Characteristics and modelling of BDS satellite inter-frequency clock bias for triple-frequency PPP

Aiming at the problem of the inter-frequency clock bias (IFCB) in Beidou Navigation Satellite System (BDS) triple-frequency observations, the variation characteristics with time are analysed in detail. The IFCB models for all the three kinds of satellites, i.e. GEO, IGSO and MEO, are proposed. Then the attenuation and long-term forecasting performance of the models are evaluated. Finally, the validity and benefit of the model are verified by triple-frequency Precise Point Positioning (PPP) experiments. The IFCB results from consecutive two-month BDS triple-frequency observation data of 44 globally distributed Multi-GNSS Experiment (MGEX) stations show that the IFCBs of GEO satellites have prominent periodic variation in general. The correlation coefficient and the determination coefficient of predicted IFCB by the model for GEO satellites are up to 0.957 and 0.915 respectively, which shows the model has a high stability and is suitable for long-term prediction. The IFCBs of IGSO satellites have the same periodicity as those of GEO satellites. Although the model is not as good as that of GEO satellites, it still performs well overall and can be applied to long-term prediction in most instances. The model of MEO satellites performs worse than GEO and IGSO due to the numerical instability and less obvious periodicity of IFCBs. Moreover, the effect of the IFCB models is better than the substitution method of using the IFCB at the corresponding moment in the previous period as the IFCB forecast value of the current period. In the triple-frequency Precise Point Positioning (PPP) experiments, the modelled IFCB correction can improve the positioning accuracy by 21.1%, 9.0% and 9.9% in the north, east and up directions and also shorten the convergence time in these three directions by 26.8%, 10.4% and 24.4% respectively compared with the observation model without IFCB correction.