Ionosphere-free Combination Research Articles

Research on Accelerating Single-Frequency Precise Point Positioning Convergence with Atmospheric Constraint

An increasing number of researchers have conducted in-depth research on the advantages of low-cost single-frequency (SF) receivers, which can effectively use ionospheric information when compared to dual-frequency ionospheric-free combination. However, SF observations are bound to increase the unknown parameters and prolong the convergence time. It is desirable if the convergence time can be reduced by external information constraints, for example atmospheric constraints, which include ionosphere- or troposphere constraints. In this study, ionospheric delay constraints, tropospheric delay constraints, and their dual constraints were considered. Additionally, a total of 18,720 test experiments were performed. First, the nearest-neighbor extrapolation (NENE), bilinear- (BILI), bicubic- (BICU), and Junkins weighted-interpolation (JUNK) method of Global Ionospheric Map (GIM) grid products were analyzed. The statistically verified BILI in the percentage of convergence time, average convergence time, and computation time consumption of them shows a good advantage. Next, the influences of global troposphere- and ionosphere-constrained on the convergence time of SF Precise Point Positioning (PPP) were analyzed. It is verified that the ionosphere-constrained (TIC2) has significant influence on the convergence time in the horizontal and vertical components, while the troposphere-constrained (TIC1) has better effect on the convergence time in the vertical components within some thresholds. Of course, the dual constraint (TIC3) has the shortest average convergence time, which is at least 46.5% shorter in static mode and 5.4% in kinematic mode than standard SF PPP (TIC0).

Unmodeled error mitigation for single-frequency multi-GNSS precise positioning based on multi-epoch partial parameterization

In global navigation satellite system (GNSS) precise positioning, some residual systematic errors inevitably exist in GNSS observations. These residual systematic errors can be regarded as the unmodeled errors that cannot be easily mitigated by traditional parameterization, modeling or other methods, mainly due to their spatiotemporal complexity. In a multi-GNSS scenario, there are enough redundant observations to reveal the unmodeled errors to a great extent, according to the behaviors of observation residuals. However, in the case of single frequency, the first-order ionospheric delays cannot be eliminated by the ionosphere-free combination in terms of two or more carrier phases with different frequencies, thus resulting in significant unmodeled effects more frequently. We therefore propose a method for real-time mitigation of the unmodeled errors in single-frequency multi-GNSS precise positioning. The proposed method has two main aspects. First, considering the estimability and property of unmodeled error parameters, the method of multi-epoch partial parameterization is proposed. Second, an iterative procedure is given, which can be applied in real time. The experiment is conducted using a four-station network of real single-frequency and combined global positioning system (GPS) and BeiDou navigation satellite system (BDS) observations, with baseline lengths from approximately 5 to 25 km. The results show that the GPS/BDS unmodeled errors are indeed mitigated and the positioning precision is improved after using the proposed method, where the precision of the up direction can be improved by a maximum of 56.90%.

Single-frequency PPP models: analytical and numerical comparison

Ionosphere delay is a key factor in the single-frequency Precise Point Positioning (SFPPP). In tradition, two SFPPP models are applied, i.e., ionosphere-corrected (IC) and ionosphere-free-half (IFH) models. The ionospheric delays are directly corrected in IC model with external ionospheric products, while they are eliminated by forming the ionosphere-free combination with code and phase in IFH model. However, almost all studies focus on the numerical performance of these two models and lack the comprehensive study on the estimability and solvability of SFPPP model with either code division multiple access (CDMA) or frequency division multiple access (FDMA) system, respectively. In this paper, we dedicate to the analytical study on SFPPP models for both CDMA and FDMA systems. To assimilate the impact of ionospheric delays on positioning, a general SFPPP model, i.e., ionosphere-weighted (IW) model, is first formulated to identify the varying situations with the different uncertainties of ionospheric constraints. Then, we mathematically show how the IC, IFH and ionosphere-float (IF) models are reduced from IW model. The numerical comparison with GPS and GLONASS data with geodetic and cost-effective receivers effectively confirms our theoretical inference on the relationship of IC, IF and IW models and indicates the best results of IW model for all situations.

Real-time precise point positioning with a low-cost dual-frequency GNSS device

In recent years, there is an increasing demand for precise positioning with low-cost GNSS devices in support of applications from self-driving cars to unmanned aerial vehicles. Although single-frequency GNSS devices are still dominant in the low-cost market to date, some GNSS manufacturers have released low-cost dual-frequency GNSS devices, which are able to track new civilian signals such as L2C or L5. With dual-frequency GNSS measurements, the ionospheric delays can be eliminated by forming ionospheric-free combinations to further improve the positioning accuracy and reliability with low-cost GNSS devices. Extensive work has been conducted in the past for precise point positioning using high-end dual-frequency GNSS receivers. For low-cost GNSS-based PPP, there are some issues that need to be addressed. One is that current low-cost dual-frequency receivers can track only civil signals but not all GPS satellites currently transmit L2C or L5 civil signals. This means that fewer dual-frequency GNSS measurements are available for position determination. Another is that the measurement quality of low-cost receivers is not as good as high-end receivers. The above will not only increase the convergence time but also affect the obtainable positioning accuracy. We propose a new method in which not only the conventional dual-frequency ionospheric-free code and phase measurements, but also the single-frequency ionosphere-corrected code measurements with precise ionospheric products, are adopted for position determination. To be more specific, ionospheric-free code and phase combinations are applied to satellites with the second civil signal, while the single-frequency ionosphere-corrected code measurement is applied to all observed satellites. Both stationary and automotive experiments have been conducted to assess the performance of the new method. The field test results show that it can quickly reach half-meter accuracy in horizontal at a much faster convergence speed than the conventional DF-PPP which would usually take several minutes to reach a similar accuracy. This indicates that the new method is more suitable for mass-market applications with low-cost GNSS devices.

Detection of Structural Vibration with High-Rate Precise Point Positioning: Case Study Results Based on 100 Hz Multi-GNSS Observables and Shake-Table Simulation.

This contribution presents and assesses the methodology aiming at the characterization of the structural vibrations with high-rate GNSS measurements. As commonly employed precise point positioning (PPP) based on ionosphere-free linear combination of undifferenced signals may not meet the high requirements in terms of displacement precision, a modified processing strategy has been proposed. The algorithms were implemented in the own-developed GNSS processing software and validated using the designed experiment. For this purpose, we have set up a field experiment taking advantage of the prototype shake-table, which simulated the dynamic horizontal displacements of the GNSS antenna. The device ensured a periodic motion of the antenna with modifiable characteristics, namely amplitude and frequency. In this experiment, we have set the amplitudes from 1.5 to 9 mm and the frequency to 3.80 Hz. As a dataset, we have used 100 Hz GPS, Galileo, and BDS measurements. The results confirmed a high applicability of the enhanced PPP processing strategy for precise displacement detection. Specifically, it was feasible to obtain the dynamic displacements with precision at the level of millimeters. The differences between the PPP-derived amplitude and the true amplitude of the simulated displacements were in the range of 0.5–1.3 mm, whereas the difference between the detected and benchmark frequency did not exceed 0.026 Hz. Hence, the proposed methodology allows meeting the specific demands of structural displacement monitoring.

GPS satellite inter-frequency clock bias estimation using triple-frequency raw observations

This study proposes a unified uncombined model to estimate GPS satellite inter-frequency clock bias (IFCB) in both triple-frequency code and carrier-phase observations. In the proposed model, the formulae of both phase-based and code-based IFCBs are rigorously derived. Specifically, satellite phase-based IFCB refers to its time-variant part and it is modeled as a periodic function related to the sun–spacecraft–earth angle. A zero-mean condition of all available GPS satellites that support triple-frequency data is introduced to render satellite code-based IFCB estimable. Three months of data from 40 globally distributed stations of the International GNSS Service Multi-GNSS Experiment are used to test our method. The results show that the four-order periodic function is suitable for eliminating the 12-h, 6-h, 4-h, and 3-h periods that exist in the a posteriori phase residuals when no periodic function is used. For comparison, the geometry-free and ionosphere-free (GFIF) phase combination and differential code bias (DCB) products released by DLR (German Aerospace Center) and IGG (Institute of Geodesy and Geophysics, China) are also used to calculate the satellite phase-based and code-based IFCBs, respectively. The results show that (1) the average root mean square (RMS) of the phase-based IFCB difference between the proposed method and the GFIF phase combination is 4.3 mm; (2) the average RMS in the eclipse period increased by 50% compared with the average RMS in the eclipse-free period; (3) the mean monthly STD for code-based IFCB from the proposed method is 0.09 ns; and (4) the average RMS values of code-based IFCB differences between the proposed method and the DCB products released by DLR and IGG are 0.32 and 0.38 ns. This proposed model also provides a general approach for multi-frequency GNSS applications such as precise orbit and clock determination.

Assessing the quality of ionospheric models through GNSS positioning error: methodology and results

Single-frequency users of the global navigation satellite system (GNSS) must correct for the ionospheric delay. These corrections are available from global ionospheric models (GIMs). Therefore, the accuracy of the GIM is important because the unmodeled or incorrectly part of ionospheric delay contributes to the positioning error of GNSS-based positioning. However, the positioning error of receivers located at known coordinates can be used to infer the accuracy of GIMs in a simple manner. This is why assessment of GIMs by means of the position domain is often used as an alternative to assessments in the ionospheric delay domain. The latter method requires accurate reference ionospheric values obtained from a network solution and complex geodetic modeling. However, evaluations using the positioning error method present several difficulties, as evidenced in recent works, that can lead to inconsistent results compared to the tests using the ionospheric delay domain. We analyze the reasons why such inconsistencies occur, applying both methodologies. We have computed the position of 34 permanent stations for the entire year of 2014 within the last Solar Maximum. The positioning tests have been done using code pseudoranges and carrier-phase leveled (CCL) measurements. We identify the error sources that make it difficult to distinguish the part of the positioning error that is attributable to the ionospheric correction: the measurement noise, pseudorange multipath, evaluation metric, and outliers. Once these error sources are considered, we obtain equivalent results to those found in the ionospheric delay domain assessments. Accurate GIMs can provide single-frequency navigation positioning at the decimeter level using CCL measurements and better positions than those obtained using the dual-frequency ionospheric-free combination of pseudoranges. Finally, some recommendations are provided for further studies of ionospheric models using the position domain method.

A comparison of three widely used GPS triple-frequency precise point positioning models

Triple-frequency signals can be transmitted by the GPS Block IIF satellites. With triple-frequency integration, more choices are available for the implementation of precise point positioning (PPP), namely the PPP using L1/L2 and L1/L5 dual-frequency ionospheric-free (IF) combinations (IF-PPP1), the PPP using the triple-frequency IF combination (IF-PPP2), and the PPP with uncombined (UC) measurements (UC-PPP). A comprehensive comparison of the three GPS triple-frequency PPP models is carried out in this study. A total of 197 3-h datasets with pure triple-frequency observables are used for analysis in the static situation. The mathematical equivalence among the three PPP models in the converged stage is proven. Only an accuracy difference of smaller than 6 mm exists among the three triple-frequency PPP models. For all the three models, the positioning accuracies are 37.2–42.9, 11.6–13.0 and 36.2–40.6 mm in east, north and vertical directions, respectively. As to the average convergence time, IF-PPP2 shows the best performance, which is 29.7, 11.0 and 29.9 min in the three directions, respectively. The improvement for IF-PPP2 on the average convergence time is 6%, 3% and 1% over IF-PPP1, and 17%, 13% and 20% over UC-PPP in the three directions, respectively. These numerical differences may be attributed to the different parameterization and noise levels for the three triple-frequency PPP models.

CASSIOPE orbit and attitude determination using commercial off-the-shelf GPS receivers

As part of the “GPS Attitude, Positioning, and Profiling experiment (GAP)” of the Canadian CASSIOPE science and technology mission, a set of four geodetic GPS receivers connected to independent antennas on the top-panel of the spacecraft can be operated concurrently to collect dual-frequency code and phase measurements on both the L1 and L2 frequencies. The qualification of the commercial off-the-shelf (COTS) GPS receivers is discussed, and flight results of precise orbit and attitude determination are presented. Pseudorange and carrier phase errors amount to roughly 65 cm and 8 mm for the ionosphere-free dual-frequency combination, which compares favorably with other missions using fully qualified space GPS receivers and is mainly limited by choice of simple patch antennas without choke rings. Precise orbit determination of CASSIOPE using GPS observations can achieve decimeter-level accuracy during continued operations but suffers from onboard and mission restrictions that limit the typical data availability to less than 50% of each day and induce regular long-duration gaps of 4–10 h. Based on overlap analyses, daily peak orbit determination errors can, however, be confined to 1 m 3D on 84% of all days, which fulfills the mission needs for science data processing of other instruments. The attitude of CASSIOPE can be determined with a representative precision of about 0.2° in the individual axes using three GAP receivers and antennas. Availability of dual-frequency measurements is particularly beneficial and enables single-epoch ambiguity fixing in about 97% of all epochs. Overall, the GAP experiment demonstrates the feasibility of using COTS-based global navigation satellite system receivers in space and the benefits they can bring for small-scale science missions.

Calibration of BeiDou Triple-Frequency Receiver-Related Pseudorange Biases and Their Application in BDS Precise Positioning and Ambiguity Resolution.

Global Navigation Satellite System pseudorange biases are of great importance for precise positioning, timing and ionospheric modeling. The existence of BeiDou Navigation Satellite System (BDS) receiver-related pseudorange biases will lead to the loss of precision in the BDS satellite clock, differential code bias estimation, and other precise applications, especially when inhomogeneous receivers are used. In order to improve the performance of BDS precise applications, two ionosphere-free and geometry-free combinations and ionosphere-free pseudorange residuals are proposed to calibrate the raw receiver-related pseudorange biases of BDS on each frequency. Then, the BDS triple-frequency receiver-related pseudorange biases of seven different manufacturers and twelve receiver models are calibrated. Finally, the effects of receiver-related pseudorange bias are analyzed by BDS single-frequency single point positioning (SPP), single- and dual-frequency precise point positioning (PPP), wide-lane uncalibrated phase delay (UPD) estimation, and ambiguity resolution, respectively. The results show that the BDS SPP performance can be significantly improved by correcting the receiver-related pseudorange biases and the accuracy improvement is about 20% on average. Moreover, the accuracy of single- and dual-frequency PPP is improved mainly due to a faster convergence when the receiver-related pseudorange biases are corrected. On the other hand, the consistency of wide-lane UPD among different stations is improved significantly and the standard deviation of wide-lane UPD residuals is decreased from 0.195 to 0.061 cycles. The average success rate of wide-lane ambiguity resolution is improved about 42.10%.

Characterizing inter-frequency bias and signal quality for GLONASS satellites with triple-frequency transmissions

The GLONASS SVNs 702K (R09), 755 (R21) and 701K (R26) satellites currently provide G1, G2 and G3 signals. The difference between satellite clocks calculated by G1/G2 and G1/G3 ionospheric-free combinations, termed inter-frequency bias (IFB), is identified. The presence of IFB limits the application of G3 signal in precise positioning. The IFB is investigated using the datasets from 70 stations with a global distribution spanning 30 consecutive days. The epoch-wise phase-specific IFB (PIFB) estimates show periodic variations with a period of eight days and an average peak-to-peak amplitude of 0.107, 0.327 and 1.663 m for the R09, R21 and R26 satellites, respectively. The daily stable code-specific IFB (CIFB) estimates also show 8-day periodic signal. The day-to-day scattering of daily stable CIFB is 0.060–0.085 m. The estimation accuracy and prediction accuracy of PIFB are 0.025 and 0.019 m, respectively, while the corresponding statistics for the daily stable CIFB are 0.452 and 0.056 m, respectively. A modified estimation approach is developed to derive the time-varying epoch-wise CIFB. The epoch-wise CIFB and PIFB shows sub-daily periodic variations with the most notable periods of 5.625 and 11.250 h, respectively. The correction rate is 32% in terms of the prediction of the time-varying part of the epoch-wise CIFB. In addition, the signal quality is assessed from such aspects as carrier-to-noise density ratio, measurement noise and multipath errors.

Characteristics of BD3 Global Service Satellites: POD, Open Service Signal and Atomic Clock Performance

The Chinese BeiDou Navigation Satellite System has provided a global-coverage service since 27 December 2018. Eighteen BD3 MEO satellites have been launched into space during 2017 and 2018. The signal constitution has been redesigned and four open service signals are used for transmission, including B1I, B1C, B2a and B3I. This paper focuses on the signal performance, Precise Orbit Determination (POD) and the atomic clock’s frequency stability issues of the BD3 satellites. The satellite-induced code bias issue found in BD2 satellites multipath combination has been proven to be eliminated in BD3 satellites. However, the pseudorange code of B1C is much noisier than that of other three frequencies, which may be related to the signal constitution and power distribution, as the minimum received power levels on the ground of B1C is 3 dB lower than that of the B2a signal. Similar results were achieved by the Ionosphere-Free combination residuals in POD using four signals, B1I-B3I, B1I-B2a, B1C-B3I and B1C-B2a, and the phase residual of B1C-B2a combination performed best. Considering the noise amplitude and compatibility with other GNSS (Global Navigation Satellite System), the B1C-B2a combination is recommended in priority for precise GNSS data processing. GFIFP combinations were also implemented to evaluate the inter-frequency phase bias of the four signals. The experimental results showed that the systematic signal with an amplitude of about 2 cm could be found in the GFIFP series. In addition, multi-GNSS POD was performed and analyzed as well, using about a hundred global-distributed IGS and iGMAS stations. Furthermore, the atomic clock’s frequency stability was estimated using the parameters of clock bias calculated in POD and the Overlap Allan Deviations showed that the frequency stability of BD3 reached approximately 2.43 × 10−14 at intervals of 10,000 s and 2.51 × 10−15 at intervals of 86,400 s, which was better than that of the GPS BLOCK IIF satellites but worse than that of Galileo satellites.

A method for precise point positioning with integer ambiguity resolution using triple-frequency GNSS data

This paper proposes a method for precise point positioning with integer ambiguity resolution (PPP-AR) using triple-frequency global navigation satellite systems (GNSS) data. Firstly, an enhanced linear combination is developed for rapid fixing of the extra wide-lane (EWL) and wide lane (WL) ambiguities. This combination has improved performance compared to the Melbourne–Wübbena linear combination, and has 6.7% lower measurement error for the GPS L1/L2 signals, 12.7% lower error for L1/L5, and 0.7% lower error for L2/L5. Comparable improvements were also determined for the Beidou and Galileo constellations.After fixing the EWL/WL ambiguities, a full-rank, triple-frequency carrier-phase-only PPP model is proposed with ionosphere constraints. The probability of AR success rate (Ps) is analysed with the LAMBDA method, using a range of carrier phase and regional ionospheric model (RIM) precisions. Results show that a Ps of 99% is achieved within four epochs of data with carrier phase std = 0.002 m and RIM std = 0.1 total electron content unit (TECU); and within six epochs when RIM std = 0.5 TECU. When the carrier phase std was increased to 0.02 m (depicting high multipath conditions), and with use of a low-precision RIM (std = 0.5 TECU), the proposed method gave significantly improved performance over the method proposed by Li et al (2014 GPS Solut. 18 429–42).The direct estimation of the more challenging narrow-lane (NL) integer ambiguity is analysed by multi-epoch averaging of a proposed geometry-free and ionosphere-free triple-frequency linear combination. Tests with GPS data showed that 65.4% of the NL ambiguities were fixed within 10 min, 90.2% within 20 min, and 95.6% within 30 min.

Performance Evaluation of the CNAV Broadcast Ephemeris

The new Global Positioning System (GPS) Civil Navigation Message (CNAV) has been transmitted by Block IIR-M and Block IIF satellites since April 2014, both on the L2C and L5 signals. Compared to the Legacy Navigation Message (LNAV), the CNAV message provides six additional parameters (two orbit parameters and four Inter-Signal Correction (ISC) parameters) for prospective civil users. Using the precise products of the International Global Navigation Satellite System Service (IGS), we evaluate the precision of satellite orbit, clock and ISCs of the CNAV. Additionally, the contribution of the six new parameters to GPS Single Point Positioning (SPP) is analysed using data from 22 selected Multi-Global Navigation Satellite System Experiment (MGEX) stations from a 30-day period. The results indicate that the CNAV/LNAV Signal-In-Space Range Error (SISRE) and orbit-only SISRE from January 2016 to March 2018 is of 0·5 m and 0·3 m respectively, which is improved in comparison with the results from an earlier period. The ISC precision of L1 Coarse/Acquisition (C/A) is better than 0·1 ns, and those of L2C and L5Q5 are about 0·4 ns. Remarkably, ISC correction has little effect on the single-frequency SPP for GPS users using civil signals (for example, L1C, L2C), whereas dual-frequency SPP with the consideration of ISCs results have an accuracy improvement of 18·6%, which is comparable with positioning accuracy based on an ionosphere-free combination of the L1P (Y) and L2P (Y) signals.

Comparison between UofC Model and Ionosphere-free Combination Model in PPP

GPS Navigation Satellite System is a satellite-based positioning system, whose accuracy and reliability depend on the number of observable satellites. The precise point positioning (PPP) in GPS navigation satellite system is a technique which can provide the centimeter-level positioning by a dual-frequency receiver and thus the research on precise point positioning is of great significance. In this paper, the ionosphere correction models of precise point positioning (PPP) are studied, including ionosphere-free combination model and UofC model. The static GPS positioning data obtained from the experiment is used and we compare the positioning accuracy of the two models. The result shows that the positioning precision using UofC model is decimeter grade, while the positioning precision using ionosphere-free combination model is centimeter level.

Characteristics of inter-system biases in Multi-GNSS with precise point positioning

The impact of inter-system bias (ISB) on integrated precise point positioning (PPP) in multi-GNSS cannot be ignored. Precise orbit/clock products of Multi-GNSS Experiment (MGEX) satellites are diverse and gradually maturing. Analysing short-term and long-term variation characteristics of different types of ISB is thus helpful in understanding their error characteristics. In this study, ISB estimation models were proposed for GPS, GLONASS, BDS, and GALILEO systems with ionosphere-free (IF) combination observations. Then, the characteristics of ISB were mostly analyzed in detail with reverse and forward filtering method. Finally, according to the diurnal variation characteristics of ISB, which were estimated using precise products from several analysis centres, different processing strategies were adopted for ISBs to determine the optional solution strategies. Preliminary results are as follows: (1) The daily mean values of ISB estimated using the same analysis centre products between GPS and GLONASS as well as between GPS and BDS differed, but they were significantly correlated with receiver types. (2) Precise products of the same satellite system exhibited systematic deviations between analysis centres, although they were consistent on the weekly scale; however, the system bias of different satellite systems of the same analysis centres and system bias of different analysis centres all differed. (3) The daily and weekly stability of ISB estimated using different analytical centre products showed similar characteristics. Furthermore, the intra-week daily stability and weekly stability of the ISB of different stations showed good consistency. (4) Considering the strength of the observed model and the stability and reliability of the positioning results, Multi-GNSS PPP with precise products of CODE, GFZ, iGMAS and WHU showed the best performance under constant estimation, 20 min piecewise constant estimation, 1 h piecewise constant estimation and constant estimation strategies, respectively.

Galileo and QZSS precise orbit and clock determination using new satellite metadata

During 2016–2018, satellite metadata/information including antenna parameters, attitude laws and physical characteristics such as mass, dimensions and optical properties were released for Galileo and QZSS (except for the QZS-1 optical coefficients). These metadata are critical for improving the accuracy of precise orbit and clock determination. In this contribution, we evaluate the benefits of these new metadata to orbit and clock in three aspects: the phase center offsets and variations (PCO and PCV), the yaw-attitude model and solar radiation pressure (SRP) model. The updating of Galileo PCO and PCV corrections, from the values estimated by Deutsches Zentrum fur Luft- und Raumfahrt and Deutsches GeoForschungsZentrum to the chamber calibrations disclosed by new metadata, has only a slight influence on Galileo orbits, with overlap differences within only 1 mm. By modeling the yaw attitude of Galileo satellites and QZS-2 spacecraft (SVN J002) according to new published attitude laws, the residuals of ionosphere-free carrier-phase combinations can be obviously decreased in yaw maneuver seasons. With the new attitude models, the 3D overlap RMS in eclipse seasons can be decreased from 12.3 cm, 14.7 cm, 16.8 cm and 34.7 cm to 11.7 cm, 13.4 cm, 15.8 cm and 32.9 cm for Galileo In-Orbit Validation (IOV), Full Operational Capability (FOC), FOC in elliptical orbits (FOCe) and QZS-2 satellites, respectively. By applying the a priori box-wing SRP model with new satellite dimensions and optical coefficients, the 3D overlap RMS are 5.3 cm, 6.2 cm, 5.3 cm and 16.6 cm for Galileo IOV, FOCe, FOC and QZS-2 satellites, with improvements of 11.0%, 14.7%, 14.0% and 13.8% when compared with the updated Extended CODE Orbit Model (ECOM2). The satellite laser ranging (SLR) validation reveals that the a priori box-wing model has smaller mean biases of − 0.4 cm, − 0.4 cm and 0.6 cm for Galileo FOCe, FOC and QZS-2 satellites, while a slightly larger mean bias of − 1.0 cm is observed for Galileo IOV satellites. Moreover, the SLR residual dependencies of Galileo IOV and FOC satellites on the elongation angle almost vanish when the a priori box-wing SRP model is applied. As for satellite clocks, a visible bump appears in the Modified Allan deviation at integration time of 20,000 s for Galileo Passive Hydrogen Maser with ECOM2, while it almost vanishes when the a priori box-wing SRP model and new metadata are applied. The standard deviations of clock overlap can also be significantly reduced by using new metadata.

Optimal Carrier-Phase Integer Combinations for Modernized Triple-Frequency BDS in Precise Point Positioning

The latest generation of Global Navigation Satellite System (GNSS) satellites are transmitting signals on three or more frequencies, it brings new opportunities and challenges for data integration in Multi–GNSS Experiment (MGEX). To reduce the convergence time, less computationally intensive method is three-carrier ambiguity resolution (TCAR). But the three combinaitons can not be found in precise point positioning (PPP) yet, especially for the third generation BeiDou Navigation Satellite System (BDS-3). This contribution concentrates on the multi-frequency carrier-phase integer combinations for BDS-3 in PPP. More specifically, the triple-frequency plane is degraded to two-dimensional plane for BDS-3 under the assumption of a low observation noise. And then third frequency coefficient should be limited as a constraint, considering that inter frequency bias (IFB) is dynamic quantity. Thereafter, a new searching algorithm based on the Satisfiability Modulo Theories (SMT) is presented to search the optimal integer combinations fast. Meanwhile, ionosphere-free combinations are exhibited and are not suitable for TCAR. Futhermore, the availability of the SMT-based searching algorithm (SMTSA) is verified. Finally, the most interesting carrier-phase combinations that can be used for PPP ambiguity resolution are screened out for later research. Meanwhile, in order to give a referenced comprehensive assessment of the integer combination, some valuable combinations for BDS-3 are displayed. It provides scientific support for the triple-frequency AR which has less convergence time than dual-frequency AR. Moreover, it also gives the important and valuable reference for future research on multi-frequency ambiguity–enabled PPP.

Undifferenced zenith tropospheric modeling and its application in fast ambiguity recovery for long-range network RTK reference stations

A large number of continuously operating reference station (CORS) networks have been established around the world to support various high-precision navigation and positioning applications. However, the presence of significant tropospheric delays makes rapid ambiguity recovery for long inter-station baselines of network real-time kinematic (RTK) systems a major challenge. Since tropospheric delays are strongly temporally correlated over short periods, we propose an undifferenced (UD) zenith tropospheric prediction model to effectively correct tropospheric errors on the subsequent epoch measurements. Using 2-h sessions of the independent baselines in a CORS network, the ambiguities are easily and reliably resolved with the conventional ionospheric-free combination method. The derived double-differenced (DD), ionospheric-free residuals are then converted to UD residuals for each satellite and all stations. The UD residuals and the corresponding wet coefficients of each satellite are used to construct the zenith tropospheric model. The model is reconstructed every 5 min for each station. The slant tropospheric errors of observations within this period can be predicted using the established models. Seven independent baselines with an average length of 97 km are used to test the ambiguity recovery performance of the proposed method. The experimental results show that the proposed tropospheric prediction model can efficiently reduce the effects of slant tropospheric errors and improve the float solution of ambiguities. The average initialization time with the proposed method is less than 111.5 s, which is a 45% improvement with respect to the conventional approach. The proposed method was shown to be effective for fast ambiguity recovery of long-range baselines between reference stations.

Monitoring Zenithal Total Delays over the three different climatic zones from IGS GPS final products: A comparison between the use of the VMF1 and GMF mapping functions

The International GNSS Service (IGS) final products (ephemeris and clocks-correction) have made the GNSS an indispensable low-cost tool for scientific research, for example sub-daily atmospheric water vapor monitoring. In this study, we investigate if there is a systematic difference coming from the choice between the Vienna Mapping Function 1 (VMF1) and the Global Mapping Function (GMF) for the modeling of Zenith Total Delay (ZTD) estimates, as well as the Integrated Precipitable Water Vapor (IPWV) estimates that are deduced from them. As ZTD estimates cannot be fully separated from coordinate estimates, we also investigated the coordinate repeatability between subsequent measurements. For this purpose, we monitored twelve GNSS stations on a global scale, for each of the three climatic zones (polar, mid-latitudes and tropical), with four stations on each zone. We used an automated processing based on the Bernese GNSS Software Version 5.2 by applying the Precise Point Positioning (PPP) approach, L3 Ionosphere-free linear combination, 7 ̊ cutoff elevation angle and 2 h sampling. We noticed an excellent agreement with the ZTD estimates and coordinate repeatability for all the stations w.r.t to CODE (the Center for Orbit Determination in Europe) and USNO (US Naval Observatory) products, except for the Antarctic station (Davis) which shows systematic biases for the GMF related results. As a final step, we investigated the effect of using two mapping functions (VMF1 and GMF) to estimate the IPWV, w.r.t the IPWV estimates provided by the Integrated Global Radiosonde Archive (IGRA). The GPS-derived IPWV estimates are very close to the radiosonde-derived IPWV estimates, except for one station in the tropics (Tahiti).