Uncombined Precise Point Positioning Research Articles

Regional Real-Time between-Satellite Single-Differenced Ionospheric Model Establishing by Multi-GNSS Single-Frequency Observations: Performance Evaluation and PPP Augmentation

The multi-global navigation satellite system (GNSS) undifferenced and uncombined precise point positioning (UU-PPP), as a high-precision ionospheric observables extraction technology superior to the traditional carrier-to-code leveling (CCL) method, has received increasing attention. In previous research, only dual-frequency (DF) or multi-frequency (MF) observations are used to extract slant ionospheric delay with the UU-PPP. To reduce the cost of ionospheric modeling, the feasibility of extracting ionospheric observables from the multi-GNSS single-frequency (SF) UU-PPP was investigated in this study. Meanwhile, the between-satellite single-differenced (SD) method was applied to remove the effects of the receiver differential code bias (DCB) with short-term time-varying characteristics in regional ionospheric modeling. In the assessment of the regional real-time (RT) between-satellite SD ionospheric model, the internal accord accuracy of the SD ionospheric delay can be better than 0.5 TECU, and its external accord accuracy within 1.0 TECU is significantly superior to three global RT ionospheric models. With the introduction of the proposed SD ionospheric model into the multi-GNSS kinematic RT SF-PPP, the initialization speed of vertical positioning errors can be improved by 21.3% in comparison with the GRAPHIC (GRoup And PHase Ionospheric Correction) SF-PPP model. After reinitialization, both horizontal and vertical positioning errors of the SD ionospheric constrained (IC) SF-PPP can be maintained within 0.2 m. This proves that the proposed SDIC SF-PPP model can enhance the continuity and stability of kinematic positioning in the case of some GNSS signals missing or blocked. Compared with the GRAPHIC SF-PPP, the horizontal positioning accuracy of the SDIC SF-PPP in kinematic mode can be improved by 37.9%, but its vertical positioning accuracy may be decreased. Overall, the 3D positioning accuracy of the SD ionospheric-constrained RT SF-PPP can be better than 0.3 m.

Investigating the Global Performance of the BDS-2 and BDS-3 Joint Real-Time Undifferenced and Uncombined Precise Point Positioning Using RTS Products from Different Analysis Centers

Compared to the traditional ionospheric-free (IF) precise point positioning (PPP) model, the undifferenced and uncombined (UU) PPP has the advantages of lower observation noise and the ability to obtain ionospheric information. Thanks to the IGS (International GNSS Service), real-time service (RTS) can provide RT vertical total electron content (VTEC) products, and an enhanced RT UU-PPP based on the RT-VTEC constraints can be achieved. The global performance of the BeiDou Navigation Satellite System-2 (BDS-2) and BDS-3 joint RT UU-PPP using different RTS products was investigated. There is not much difference in the RTS orbit accuracy of medium earth orbit (MEO) satellites among all analysis centers (ACs), and the optimal orbit accuracy is better than 5, 9, and 7 cm in the radial, along-track, and cross-track directions, respectively. The orbit accuracy of inclined geosynchronous orbit (IGSO) satellites is worse than that of MEO satellites. Except for CAS of 0.46 ns, the RTS clock accuracy of MEO satellites for other ACs achieves 0.2–0.27 ns, and the corresponding accuracy is about 0.4 ns for IGSO satellites. In static positioning, due to the limited accuracy of RT-VTEC, the convergence time of the enhanced RT UU-PPP is longer than that of RT IF-PPP for most ACs and can be better than 25 and 20 min in the horizontal and vertical components, respectively. After convergence, the 3D positioning accuracy of the static RT UU-PPP is improved by no more than 8.7%, and the optimal horizontal and vertical positioning accuracy reaches 3.5 and 7.0 cm, respectively. As for the kinematic mode with poor convergence performance, with the introduction of RT-VTEC constraints, the convergence time of RT UU-PPP can be slightly shorter and reaches about 55 and 60 min in the horizontal and vertical components, respectively. Both the horizontal and vertical positioning accuracies of the kinematic RT UU-PPP can be improved and achieve around 7.5 and 10 cm, respectively.

BDS-3 six-frequency precise point positioning: effects of receiver-dependent inter-frequency clock bias and multi-frequency contributions

We find that there is time-varying bias at the receiver in BDS-3 multi-frequency phase observations, which will lead to receiver-dependent inter-frequency clock bias (RIFCB) and damage the rigor of BDS-3 multi-frequency precise point positioning (PPP) model. A unified RIFCB correction method compatible with uncombined and ionospheric-free combined PPP models using any BDS-3 frequencies is proposed. The contributions of multi-frequency integration to BDS-3 static and kinematic PPP performance are evaluated. The results indicate that the RIFCB amplitude can reach 1 dm. If the RIFCB correction is ignored, the phase observation residuals present a systematic bias. RIFCB has potential effects on precise time transfer, ionospheric monitoring, and fractional cycle bias estimation. Experimental results show that the long-term frequency stability of time transfer can be improved by correcting RIFCB. The joint use of all BDS-3 six-frequency signals can significantly shorten the convergence time, and the positioning accuracy can also be slightly improved.

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.

Estimating BDS-3 Satellite Differential Code Biases with the Single-Frequency Uncombined PPP Model.

Differential Code Bias (DCB) is a crucially systematic error in satellite positioning and ionospheric modeling. This study aims to estimate the BeiDou-3 global navigation satellite system (BDS-3) satellite DCBs by using the single-frequency (SF) uncombined Precise Point Positioning (PPP) model. The experiment utilized BDS-3 B1 observations collected from 25 International GNSS Service (IGS) stations located at various latitudes during March 2023. The results reveal that the accuracy of estimating B1I-B3I DCBs derived from single receiver exhibits latitude dependence. Stations in low-latitude regions show considerable variability in the root mean square (RMS) of absolute offsets for satellite DCBs estimation, covering a wide range of values. In contrast, mid- to high-latitude stations demonstrate a more consistent pattern with relatively stable RMS values. Moreover, it has been observed that the stations situated in the Northern Hemisphere display a higher level of consistency in the RMS values when compared to those in the Southern Hemisphere. When incorporating estimates from all 25 stations, the RMS of the absolute offsets in satellite DCBs estimation consistently remained below 0.8 ns. Notably, after excluding 8 low-latitude stations and utilizing data from the remaining 17 stations, the RMS of absolute offsets in satellite DCBs estimation decreased to below 0.63 ns. These enhancements underscore the importance of incorporating a sufficient number of mid- and high-latitude stations to mitigate the effects of ionospheric variability when utilizing SF observations for satellite DCBs estimation.

Single-Epoch Decimeter-Level Precise Point Positioning with a Galileo Five-Frequency Ionosphere-Reduced Combination

Currently, there are two main methods for single-epoch decimeter-level precise point positioning (PPP); one is a model based on ambiguity-fixed ionosphere-free (AFIF) observations, and the other is based on uncombined (UC) PPP. The implementation of these two models requires both extra-wide-lane (EWL) and wide-lane (WL) ambiguity fixing. Different from the existing methods, this paper proposes a multi-frequency ionosphere-reduced (IR) PPP model suitable for single-epoch decimeter-level positioning. Based on Galileo five-frequency data, the optimal selection strategy of IR combinations is first studied with ionosphere, noise level and wavelength factors considered. Then, based on the selected IR combination, two IR PPP models, namely IR(EST) and IR(IGN), are established according to whether ionosphere parameters are estimated or ignored. Finally, the proposed models are verified with real tracked data from globally distributed stations, and further compared with the existing AFIF/UC models in terms of positioning performance and time consumption. The relationship between the ionosphere equivalent ranging error and satellite elevation in the IR models is analyzed. The lower the elevation is, the larger the residual ranging error is, and its impact on positioning is weakened by downweighting its observations and adjusting the cut-off elevation during the partial ambiguity fixing (PAF) process. The results show that the performance of the two IR models is basically the same, and both can achieve horizontal and vertical accuracies better than 20 cm and 40 cm, respectively. Compared with the existing AFIF/UC models, the proposed IR models can achieve similar decimeter-level accuracy with only one step of EWL ambiguity fixing, and at the same time, the IR models have varying degrees of improvement in time consumption: 38% shorter than the AFIF model and 97% shorter than the UC model.

Precise point positioning with mixed single- and dual-frequency GNSS observations from Android smartphones considering code-carrier inconsistency

The open access to raw Global Navigation Satellite System (GNSS) measurements from Android smartphones brings new possibilities to high-accuracy positioning with precise point positioning (PPP) technique. However, decimeter or even centimeter level positioning accuracy is still challengeable mainly due to the low completeness of dual-frequency observations and specific observation errors from a smartphone. Huawei P40 released in 2020 is the first module that can track multi-GNSS multi-frequency signals. Based on GNSS observations from P40 and a co-located geodetic-grade receiver, a method is proposed to extract and model the P40 code-carrier inconsistency errors. The P40 observation inconsistency presents a linear drift pattern with respect to time at each frequency and can even reach hundreds of meters. A correction model is then established which reduces the code-carrier differences to a level of few meters. Regarding to the P40 code-carrier inconsistency error signature, an uncombined precise point positioning (PPP) algorithm is developed with the inconsistency error treated with different schemes, including correction using established model, parameterization as a linear function, or estimated as an independent clock offset for carrier-phase observation. Meanwhile, to cope with the dual-frequency data deficiency of P40, the PPP algorithm is designed based on pure single-frequency processing which can handle mixed single- and dual-frequency observations. Positioning performances of three schemes are validated in both static and kinematic scenarios. With applying the code-carrier inconsistency schemes, P40 positioning precision reaches 0.2 m and 1.0 m for static and kinematic experiments, respectively, which is improved by an order of magnitude against the case without. Furthermore, precision can be even improved when P40 is equipped with an external geodetic-grade antenna. It reaches 5.8 and 18.2 cm in the static and kinematic tests, respectively, revealing improvements of 28.7% and 54.4% against the case without considering the inconsistencies, respectively. Such result is very encouraging and indicate a positioning accuracy level close to a geodetic receiver.

TECX-TCN: Prediction of ionospheric total electron content at different latitudes in China based on XGBoost algorithm and temporal convolution network

The correction of ionospheric delay error through the estimation of ionospheric total electron content (TEC) parameters is very important for the accuracy and rapid convergence of precise point positioning (PPP), especially for single frequency PPP position and double frequency undifferenced and uncombined PPP models. This paper studies the prediction method of ionospheric TEC by analyzing various influencing factors. Firstly, considering the influence of latitude, the TEC values of different observation stations in China at middle and low latitudes are used for research. Secondly, due to the influence of solar storms, the years with and without solar storms are selected for comparative study. At the same time, the moving average value of TEC and the lag value of each solar and geomagnetic index are also added. Because the influence of each feature on ionospheric TEC is not clear, the importance score of each feature is obtained according to the EXtreme Gradient Boosting (XGBoost) algorithm to establish a reasonable feature combination. Then, the temporal convolution network (TCN) is used to train according to data of the collected solar cycle and predict data for the next year. Through experiments, the prediction model with feature selection can obtain higher prediction accuracy, the error reaches ∼2TECU at middle latitude region and ∼3 TECU at low latitude region. Finally, compared with other algorithms, our ionospheric TEC prediction algorithm based on XGBoost and TCN (TECX-TCN) significantly has better performance. When the solar activity is low, its RMSE can reach ∼2.5TECU, the correlation coefficient can reach ∼0.9, and even in the low latitude area with high solar activity, its RMSE can reach ∼5TECU, the correlation coefficient can reach ∼0.85.

Numerical Modification to Speed Up the Convergence of Uncombined Precise Point Positioning

Numerical Modification to Speed Up the Convergence of Uncombined Precise Point Positioning

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

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

Integrity monitoring scheme for single-epoch GNSS PPP-RTK positioning

Integrity monitoring for precise point positioning is critical for safety-related applications. With the increasing demands of high-accuracy autonomous navigation for unmanned ground and aerial vehicles, the integrity monitoring method of high-precision positioning has become an essential requirement. While high precision Global Navigation Satellite Systems (GNSS) positioning is widely used in such applications, there are still many difficulties in the integrity monitoring method for the multi-frequency multi-GNSS undifferenced and uncombined Precise Point Positioning (PPP). The main difficulties are caused by using the measurements of multiple epochs in PPP. Based on the baseline Multiple Hypothesis Solution Separation (MHSS) Advanced Receiver Autonomous Integrity Monitoring (ARAIM) algorithm, this paper discusses the feasibility of the pseudorange-based baseline ARAIM method on the single-epoch PPP based on Real-Time Kinematic (RTK) networks (PPP-RTK) framework to overcome these difficulties. In addition, a new scheme is proposed to transfer the conventional PPP process into the single-epoch PPP-RTK framework. The simulation results using the proposed model are analyzed in this study. The Protection Levels (PLs) estimated by PPP Wide-lane Ambiguity Resolution (PPP-WAR) model with regional corrections can reach the meter level and the PLs estimated by PPP Ambiguity Resolution (PPP-AR) and PPP-RTK models are usually the sub-meter level. Given a horizontal Alert Limit (AL) of 1.5 m, the global coverage of availability above 99.9% for PPP-WAR, PPP-AR, and PPP-RTK can reach 92.6%, 99.4%, and 99.7% respectively. The results using real kinematic data also show that tight PLs can be achieved when the observation conditions are good.

Multilayer Ionospheric Model Constrained by Physical Prior Based on GNSS Stations

The need for accurate modeling of the ionosphere plays an important role in the global navigation satellite system (GNSS) positioning. The traditional multi-layer VTEC model without prior has been used for modeling the ionospheric delay error. However, it is assumed that the electron density of the ionosphere is compressed into multiple thin layers at fixed heights in the lack of capturing ionospheric physics. In this paper, the data enhancement method by virtual observations is proposed to build the constrained multi-layer VTEC model to capture physical features from empirical ionospheric models. The extraction methods of physical knowledge have been developed by prior VTEC based on principal component analysis and model coefficients based on emulated basis function. The constrained multi-layer modeling has been verified based on simulation and real measurement of GNSS data in Yunnan, China collected from Qianxun Ground-based Augmentation System on November 3, 2021. The receiver DCB error estimated by the multi-layer model with prior constraint is significantly lower than that of the single-layer model and the traditional multi-layer model. The experimental test shows that the constrained multi-layer model achieves the accuracy of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.5 \,\rm {TECU}$</tex-math></inline-formula> for the independent reference station. The dSTEC of the proposed two multi-layer models are significantly lower than those of the single-layer model for low elevation angles, and the RMSE of dSTEC is reduced by 63 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> with the cutoff elevation angle of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{\circ }$</tex-math></inline-formula> . The spatial distribution of the multi-layer VTEC model shows consistency with the tomography model to verify vertical feature capturing capability. Compared with the Un-differenced and Un-combined Precise Point Positioning (UCPPP) without ionospheric constraint, the multi-layer constrained model based on the test data improves the convergence time approximately by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$36.55\%$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$18.78\%$</tex-math></inline-formula> in the horizontal (H), up (U) directions, respectively. These results demonstrate that the proposed multi-layer models not only improve ionospheric delay estimation precision but also can obtain the VTEC distribution capturing the physical characteristics of the ionosphere. The proposed multi-layer models may be valuable for the ionospheric delay modeling of satellite navigation systems in harsh variable ionospheric conditions.

An Improved Strategy for Real-Time Troposphere Estimation and Its Application in the Severe Weather Event Monitoring

The water vapor content in the atmosphere is highly correlated with rainfall events, which can be used as a data source for rainfall prediction or drought monitoring. The GNSS PPP (Precise Point Positioning) technique can be used to estimate the troposphere ZWD (Zenith Wet Delay) parameter which can be converted into precipitable water vapor (PWV). In this study, we first investigate the impacts of the weighting strategies, observation noise settings, and gradient estimation on the accuracy of ZWD and positions. A refined strategy is proposed for the troposphere estimation with uncombined raw PPP model, down-weighting of Galileo/GLONASS/BDS code observation by a factor of 3, using a sine2-type elevation-dependent weighting function and estimating the horizontal gradients. Based on the strategy, the correlation of the estimated tropospheric parameters with the rainfall is analyzed based on the data from the “7.20” rainstorm in Henan Province, China. The PWV is first calculated based on the hourly global pressure and temperature (HGPT) model and compared with the results from ERA5 products. Results show their good consistency during the rainfall period or the normal period with a standard deviation of 3 mm. Then, the high correlation between the PWV and the heavy rain rainfall event is validated. Results show that the accumulated PWV maintains a high level before the rainstorm if a sustainable water supply is available, while it decreased rapidly after the rainfall. In comparison, the horizontal gradients and the satellite residuals are less correlated with the water vapor content. However, the gradients can be used to indicate the horizontal asymmetry of the water vapor in the atmosphere.

Regional Ionospheric Maps with Quad-Constellation Raw Observations as Applied to Single-Frequency PPP

Ionospheric delay is one of the most problematic errors in single-frequency (SF) global navigation satellite system (GNSS) data processing. Global/regional ionospheric maps (GIM/RIM) are thus vitally important for positioning users. Given the coexistence of multi-GNSS, the integration of quad-constellation observations is essential for improving the distribution of ionospheric penetration points (IPPs) and increasing redundant observations compared with the existing GIM products from the IGS analysis center. In this paper, quad-constellation (GPS/GLONASS/Galileo/BDS) observations are applied to set up the RIM over Australia with uncombined precise point positioning (UC-PPP) and a low-order spherical harmonic function. The generated RIMs are then introduced to ionosphere-corrected (IC) and ionosphere-weighted (IW) single-frequency PPP (IC-SFPPP and IW-SFPPP) to verify their performance in terms of positioning accuracy and convergence time. Taking the CODE GIM as a reference, the results show that the mean root mean square (RMS) of VTEC differences is 0.867 TECUs, and the quad-constellation RIM (referred as ‘RIM4′) can improve the RMS of RIMs compared to single-constellation mode at the edge of regional experiment area. The application of the RIM4 in the BDS IC-SFPPP results in a 18.38% improvement (from 100.47 cm to 82.00 cm) of 3D positioning RMS compared to the CODE-GIMs, whereas 35.36% enhancement (from 115.92 cm to 74.62 cm) of 3D positioning RMS is achievable during an active ionospheric period. Moreover, if the criterion of the convergence time is defined as when positioning errors in the horizontal and vertical directions are less than 0.3 m and 0.6 m for 20 consecutive epochs, the IW-SFPPP can significantly speed up the convergence time compared to the uncombined SFPPP; that is, the convergence time is reduced by 52.7% (from 37 min to 17.5 min), 37.2% (from 72.5 min to 45.5 min), and 37.1% (from 62.0 min to 39.0 min) in the north, east and up direction, respectively, at the 68% confidence level.

IMU-Aided Precise Point Positioning Performance Assessment with Smartphones in GNSS-Degraded Urban Environments

The tracking of satellite signals with the passive linearly polarized embedded global navigation satellite system (GNSS) antenna of smartphones in dynamic scenarios is susceptible to the changing multipath and obstructions in urban environments, which lead to a significant decrease in the availability and reliability of GNSS solutions. Accordingly, based on the characteristics of smartphone GNSS and inertial measurement unit (IMU) sensors data in GNSS-degraded environments, we established an IMU-aided uncombined precise point positioning (PPP) mathematical model that is suitable for smartphones. To enhance the reliability of initial alignment in dynamic mode, the step function variances depending on carrier-to-noise density ratio were established with the variances of GNSS measurements, and the inertial navigation system (INS) parameters were initialized while both the velocity of smartphones and the position dilution of precision (PDOP) reached corresponding thresholds. Considering the measurement noise and observations gaps of smartphones, the robust Kalman filter (RKF) with equivalent variance matrix was used for parameter estimation to improve the convergence efficiency of the coupled PPP/INS model. Experimental results indicated that the proposed PPP/INS method can effectively improve the positioning performance of smartphones in GNSS-degraded environments. Compared with the conventional smartphone PPP scheme, the PPP/INS horizontal errors in the eastern and western areas of the long trajectory experiment decreased by 49.37% and 48.29%, respectively. Meanwhile, the trajectory deviation of smartphones can remain stable in the tunnel where GNSS signals are blocked.

A Short-Term Forecasting Method of Inter-Frequency Clock Bias for GPS Block IIF Satellites

The time-varying biases within carrier phase observations are integrated into satellite clock offset parameters for precise clock estimation. Consequently, when the precise satellite clock bias is applied to the third frequency observation for precise point positioning (PPP), a new type of inter-frequency clock bias (IFCB) with satellite dependence should be noticed. If the IFCB is estimated together with the receiver coordinates, tropospheric wet delay, ambiguity and other parameters, it will increase the computational burden and lead to more time consumption. In order to solve this problem, the IFCB of GPS Block IIF satellites were estimated using 162 global uniformly distributed Multi-GNSS Experiment (MGEX) stations. By analyzing the time-varying characteristic of each satellite IFCB and combining the lag characteristics of the final ephemeris products, a modeling method of short-term IFCB prediction based on the epoch-by-epoch sliding Pearson autocorrelation function is proposed. The feasibility of this method was verified through the Student’s t-distribution, comparison with the measured IFCB, the posteriori residual of the third frequency carrier phase and the kinematic/static PPP solutions. The results showed that since the IFCB period was not a complete 24 h, the difference in the IFCBs time series on different days was increasingly significant with the passage of lag time, and the correlation constantly decreased. The peak-to-peak amplitudes of the IFCB difference reached 1.13, 3.44, 6.86 and 11.25 cm when the lag time was 1, 9, 19 and 29 days, respectively. In addition, based on the lag characteristic of final precise ephemerides released by the International GNSS Service (IGS) analysis centers, the prediction accuracy of the IFCB was evaluated with a time lag of 7 days. The root mean square of the posteriori residuals at the third-frequency observation decreased by approximately 51.3% compared to that without considering for IFCB correction. The triple-frequency uncombined PPP in the horizontal and vertical directions improved by approximately 33.2% and 17.2% for the static PPP solutions and 50.2% and 39.7% for the kinematic PPP solutions, respectively. In general, the accuracy and convergence time of the triple-frequency uncombined PPP were equivalently improved when the predicted IFCB and the measured IFCB were used.

Identifying spurious cycle slips based on iterative filtering under disturbed ionospheric conditions for undifferenced GNSS observations

The TurboEdit method is widely used to detect cycle slips on the global navigation satellite system (GNSS) carrier-phase measurements. However, it leads to an increasing number of false alarms in detecting cycle slips under disturbed ionospheric conditions. Besides, once the method detects a cycle slip at one satellite, it treats dual frequencies with cycle slips rather than at one frequency. Considering these two challenges, we developed a solution-based iterative filter detection method to reduce the number of spurious cycle slip detection under disturbed ionospheric conditions. The method initially assumes that there is no cycle slip at each frequency. We then estimate the solutions without cycle slips. A decision of exiting cycle slips is made by examining and comparing the two results solutions with or without cycle slips in terms of usable satellites, ambiguities, and residuals. The uncombined precise point positioning (PPP) during disturbed ionospheric conditions on 17 March 2015 at high latitude was studied to validate the proposed method. Results showed that the detected number of spurious cycle slips was reduced significantly. With fewer marked cycle slips, more stable and smoother positioning performance was achieved when fewer ambiguity parameters were reinitialized.

Effect of Stochastic Modeling for Inter-Frequency Biases of Receiver on BDS-3 Five-Frequency Undifferenced and Uncombined Precise Point Positioning

The third generation of the Beidou navigation satellite system (BDS-3) broadcasts navigation signals of five frequencies. Focusing on the deep integration of five-frequency signals, we applied the joint BDS-3 five-frequency undifferenced and uncombined precise point positioning (UC-PPP) to analyze the receiver inter-frequency biases (IFB). Firstly, 12 Multi-GNSS Experiment tracking (MGEX) stations are selected to investigate the time-varying characteristics of receiver IFB and, according to random characteristics, three random modeling schemes are proposed. Secondly, the effects of three stochastic modeling methods on zenith tropospheric delay, ionospheric delay, floating ambiguity, and quality control are analyzed. Finally, the effects of three IFB stochastic modeling methods on positioning performance are evaluated. The results showed that the amplitude in the IFB for B2b is 5.139 m, B2a is 1.964 m, and B1C is 0.950 m by measuring one week’s observation data. The IFB stochastic modeling method based on random walks can shorten the PPP convergence time by 4~12%, diminish the false alarm of quality control, and improve the positioning accuracy. The random walk model is recommended to simulate the variation of IFB, which can not only overcome the disadvantage of the time constant model being unable to accurately describe the time-varying characteristics of the IFB, but also avoid reducing the strength of the kinematic PPP positioning model due to the large process noise of the white noise model.

Galileo Time Transfer with Five-Frequency Uncombined PPP: A Posteriori Weighting, Inter-Frequency Bias, Precise Products and Multi-Frequency Contribution

Galileo satellites can broadcast signals on five frequencies, namely E1, E5A, E5B, E5 (A+B), and E6. The multi-frequency integration has become an emerging trend in Global Navigation Satellite System (GNSS) data processing. This study focused on the precise time transfer based on Galileo five-frequency uncombined precise point positioning (PPP), including the performance comparison of PPP time transfer with a priori and a posteriori weighting strategies, with different inter-frequency bias (IFB) dynamic models, and with the precise satellite products from different analysis centers, as well as the contribution of multi-frequency observations for time transfer. Compared with the a priori weighting strategy, the short-term frequency stability of time transfer adopting the Helmert variance component estimation method can be improved by 28.9–37.6% when the average time is shorter than 100 s. The effect of IFB dynamic models on Galileo five-frequency PPP time transfer is not obvious. When employing the post-processed precise satellite products from seven analysis centers, the accuracy of time transfer can be better than 0.1 ns, while an accuracy of 0.253 ns can be obtained in the real-time mode. At an average time of approximately 10,000 s, the post-processed time transfer with Galileo five-frequency PPP can provide a frequency stability of 3.283 × 10−14 to 3.459 × 10−14, while that in real-time mode can be 3.541 × 10−14. Compared with dual-frequency PPP results, the contribution of multi-frequency combination to the accuracy and frequency stability of time transfer is not significant, but multi-frequency PPP can achieve more reliable time transfer results when the signal quality is poor.

Integrity monitoring for undifferenced and uncombined PPP under local environmental conditions

Integrity monitoring of precise point positioning (PPP) can provide tightly guaranteed absolute position error bounds for safety-critical applications. However, a complex local environment makes PPP integrity monitoring much more challenging, such as urban canyons. Significant prone multipaths and low observation redundancy are the main difficulties in the accuracy and reliability of PPP. Therefore, we proposed a solution separation-based integrity monitoring algorithm, which is based on a single and dual frequency undifferenced and uncombined PPP model considering compensation for the multipath error distortion by Gaussian overbounding. Both the static and the kinematic data are utilized to test the proposed algorithm. The results show that the proposed algorithm can produce an adequate protection level (PL) in horizontal and vertical directions. Furthermore, the proposed algorithm can obtain a smoother PL and positioning error in a dynamic local environment, and effectively suppress misleading information.