In consideration of contributions from both carrier phase and pseudorange observations during computation, this study introduces a reconstructed method for estimating the multi-frequency global navigation satellite system (GNSS) inter-frequency clock biases (IFCB). Diverging from conventional approaches that separately calculate the time-varying and constant parts of IFCBs using carrier phase and pseudorange observations, the reconstructed method directly utilizes their combination to estimate satellite IFCBs. To validate the efficacy of the presented approach, 7 d observations from 87 International GNSS Service (IGS) stations are analyzed, with a specific focus on triple-frequency GPS, multi-frequency BDS-3, and Galileo IFCBs, aiming to scrutinize their distinctive characteristics. Furthermore, the performances of satellite IFCB estimation are investigated in both precise point positioning (PPP) and single-point positioning (SPP) using 35 IGS stations over 3 d. The results demonstrate that IFCBs of GPS BLOCK IIIA satellites exhibit centimeter-level variations, distinguishing them from BLOCK IIF counterparts. The Galileo IFCBs vary from millimeter to centimeter level, while those of BDS-3 reach a centimeter level. These variations significantly impact GPS PPP convergence performances but have minimal effects on Galileo and BDS-3 PPP. SPP performances are slightly enhanced when the time-varying IFCB part are taken into account. Additionally, we note disparities in the constant parts of satellite IFCBs computed with differential code bias (DCB) products, reconstructed, and pseudorange observation-based methods, particularly for BDS-3 C2I/C6I-C1P/C5P and C2I/C6I-C1P/C6I combinations. The differences of IFCB estimated with different strategies, SPP and PPP performances show that the reconstructed method is better than others, and the IFCB accuracy decreases when computed with satellite DCB products.

Based on the time-varying characteristics of GPS satellite differential code biases (DCB) between C1 and P1 observations, a method for directly estimating its model coefficients is proposed to streamline processing and enhance product accuracy. Evaluation over three days compares estimated DCB (C1–P1) values and assesses real-time service capabilities in single point positioning (SPP). Three-day DCB (C1–P1) estimates match single-epoch time-varying DCB (C1–P1) series fitting accuracy. SPP results show significant improvement over Orbit Determination in Europe (CODE), reducing average root mean square error in three dimensions by 12 cm and by 4 cm compared to Chinese Academy of Sciences. Real-time DCB (C1–P1) service maintains stable and accurate performance, comparable to fitted values of single-epoch time-varying series. Real-time SPP application shows the method enhances positioning accuracy compared to CODE by up to 35%, confirming the feasibility and effectiveness of the proposed model estimation method.

Aiming to address poor self-consistency of the satellite clock and orbit interpolations at the day boundary, which is caused by the discontinuities of their International GNSS Service (IGS) products for two consecutive days, an effective method is proposed to improve the precise point positioning (PPP) performances at the day boundary. According to different orders of Lagrange interpolator and different IGS Analysis Centers products of Center for Orbit Determination in Europe (CODE), GeoForschungsZentrum (GFZ) and Wuhan University (WUH), biases at the day boundaries are estimated and analyzed using a 4 day (DOY 94-97, 2022) data set of GPS, BDS-3 and Galileo from 123 IGS stations. These estimated biases show the time-varying characteristics. The differences in biases across 9th-11th orders of Lagrange interpolator are minimal, and these variances have a negligible impact on positioning. The results show that this poor self-consistency at the day boundary has an obvious influence on the kinematic PPP positioning, especially there is a centimeter-level variation at time of 24:00:00/00:00:00. This influence on the Up direction of kinematic PPP positioning is more serious than other directions. When the bias is estimated and corrected, the kinematic PPP positioning accuracies at the day boundary have a mean improvement of 0.043, 0.064 and 0.027 m for WUH, GFZ and CODE, respectively. The mean improvements for GPS, BDS-3 and Galileo are 0.021, 0.062 and 0.051 m. Additionally, the static PPP performances at the day boundary show the convergence times are shortened by 3.2, 6.2, and 2.5 min for WUH, GFZ and CODE, respectively, when the poor self-consistency of the satellite clock and orbit interpolations is estimated and corrected. Meanwhile, its 0.5 and 1 h positioning accuracies are improved.

The Global Navigation Satellite System (GNSS) carrier phase and pseudorange measurements exhibit different levels of accuracy, and there are differences in observation accuracy among different observation systems, satellites and receivers. A reasonable stochastic model is crucial for improving Precise Point Positioning (PPP) results. We propose a method for constructing stochastic models in PPP using triple-frequency observations. By forming geometry-free and ionosphere-free (GFIF) combinations and subtracting inter-frequency clock biases (IFCB), the accuracies of phase and pseudorange observations are evaluated using residuals to determine weights. Data from 278 International GNSS Service (IGS) stations on September 1, 2023 are processed to calculate the accuracies of carrier phase and pseudorange observations in GPS, Galileo, and BDS-3 systems. The results show that the accuracies of carrier phase observations among different receivers are relatively close, while there are obvious disparities in pseudorange observation accuracies. The average Root Mean Square (RMS) of Galileo carrier phase and pseudorange observations is the smallest among three systems. In BDS-3 system, there are certain disparities in carrier phase observation accuracies among different types of satellites. Specifically, for LEICA and SEPT receivers, the average RMS of IGSO satellites is greater than that of MEO satellites. To validate the effectiveness of the proposed model, the static and dynamic PPP solutions are conducted using 80 IGS stations. The results demonstrate that compared to the empirical model, the proposed model shortens the average convergence time by 22.5%, 31.5%, and 24.1% in static PPP for GPS, Galileo, and BDS-3 systems, respectively. At 0.5h, the average three dimensions (3D) positioning accuracies improve by 2.1, 3.5 and 2.9 cm, respectively. For dynamic PPP within the range of 0.5 to 1 hour, the average RMS of the 3D positioning are reduced by 2.6, 4.7 and 3.0 cm, respectively. Furthermore, for multi-system PPP of GPS+Galileo+BDS-3, the average convergence time and positioning accuracy are also improved with the proposed stochastic model.