P-Order Secant Method for Rapidly Solving the Ray Inverse Problem of Underwater Acoustic Positioning

Abstract The computational efficiency of underwater acoustic positioning based on the ray tracing is mainly limited to a great amount of calculation of ray inverse problem. We propose two kinds of p-order secant methods to improve the efficiency of traditional method, and the proposed methods can be regarded as a generalization of the traditional secant method from two points to p points for rapidly solving the inverse problem. In the proposed methods, the calculation information in previous iterations is utilized to fit a polynomial model to speed up the algorithm convergence. In the first-kind method, the inverse problem is calculated by solving a polynomial equation approximating the function mapping from the emission angle to the radial distance of the ray. In the second-kind method, the inverse problem is however directly solved by approximating the function mapping from the radial distance to the emission angle. As the first-kind method needs to solve a p-order polynomial equation, the practicability of this method is limited to the complexity of solving the high-order equation, while the second-kind method can directly approximate the solution of the inverse problem, which is more practical and flexible. The proposed methods have been verified in deep-sea trial. It shows that, the proposed methods can precisely produce the solution of the acoustic ray inverse problem within one iteration, and the computational efficiency of proposed method is about 6 times faster than that of the traditional method.