。通过大量合成数据的三维数值实验，验证了该新算法的有效性与可行性。