Multi-frequency GNSS incomplete observation repair method for combined positioning based on epoch-differenced observation constraints

Abstract Multi-frequency Global Navigation Satellite System (GNSS) positioning offers significant advantages in positioning accuracy, convergence speed, and carrier-phase ambiguity resolution. However, in real-world operating environments, partial signal outages across frequencies frequently occur, resulting in incomplete multi-frequency observations. Such observation deficiencies weaken the functional and stochastic models and consequently degrade positioning performance. To address this issue, this paper proposes a multi-frequency GNSS incomplete observation repair method based on epoch-differenced observation constraints. A single-satellite, epoch-differenced observation model is constructed and constrained by a rigorous variance-covariance formulation, enabling the recovery of missing measurements while preserving a consistent and reliable precision characterization. To ensure carrier-phase continuity, a robust cycle-slip detection scheme is integrated into the recovery process. The recovered observations are subsequently incorporated into extra-wide-lane combination models to enhance single-epoch positioning capability. Experimental validation using 24-h BDS-3 and Galileo datasets demonstrates that the proposed method achieves a precision of 1–2 mm at a 1 s sampling interval under both ionosphere-ignored and ionosphere-estimated strategies. When the sampling interval is increased to 30 s, neglecting ionospheric variation introduces residuals of approximately 3–5 mm, whereas explicit ionospheric estimation maintains millimeter-level accuracy. In a 197.2 km baseline experiment, excluding one to three frequencies per satellite degrades extra-wide-lane positioning accuracy from 0.137/0.144/0.369 m (N/E/U) to 0.172/0.191/0.606 m. After applying the proposed recovery method, the positioning performance is restored to a level highly consistent with that obtained using complete observations, with deviations limited to 0.007 %, 0.0002 %, and 0.049 % of the original solution in the north, east, and up components, respectively. These results confirm that the proposed method effectively preserves ambiguity fixability and positioning accuracy under incomplete-frequency conditions, providing practical and reliable support for high-precision GNSS applications in challenging environments.