Approximate formulas for acoustic ray-travel time and applicable conditions

To address the balance between accuracy and computational efficiency in travel time calculation for constant-gradient acoustic ray tracing in underwater positioning, this study systematically analyzes the performance of exact solutions and their approximations. Based on the harmonic mean sound speed model, exact solutions alongside their 1st to 3rd order Taylor series expansions and Padé approximations are derived as closed-form alternatives. These solutions are compared through two simulation experiments featuring distinct acoustic ray-bending scenarios with environmental errors, evaluating positioning accuracy, time calculation errors, and computational efficiency. Implementation guidelines recommend the exact solution for high-precision positioning or scenarios with grazing angles below 25°, the 2nd order approximation for efficient processing of datasets containing multiple small grazing angles (<35°), and the 3rd order/Padé approximations for datasets dominated by large grazing angles.