A high-precision analytical algorithm based on the second-order time difference of arrival for the period-drifted acoustic sources

The existing single-platform underwater acoustic localization methods based on the second-order time difference of arrival (STDOA) rely on numerical solution techniques and are prone to convergence difficulties, degraded positioning accuracy, and insufficient computational efficiency when prior information of the underwater target is unavailable. An analytical algorithm based on the STDOA is proposed in this paper. Building upon the classical Fang algorithm, it algebraically transforms the STDOA measurement equations into a linear system that matches the single-platform observation geometry and subsequently derives a closed-form expression for the localization solution. All steps of the algorithm involve explicit analytic operations without iterative loops, thus eliminating the dependence on initial guesses required by conventional numerical iterations and theoretically avoiding the convergence failures and accuracy losses caused by poor initialization. The results of real lake trial demonstrate that the proposed algorithm significantly outperforms the conventional numerical method in localization accuracy in the actual underwater environments with lacking prior position knowledge of the non-cooperative acoustic beacon, while its computational speed is sufficient for real-time processing requirements.