Uncalibrated Phase Delay Research Articles

UPDs estimation and ambiguity resolution performance evaluation of LEO navigation system

Low-Earth orbit (LEO) satellites have fast motion and thus provide rapid geometric change relation to the ground stations in a short period of time, which are expected to improve the positioning performance. In this study, a LEO constellation of 160 LEO satellites was simulated and simulated LEO observation data at global, European, and American stations were used to estimate the uncalibrated phase delays (UPDs) and achieve static and kinematic precise point positioning (PPP) ambiguity resolution (AR). The quality of the estimated UPD products were evaluated. More than 89 % and 96 % of the wide- and narrow-lane UPDs had a posteriori residual of less than 0.15 and 0.25 cycles, respectively. In static PPP, all ambiguity fixing rates were larger than 63 %. Compared with float solutions, the average convergence time of the fix solutions at the global, European and American stations was accelerated by 8 %, 6 % and 7 %, respectively; the average root-mean-square errors (RMSEs) in the east, north, and up components at the global, European and American stations were decreased by (22 %, 20 %, 15 %), (36 %, 34 %, 28 %) and (44 %, 31 %, 29 %), respectively. In kinematic PPP, all ambiguity fixing rates were larger than 40 %. Compared with float solutions, the average convergence time of the fix solutions at the global, European and American stations was accelerated by 4 %, 19 % and 14 %, respectively; the average RMSEs in the east, north, and up components at the global, European and American stations were decreased by (10 %, 10 %, 7 %), (27 %, 22 %, 11 %) and (32 %, 20 %, 10 %), respectively. In both static and kinematic PPP, AR accelerated the convergence time and improved the positioning accuracy.

Elevating performance in BDS-3 Precise Point Positioning Ambiguity Resolution through LEO augmented constellations

With the help of constellations combined BDS-3 (BeiDou Navigation Satellite System) and LEO (Low Earth Orbit), we aim to improve the performances of PPP (Precise Point Positioning) and PPP Ambiguity Resolution (PPP-AR). First, LEO constellations with 66, 96, 156, 192, 270, and 288 satellites are designed to establish six LEO augmented constellations. Based on four challenge scenarios, PPP of six LEO augmented constellations are performed. Even in the strip-shaped obstruction area losing more than 60% of observations, the positioning accuracies of static PPP are improved by (1.0%, 5.7%, 0.2%) for BDS+66LEO, and (14.9%, 33.7%, 7.5%) for BDS+288LEO in the east, north, and up components. The positioning accuracy and convergence time improve quickly as the number of LEO satellites increased from 66 to 156, and slowdown from 192 to 288. Two constellations, BDS+156LEO and BDS+192LEO, are considered to strike the balance of acceptable PPP performance and constellation load. Then, the performance of PPP-AR is realized with the estimated UPD (Uncalibrated Phase Delay). The correlations among geodetic parameters and ambiguities deriving from the post-fit variance–covariance matrix are analyzed to illustrate the enhancement of LEO augmented constellations. Under the condition of 20°elevation angle occlusion with azimuth ranging from 0 to 360°, BDS+156LEO can achieve an ambiguity fix rate of up to 98.37% with a faster mean TTFF (Time To First Fix) of 12.76 min, and the positioning accuracies in the east, north, and up components are improved by 28.2%, 21.4%, and 15.1% respectively, compared with BDS-3. BDS+192LEO realized an 98.42% ambiguity fixing rate, and improved BDS-3 by 31.4%, 26.7% and 11.5% with its TTFF been 12.25 min.

Real-Time Estimation of BDS-3 Satellite Clock Offset with Ambiguity Resolution Using B1C/B2a Signals

The third generation of the BeiDou navigation satellite system (BDS-3) can transmit five-frequency signals. The real-time satellite clock offset of BDS-3 is typically generated utilizing the B1I/B3I combination with the ambiguity-float solutions. By conducting the ambiguity resolution (AR), the reliability of the satellite clock offset can be improved. However, the performance of BDS-3 ambiguity-fixed real-time satellite clock offset with B1C/B2a signals remains unknown and unrevealed. In this contribution, the performance of the BDS-3 ambiguity-fixed satellite clock offset with the new B1C/B2a signals is investigated. One week of observation data from 85 stations was used to perform ambiguity-fixed satellite clock offset estimation. For B1I/B3I and B1C/B2a signals, the wide-lane (WL) uncalibrated phase delay (UPD) on the satellite end is fairly stable for one day, while the narrow-lane (NL) UPD standard deviation (STD) amounts to 0.122 and 0.081 cycles, respectively. The mean ambiguity fixing rate is 80.7% and 78.0% for these two signal combinations, and the time to first fix (TTFF) for the B1C/B2a signals is remarkably shorter than that of the B1I/B3I signals. The STDs of the ambiguity-float and -fixed satellite clock offsets are 0.033 and 0.026 ns, respectively, for the B1I/B3I combination, and it is reduced to 0.024 and 0.023 ns for B1C/B2a signals, respectively. Using the estimated UPD and clock offset products, the positioning performance of the kinematic Precise Point Positioning (PPP)-AR results amounts to 1.56, 1.23, and 4.46 cm in the east, north, and up directions for B1I/B3I signals, respectively. It is improved to 1.36, 1.16, and 4.25 cm using the products estimated with the B1C/B2a signals, with improvements of 12.8%, 5.7%, and 4.7% in three directions, respectively. The experiments showed that the performances of the ambiguity-fixed satellite clock offsets and the PPP-AR results using B1C/B2a signals are better than those of B1I/B3I.

Orbit and clock products for quad-system satellites with undifferenced ambiguity fixing approach

Integer Ambiguity Resolution (IAR) can significantly improve the accuracy of GNSS Precise Orbit Determination (POD). Traditionally, the IAR in POD is achieved at the Double Differenced (DD) level. In this contribution, we develop an Un-Differenced (UD) IAR method for Global Positioning System (GPS)+ BeiDou Navigation Satellite System (BDS) + Galileo navigation satellite system (Galileo)+ Global'naya Navigatsionnaya Sputnikovaya Sistema (GLONASS) quad-system POD by calibrating UD ambiguities in the raw carrier phase and generating the so-called carrier range. Based on this method, we generate the UD ambiguity-fixed orbit and clock products for the Wuhan Innovation Application Center (IAC) of the International GNSS Monitoring and Assessment System (iGMAS). One-year observations in 2020 from 150 stations are employed to investigate performance of orbit and clock products. Notably, the UD Ambiguity Resolution (AR) yields more resolved integer ambiguities than the traditional DD AR, scaling up to 9%, attributable to its avoidance of station baseline formation. Benefiting from the removal of ambiguity parameters, the computational efficiency of parameter estimation undergoes a substantial 70% improvement. Compared with the float solution, the orbit consistencies of UD AR solution achieve the accuracy of 1.9, 5.2, 2.8, 2.1, and 2.7 cm for GPS, BeiDou-2 Navigation Satellite System (BDS-2), BeiDou-3 Navigation Satellite System (BDS-3), Galileo, and GLONASS satellites respectively, reflecting enhancements of 40%, 24%, 54%, 34%, and 42%. Moreover, the standard deviations of Satellite Laser Ranging (SLR) residuals are spanning 2.5–3.5 cm, underscoring a comparable accuracy to the DD AR solution, with discrepancies below 5%. A notable advantage of UD AR lies in its capability to produce the Integer Recovered Clock (IRC), facilitating Precise Point Positioning (PPP) AR without requiring additional Uncalibrated Phase Delay (UPD) products. To assess the performance of quad-system kinematic PPP based on IRC, a network comprising 120 stations is utilized. In comparison to the float solution, the IRC-based PPP AR accelerates convergence time by 31% and enhance positioning accuracy in the east component by 54%.

Uncalibrated phase delay estimation and ambiguity resolution for BDS-3 six-frequency uncombined PPP with legacy B1I/B3I and new B1C/B2a/B2b/B2a + b signals

Multi-frequency integration can accelerate the ambiguity resolution (AR) and shorten the convergence time of precise point positioning (PPP). BDS-3 satellites can broadcast legacy B1I/B3I and new B1C/B2a/B2b/B2a + b signals, which facilitates the multi-frequency PPP. An uncalibrated phase delay (UPD) estimation method and a fast AR method are extended for BDS-3 six-frequency uncombined PPP. The characteristics of UPD estimates are analyzed in terms of stability, ambiguity residuals, and usage rate. The data sets on seven consecutive days from 224 globally distributed stations are selected for UPD estimation, and those from 21 stations are employed for PPP AR. The results indicate that the static convergence time of BDS-3 six-frequency PPP ambiguity-fixed solutions (with a threshold of 10 cm) can reach 8.2, 4.0, and 13.4 min in the east, north, and up directions, respectively, and the three-dimensional (3D) convergence time is shortened by 21.1 % and 26.5 % compared with the six-frequency ambiguity-float solutions and the dual-frequency ambiguity-fixed solutions, respectively. The corresponding kinematic convergence time can reach 13.0, 7.0, and 33.2 min in three directions, respectively, and the corresponding improvement rates are 21.0 % and 33.7 %, respectively. In addition, the convergence time of BDS-3 six-frequency PPP ambiguity-fixed solutions can be slightly shortened even compared with the five-frequency case. The static and kinematic positioning accuracy of BDS-3 six-frequency PPP ambiguity-fixed solutions can reach 4, 3, and 9 mm, and 18, 14, and 38 mm in three directions, respectively, which is slightly better than other cases.

Properties of multi-GNSS uncalibrated phase delays with considering satellite systems, receiver types, and network scales

The Wide-Lane (WL) and Narrow-Lane (NL) Uncalibrated Phase Delays (UPDs) are the prerequisites in the traditional Precise Point Positioning (PPP) Ambiguity Resolution (AR). As the generation mechanism of various biases becomes more complex, we systematically studied the impact factors of four satellite systems WL and NL UPDs from the perspective of parameter estimation. Approximately 100 stations in a global network are used to generate the UPDs. The results of different satellite systems show that the estimation method, update frequency, and solution mode need to be treated differently. Two regional networks with different receiver types, JAVAD, and Trimble, are also adopted. The results indicate that the receiver-dependent bias has an influence on UPD estimation. Also, the hardware delays can inhibit the satellite-side UPDs if these receiver-specific errors are not fully deployed or even misused. Furthermore, the temporal stability and residual distribution of NL UPDs are significantly enhanced by utilizing a regional network, with the improvements by over 68% and 40%, respectively. It demonstrates that different network scales exhibit the different implication of unmodeled errors, and the unmodeled errors cannot be ignored and must be handled in UPD estimation.

BDS-3 new signals observable-specific phase biases estimation and PPP ambiguity resolution

Nowadays, the BDS-3 satellites transmit new signals B1c and B2a that are compatible with GPS and Galileo, which is meaningful to fully take advantage of the BDS positioning capability. However, further research and analysis are needed to investigate the precise point positioning ambiguity resolution (PPP-AR) performance of the new BDS-3 signal combination. In this manuscript, the characteristics and quality of differential code bias (DCB), inter frequency clock bias (IFCB) and uncalibrated phase delay (UPD) of BDS-3 new signals are analyzed based on 30 days observation data from 70 multi GNSS experiment (MGEX) stations. The results show that the mean value of IFCB between B1I/B3I and B1C/B2a is 1.18 cm, which indicates that the IFCB can be ignored when performing the clock errors reference conversion of BDS-3 satellites. Besides, the DCB of the new BDS-3 signals varies within 0.2 ns over 30 days, and with a difference of 0.08 ns compared to the DCB product provided by chinese academy of science (CAS). The percentage of the estimated residuals less than 0.25 cycles for new signals WL UPD and less than 0.15 cycles for new signals NL UPD are 99.76% and 97.19%, respectively. The above results demonstrate the reliability of the DCB and UPD products, which can be used to estimate the carrier phase observable specific signal bias (OSB) product for the new signals. The average positioning errors of the BDS-3 new signals static PPP-AR in the N, E and U directions is 0.50 cm, 0.43 cm and 1.62 cm respectively, with an improvement of 18.3%, 28.3% and 11.5% compared to the ambiguity float static PPP. The average positioning errors of the kinematic PPP-AR are 2.09 cm, 2.67 cm and 4.79 cm in three directions, respectively, with an improvement of 17.1%, 23.9% and 15.7% compared to the ambiguity float PPP. The convergence time of the static and kinematic PPP-AR are 26.36 min and 29.12 min, respectively.

Continuous Decimeter-Level Positioning in Urban Environments Using Multi-Frequency GPS/BDS/Galileo PPP/INS Tightly Coupled Integration

Time- and precision-critical applications, such as autonomous driving, have extremely rigorous standards for continuous high-precision positioning. The importance of precise point positioning (PPP) technology for self-navigation equipment is self-evident by delivering decimeter-/centimeter-level absolute position accuracy in a global coordinate framework. Nevertheless, the prolonged initialization period renders PPP almost too difficult to be widely used for time-critical applications in real transportation scenarios, for example, in city canyons and overpasses. We proposed a method to accomplish continuous decimeter-level positioning by leveraging triple-frequency GNSS observations with an Inertial Navigation System (INS) aiding in urban environments. In the proposed method, the inertial measurements and the original tri-frequency pseudorange and carrier phase measurements are tightly fused in an extended Kalman filter to obtain the optimal state estimate. Afterwards, with precise extra-wide-lane (EWL) and wide-lane (WL) Uncalibrated Phase Delay (UPD) products, the EWL and WL ambiguities can be resolved sequentially to implement instantaneous decimeter-level positioning. Exploiting the short-term high-precision characteristic of the INS, the continuity of the PPP solution can be ameliorated noticeably, and the ability of decimeter-level positioning can be maintained effortlessly throughout the navigation process. With land vehicle data collected, several experiments are undertaken to comprehensively assess the capability of the proposed system in urban scenarios, taking the solution obtained by a commercial post-processing software as the reference. The single-epoch decimeter-level position estimation can be captured instantaneously for WL ambiguity-fixed PPP solutions. Furthermore, for WL ambiguity-fixed PPP/INS-integrated solutions, the position estimation is better than 0.2 m for horizontal components and improved by 40–90% compared with that of the WL ambiguity-fixed PPP solutions. More importantly, the availability rate of positioning accuracy is better than 0.3 m in the horizontal direction and 0.5 m in the up direction reaching 93.91%, whereas it is only 53.78% for WL ambiguity-fixed PPP solutions. Overall, the WL ambiguity-fixed PPP/INS-integrated solutions have the favorable performance to maintain continuous decimeter-level positioning for self-navigation equipment, even if full GNSS outages are encountered.

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.

Multipath mitigation for improving GPS narrow-lane uncalibrated phase delay estimation and speeding up PPP ambiguity resolution

Precise point positioning (PPP) has been recognized as a powerful tool for various geophysical applications. However, the long convergence time required to resolve a reliable ambiguity impedes its further application in time-critical scenarios. Although PPP ambiguity resolution (AR) can shorten the convergence time, its performance is subject to the quality of float ambiguity estimates and the uncalibrated phase delay (UPD), which can be contaminated by multipath errors. Furthermore, the observation residuals derived from PPP are very likely to be affected by the common-mode error (CME), thereby deteriorating the multipath modeling accuracy. The principal component analysis (PCA) is employed to mitigate the CME effect, and the multipath is modeled using a multipath hemispherical map (MHM). Consequently, the narrow-lane (NL) UPDs with multipath correction have better temporal stability and residual distributions than those without correction. Compared with sidereal filtering (SF), the MHM0.5 has comparable residual variance reduction percentages, indicating its capability of capturing high-frequency multipath. For static PPP AR, the averaged time to first fix (TTFF) can be reduced by 24.2% to about 26 min and the convergence time can be achieved within 16.2 min after multipath correction. The pseudorange multipath correction significantly contributes to shortening the TTFF and convergence time. Reducing the resolution of MHM increases the risk of extending the TTFF. For kinematic PPP AR with MHM0.5, the convergence time exhibits a remarkable improvement when compared with that of the uncorrected case (21.7 min versus 40.2 min), and 20% of the stations achieve convergence within 10 min. Meanwhile, a few stations only take one minute to achieve convergence. The contribution of the multipath correction to the fixing rate is comparatively small. After applying MHM0.5, the kinematic positioning accuracies are improved by 35.7%, 12.6%, and 24.4% to 1.26, 1.39, and 2.73 cm for the east, north, and up components, 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.

Inter-Satellite Single-Difference Ionospheric Delay Interpolation Model for PPP-RTK and Its Positioning Performance Verification

In PPP-RTK, obtaining accurate atmospheric delay information for the user through interpolation is one of the keys to achieving high-precision real-time positioning. The ionospheric delay that is extracted by a reference network based on uncalibrated phase delay (UPD) products is often difficult to separate from errors such as receiver code hardware delay and UPD reference error. Inter-satellite single-difference (SD) ionospheric delay information is typically provided to the user. This paper proposes an interpolation model that uses the atmospheric delay coefficient to represent the SD ionospheric delay, based on the mean position of the ionospheric pierce point (IPP) of each satellite pair and the center position of the network, which is called the differenced surface model (DSM). We chose four scenarios to compare the interpolation accuracy of the proposed model with the inverse distance-based linear interpolation method (DIM) and USM based on the difference between the longitude and latitude of the reference and ionospheric pierce point (IPP) of every satellite (here, we call it USM for short). The four scenarios involve a medium-scale reference network with an average distance to the reference station of 41 km, a large-scale reference network with an average distance to the reference station of 98 km, and out-of-network users, and a network with a common minimum of three reference stations. The results show that the root mean square (RMS) of the SD residuals of ionospheric delay for DSM were 1.4, 3.2, 2.2, and 1.4 cm, respectively, for the four scenarios that were considered, which are slightly better delay values than those that were achieved using DIM and USM. For the scenario with three reference stations, the interpolation accuracies of DIM and DSM were no different from those for four reference stations, indicating that the server can still try to provide ionospheric correction service under the condition of fewer reference stations. In contrast, USM could not provide service because it lacked the sufficient number of reference stations. DSM was used as the ionospheric delay interpolation model to analyze GPS and Galileo dual-system PPP-RTK positioning performance. In addition, the atmospheric parameter constraint method of users was used in PPP-RTK in reference networks of different scales. For the 41-km and 98-km reference networks, the time to first fix (TTFF) were 14.5 s and 33.1 s, respectively, and the mean RMS values for the east (E), north (N), and up (U) directions were 0.80, 0.93, and 2.72 cm, respectively, and 1.0, 1.1, and 4.0 cm, respectively, for a period of 5 min after convergence. The fixing rate and positioning accuracy of DSM during the 5-min period were better than those of DIM when the same empirical model was used to determine the mean square error of atmospheric delay.

Toward BDS/Galileo/GPS/QZSS triple-frequency PPP instantaneous integer ambiguity resolutions without atmosphere corrections

Multi-frequency precise point positioning (PPP) has drawn attention along with the modernization of the Global Navigation Satellite Systems. There are now nearly 90 satellites providing multi-frequency signals. This contribution aims to achieve fast convergence of a few seconds for BDS/Galileo/GPS/QZSS integrated triple-frequency PPP with integer ambiguity resolution (IAR) without atmosphere corrections. A unified model of an uncombined and undifferenced manner for PPP-IAR with dual- and triple-frequency observations is presented. The uncalibrated phase delays (UPD) of extra wide-lane (EWL), wide-lane (WL), and N1 ambiguities for triple-frequency PPP are estimated with standard deviations of 0.02, 0.05, and 0.10 cycles achieved, respectively. The PPP-IAR validation based on 20 stations evenly distributed in China is conducted using UPD products generated from a regional network covering a large part of China. The EWL, WL, and N1 ambiguities are sequentially fixed utilizing the least-squares ambiguity decorrelation adjustment (LAMBDA) technique. In terms of convergence time, PPP instantaneous IAR is achievable without using atmosphere corrections, thanks to the contribution of the multi-frequency and multi-constellation observations. This has been proved by performing PPP-IAR restart every 10-min over 2520 times in our case study. For PPP-IAR solutions produced with BDS/Galileo/GPS/QZSS triple-frequency observations with an interval of 1 s, the convergence is fulfilled within 1 s for the horizontal components with an accuracy of better than 5 cm, while 2 s for the vertical component with better than 10 cm accuracy, and both are at 95% confidence level.

PPP-RTK considering the ionosphere uncertainty with cross-validation

With the high-precision products of satellite orbit and clock, uncalibrated phase delay, and the atmosphere delay corrections, Precise Point Positioning (PPP) based on a Real-Time Kinematic (RTK) network is possible to rapidly achieve centimeter-level positioning accuracy. In the ionosphere-weighted PPP–RTK model, not only the a priori value of ionosphere but also its precision affect the convergence and accuracy of positioning. This study proposes a method to determine the precision of the interpolated slant ionospheric delay by cross-validation. The new method takes the high temporal and spatial variation into consideration. A distance-dependent function is built to represent the stochastic model of the slant ionospheric delay derived from each reference station, and an error model is built for each reference station on a five-minute piecewise basis. The user can interpolate ionospheric delay correction and the corresponding precision with an error function related to the distance and time of each reference station. With the European Reference Frame (EUREF) Permanent GNSS (Global Navigation Satellite Systems) network (EPN), and SONEL (Système d'Observation du Niveau des Eaux Littorales) GNSS stations covering most of Europe, the effectiveness of our wide-area ionosphere constraint method for PPP-RTK is validated, compared with the method with a fixed ionosphere precision threshold. It is shown that although the Root Mean Square (RMS) of the interpolated ionosphere error is within 5 cm in most of the areas, it exceeds 10 cm for some areas with sparse reference stations during some periods of time. The convergence time of the 90th percentile is 4.0 and 20.5 min for horizontal and vertical directions using Global Positioning System (GPS) kinematic solution, respectively, with the proposed method. This convergence is faster than those with the fixed ionosphere precision values of 1, 8, and 30 cm. The improvement with respect to the latter three solutions ranges from 10 to 60%. After integrating the Galileo navigation satellite system (Galileo), the convergence time of the 90th percentile for combined kinematic solutions is 2.0 and 9.0 min, with an improvement of 50.0% and 56.1% for horizontal and vertical directions, respectively, compared with the GPS-only solution. The average convergence time of GPS PPP-RTK for horizontal and vertical directions are 2.0 and 5.0 min, and those of GPS + Galileo PPP-RTK are 1.4 and 3.0 min, respectively.

Realization and evaluation of real-time uncombined GPS/Galileo/BDS PPP-RTK in the offshore area of China’s Bohai sea

The real-time kinematic (RTK) technology has been widely used as the high-precision positioning method in the offshore area. However, RTK requires a bi-directional communication and groups measurement errors together, thereby limiting its mass-market applications. Combining the advantages of precise point positioning (PPP) and RTK, PPP-RTK has become one of the hotspot technologies in the mass market. In this contribution, we propose the uncombined multi-GNSS PPP-RTK model using uncalibrated phase delays (UPDs) estimated from the legacy ionosphere-free and Melbourne-Wübbena combination. With the UPDs estimated based on 14 regional stations, we conduct PPP ambiguity resolution (AR) at 3 augmentation stations and derive precise atmospheric corrections, i.e., RMS of zenith tropospheric and slant ionospheric correction can be up to 4.89 mm and 2.20 cm, respectively. After applying atmospheric correction, the correct fixed solution of four on-board kinematic experiments can be better than 95% and the positioning accuracy can be better than 5 cm in both horizontal and vertical direction, showing the encouraging performance similar to RTK in the offshore area.

Uncertainty estimation of atmospheric corrections in large-scale reference networks for PPP-RTK

Regional Augmented Precise Point Positioning (RA-PPP) is a method used to support PPP-Real Time Kinematic (PPP-RTK) based on state space representation. In case of RA-PPP, uncalibrated phase delays and undifferenced ionospheric and tropospheric corrections generated by the reference station network are interpolated at the rover in real time. This study proposes a set of algorithms for estimating the uncertainties of atmospheric corrections that are compatible for use with the RA-PPP model. There are two different error sources of interpolated atmospheric corrections: one relates to the error propagation law and the other to the systematic bias of the interpolation model. We discuss the spatial changes in systemic errors of the interpolation model and present a new interpolation method and its use in calculating the uncertainties of interpolated atmospheric corrections. In addition, we verify the existence of systematic errors in the interpolation model and discuss the correlation between zenith tropospheric delay and slant ionospheric corrections. The results of PPP ambiguity resolution experiments using GPS/Galileo observations are presented. For a reference station network with an interstation distance of more than 110 km, when the proposed method is used, the rate of successfully fixing ambiguities reaches 67.8% in 5 min, and the fixing rate is higher than 90%. It is considered that our method has enhanced applicability for use with large networks and in complex environments.

Precise orbit determination for LEO satellites: single-receiver ambiguity resolution using GREAT products

ABSTRACT In recent years, the large Low Earth Orbit (LEO) constellations have become a hot topic due to their great potential to improve the Global Navigation Satellite Systems (GNSS) positioning performance. One of the important focus is how to obtain the accurate and reliable orbits for these constellations with dozens of LEO satellites. The GNSS-based Precise Orbit Determination (POD) will be exclusively performed to achieve this goal, where the Integer Ambiguity Resolution (IAR) plays a key role in acquiring high-quality orbits. In this study, we present a comprehensive analysis of the benefit of the single-receiver IAR in LEO POD and discuss its implication for the future LEO constellations. We perform ambiguity-fixed LEO POD for four typical missions, including Gravity Recovery and Climate Experiment (GRACE) Follow-On (GRACE-FO), Swarm, Jason-3 and Sentinel-3, using the Uncalibrated Phase Delay (UPD) products generated by our GREAT (GNSS+ REsearch, Application and Teaching) software. The results show that the ambiguity fixing processing can significantly improve the accuracy of LEO orbits. There are negligible differences between our UPD-based ambiguity-fixed orbits and those based on the Observable Signal Bias (OSB) and Integer Recovery Clock (IRC) products, indicating the good-quality of UPD products we generated. Compared to the float solution, the fixed solution presents a better consistency with the external precise science orbits and the largest accuracy improvement of 5 mm is achieved for GRACE-FO satellites. Meanwhile, the benefit can be observed in laser ranging residuals as well, with a Standard Deviation (STD) reduction of 3–4 mm on average for the fixed solutions. Apart from the absolute orbits, the relative accuracy of the space baseline is also improved by 20–30% in the fixed solutions. The result demonstrates the superior performance of the ambiguity-fixed LEO POD, which appears as a particularly promising technique for POD of future LEO constellations.

Improve real-time GNSS orbit with epoch-independent undifferenced ambiguity resolution

AR (Ambiguity Resolution) is one of the critical issues in high-accuracy GNSS (Global Navigation Satellite System) applications. Commonly, the double-difference AR technique is employed in GNSS network data processing. With the demand of high-accuracy, low-latency and high-integrity GNSS orbit, the orbit estimation in sequential mode becomes a candidate of that in batch mode. Unlike orbit estimation with double-difference AR in either batch or sequential mode, researches on real-time orbit estimation with undifferenced AR in sequential mode is still limited. In this study, real-time orbit determination with epoch-independent undifferenced AR in sequential mode is assessed. With one-week observation data from about 100 global distributed IGS (International GNSS Service) network stations, simulated real-time Galileo orbit determination are conducted with GFZ post-processed solution as reference for validation. Numerical results indicate that, first, the 3D error RMS (Root Mean Square) of fixed orbit is about 30% smaller than that of float orbit; second, the Galileo orbit error RMS reach 3.66, 3.60 and 3.09 cm for the radial, tangential and normal directions respectively for the fixed solution; third, small jumps exist in the real-time orbit and may be attributed to the temporal variations of UPD (Uncalibrated Phase Delay) and wrongly fixed ambiguities. In summary, the orbit quality is comparable to that of IGS real-time solutions, and further improvements may be achieved with improved AR constraint strategy and background orbit model.

High-Accuracy Attitude Determination Using Single-Difference Observables Based on Multi-Antenna GNSS Receiver with a Common Clock

A global navigation satellite system (GNSS) receiver with multi-antenna using clock synchronization technology is a powerful piece of equipment for precise attitude determination and reducing costs. The single-difference (SD) can eliminate both the satellites and receiver clock errors with the common clock between antennas, which benefits the GNSS short-baseline attitude determination due to its lower noise, higher redundancy and stronger function model strength. However, the existence of uncalibrated phase delay (UPD) makes it difficult to obtain fixed SD attitude solutions. Therefore, the key problem for the fixed SD attitude solutions is to separate the SD UPD and fix the SD ambiguities into integers between antennas. This article introduces the one-step ambiguity substitution approach to separate the SD UPD, through which we merge the SD UPD parameter with the SD ambiguity of the reference satellite ambiguity as the new SD UPD parameter. Reconstructing the other SD ambiguities, the rank deficiency can be remedied by nature, and the new SD ambiguities can have a natural integer feature. Finally, the fixed SD baseline and attitude solutions are obtained by combining the ambiguity substitution approach with integer ambiguity resolution (IAR). To verify the effect of the ambiguity substitution approach and the advantages of the SD observables with a common clock in practical applications, we conducted static, kinematic, and vehicle experiments. In static experiments, the root mean squared errors (RMSEs) of the yaw and pitch angles obtained by the SD observables with a common clock were improved by approximately 80% and 93%, respectively, compared to double-difference (DD) observables with a common clock in multi-day attitude solutions. The kinematic results show that the dispersion of the SD-Fix in the pitch angle is two times less that of the DD-Fix, and the standard deviations (STDs) of the pitch angle for SD-Fix can reach 0.02°. Based on the feasibility, five bridges with low pitch angles in the vehicle experiment environment, which the DD observables cannot detect, were detected by the SD observables with a common clock. The attitude angles obtained by the SD observables were also consistent with the fiber optic gyroscope (FOG) inertial navigation system (INS). This research on the SD observables with a common clock provides higher accuracy.

Calibrating receiver-type-dependent wide-lane uncalibrated phase delay biases for PPP integer ambiguity resolution

Wide-lane (WL) uncalibrated phase delay (UPD) is usually derived from Melbourne–Wübbena (MW) linear combination and is a prerequisite in Global Navigation Satellite Systems (GNSS) precise point positioning (PPP) ambiguity resolution (AR). MW is a linear combination of pseudorange and phase, and the accuracy is limited by the larger pseudorange noise which is about one hundred times of the carrier phase noise. However, there exist inconsistent pseudorange biases which may have detrimental effect on the WL UPD estimation, and further degrade user-side ambiguity fixing. Currently, only the large part of pseudorange biases, e.g., the differential code bias (DCB), are available and corrected in PPP-AR, while the receiver-type-dependent biases have not yet been considered. Ignoring such kind of bias, which could be up to 20 cm, will cause the ambiguity fixing failure, or even worse, the incorrect ambiguity fixing. In this study, we demonstrate the receiver-type-dependent WL UPD biases and investigate their temporal and spatial stability, and further propose the method to precisely estimate these biases and apply the corrections to improve the user-side PPP-AR. Using a large data set of 1560 GNSS stations during a 30-day period, we demonstrate that the WL UPD deviations among different types of receivers can reach ± 0.3 cycles. It is also shown that such kind of deviations can be calibrated with a precision of about 0.03 cycles for all Global Positioning System (GPS) satellites. On the user side, ignoring the receiver-dependent UPD deviation can cause significant positioning error up to 10 cm. By correcting the deviations, the positioning performance can be improved by up to 50%, and the fixing rate can also be improved by 10%. This study demonstrates that for the precise and reliable PPP-AR, the receiver-dependent UPD deviations cannot be ignored and have to be handled.