Acoustic Doppler localisation and tracking in 3D space with retardation correction

This study extends Doppler-based acoustic localisation from 2D to a complete 3D framework for UAV tracking. A key methodological contribution is the explicit correction for the retardation effect (signal propagation delay), solved numerically using the Newton-Raphson method. This correction proved essential, as the average localisation error was reduced from approximately 50 m (without correction) to about 15 m (with correction), confirming the practical necessity of including retardation in 3D acoustic models. The VarPro method was implemented to exploit the separable structure of the least-squares problem. Under stabilised conditions (with velocity fixed to resolve the V/f identifiability issue), the method demonstrated convergence and provided reasonable trajectory estimates with mean trajectory errors below 5%. At the same time, the diagnostic analysis revealed fundamental limitations of the VarPro method. First, the strong correlation between source velocity (V) and source frequency (f) makes unconstrained optimisation unstable and prone to divergence. Second, even in stabilised runs, the simplified analytical trajectory models (straight-line or their 7-parameter extension) are structurally inadequate for representing stochastic UAV motion, thereby forcing the optimiser toward non-physical solutions. These findings suggest that while VarPro can be applied successfully to simplified scenarios with constrained parameters, it is not a suitable general solution for localizing UAVs with complex, random trajectories. Future research will therefore focus on recursive state estimation methods such as the extended Kalman filter, which are better suited for dynamic stochastic motion and time-varying source frequencies.