Imaging a moving point source in R^3 from the time of arrival at sparse observation points

TL;DR

This paper presents a new numerical method for reconstructing the 3D trajectory and emission time of a moving point source using only time-of-arrival data from a small number of observation points, with proven uniqueness and stability.

Contribution

The paper introduces a novel approach that reconstructs source trajectories in 3D space from minimal measurements, with rigorous proofs of uniqueness and stability.

Findings

Method accurately reconstructs source trajectories in simulations.

Uniqueness of the solution is established mathematically.

Numerical experiments confirm the method's effectiveness.

Abstract

In this paper, we introduce a novel numerical method for reconstructing the trajectory within three-dimensional space, where both the emission moment and spatial location of the point source are unknown. Our approach relies solely on measuring the time of arrival at five or seven properly chosen observation points. By utilizing the distinctive geometric configuration of these five or seven observation points, we establish the uniqueness of the trajectory and emission moment of the point source through rigorous mathematical proofs. Moreover, we analyze the stability of our proposed method. The effectiveness of the method is also verified by numerical experiments.