Estimation and universal formulation approach of BDS-3/GNSS inter-frequency bias

Global navigation satellite system (GNSS) multi-frequency integration can form various linear combinations, enriching precise point positioning (PPP) models, especially for BDS-3 six-frequency observations. However, the ways of establishing multi-frequency PPP models are various according to different user requirements, resulting in the increasingly complex formulation and processing of inter-frequency bias (IFB). In this study, the estimation and universal formulation approach of BDS-3/GNSS IFB is proposed, which can facilitate the IFB calibration and the provision of unified IFB products. The unified formulation of IFB with the universal multi-frequency uncombined PPP model without assigning reference frequencies, and with the universal multi-frequency ionospheric-free (IF) combined model without assigning specific observation combinations and a reference IF combination is first conducted. Most importantly, the theoretical intra-relationships of IFB estimates among various multi-frequency PPP models are derived and experimentally verified. In addition, the effects of four different IFB handling strategies, including ignoring IFB, and modeling IFB as white noise process, random-walk process, and constants, on position solutions from various models are analyzed. For IF combined models, as more signals are used to form a single IF combination, the convergence time of the strategy ignoring IFB gradually approaches that of three IFB estimation strategies due to decreasing numerical values of estimated IFB. Also, the IFB estimates are characterized. Both receiver type and firmware version can affect the numerical values of IFB estimates. The single-day stability of epoch-wise IFB estimates is 0.059-0.106 ns, while the monthly stability of daily IFB estimates ranges from 0.087 to 0.097 ns.