Real-time Detection and Repair Method for Cycle Slips in Optimal Linear Combination of BDS-3 Five-Frequency Signals

Abstract This paper proposes an evaluation method for selecting the optimal linear combination coefficients for five-frequency cycle slip detection in BeiDou-3. First, the essence of the optimal linear combination problem for cycle slip detection is analyzed, and two linear combination constraint strategies are introduced: geometry-free ionosphere-free (GFIF) and geometry-free (GF) constraints. For the GFIF combination, a linear combination index (LCI) is proposed to assess the cycle slip detection performance of different linear combinations, providing a quantitative measure for evaluating the performance of various combination coefficients. By solving the constrained optimization problem, the optimal linear combinations for phase‑only geometry‑free ionospheric‑free (PGFIF) and phase-code geometry-free ionospheric-free (PCGFIF) are determined. For the GF combination, based on phase biases induced by cycle slips, standard deviations of cycle slip detection measurements, ionospheric delay amplification factors, and the success rate of detecting insensitive cycle slips using GFIF, suitable phase-only geometry-free combination (PGF) combinations are selected. The optimal linear combination for PGF is then determined based on the principle of minimizing the condition number. For cycle slip repair, the least squares method is used to solve for the float solution and its covariance matrix, while the LAMBDA algorithm is employed to search for the integer solution of the cycle slip. Finally, the effectiveness of the proposed method is validated through experiments using static, shipborne kinematic, ionospheric active period, and low-sampling-rate data. The experimental results demonstrate that, under five-frequency conditions, the proposed algorithm can accurately detect and repair all simulated cycle slips, confirming the feasibility and effectiveness of the method.