Three-dimensional tracking using one-dimensional bearings from multiple horizontal linear arrays under isogradient sound speed profile

Three-dimensional tracking that relies solely on one-dimensional bearing measurements from multiple horizontal linear arrays is particularly challenging in underwater environments. Depth-dependent variations in sound speed cause acoustic rays to bend, leading to deviations in the incident bearing angles. Moreover, the relatively low sound speed in water makes the propagation delay non-negligible, a bearing angle measured at the current time instant typically corresponds to an earlier target state. To address these issues, we formulate a bearing measurement model under an isogradient sound speed profile that jointly accounts for ray bending and propagation delay. Bearing angles measured by multiple arrays at a common time correspond to target states at different emission times, which are coupled with unknown propagation delays through an implicit constraint. To handle this coupling, we propose a centralized particle filter with an iterative procedure to jointly estimate the propagation delay and the target state at the emission time for each array. The resulting emission-time state estimates are then marginalized to obtain the posterior estimate of the target state at the common measurement time. Finally, we derive the posterior Cramér–Rao lower bound to provide a performance benchmark. Simulation studies and sea-trial data demonstrate the effectiveness of the proposed approach.