Analytical solvability and numerical calculation of acoustic ray tracing inverse problems

Rapidly solving the acoustic ray inverse problem, which involves determining the acoustic ray tracking and travel time between two known points underwater, is fundamental for efficient acoustic ray tracking. Therefore, exploring the basic equations and solution algorithms of the this inverse problem has significant theoretical value and practical significance. Based on Fermat’s principle, this paper systematically investigates the solvability of the acoustic ray tracing inverse problem under assumptions of inter-layer constant sound speed and constant gradient between layers, derives its calculation formulas, and establishes an algorithmic framework applicable to different scenarios. Theoretical derivation demonstrates that a two-layer sound speed structure yields an analytical solution, whereas structures with three or more layers lack analytical solutions due to the corresponding polynomial equations exceeding fifth order. For application scenarios with a two-layer structure, the computational efficiency of the analytical method is compared with that of numerical methods (bisection, secant, and Newton’s method). Experimental results indicate that the analytical method achieves significantly higher computational efficiency, with improvements of approximately 97.39%, 88.94%, and 85.87% compared to the bisection, secant, and Newton’s methods, respectively.