Analysis and Forecasting of Clock Product Deviation for BDS-3 PPP-B2b Based on the Normalized Time-Frequency Transform.

Real-time BDS-3 PPP-B2b satellite clock products exhibit satellite-dependent clock product deviations (CPDs) relative to post-processed precise clocks, an important error source for real-time precise positioning and timing. Previous studies largely treat CPDs as static statistics or estimable parameters, leaving their nonstationary periodic structure and prediction potential unexplored. A hierarchical analytical framework based on the normalized time-frequency transform (NTFT) is therefore established, whose dynamic ridge model tracks the joint time-varying trajectories of amplitude, frequency, and phase without period priors. The modeling residual standard deviation is reduced from 5.273 ns to 0.429 ns, and three cross-satellite common components are identified. Building on this structure, a unified ridge envelope extrapolation (uREE) method converts the envelope trend and phase evolution into short-term forecasts through three satellite adaptive modes. On an independent single-day test, uREE reduces the overall RMS from 3.649 ns to 1.665 ns, corresponding to a 54.4% improvement, and outperforms all four benchmark models. Bias-dispersion decomposition reveals that this gain stems from the mitigation of systematic bias (whose contribution falls from above 83% to 18.5%) rather than from random noise suppression. These results shift CPD research from static bias estimation to physically interpretable time-varying modeling and forecasting.