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Underwater acoustic localization usually relies on time-of-arrival (ToA) measurements, which are then converted into range estimates. However, the water medium is inhomogeneous and the sound speed varies depending on several parameters, e.g., the temperature, pressure and salinity. As a result, sound waves do not necessarily travel in straight lines. Ignoring this stratification effect could lead to considerable bias in the range estimates. We propose a depth-based approach to compensate the stratification effect for improved underwater ranging. We assume that the sound velocity profile (SVP) is only vertically stratified, the position of the sender is known, and the receiver has a noisy depth estimate via a depth sensor. We find a numerically simple range estimator, based on reconstructing the slanted path using Fermat's Principle and calculus of variations. This estimator removes the bias and is asymptotically efficient. We compare our solution to the simplistic linear estimator that assumes straight-line propagation in a shallow-water example where the sound speed decreases monotonically with depth. We find that the bias of the linear estimator increases with range and is non-negligible when the ToA measurements have a small variance, while our solution is bias-free and meets the Cramer-Rao lower bound (CRLB).
Berger et al. (Thu,) studied this question.