Formulas for three-dimensional source localization using multipath time delays measured by asynchronous distributed sensors in deep water

In this study, three-dimensional source localization is investigated based on time delays between the direct (D) and surface-reflected (SR) paths measured using asynchronous distributed sensors in the deep-water reliable-acoustic-path environment. Instead of matching replicas, we develop computationally efficient formulas to locate the source. The formulas are derived by first pseudolinearizing the measurement model of the D–SR time delay and then solving the obtained pseudolinear system of equations using the instrumental variable technique. Furthermore, the Cramér–Rao lower bound of the investigated problem is derived theoretically. Monte Carlo simulations are performed to demonstrate the performance of the proposed formulas. The results show that the formulas can provide a localization accuracy comparable to that of the matching algorithm when the measurement noise level is low, while exhibiting an acceptable inferiority at high noise levels. Finally, the feasibility of the proposed formulas is verified using experimental data acquired in the western Pacific. In all experimental test groups, the localization biases of the proposed formulas do not exceed 250 m in the horizontal plane and are less than 5% for depth estimation.