Detection of a moving point-like scatter by using moving receivers and emitter

We consider an inverse scattering problem for the scalar wave equation in which the point emitter, a point-like scatterer, and multiple receivers are all moving. The goal is to reconstruct the trajectory of a moving point-like scatterer from time-dependent measurements of the scattered field. To this end, we establish a rigorous point-interaction model for the moving scatterer in the time domain, proving well-posedness of the forward scattering problem, including existence, uniqueness and continuous dependence of the scattered field on the scatterer trajectory and scattering parameter. By exploiting the retarded-time structure, we introduce distance functions that connect emission, scattering, and observation processes, and show that they satisfy a coupled system of nonlinear ordinary differential equations determined by the measurement data. Based on this formulation, we propose a reconstruction algorithm that combines initial localization with the solution of the derived ODE system. Numerical experiments demonstrate that the method is accurate and robust with respect to noise, with reconstruction errors remaining stable under moderate perturbations.