Imaging a Moving Point Source from Multifrequency Data Measured at One and Sparse Observation Directions (Part I): Far-Field Case

We propose a multifrequency algorithm for recovering partial information on the trajectory of a moving point source from one and sparse far-field observation directions in the frequency domain. The starting and terminal time points of the moving source are both supposed to be known. We introduce the concept of observable directions (angles) in the far-field region and derive all observable directions (angles) for straight and circular motions. The existence of nonobservable directions makes this paper much different from inverse stationary source problems. At an observable direction, it is verified that the smallest strip containing the trajectory and perpendicular to the direction can be imaged, provided the angle between the observation direction and the velocity vector of the moving source lies in . If otherwise, one can only expect to recover a strip thinner than this smallest strip for straight and circular motions. The far-field data measured at sparse observable directions can be used to recover the -convex domain of the trajectory. Both two- and three-dimensional numerical examples are implemented to show effectiveness and feasibility of the approach.