Impact of the third frequency GNSS pseudorange and carrier phase observations on rapid PPP convergences

New GNSS signals have significantly augmented positioning service and promoted algorithmic innovations such as rapid PPP convergence. With the emerging of multifrequency signals, it becomes essential to thoroughly explore the contribution of third frequency pseudorange and carrier phase toward PPP. In this study, we research the role of the third frequency observations on accelerating PPP convergence, commencing from both stochastic and functional models. We first constructed the stochastic model depending on the observation noise and then introduced two uncombined functional models with respect to different inter-frequency bias (IFB) estimation strategies. The double-differenced residuals based on a zero baseline were used to evaluate the signal noises, which were 0.09, 0.07, 0.11, 0.01 and 0.09 m for Galileo E1/E5a/E5b/E5/E6 pseudorange and 0.24, 0.31 and 0.05 m for BeiDou B1/B2/B3. Besides, carrier phase observations E5a/E5/E6/B1I/B3I shared a comparable signal noise of 0.002 m, while the signal noises of E1/E5b/B2I were 0.003 m. Both BeiDou-2/Galileo and Galileo-only float PPP were implemented based on the dataset collected from 25 stations, spanning 30 days. Triple-frequency Galileo PPP achieved convergence successfully in 19.9 min if observations were weighted according to observation precision, showing a comparable performance of dual-frequency PPP. Meanwhile, the convergence time of triple-frequency float PPP was further shortened to 19.2 min when satellite pair IFBs were eliminated by estimating a second satellite clock. While the improvement of triple-frequency float PPP was marginal, triple-frequency PPP-AR using signals E1/E5a/E6 shortened the initialization time of the dual-frequency counterpart by 38%. Moreover, the performance of triple-frequency PPP-AR kept almost unchanged after we excluded the third frequency pseudorange observations. We thus suggest that the contribution of the third frequency to PPP mainly rests on ambiguity resolution, favored by the additional carrier phase observations.