GNSS satellite inter-frequency clock bias estimation and correction based on IGS clock datum: a unified model and result validation using BDS-2 and BDS-3 multi-frequency data

To deal with inconsistency between Global Navigation Satellite System (GNSS) multi-frequency data and International GNSS Service (IGS) clock product, we propose a unified model for GNSS satellite inter-frequency clock bias (IFCB) estimation and correction based on the IGS clock datum. The proposed model is rigorously derived by three sequential steps. First, by means of accurate modeling of the time-variant part of satellite phase-based IFCB (PIFCB) and introducing a zero-mean condition for satellite code-based IFCB (CIFCB), a set of independent satellite IFCBs are estimated in a full-rank multi-frequency uncombined model where IGS orbit and clock products are taken as input. The independent satellite IFCB refers to hardware delay difference between a specified frequency and two reference frequencies that are used for generating the IGS product. In the second step, a linear equation between satellite IFCBs from any three frequencies and the independent satellite IFCB estimates is derived. Based on this equation, the third step aims to establish a general satellite IFCB correction model which is used to align data on any two frequencies to the IGS clocks. The proposed model is implemented and validated using one month of multi-frequency data from BeiDou regional system (BDS-2) and BeiDou global system (BDS-3), i.e., B1I, B2I, B3I, B1C, B2a and B2b. By choosing B1I and B3I as reference frequencies, periodic analysis results suggest that a second-order periodic function is suitable for modeling BDS-2 independent satellite PIFCB variations. Yet there is no need to introduce periodic function into BDS-3 independent satellite PIFCB variations at all frequencies. For observation types of 1X (B1C), 5X (B2a) and 7Z (B2b), the standard deviation (STD) of three independent CIFCB estimates at each BDS-3 new frequency (i.e., B1C, B2a and B2b) and one CIFCB from the three new frequencies for all BDS-3 satellites are 3.14, 0.09, 0.12 and 0.09 ns, while those for the observation types of 1P (B1C), 5P (B2a) and 7D (B2b) are 2.52, 0.08, 0.08 and 0.04 ns, respectively. A major reason for the large noise of the independent satellite CIFCB at frequency B1C is that the frequency value of B1C is very close to that of B1I. Using the satellite IFCB estimates as corrections, the average RMS of kinematic precise point positioning (PPP) errors using ionosphere-free combination of BDS-3 multi-frequency data is 2.7 and 5.0 cm on horizontal and up directions, respectively, showing a same level with that of B1I/B3I-based PPP. In comparison, the accuracy of PPP is decreased by several millimeters during convergence when the differential code bias (DCB) product is used for correction.