The GFZ real-time GNSS precise positioning service system and its adaption for COMPASS

Motivated by the IGS real-time Pilot Project, GFZ has been developing its own real-time precise positioning service for various applications. An operational system at GFZ is now broadcasting real-time orbits, clocks, global ionospheric model, uncalibrated phase delays and regional atmospheric corrections for standard PPP, PPP with ambiguity fixing, single-frequency PPP and regional augmented PPP. To avoid developing various algorithms for different applications, we proposed a uniform algorithm and implemented it into our real-time software. In the new processing scheme, we employed un-differenced raw observations with atmospheric delays as parameters, which are properly constrained by real-time derived global ionospheric model or regional atmospheric corrections and by the empirical characteristics of the atmospheric delay variation in time and space. The positioning performance in terms of convergence time and ambiguity fixing depends mainly on the quality of the received atmospheric information and the spatial and temporal constraints. The un-differenced raw observation model can not only integrate PPP and NRTK into a seamless positioning service, but also syncretize these two techniques into a unique model and algorithm. Furthermore, it is suitable for both dual-frequency and sing-frequency receivers. Based on the real-time data streams from IGS, EUREF and SAPOS reference networks, we can provide services of global precise point positioning (PPP) with 5–10 cm accuracy, PPP with ambiguity-fixing of 2–5 cm accuracy, PPP using single-frequency receiver with accuracy of better than 50 cm and PPP with regional augmentation for instantaneous ambiguity resolution of 1–3 cm accuracy. We adapted the system for current COMPASS to provide PPP service. COMPASS observations from a regional network of nine stations are used for precise orbit determination and clock estimation in simulated real-time mode, the orbit and clock products are applied for real-time precise point positioning. The simulated real-time PPP service confirms that real-time positioning services of accuracy at dm-level and even cm-level is achievable with COMPASS only.     

A novel predictive algorithm for double difference observations of obstructed BeiDou geostationary earth orbit (GEO) satellites

Transmission link disturbances and device failure cause global navigation satellite system (GNSS) receivers to miss observations, leading to poor accuracy in real-time kinematic (RTK) positioning. Previously described solutions for this problem are influenced by the length of the prediction period, or are unable to account for changes in receiver state because they use information from previous epochs to make predictions. We propose an algorithm for predicting double difference (DD) observations of obstructed BeiDou navigation system (BDS) GEO satellites. Our approach adopts the first-degree polynomial function for predicting missing observations. We introduce a Douglas-Peucker algorithm to judge the state of the rover receiver to reduce the impact of predictive biases. Static and kinematic experiments were carried out on BDS observations to evaluate the proposed algorithm. The results of our navigation experiment demonstrate that RTK positioning accuracy is improved from meter to decimeter level with fixed ambiguity (horizontal?<?2?cm, vertical?<?18?cm). Horizontal accuracy is improved by over 50%, and the vertical accuracies of the results of the static and kinematic experiments are increased by 47% and 27% respectively, compared with the results produced by the classical approach. Though as the baseline becomes longer, the accuracy is weakened, our predictive algorithm is an improvement over existing approaches to overcome the issue of missing data.     

Performance assessment of real-time orbit determination for the Haiyang-2D using onboard BDS-3/GPS observations

The Global Navigation Satellite System (GNSS) receivers equipped on the Haiyang-2D (HY-2D) satellite is capable of tracking the signals of both the third generation of BeiDou satellite navigation System (BDS-3) and the Global Positioning System (GPS), which make it feasible to assess the performance of real-time orbit determination (RTOD) for the HY-2D using onboard GNSS observations. In this study, the achievable accuracy and convergence time of RTOD for the HY-2D using onboard BDS-3 and GPS observations are analyzed. Benefiting from the binary-offset-carrier (BOC) modulation, the BDS-3 C1X signal includes less noise than the GPS C1C signal, which has the same signal frequency and chipping rate. The root mean squares (RMS) of the noises of C1X and C1C code measurements are 0.579 m and 1.636 m, respectively. Thanks to a ten-times higher chipping rate, the code measurements of BDS-3 C5P, GPS C1W and C2W are less noisy. The RMS of code noises of BDS-3 C5P, GPS C1W, and C2W are 0.044 m, 0.386 m, and 0.272 m, respectively. For the HY-2D orbit, the three-dimensional (3D) and radial accuracies can reach 31.8 cm and 7.5 cm with only BDS-3 observations, around 50 % better than the corresponding accuracies with GPS. Better performance of the BDS-3 in RTOD for the HY-2D is attributed to the high quality of its broadcast ephemeris. When random parameters are used to absorb ephemeris errors, substantial improvement is seen in the accuracy of HY-2D orbit with either BDS-3 or GPS. The 3D RMS of HY-2D orbit errors with BDS-3 and GPS are enhanced to 23.1 cm and 33.6 cm, and the RMS of the radial components are improved to 6.1 cm and 13.3 cm, respectively. The convergence time is 41.6 and 75.5 min for the RTOD with BDS-3 and GPS, while it is reduced to 39.2 and 27.4 min after the broadcast ephemeris errors are absorbed by random parameters. Overall, the achievable accuracy of RTOD with BDS-3 reaches decimeter level, which is even better than that with GPS, making real-time navigation using onboard BDS-3 observations a feasible choice for future remote sensing missions.     

Network based real-time precise point positioning

Network based real-time precise point positioning system includes two stages, i.e. real-time estimation of satellite clocks based on a reference network and real-time precise point positioning thereafter. In this paper, a satellite- and epoch-differenced approach, adopted from what is introduced by Han et al. (2001), is presented for the determination of satellite clocks and for the precise point positioning. One important refinement of our approach is the implementation of the robust clock estimation. A prototype software system is developed, and data from the European Reference Frame Permanent Network on September 19, 2009 is used to evaluate the approach. Results show that our approach is 3 times and 90 times faster than the epoch-difference approach and the zero-difference approach, respectively, which demonstrates a significant improvement in the computation efficiency. The RMS of the estimated clocks is at the level of 0.1 ns (3 cm) compared to the IGS final clocks. The clocks estimates are then applied to the precise point positioning in both kinematic and static mode. In static mode, the 2-h estimated coordinates have a mean accuracy of 3.08, 5.79, 6.32 cm in the North, East and Up directions. In kinematic mode, the mean kinematic coordinates accuracy is of 4.63, 5.82, 9.20 cm.     

GNSS global real-time augmentation positioning: Real-time precise satellite clock estimation,prototype system construction and performance analysis

Lots of ambiguities in un-differenced (UD) model lead to lower calculation efficiency, which isn’t appropriate for the high-frequency real-time GNSS clock estimation, like 1 Hz. Mixed differenced model fusing UD pseudo-range and epoch-differenced (ED) phase observations has been introduced into real-time clock estimation. In this contribution, we extend the mixed differenced model for realizing multi-GNSS real-time clock high-frequency updating and a rigorous comparison and analysis on same conditions are performed to achieve the best real-time clock estimation performance taking the efficiency, accuracy, consistency and reliability into consideration. Based on the multi-GNSS real-time data streams provided by multi-GNSS Experiment (MGEX) and Wuhan University, GPS + BeiDou + Galileo global real-time augmentation positioning prototype system is designed and constructed, including real-time precise orbit determination, real-time precise clock estimation, real-time Precise Point Positioning (RT-PPP) and real-time Standard Point Positioning (RT-SPP). The statistical analysis of the 6 h-predicted real-time orbits shows that the root mean square (RMS) in radial direction is about 1–5 cm for GPS, Beidou MEO and Galileo satellites and about 10 cm for Beidou GEO and IGSO satellites. Using the mixed differenced estimation model, the prototype system can realize high-efficient real-time satellite absolute clock estimation with no constant clock-bias and can be used for high-frequency augmentation message updating (such as 1 Hz). The real-time augmentation message signal-in-space ranging error (SISRE), a comprehensive accuracy of orbit and clock and effecting the users’ actual positioning performance, is introduced to evaluate and analyze the performance of GPS + BeiDou + Galileo global real-time augmentation positioning system. The statistical analysis of real-time augmentation message SISRE is about 4–7 cm for GPS, whlile 10 cm for Beidou IGSO/MEO, Galileo and about 30 cm for BeiDou GEO satellites. The real-time positioning results prove that the GPS + BeiDou + Galileo RT-PPP comparing to GPS-only can effectively accelerate convergence time by about 60%, improve the positioning accuracy by about 30% and obtain averaged RMS 4 cm in horizontal and 6 cm in vertical; additionally RT-SPP accuracy in the prototype system can realize positioning accuracy with about averaged RMS 1 m in horizontal and 1.5–2 m in vertical, which are improved by 60% and 70% to SPP based on broadcast ephemeris, respectively.     

BDS-2/BDS-3 uncalibrated phase delay estimation considering the intra-system bias

Intra-system biases (ISBs) between BDS-2 and BDS-3 are of critical importance when combining observations from the BDS-2 and BDS-3 systems, which is meaningful to fully take advantage of the BDS positioning capability. Meanwhile, ISBs should also be considered in the estimation of BDS uncalibrated phase delays (UPDs). In this research, we present a BDS-2/BDS-3 joint-processing scheme, as well as a method for estimating BDS UPDs. The characteristics of ISBs and the quality of BDS UPDs are analyzed based on 30-day data from 130 multi-GNSS experimental (MGEX) stations. Our results indicate that the ISBs are related to the type and version of the receiver. The ISBs can be regarded as constant across the course of a given day, and the mean standard deviation (STD) values of ISBs over one month for different types of receivers are generally within 0.2 m. Moreover, to assess the quality of UPD products, the residuals of the estimated UPDs and the utilization rates of the observation data are computed. The results show that the quality of BDS UPDs can be improved by correcting the satellite-induced pseudo-range variations, and by estimating the wide-lane (WL) UPD difference between BDS-2 and BDS-3. The average RMS values of the estimated residuals of WL UPD and narrow-lane (NL) UPD are 0.07 and 0.09 cycles, respectively; moreover, the utilization rate of the observation data of WL UPD and NL UPD can reach above 90 %. The performance of BDS precise point positioning (PPP) and PPP ambiguity resolution (PPP-AR) is analyzed in terms of positioning accuracy and convergence performance in both the static and kinematic modes. Compared with PPP ambiguity-float solutions, the positioning accuracy of PPP-AR is significantly improved, especially in the east direction. The impact of ISBs on PPP and PPP-AR is also analyzed, and the results indicate that ISBs can improve the convergence speed of float PPP, but can be disregarded in PPP-AR.     

Precise orbit determination and baseline consistency assessment for Swarm constellation

Precise orbit determination (POD) and precise baseline determination (PBD) of Swarm satellites with 4 years of data are investigated. Ambiguity resolution (AR) plays a crucial role in achieving the best orbit accuracy. Swarm POD and PBD based on single difference (SD) AR and traditional double difference (DD) AR methods are explored separately. Swarm antenna phase center variation (PCV) corrections are developed to further improve the orbit determination accuracy. The code multipath of C1C, C1W and C2W observations is first evaluated and clear variations in code noise related to different receiver settings are observed. Carrier phase residuals of different time periods and different loop tracking settings of receiver are studied to explore the effect of ionospheric scintillation on POD. The reduction of residuals in the polar and geomagnetic equator regions confirms the positive impact of the updated carrier tracking loops (TLs) on POD performance. The SD AR orbits and orbits with float ambiguity (FA) are compared with the Swarm precise science orbits (PSOs). An average improvement of 27 %, 4 % and 16 % is gained in along-track, cross-track and radial directions by fixing the ambiguity to integer. For Swarm-A/B and Swarm-B/C formations, specific days are selected to perform the DD AR-based POD during which the average distance of the formation satellites is less than 5000 km. Satellite laser ranging (SLR) observations are employed to validate the performance of FA, SD AR and DD AR orbits. The consistency between the SD AR orbits and SLR data is at a level of 10 mm which shows an improvement of 25 % when comparing with the FA results. An SLR residuals reduction of 15 % is also achieved by the DD AR solution for the selected days. Precise relative navigation is also an essential aspect for spacecraft formation flying missions. The closure error method is proposed to evaluate the baseline precision in three dimensions. A baseline precision of 1–3 mm for Swarm-A/C formation and 3–5 mm for Swarm-A/B and Swarm-B/C satellite pairs is verified by both the consistency check and closure error method.     

A real-time combined quality control method for GNSS precise positioning in harsh environments

Global Navigation Satellite System (GNSS) precise positioning can be significantly affected by severe multipath effects and outliers in harsh environments, and highly relies on quality control strategies. Previous studies mainly focus on the posterior residuals to check and exclude the outliers in GNSS observations, limited work emphasizes the combined quality control method considering both the prior and posterior knowledge simultaneously. This paper proposed a real-time combined quality control method to process the multipath effects and outliers in harsh environments simultaneously. Specifically, in the prior stage, a modified multipath processing strategy is proposed for both phase and code observations, then a modified detection, identification, and adaptation (DIA) method considering the maximum times of data snooping is studied in the posterior stage. Two dedicated experiments in real harsh environments were carried out to evaluate the performance of the proposed combined quality control method. For the static experiment, the proposed method exhibits smaller positioning errors, the best positioning accuracy, and the highest availability in this study. Specifically, the proposed method exhibits an improved percentage of 55.4 %, 56.3 %, and 59.7 % for positioning accuracy compared to those without the quality control method in the E, N, and U directions, respectively. Besides, the proposed method can further improve the performance of ambiguity resolution with an improved percentage of 32.2 %. For the kinematic experiment, the three-dimensional positioning accuracy of the proposed method is 0.577 m, which exhibits a 40.0 % improvement compared to those without the quality control method. Also, the proposed method exhibits better performance under relatively strong multipath effects. In this sense, the proposed real-time combined quality control method is highly appreciated in terms of positioning availability, accuracy, and ambiguity resolution for GNSS precise positioning, especially in harsh environments.     

Tightly combined GPS/Galileo RTK for short and long baselines: Model and performance analysis

To ensure the compatibility and interoperability with modernized GPS, Galileo satellites are capable of broadcasting navigation signals on carrier phase frequencies that overlap with GPS, i.e., GPS/Galileo L1-E1/L5-E5a. Moreover, the GPS/Galileo L2-E5b signals have different frequencies with wavelength differences smaller than 4.2?mm. Such overlapping and narrowly spaced signals between GPS and Galileo bring the opportunity to use the tightly combined double-differenced (DD) model for precise real-time kinematic (RTK) positioning, resulting in improved performance of ambiguity resolution and positioning with respect to the classical standard or loosely combined DD model. In this paper, we focus on the model and performance assessment of tightly combined GPS/Galileo L1-E1/L2-E5b/L5-E5a RTK for short and long baselines. We first investigate the tightly combined GPS/Galileo DD observational model for both short and long baselines with simultaneously considering the GPS/Galileo overlapping and non-overlapping frequencies. Particularly, we introduce a reparameterization approach to solve the rank deficiency that caused by the correlation between the DISB parameters and the DD ionospheric parameters for both overlapping and non-overlapping frequencies. Then we present performance assessment for the tightly combined GPS/Galileo RTK model with real-time estimation of the differential inter-system bias (DISB) parameters for short and long baselines in terms of ratio value, ambiguity dilution of precision (ADOP), ambiguity conditional number, decorrelation number, search count, empirical success rate, time-to-first-fix (TTFF), and positioning accuracy. Results from both static and kinematic experiments demonstrated that compared to the loosely combined model, the tightly combined model can deliver improved performance of ambiguity resolution and precise positioning with different satellite visibility. For the car-driven short baseline experiment with 10° elevation cut-off angle, the tightly combined model can not only significantly increase the ratio value by approximately 27.5% (from 16.0 to 20.4), but also reduce the ambiguity ADOP, the conditional number, and the search count in LAMBDA by approximately 22.2% (from 0.027 to 0.021 cycles), 14.9% (from 199.2 to 169.6), and 25.4% (from 150.1 to 112.0), respectively. Comparable decorrelation number, empirical success rate, and positioning accuracy are also obtained. For the car-driven long baseline experiment, it is also observed that the ambiguity resolution performance in terms of the ratio value, the decorrelation number, the condition number, and the search count are significantly improved by approximately 18.5% (from 2.7 to 3.2), 22.0% (from 0.186 to 0.227), 55.9% (from 937.6 to 413.7), and 10.3% (from 43.8 to 39.3), respectively. Moreover, comparable ADOP, empirical success rate, and positioning accuracy are obtained as well. Additionally, the TTFF can be reduced (from 54.1 to 51.8 epochs with 10° elevation cut-off angle) as well from the results of static experiments.     

Analysis of the code and phase between-receiver inter-system biases of the overlapping frequencies for GPS/Galileo/BDS

The overlapping-frequency signals from different GNSS constellations are interoperable and can be integrated as one constellation in multi-GNSS positioning when inter-system bias (ISB) is properly disposed. The look-up table method for ISB calibration can enhance the model strength, maximize the number of integer-estimable ambiguities, and thus is preferred. However, the characteristics and magnitudes of the receiver code ISB and phase fractional ISB (F-ISB) are not well known and the wrong values of the biases can seriously degrade the positioning results. In this contribution, we first estimate the between-receiver code ISB and phase F-ISB of hundreds of the baselines up to around 25km in the European Permanent GNSS Network (EPN) and the Multi-GNSS Experiment (MGEX) for the overlapping frequencies L1-E1 (L1), L5-E5a (L5) and E5b-B2b (L7). The data collected from 1st January 2016 to 1st January 2019. Second, the receiver-type and firmware-version combinations for the receivers of Trimble, Leica, Javad, Septentrio and NovAtel are carefully classified. Results show that the Septentrio receivers have consistent code and phase ISB values for the three overlapping frequencies i.e. only one value for each frequency and no receivers are different. The Leica, Trimble and Javad receivers have two or more ISB values for at least one of the three frequencies. A few receivers with biases to the groups are also found and listed. Third, the code ISB and phase F-ISB of the groups are adjusted by the least-squares method. The root mean square errors (RMSE) of the least square adjustment are 0.240 m, 0.250 m and 0.200 m for code of L1, L5 and L7 frequencies, respectively, and are 0.0009 m, 0.0015 m and 0.0031 m for phase of L1, L5 and L7 frequencies, respectively. Finally, the effects of code ISB errors on code positing are investigated with the zero-baseline MAT1_MATZ. The distance root mean square error (DRMS) of L1-E1 code positioning can be reduced by 48.2% with 5 GPS and Galileo satellites and the DRMS degrades quickly when the code ISB error is larger.     

Galileo real-time orbit determination with multi-frequency raw observations

Real-time GNSS-based applications require corresponding real-time orbit products. While traditional GNSS orbits are generated with the dual-frequency IF (Ionosphere-Free) model, the increase of multi-frequency signal satellites brings new challenges for the data processing. Therefore, real-time orbit determination with the multi-frequency UC (Uncombined) model is introduced in this study considering its flexibility. With the derived mathematical model conforming to IGS (International GNSS Service) dual-frequency clock definition and one-week triple-frequency Galileo observation data from 90 IGS network stations, the convergence and accuracy of real-time orbits is assessed and the characteristics of satellite IFCB (Inter-Frequency Clock Bias) are analyzed. Results indicate that the model differences, including dual-frequency IF model, dual-frequency UC model and triple-frequency UC model, contribute to only cm-level differences with CODE (Center for Orbit Determination in Europe) final orbits after a convergence time of around 12 h. The constellation-mean RMS (Root Mean Square) differences of the converged real-time orbits with the CODE final orbits reaches about 5.0 cm, 7.0 cm and 5.0 cm for the radial, tangential and normal directions. The convergence of satellite IFCB is much faster than that of satellite orbit, which reflects a loose correlation between these two parameters. While the Galileo satellite IFCB are temporally stable, the modeling of satellite IFCB may be unreliable when over constrained and becomes even more unstable with commonly encountered datum changes. In summary, real-time GNSS orbit determination with multi-frequency raw observations is feasible and extendable with proper treatment of IFCB.     

Precision orbit determination for TOPEX/Poseidon using TDRSS doppler tracking data

Precision orbit determination on the TOPEX/Poseidon (T/P) altimeter satellite is now being routinely achieved with sub-5cm radial and sub-15 cm total positioning accuracy using state-of-the-art modeling with precision tracking provided by a combination of: (a) global Satellite Laser Ranging (SLR) and Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), or (b) the Global Positioning System (GPS) Constellation which provides pseudo-range and carrier phase observations. The geostationary Tracking and Data Relay Satellite System (TDRSS) satellites are providing the operational tracking and communication support for this mission. The TDRSS Doppler data are of high precision (0.3 mm/s nominal noise levels). Unlike other satellite missions supported operationally by TDRSS, T/P has high quality independent tracking which enables absolute orbit accuracy assessments. In addition, the T/P satellite provides extensive geometry for positioning a satellite at geostationary altitude, and thus the TDRSS-T/P data provides an excellent means for determining the TDRS orbits. Arc lengths of 7 and 10 days with varying degrees of T/P spacecraft attitude complexity are studied. Sub-meter T/P total positioning error is achieved when using the TDRSS range-rate data, with radial orbit errors of 10.6 cm and 15.5 cm RMS for the two arcs studied. Current limitations in the TDRSS precision orbit determination capability include mismodeling of numerous TDRSS satellite-specific dynamic and electronic effects, and in the inadequate treatment of the propagation delay and bending arising from the wet troposphere and ionosphere.     

RTK model and positioning performance analysis using Galileo four-frequency observations

This paper proposes a real-time kinematic (RTK) model that uses one common reference satellite for the Galileo system with four frequency observations. In the proposed model, the double-differenced (DD) pseudorange and carrier phase biases among the different frequencies are estimated as unknown parameters to recover the integer features of the DD ambiguities among the different frequencies for ambiguity resolution and precise positioning. Analysis results show that the E5a, E5b, and E5 frequencies have virtually the same performance in terms of the positioning accuracy, observation residuals, and ratio values of ambiguity resolution. However, the E1 frequency performs worse than the E5a, E5b, and E5 frequencies. The RTK results for the combination of multiple frequencies are much better than those for a single-frequency observation, the coordinates’ standard deviation is improved about 20–30%, and the ambiguity fix time is improved about 10%.     

Orbit determination of BDS-3 satellite based on regional ground tracking station and inter-satellite link observations

The BeiDou global navigation satellite system (BDS-3) has established the Ka-band inter-satellite link (ISL) to realize a two-way ranging function between satellites, which provides a new observation technology for the orbit determination of BDS-3 satellites. Therefore, this study presents a BDS satellite orbit determination model based on ground tracking station (GTS) observations and ISL ranging observations firstly to analyze the impact of the ISL ranging observations on the orbit determination of BDS-3 satellites. Subsequently, considering the data fusion processing, the variance component estimation (VCE) algorithm is applied to the parameter estimation process of the satellite orbit determination. Finally, using the measured data from China’s regional GTS observations and BDS-3 ISL ranging observations, the effects of ISL ranging observations on the orbit determination accuracy of BDS-3 satellites are analyzed. Moreover, the impact of the VCE algorithm on the fusion data processing is evaluated from the aspects of orbit determination accuracy, Ka-band hardware delay parameter stability, and ISL ranging observation residuals. The results show that for China’s regional GTSs, the addition of BDS-3 ISL ranging observations can significantly improve the orbit determination accuracy of BDS-3 satellites. The observed orbit determination accuracy of satellite radial component is improved from 48 cm to 4.1 cm. In addition, when the initial weight ratio between GTS observations and ISL ranging observations is not appropriate, the various indicators which include orbit determination accuracy, ISL hardware delay, and ISL observation residuals were observed to have improved after the adjustment of the VCE algorithm. These results validate the effectiveness of the VCE algorithm for the fusion data processing of the GTS observations and ISL ranging observations.     

Initial accuracy and reliability of current BDS-3 precise positioning,velocity estimation,and time transfer (PVT)

The primary system of Chinese global BeiDou satellite system (BDS-3) was completed to provide global services on December 27, 2018; this was a key milestone in the development process for BDS in terms of its provision of global services. Therefore, this study analyzed the current performance of BDS-3, including its precise positioning, velocity estimation, and time transfer (PVT). The datasets were derived from international GNSS monitoring and assessment system (iGMAS) tracking networks and the two international time laboratories in collaboration with the International Bureau of Weights and Measures (BIPM). With respect to the positioning, the focus is on the real-time kinematic (RTK) positioning and precise point positioning (PPP) modes with static and kinematic scenarios. The results show that the mean available satellite number is 4.8 for current BDS-3 system at short baseline XIA1–XIA3. The RTK accuracy for three components is generally within cm level; the 3D mean accuracy is 8.9 mm for BDS-3 solutions. For the PPP scenarios, the convergence time is about 4 h for TP01 and BRCH stations in two scenarios. After the convergence, the horizontal positioning accuracy is better than cm level and the vertical accuracy nearly reaches the 1 dm level. With respect to kinematic scenarios, the accuracy stays at the cm level for horizontal components and dm level for the vertical component at two stations. In terms of velocity estimation, the horizontal accuracy stays at a sub-mm level, and the vertical accuracy is better than 2 mm/s in the BDS-3 scenario, even in the Arctic. In terms of time and frequency transfer, the noise level of BDS-3 time links can reach 0.096 ns for long-distances link NT01–TP02 and 0.016 ns for short-distance links TP01–TP02. Frequency stability reaches 5E–14 accuracy when the averaging time is within 10,000 s for NT01–TP02 and 1E–15 for TP01–TP02.     

Differential code bias estimation based on uncombined PPP with LEO onboard GPS observations

Differential Code Bias (DCB) is an essential correction that must be provided to the Global Navigation Satellite System (GNSS) users for precise position determination. With the continuous deployment of Low Earth Orbit (LEO) satellites, DCB estimation using observations from GNSS receivers onboard the LEO satellites is drawing increasing interests in order to meet the growing demands on high-quality DCB products from LEO-based applications, such as LEO-based GNSS signal augmentation and space weather research. Previous studies on LEO-based DCB estimation are usually using the geometry-free combination of GNSS observations, and it may suffer from significant leveling errors due to non-zero mean of multipath errors and short-term variations of receiver code and phase biases. In this study, we utilize the uncombined Precise Point Positioning (PPP) model for LEO DCB estimation. The models for uncombined PPP-based LEO DCB estimation are presented and GPS observations acquired from receivers onboard three identical Swarm satellites from February 1 to 28, 2019 are used for the validation. The results show that the average Root Mean Square errors (RMS) of the GPS satellite DCBs estimated with onboard data from each of the three Swarm satellites using the uncombined PPP model are less than 0.18 ns when compared to the GPS satellite DCBs obtained from IGS final daily Global Ionospheric Map (GIM) products. Meanwhile, the corresponding average RMS of GPS satellite DCBs estimated with the conventional geometry-free model are 0.290, 0.210, 0.281 ns, respectively, which are significantly larger than those obtained with the uncombined PPP model. It is also noted that the estimated GPS satellite DCBs by Swarm A and C satellites are highly correlated, likely attributed to their similar orbit type and space environment. On the other hand, the Swarm receiver DCBs estimated with uncombined PPP model, with Standard Deviation (STD) of 0.065, 0.037 and 0.071 ns, are more stable than those obtained from the official Swarm Level 2 products with corresponding STD values of 0.115, 0.101, and 0.109 ns, respectively. The above indicates that high-quality DCB products can be estimated based on uncombined PPP with LEO onboard observations.     

Evaluation of rapid phase clock/bias products for PPP ambiguity resolution and its application to the M7.1 2019 Ridgecrest,California earthquake

Precise point positioning with ambiguity resolution (PPP-AR) is a useful tool for high-precision geodetic and geophysical applications, while phase bias products are the prerequisite to implement PPP-AR. Wuhan University has been providing the final (the best operationally post-processing solution based) phase clock/bias products with a latency of two weeks since March of 2019, while a dedicated open-source software package PRIDE PPP-AR is released to leverage these products for high-precision positioning. In order to satisfy some both time and precision critical applications, such as rapid earthquake response, Wuhan University also released rapid (with comparable quality but with much shorter delivery latency) phase clock/bias products with a latency of less than 24 h and updated PRIDE PPP-AR in July 2019. We first introduce the phase clock/bias generation and validation schemes and the maintenance of routine products provision. Then, with 14 days (July 2 to July 15 in 2019) of GPS data collected from 146 globally distributed IGS (International GNSS Service) stations, we evaluated the positioning performance of the rapid products with respect to their final counterparts. It is found that positioning precision of PPP-AR using rapid products is comparable to that using final products, especially in kinematic positioning mode. When rapid products are used, the RMS of PPP-AR in static mode with respect to IGS weekly solutions can reach 1.7 mm, 1.8 mm and 5.5 mm in the east, north and up components, respectively. Furthermore, the RMS of epoch-wise positions with respect to daily solutions for the east, north and up components are 0.51 cm, 0.57 cm and 1.51 cm for PPP-AR with rapid products in kinematic mode. It demonstrates that the rapid phase clock/bias products can sufficiently meet the precision requirement of most geodetic and geophysical applications yet with much shorter time delay. Finally, we study the July 6th M7.1 2019 Ridgecrest, California earthquake using the rapid phase clock/bias products and demonstrate their comparable performance against the final products.     

Comparative analysis of four different single-frequency PPP models on positioning performance and atmosphere delay retrieval

Single-frequency precise point positioning (SF-PPP) has attracted increasing attention due to its high precision and cost effectiveness. With various strategies to handle the dominant error, i.e., ionosphere delay, the ionosphere-float (IF), ionosphere-free-half (IFH), ionosphere-corrected (IC), and ionosphere-weighted (IW) SF-PPP models are certain to possess different characteristics and performance levels. This study is dedicated to assessing and comparing the four models from model characteristics, positioning performance, and atmosphere delay retrieval. The model comparison shows that IC and IW models are full-rank while IF and IFH models have a rank deficiency of size one that will result in biased estimations, which means the better solvability of IC and IW models. The experiments are carried out based on the 7-day Global Positioning System (GPS) observations collected at 57 global Multi-GNSS Experiment (MGEX) stations and Global Ionosphere Map (GIM) products. The results indicate that the IW model can accelerate SF-PPP convergence and achieve higher positioning accuracy compared to the other three SF-PPP models, especially in kinematic mode. With convergence criteria of 0.25 m in horizontal and 0.5 m in vertical, the east/north/up convergence times of IW model are 0.5/15.0/25.0 min and 0.5/16.0/36.5 min for static and kinematic modes, respectively. The IW model is able to achieve an instantaneous positioning accuracy of 0.28/0.35/0.75 m. In addition, a real kinematic test also demonstrates the best positioning solutions of IW model. Regarding troposphere delay retrieval, the IF, IFH, and IW models obtain a comparable daily accuracy of 3.0 cm on average, while the IC model achieves the worst accuracy of 8.0 cm. For precise ionosphere delay estimation, IW model only needs an average initialization time of 34.3 min, but a longer initialization time of 51.6 min is required for IF model. The daily precision of ionosphere delay estimation for IW model can reach up to 10.8 cm. At the present accuracy of GIM products, it is suggested that the IW model should be adopted for SF-PPP first due to its superior performance in positioning and atmosphere delay retrieval.     

Accuracy assessment of single and double difference models for the single epoch GPS compass

The single epoch GPS compass is an important field of study, since it is a valuable technique for the orientation estimation of vehicles and it can guarantee a total independence from carrier phase slips in practical applications. To achieve highly accurate angular estimates, the unknown integer ambiguities of the carrier phase observables need to be resolved. Past researches focus on the ambiguity resolution for single epoch; however, accuracy is another significant problem for many challenging applications. In this contribution, the accuracy is evaluated for the non-common clock scheme of the receivers and the common clock scheme of the receivers, respectively. We focus on three scenarios for either scheme: single difference model vs. double difference model, single frequency model vs. multiple frequency model and optimal linear combinations vs. traditional triple-frequency least squares. We deduce the short baseline precision for a number of different available models and analyze the difference in accuracy for those models. Compared with the single or double difference model of the non-common clock scheme, the single difference model of the common clock scheme can greatly reduce the vertical component error of baseline vector, which results in higher elevation accuracy. The least squares estimator can also reduce the error of fixed baseline vector with the aid of the multi-frequency observation, thereby improving the attitude accuracy. In essence, the “accuracy improvement” is attributed to the difference in accuracy for different models, not a real improvement for any specific model. If all noise levels of GPS triple frequency carrier phase are assumed the same in unit of cycles, it can be proved that the optimal linear combination approach is equivalent to the traditional triple-frequency least squares, no matter which scheme is utilized. Both simulations and actual experiments have been performed to verify the correctness of theoretical analysis.     

Inertial aided BDS triple-frequency integer ambiguity rounding method

The integer ambiguity resolution (AR) of carrier phase is significant for Global Navigation Satellite System (GNSS) precise positioning. However, in kinematic case, single-epoch AR methods based on alone GNSS are usually not reliable due to the instable pseudorange accuracy. Moreover, the computation of classical AR method Least Squares Ambiguity Decorrelation Adjustment (LAMBDA) is large. Thus, the inertial measurement unit (IMU) is introduced, a new inertial-aided AR method that directly rounds the float ambiguity of BeiDou triple-frequency combined observations, which is characterized by long wavelength, low carrier-phase noise and ionospheric delay, is proposed. The mathematical model of the new method is derived first. Then the impacts of the carrier-phase noise, ionospheric delay and inertial navigation system (INS) position error on the AR success ratio of combined observation are analyzed through probabilistic approach. Based on above investigation, the combinations (0, ?1, 1), (1, 4, ?5) and (4, ?2, ?3) are selected to resolve the original ambiguity. A vehicular integrated navigation test is performed to demonstrate the proposed method. The results show that the average AR success ratios of the three selected combinations, whose float ambiguity errors are 0.041, 0.146, 0.279 cycle respectively, are above 97.25% without regard to low-elevation C05. With respect to positioning accuracy based on our AR method when compared with IE software, the east, north, up error RMS of position are 0.042, 0.024, 0.069 m, respectively. In terms of the AR recover after the BeiDou signals outage, as long as 62 s BeiDou signal complete outage, all the ambiguities of all satellites could be re-fixed immediately. Besides, during the 90 s signals partial outage, the AR is not influenced by the position error, since the float ambiguity errors are all below half-cycle. The research of this contribution demonstrates the effectiveness of the proposed new method, which indicates it is applicable to kinematic positioning, even in BDS degraded and denied environments.