Inter-system Bias Research Articles

LEO enhanced GNSS (LeGNSS) precise point positioning with emphasis on model comparison

With the rapid construction of the low-earth-orbit (LEO) constellation, LEO Enhanced Global Navigation Satellite System (LeGNSS) has become a hotspot in the research of GNSS applications and services, especially for fast convergence of precise point positioning (PPP). In this study, we analyze the impact of LEO satellites on the spatial geometry based on simulated LEO and real GNSS data, and evaluate the LeGNSS PPP performance with ionosphere-free (IF), undifferenced and uncombined (UDUC), and ionosphere-weighted (IW) models. Eight Multi-GNSS Experiment (MGEX) stations are selected for static and kinematic experiments. Results show that Position Dilution of Precision (PDOP) values of LeGNSS are about 0.5–1.2, which are improved by 38 % and 22 % compared with GPS-only and combined GPS, Galileo, and BDS (G\\E\\C) solutions. With LEO satellites, the positioning accuracy in east (E), north (N), and up (U) directions are improved by about 50 %, 30 %, and 50 % for the static mode, and about 35 %, 30 %, and 67 % for the kinematic mode. Moreover, the positioning accuracies of the IF, UDUC, and IW models are comparable. For convergence time, the static and kinematic results with LEO satellites are about 1.1 and 1.3 min, respectively, while those without LEO satellites are longer than 10.0 min. In addition, the model with inter-system bias (ISB) constraints can reduce the convergence time by within 1.0 min.

Influence of Inter-System Biases on Combined Single-Frequency BDS-2 and BDS-3 Pseudorange Positioning of Different Types of Receivers

The BeiDou Navigation Satellite System (BDS) has developed rapidly, and the combination of BDS Phase II (BDS-2) and BDS Phase III (BDS-3) has attracted wide attention. It is found that there are code ISBs between BDS-2 and BDS-3, which may have a certain impact on the BDS-2 and BDS-3 combined positioning. This paper focuses on the performance of BDS-2/BDS-3 combined B1I single-frequency pseudorange positioning and investigates the positioning performance with and without code ISBs correction for different types of receivers, include geodetic GNSS receivers and low-cost receivers. The results show the following: (1) For geodetic GNSS receivers, the code ISBs of each receiver is about −0.3 m to −0.8 m, and the position deviation is reduced by 7% after correcting code ISBs. The code ISBs in the baseline with homogeneous receivers has a little influence on the positioning result, which can be ignored. The code ISBs in the baseline with heterogeneous receivers is about −0.5 m, and the position deviation is reduced by 4% after correcting code ISBs. (2) The code ISBs in the low-cost receivers are significantly larger than those in the geodetic GNSS receivers, and the impact on the positioning performance of the low-cost receivers is significantly greater than that on the geodetic GNSS receivers. After correcting the code ISBs, the position deviation of low-cost receivers can be reduced by around 12% for both undifferenced and differenced modes. (3) For low-cost receivers, correcting the code ISBs can increase the number of epochs successfully solved, which effectively improves the low-cost navigation and positioning performance. (4) The carrier-phase-smoothing method can effectively reduce the distribution dispersion of code ISBs and make the estimation of ISBs more accurate. The STD values of estimated code ISBs in geodetic GNSS receivers are reduced by about 40% after carrier-phase smoothing, while the corresponding values are reduced by about 7% in low-cost receivers due to their poor carrier-phase observation quality.

Performance analysis of the tight combination of GPS, BDS, GALILEO, and QZSS mixed frequencies signals

With the gradual development and modernization of satellite navigation systems, using observation information from multi-GNSS has become one of the hot-spot issues in recent years. Multi-system loose combinations form double-difference observation equations within their respective systems, and the positioning effect is improved. However, the interchangeability and compatible interoperability between global navigation satellite systems (GNSS) cannot be truly realized. At the same time, when the number of visible satellites decreases abruptly, the positioning performance deteriorates sharply. This paper focuses on the GNSS multi-system tight combination relative positioning technique, gives a mathematical model of multi-system tight combination relative positioning considering differential inter-system bias (DISB), and analyzes the time-varying characteristics of DISB at overlapping and non-overlapping frequencies among GPS/Galileo, GPS/BDS, and GPS/QZSS in terms of receiver brand, temperature, and receiver restart. The GNSS tight combination relative positioning performance is verified by static data from Curtin University and dynamic data measured at Taiyuan University of Technology. The results show that compared with loose combination, the ambiguity-fixed rate increases from 62.18% to 97.60% for static data and from 74.97% to 99.53% for dynamic data when the elevation mask angle is 50°, resulting in a significant improvement in positioning performance.

Characteristics of Inter-System Bias between BDS-2 and BDS-3 and Its Impact on BDS Orbit and Clock Solutions

The inter-system-like bias between the regional (BDS-2) and global (BDS-3) constellation of the BeiDou Navigation Satellite System (BDS) has been identified on common B1I pseudo-range observations. In this study, its characteristics are investigated with tracking data from the International GNSS Service (IGS) and International GNSS Monitoring and Assessment System (iGMAS) network. Firstly, the satellite-specific inter-system-like bias is calculated and the dependency on satellite is observed. Clearly noticeable discrepancies on BDS-2 and BDS-3 can be identified. Hence, the constellation-specific inter-system-like bias is estimated. Biases for all receivers are quite stable, with standard derivation (STDev) less than 0.2 m in average. The bias shows clear dependence on the receiver, while the firmware and antenna have limited but not negligible impacts, particularly for Trimble NetR9 and Alloy receivers. The Trimble NetR9 with TRM59800.00 antenna shows noticeable discrepancy up to about 1.5 m with different antenna, and the bias of the Trimble Alloy 5.37 jumps about 2.4 m with respect to later firmware. In addition, clear annual variations are observed for stations ABPO and MIZU with Septentrio POLARX5 5.3.2 and ASTERX4 4.4.2 receivers, respectively. Furthermore, the impacts of the biases on the BDS orbit and clock solutions are analyzed. Once BDS-2 and BDS-3 are treated as two independent systems, the root mean square (RMS) of code and carrier phase residuals can be reduced by around 9.3 cm and 0.23 mm, respectively, while the three-dimensional orbit consistency is improved by 6.8%, mainly in the tracking direction. Satellite laser ranging (SLR) shows marginal impacts on IGSO and MEO satellites. However, the SLR residual of C01 shifts −13.2 cm, resulting in a smaller RMS value. In addition, the RMS of linear clock fitting is reduced from 0.050 ns to 0.038 ns for BDS-3 MEO satellites in average.

Real-time clock estimation using system time offset maintenance aiming at multi-GNSS time interoperability

Time interoperability is one of the important aspects of Global Navigation Satellite System (GNSS) compatibility and interoperability, which will greatly benefit the accuracy and availability of the positioning, navigation and timing (PNT) solution. In this paper, we proposed a real-time satellite clock estimation method using system time offset (STO) maintenance to help users realize GNSS time interoperability. First, receiver inter-system bias (ISB) was investigated and modeled for STO maintenance. Then, we presented clock estimation assessment results, which demonstrate that the new method can make the satellite clock product maintain a stable STO. Finally, precise point positioning (PPP) was used for further validation. The results show that in the condition of a 45° cutoff elevation angle, fixing the ISB can significantly reduce the convergence time, increase the availability by 21.5%, and improve the average positioning accuracy in horizontal and vertical by 37.7% and 44.4%, respectively.

Stochastic modeling of the receiver clock parameter in Galileo-only and multi-GNSS PPP solutions

In Precise Point Positioning (PPP), the receiver clock parameter is typically estimated independently in each observation epoch, which increases the noise of the estimated station coordinates and troposphere parameters due to correlations. Applying stochastic modeling to the receiver clock parameter stabilizes PPP solutions and reduces clock noise for the time transfer. However, the receiver clock modeling is possible only for the GNSS receivers connected to the utmost stable atomic clocks. We propose receiver clock modeling that involves the Markov stochastic process in the form of a random walk. We test different levels of random walk constraints for GNSS stations equipped with different types of clocks for Galileo-only and multi-GNSS solutions in kinematic and static PPP. In multi-GNSS solutions, the common clock parameter is derived with inter-system biases (ISBs). This raises the question of the constraints that should be imposed on the common clock only or also on the ISBs. We found that similar results can be achieved by imposing constraints on the common clock parameter and estimating ISB as a constant parameter and when constraining the common clock parameter and ISBs with a ratio of 1:100. Other ratios of clock-to-ISB constraints, such as 1:1 and 1:10, give inferior results. In the kinematic PPP, stochastic clock modeling has a marginal impact on the North and East coordinate components, whereas the Up component is substantially improved for GNSS receivers equipped with hydrogen masers. In the static PPP, the clock modeling improves the time transfer, due to the reduced noise of the clocks.

Characteristic analysis and calibration of CDMA and FDMA intra-system and inter-system biases in multi-GNSS precise point positioning

Characteristic analysis and calibration of CDMA and FDMA intra-system and inter-system biases in multi-GNSS precise point positioning

A new inter-system double-difference RTK model applicable to both overlapping and non-overlapping signal frequencies

Aiming at the problem that the traditional inter-system double-difference model is not suitable for non-overlapping signal frequencies, we propose a new inter-system double-difference model with single difference ambiguity estimation, which can be applied for both overlapping and non-overlapping signal frequencies. The single difference ambiguities of all satellites and Differential Inter-System Biases (DISB) are first estimated, and the intra-system double difference ambiguities, which have integer characteristics, are then fixed. After the ambiguities are successfully fixed, high-precision coordinates and DISB can be obtained with a constructed transformation matrix. The model effectively avoids the DISB parameter filtering discontinuity caused by the reference satellite transformation and the low precision of the reference satellite single difference ambiguity calculated with the code. A zero-baseline using multiple types of receivers is selected to verify the stability of the estimated DISB. Three baselines with different lengths are selected to assess the positioning performance of the model. The ionospheric-fixed and ionospheric-float models are used for short and medium-long baselines, respectively. The results show that the Differential Inter-System Code Biases (DISCB) and Differential Inter-System Phase Biases (DISPB) have good stability regardless of the receivers type and the signal frequency used and can be calibrated to enhance the strength of the positioning model. The positioning results with three baselines of different lengths show that the proposed inter-system double-difference model can improve the positioning accuracy by 6–22% compared with the intra-system double-difference model which selects the reference satellite independently for each system. The Time to First Fix (TTFF) of the two medium-long baselines is reduced by 30% and 29%, respectively.

Evaluation of Ionospheric Delay Extraction Model Using Dual-Frequency Multisystem Observations

Ionospheric delay presents complex spatiotemporal variation characteristics and is a critical error source restricting the improvement of global navigation satellite system (GNSS) positioning, navigation, and timing performance. It is a prerequisite to constructing an accurate ionospheric delay extraction model that supports the study of the ionospheric delay characteristics. By using modernized dual-frequency and multi-system signals and inter-system bias (ISB) calibration, the performance of the ionospheric delay extraction model can be enhanced further. This study focuses on quantitatively analyzing the influence of the geometric dilution of precision (GDOP) and the redundancy of observations on the accuracy of ionospheric delay extraction, by constructing two state-of-the-art high-precision dual-frequency multi-system ionospheric delay extraction models using the tightly coupled un-difference and un-combination precise point positioning (UPPP) and carrier-to-code leveling (CCL) methods. GNSS datasets from eight international GNSS service (IGS) stations were selected to evaluate their accuracy and analyze their influencing factors. The results show that the ionospheric delay extraction model using the UPPP method is more accurate than the CCL method for both single and multiple systems. The ionospheric delay extraction model using the CCL method is insensitive to time and coordinates. The ionospheric delay extraction model using the UPPP method shows inconsistent accuracy at different times and coordinates, which reflects a correlation with the GDOP effect.

Multi-GNSS PPP solutions with different handling of system-specific receiver clock parameters and inter-system biases

In the multi-GNSS solutions integrating GPS, GLONASS, Galileo, and BeiDou, the receiver clock may be treated twofold; the clock parameter may be estimated for each GNSS separately or the common clock can be estimated, e.g., for GPS, with inter-system biases (ISBs) for other systems. The latter strategy reduces the number of estimated independent clock parameters per epoch almost by a factor of four because the clock parameters are estimated epoch-wise, whereas ISBs are estimated as constant values for the entire day or month. Due to the discontinuities in reference satellite clocks, the estimated ISBs and receiver clock parameters have also to be reinitialized at day boundaries. This raises questions about whether only the common clock has to be reset or all ISB values and what is the impact of the reinitialization of clock parameters with covariance values when estimating system-specific clock parameters. We analyze the effects of different types of stochastic modeling applied to the parameters of clocks and ISBs. In this study, we test five different strategies to clock handling in multi-GNSS kinematic Precise Point Positioning derived continuously for one month. We found that two solutions can be considered equivalent: (1) estimating system-specific clocks and (2) estimating the common clock with ISB and resetting at day boundaries the common clock parameter and ISBs. Oppositely, resetting only the common clock parameter or assuming that the ISB keep their stabilities over long periods is insufficient to obtain superior results of station coordinates and reliable time transfer results.

GNSS/RNSS Integrated PPP Time Transfer: Performance with Almost Fully Deployed Multiple Constellations and a Priori ISB Constraints Considering Satellite Clock Datums

Currently, the space segment of all the five satellite systems capable of providing precise time transfer services, namely BDS (including BDS-3 and BDS-2), GPS, GLONASS, Galileo, and Quasi-Zenith Satellite System (QZSS), has almost been fully deployed, which will definitely benefit the precise time transfer with satellite-based precise point positioning (PPP) technology. This study focuses on the latest performance of the BDS/GPS/GLONASS/Galileo/QZSS five-system combined PPP time transfer. The time transfer accuracy of the five-system integrated PPP was 0.061 ns, and the frequency stability was 1.24 × 10−13, 2.28 × 10−14, and 8.74 × 10−15 at an average time of 102, 103, and 104 s, respectively, which significantly outperforms the single-system cases. We also verified the outstanding time transfer performance of the five-system integrated PPP at locations with limited sky view. In addition, a method is proposed to mitigate the day-boundary jumps of inter-system bias (ISB) estimates by considering the difference in the satellite clock datums between two adjacent days. After applying a priori ISB constraints, the time transfer accuracy of the five-system integrated PPP can be improved by 37.9–51.6%, and the frequency stability can be improved by 14.8–21.6%, 5.3–7.6% and 20.0–29.6% at the three average times, respectively.

Analysis of Characteristics for Inter-System Bias on Multi-GNSS Undifferenced and Uncombined Precise Point Positioning

Multi Global Navigation Satellite System (GNSS) Precise Point Positioning (PPP) has become the mainstream of PPP technology. Due to the differences in the coordinates and time references of each GNSS, multi-GNSS PPP must include additional Inter-System Bias (ISB) parameters to ensure compatibility between different GNSSs. Therefore, research on the characteristics of ISB is also essential. To analyze the short- and long-term time characteristics of multi-GNSS ISBs, as well as their relationship with receiver type and receiver antenna type, the Undifferenced and Uncombined (UDUC) PPP model of Global Positioning System (GPS), BeiDou navigation satellite system (BDS), and Galileo satellite navigation system (Galileo) was rigorously derived, and the physical of ISBs was elaborated in depth. ISB parameters were estimated and analyzed using 31 days of data from the 31 Multi-GNSS Experimental stations (MGEX). The results indicate that: (1) the ISB value is dependent on the station receiver type, receiver antenna type, analysis center product utilized, and GNSS system. (2) The short-term time characteristics of ISB-COM, ISB-WUM, and ISB-GBM are similar for the same station but not for the long term. In addition, ISBs are more stable in the short term. (3) There is little correlation between the ISB time characteristics, the receiver type, and the receiver antenna type, and the day-boundary discontinuity(DBD) on the ISB can be ignored for the concussive days’ process.

Multi-GNSS Differential Inter-System Bias Estimation for Smartphone RTK Positioning: Feasibility Analysis and Performance

An inter-system model for multi-GNSSs (global navigation satellite systems) makes the interoperability among different GNSS constellations possible. In recent years, inter-system models for geodetic receivers have been extensively studied. However, the precise positioning of smartphones suffers from various problems, and the current research mostly focuses on how to achieve the GNSS ambiguity resolution. Based on the research of receiver channel-dependent bias, in this study, we will research the temporal behaviors of differential inter-system bias (DISB) and implement an inter-system model for smartphones. A representative Huawei P40 (HP40) smartphone was used in the experiments, and the results show the following: (1) For the HP40, the frequencies of Code Division Multiple Access (CDMA) systems are free of receiver channel-dependent phase bias, which provides the chances for further interoperability among these systems. However, the code observations of the HP40 are influenced by receiver channel-dependent code bias; it is therefore suggested to set a large initial standard deviation (STD) value for code observations in the positioning. (2) GPS L1/QZSS L1 and BDS-2 B1I /BDS-3 B1I are free of phase DISB, and there is obvious phase DISB between GPS L1 and Galileo E1; even so, the valuations are sufficiently stable and the STD is close to 0.005 cycles. However, the phase DISB of GPS L1/BDS B1I is unstable. (3) For kinematic positioning, when the stable phase DISB is introduced, a 3–38.9% improvement in the N/E/U directions of the positioning accuracies in the inter-system differencing is achieved compared with the intra-system differencing.

Long-Term Performance Evaluation of BeiDou PPP-B2b Products and Its Application in Time Service

Precise Point Positioning (PPP) is an official service of the BeiDou Global Navigation Satellite System (BDS-3) through the PPP-B2b signal. In this paper, we mainly focus on the long-term performance evaluation of BDS-3 PPP-B2b products and their application in time service. Since the PPP-B2b product is only available in and around China area, the arcs of PPP-B2b products are about several hours. We propose to evaluate the time datum stability by using all available satellites. Then, 557 day PPP-B2b products are collected for this experiment. The results show that there are large jumps in the GPS satellite clock time datum series. However, the BDS-3 satellite clock datum stability is almost at the same level with current Space State Representation (SSR) corrections from the International Global navigation satellite system Service (IGS). The difference between PPP-B2b GPS and BDS-3 satellite clock time datum will be absorbed into the Inter System Bias (ISB) parameter. Thus, it should be specially noted that the ISB parameter cannot be estimated as constant values if users use PPP-B2b products. In addition, the accuracy of the BDS-3 satellite clock is significantly better than that of the GPS for both the Root Mean Square Error (RMSE) and standard deviation (STD). The average Signal in Space Range Errors (SISREs) is 0.22 ns and 0.13 ns for GPS and BDS-3, respectively. The one-way timing experiment shows BDS-3 timing stability is 2.9 × 10−14@104 s. In addition, 10 baselines from 13 km to 4494 km are formed for time synchronization evaluation by using PPP-B2b products. The average RMSEs of time synchronization is from 0.46 ns to 1.58 ns and from 0.66 ns to 1.19 ns for GPS and BDS-3, respectively. As for STD, the average values are from 0.27 ns to 0.74 ns and from 0.27 ns to 0.47 ns for GPS and BDS-3, respectively. Overall, the results show that the time datum stability, accuracy, and service performance of BDS-3 PPP-B2b products has been stable over the past two years.

Some Key Issues on Pseudorange-Based Point Positioning with GPS, BDS-3, and Galileo Observations

Nowadays, BDS-3 and Galileo are still developing and have global service capabilities. This study aims to provide a comprehensive analysis of pseudorange-based/single point positioning (SPP) among GPS, BDS-3, and Galileo on a global scale. First, the positioning accuracy distribution of adding IGSO and GEO to the MEO of BDS-3 is analyzed. The results show that after adding IGSO and GEO, the accuracy of 3D in the Asia-Pacific region is significantly improved. Then, the positioning accuracy of the single-system and single-frequency SPP was validated and compared. The experimental results showed that the median RMS values for the GPS, Galileo, and BDS-3 are 1.10/1.10/1.30 m and 2.57/2.69/2.71 m in the horizontal and vertical components, respectively. For the horizontal component, the GPS and Galileo had better positioning accuracy in the middle- and high-latitude regions, while BDS-3 had better positioning accuracy in the Asia-Pacific region. For the vertical component, poorer positioning accuracy could be seen near the North Pole and the equator for all three systems. Meanwhile, in comparison with the single-system and single-frequency SPP, the contribution of adding pseudorange observations from the other satellite system and frequency band was analyzed fully. Overall, the positioning accuracy can be improved to varying degrees. Due to the observation of noise amplification, the positioning errors derived from dual-frequency SPP were much noisier than those from single-frequency SPP. Moreover, the positioning performance of single-frequency SPP with the ionosphere delay corrected with CODE final (COD), rapid (COR), 1-day predicted (C1P), and 2-day predicted (C2P) global ionospheric map (GIM) products was investigated. The results showed that SPP with COD had the best positioning accuracy, SPP with COR ranked second, while C1P and C2P were comparable and slightly worse than SPP with COR. SPP with GIM products demonstrated a better positioning accuracy than that of the single- and dual-frequency SPP. The stability and variability of the inter-system biases (ISBs) derived from the single-frequency and dual-frequency SPP were compared and analyzed, demonstrating that they were stable in a short time. The differences in ISBs among different receivers with single-frequency SPP are smaller than that of dual-frequency SPP.

Tight Integration Kinematic PPP-AR Using GPS/Galileo/QZSS Overlapping Frequency Signals and Its Performance in High-Shade Environments

Several global navigation satellite systems (GNSS) broadcast overlapping frequencies to enhance interoperability, allowing tight integration with only one reference satellite for every system with the same frequency. The key to realizing tight integration is estimating and utilizing differential intersystem biases, which allows the integer characteristic of the differenced ambiguity of two satellites from different systems to be retrieved. In this study, a detailed algorithm flow of a tight integration kinematic uncalibrated phase delay (UPD)-based PPP ambiguity resolution (PPP-AR) method, which includes multiple parts, is introduced. Subsequently, PPP-AR numerical experiments were conducted in a high-shade observation environment to determine the performance. In comparison with traditional methods, our results indicate that the average success fix rate improves from 83% to 100% after using the tight integration method in an environment where only three satellites are observed for each system. Moreover, for fewer than nine satellites, the tight integration method can still consistently maintain a fixed state. However, for comparison, the traditional loose integration method could no longer be implemented.

An extended inter-system biases model for multi-GNSS precise point positioning

The inter-system bias (ISB) is an important parameter in multi-GNSS precise point positioning (PPP). However, on the one hand, the generation mechanism and error components of ISB are not clear. On the other hand, it is unclear whether the ISB parameter should be added to the BDS-2/BDS-3 combined PPP. First, in order to solve these problems, an extended ISB mathematical model is proposed, which unifies the common errors between receiver and satellite, and extends the original ISB model. Second, to demonstrate the correctness of the new model, the components of the new ISB model are verified, and then it is used to explain whether the ISB parameter should be added to BDS-2/BDS-3 combined PPP. Furthermore, 41 stations from the MGEX network and precise products from COD (Center for Orbit Determination in Europe), GBM (Deutsches GeoForschungsZentrum) and WUM (Wuhan University) are used to calculate and analyze the multi-GNSS PPP and ISB during day of year (DOY) 307–365, 2020. Finally, we propose to use a different estimation method of ISB with different precise products to improve the positioning accuracy and shorten the convergence time. The experimental results show that: (1) ISB parameter is composed of five parts: time system error, receiver hardware delay, signal distortion biases (SDB), MGEX-realized (multi-GNSS experiment) time scale and other unmodeled deviations. (2) Due to different receiver hardware delay, SDB and MGEX-realized time scale between BDS-2 and BDS-3, it is necessary to add an ISB parameter in BDS-2/BDS-3 PPP. (3) In multi-GNSS PPP, if the ISB changes greatly but the traditional constant method is used to estimate the ISB parameter, the impact on single station PPP coordinates can reach decimeters. The statistical results demonstrate that the RMS of GBM(CON(Constant)) in East (E), North (N) and Up (U) directions are 2.34 cm, 0.60 cm, and 1.59 cm, respectively, while the RMS of GBM(RWK(Random walk)) decreased by 59.8 %, 13.3 %, and 18.2 %, respectively.

Tight integration of BDS-3/BDS-2/GPS/Galileo observations considering the new overlapping DISBs and its application in obstructed environments

Tight integration can enhance the model strength and positioning performance by considering the characteristic of differential inter-system bias (DISB), especially in obstructed environments. However, limited work emphasizes the comprehensive analysis of five-frequency DISBs between BDS-3 and other systems considering the receiver type, receiver configuration, and antenna type. In addition, the overlapping DISBs between BDS-3 and BDS-2 are also in great demand for further investigation since they are often regarded as one system. In this study, one DISB-float model is introduced to estimate the DISBs, and one DISB-fixed model and one DISB-free model are formulated to enhance the model strength of tight integration. Four dedicated datasets were collected to estimate the DISBs, which are also comprehensively analyzed considering the receiver type, receiver configuration, and antenna type. The results show that the DISBs between BDS-3 and other systems are rather stable over a certain period and are related to the receiver type and receiver configuration, whereas are not related to the antenna type. More interestingly, the B1I code DISB between BDS-3 and BDS-2 exhibits significant magnitude with a mean value of −1.44 m for the baseline composed of two different receivers. In this case, the B1I code DISB must be considered and the tight integration between BDS-3 and BDS-2 considering its calibration can improve the positioning performance. Besides, the tight integration of the DISB-fixed model can significantly improve the positioning accuracy between multiple GNSS. Compared to the loose integration, the improvement of 60.6 %, 56.6 %, and 61.2 % can be obtained in the E, N, and U directions, when only two satellites are available for each system. In real obstructed environments, the tight integration of the DISB-free model can also improve the positioning performance in terms of positioning availability and accuracy, as well as the ambiguity resolution performance.

A mixed multi-frequency precise point positioning strategy based on the combination of BDS-3 and GNSS multi-frequency observations

Due to the traditional fixed model used in precise point positioning (PPP) solutions, multi-frequency and multi-Global Satellite Navigation System (GNSS) observations have not been fully introduced into positioning services. In consideration of the BDS-3 multi-frequency signals and the new development of other GNSS systems, a new multi-frequency and multi-GNSS PPP solution strategy should be proposed to flexibly model and use all observations. In this study, a preliminary mixed multi-frequency PPP solution strategy is analyzed and tested based on a combination of BDS-3 and GNSS observations. First, the multi-frequency observations are combined and their coefficients are rapidly estimated by least squares; then, the inter-system bias parameter and the stochastic model are introduced into the function model; and finally, the mixed PPP solution and its software are developed and verified by three groups of experiments. According to the experimental results of 96 stations and ten-day multi-GNSS experiment observations, it is indicated that the root-mean-square error of positioning and the convergence time are significantly optimized with the aid of additional frequencies, where the accuracy improvements of multi-frequency and multi-GNSS scheme in the east (E), north (N) and up (U) directions can respectively reach up to 23.2%, 13.3% and 23.8% compared with the traditional BDS-3 dual-frequency ionosphere-free (IF) PPP model; and the corresponding convergence time is reduced from 18.54 min to 13.18 min. Meanwhile, from the results of multi-frequency BDS-3 PPP experiments based on 53 stations, it is suggested that a better performance of positioning and convergence can be obtained by the mixed PPP solution, where the position RMS of the E, N and U directions are reduced by 38.2%, 23.9% and 26.3%, and the convergence time is decreased from 23.86 min to 12.43 min for the combined BDS-3 of all observations, compared with the BDS-3-only solution. Furthermore, in the vehicle experiment of multi-frequency kinematics PPP, a convergence process can be found for different scenarios of BDS-3 combination with other observations. Moreover, the residual series are different for each solution, in which reductions of 71.1%, 33.3% and 77.1% in the E, N and U directions, respectively, can be obtained compared with the traditional BDS-3 dual-frequency IF model in kinematics experiments based on multi-GNSS and multi-frequency scenarios. Therefore, it is meaningful to recommend the mixed PPP solution in the GNSS community to fully use multi-frequency and multi-GNSS observations by the adaptive combination of different observations.

INS-assisted inter-system biases estimation and inter-system ambiguity resolution in a complex environment

INS-assisted inter-system biases estimation and inter-system ambiguity resolution in a complex environment