An efficient method for undifferenced BDS-2/BDS-3 high-rate clock estimation

Due to the three-step implementation of the BeiDou Navigation Satellite System (BDS), users face the coexistence of BDS-2 and BDS-3 for a period of time. In particular, the real-time BDS-2/BDS-3 clock is the prerequisite for real-time precise point positioning (PPP) and precise orbit determination of low earth orbit satellites. Although the undifferenced clock estimation model can obtain all useful information in the parameter domain, many phase ambiguities make the clock estimation time-consuming. We develop a high-rate clock estimation algorithm without external complex matrix library, which can be easily implemented in cross-platforms. First, the recursive Kalman filter based on a single observation is used to avoid the complex matrix inversion. In addition, parallel computing is also employed to reduce the complexity of the update of the covariance matrix and the data preprocessing of multiple stations. Considering the continuous update of receiver firmware and that 60 globally distributed stations cover two different periods, the algorithm is validated in terms of computational efficiency, clock precision and accuracy of kinematic PPP. The results illustrated that the calculation time per epoch is approaching 0.5 s when the filter's matrix dimension reaches 1300. With the benefit of upgraded receivers, the average precision of BDS-3 C19–C37 is improved by 13.8%, while the precision of BDS-3 C38–C46 is 0.124 ns. The accuracy of kinematic PPP using all estimated BDS-2/BDS-3 satellites was 2.78 cm, 4.44 cm and 7.48 cm in north, east and up directions, respectively.