A Fast Fractional Cycle Bias Estimation Method Based on Matrix Dimensionality Reduction

Efficient fractional cycle bias (FCB) estimation is critical for real-time precise point positioning with ambiguity resolution (PPP-AR). However, conventional undifferenced models simultaneously estimate receiver and satellite parameters, causing the dimension of normal equations to expand rapidly in large-scale networks. To address this computational bottleneck, this paper proposes a fast FCB estimation algorithm based on matrix dimensionality reduction (MDR). By leveraging the theoretical equivalence between differenced and undifferenced estimators, the algorithm analytically eliminates receiver parameters from the observation equations. This reduces the unknowns from m  +  n to m (satellites), drastically decreasing computational complexity without compromising precision. Validation using 305 global IGS stations (GPS/Galileo/BDS) confirms that the MDR method is numerically equivalent to the classical solution within double-precision limits. Computational tests demonstrate an acceleration factor of 2.5-4.0× per epoch and a cumulative time reduction of over 90% for continuous processing. Independent PPP-AR experiments verify significantly improved positioning accuracy and stability compared to float solutions. Furthermore, a maritime buoy experiment in the South China Sea confirms the method's robustness under dynamic offshore conditions. The proposed MDR method significantly enhances efficiency while preserving rigor, making it ideal for large-scale real-time multi-GNSS applications.