A multi-station shaft-rate magnetic field trajectory estimation method based on spectral-direction confidence weighting

Shaft-rate electromagnetic fields are generated by the coupling between ship electrochemical currents and the periodic motion of the propulsion shaft system. Owing to their stable line-spectrum characteristics and spatial directional information related to target bearing, they provide an important information source for non-acoustic passive ship localization and continuous trajectory estimation. However, existing shaft-rate electromagnetic localization and tracking methods generally rely on relatively accurate source and propagation models and often implicitly assume comparable reliability among observations obtained from different stations and frequencies. In practical marine environments, propagation attenuation, background electromagnetic interference, and differences in harmonic observability lead to pronounced quality heterogeneity among multi-station and multi-harmonic bearing observations. Consequently, directly incorporating low-quality observations into the fusion process may introduce localization bias and trajectory instability. To address these issues, a spectral-direction confidence-weighted multi-station and multi-harmonic trajectory estimation method for shaft-rate magnetic fields is proposed. Under the assumption that the shaft-rate fundamental frequency and major harmonics are identifiable, the horizontal magnetic-field components Hx and Hy recorded at multiple stations are used to construct directional projection spectra, from which bearing-line observations corresponding to different stations and harmonics are extracted. This approach reduces the dependence of trajectory estimation on complete source-parameter inversion. A spectral-peak prominence index is constructed to characterize the sharpness of the main peak, which is combined with multi-frequency bearing consistency to form station-level azimuth-angle confidence. The resulting confidence is then mapped to the weights of multi-station and multi-harmonic bearing lines, and a weighted least-squares trajectory estimation model is developed so that highly reliable observations provide stronger geometric constraints while degraded observations are adaptively suppressed. Simulation results show that, at a signal-to-noise ratio (SNR) of 5 dB, the proposed method achieves a mean absolute error (MAE), root mean square error (RMSE), and 95th-percentile error (P95) of 11.22, 13.75, and 26.45 m, respectively, substantially lower than the corresponding values of 23.48, 32.39, and 70.98 m obtained using the equal-weight multi-station least-squares method. Stable trajectory recovery is maintained over the SNR range from 0 to 20 dB, with the RMSE remaining below 10 m when the SNR reaches 10 dB or higher. The method is further validated using a towed simulated-source experiment conducted in the Yellow Sea with four seafloor electromagnetic observation stations. For two experimental track segments, the proposed method achieves MAEs of 15.09 and 14.94 m and RMSEs of 20.89 and 20.42 m, respectively, outperforming the comparative localization methods.These results demonstrate that the proposed spectral-direction confidence effectively characterizes reliability differences among multi-station and multi-harmonic bearing observations and improves the accuracy and stability of shaft-rate magnetic-field trajectory estimation through adaptive reliability-aware weighting. The proposed framework therefore provides an interpretable and engineering-oriented approach for passive ship localization and trajectory estimation under low-SNR conditions and heterogeneous observation quality.