Delay Correction Method Based on VLF Timing Signal Phase Variation Model

Positioning, navigation, and timing (PNT) services require stable time transfer, but satellite-based PNT signals are vulnerable to interference, attenuation, and limited availability in constrained environments. Very-low-frequency (VLF) signals propagate over long distances in the Earth-ionosphere waveguide and can serve as a terrestrial complement to satellite-based timing systems. Their timing performance, however, is affected by propagation-delay variation, especially the diurnal component associated with changes in the effective ionospheric reflection height. This study presents a propagation-delay correction method for VLF timing signals based on a phase-variation model. The total delay error is separated into primary path delay, secondary propagation delay, and residual random error. The primary delay is calculated from the transmitter-receiver path, while the periodic secondary delay is corrected using the predicted phase variation. Historical Alpha observations recorded at Chongqing and Guilin were used to evaluate the correction performance. The results show that the corrected standard deviation is reduced to 2.0054-2.2500 μs for the Chongqing paths and 2.7987-4.4792 μs for the Guilin paths. The corrected root mean square error (RMSE) ranges from 2.1316 μs to 4.5641 μs across the six Alpha propagation paths. These results indicate that the proposed method can suppress the main diurnal propagation-delay component in the selected historical Alpha datasets, although further validation with contemporary and multi-season VLF observations is still needed.