Accounting for Signal Distortion Biases for Wide-Lane and Narrow-Lane Phase Bias Estimation with Inhomogeneous Networks

Due to different designs of receiver correlators and front ends, receiver-related pseudorange biases, called signal distortion biases (SDBs), exist. Ignoring SDBs that can reach up to 0.66 cycles and 10 ns in Melbourne-Wübbena (MW) and ionosphere-free (IF) combinations can negatively affect phase bias estimation. In this contribution, we investigate the SDBs and evaluate the impacts on wide-lane (WL) and narrow-lane (NL) phase bias estimations, and further propose an approach to eliminating these SDBs to improve phase bias estimation. Based on a large data set of 302 multi-global navigation satellite system (GNSS) experiment (MGEX) stations, including 5 receiver brands, we analyze the characteristics of these SDBs The SDB characteristics of different receiver types for different GNSS systems differ from each other. Compared to the global positioning system (GPS) and BeiDou navigation satellite system (BDS), SDBs of Galileo are not significant; those of BDS-3 are significantly superior to BDS-2; Septentrio (SEPT) receivers show the most excellent consistency among all receiver types. Then, we apply the corresponding corrections to phase bias estimation for GPS, Galileo and BDS. The experimental results reveal that the calibration can greatly improve the performance of phase bias estimation. For WL phase biases estimation, the consistencies of WL phase biases among different networks for GPS, Galileo, BDS-2 and BDS-3 improve by 89%, 77%, 76% and 78%, respectively. There are scarcely any improvements of the fixing rates for Galileo due to its significantly small SDBs, while for GPS, BDS-2 and BDS-3, the WL ambiguity fixing rates can improve greatly by 13%, 27% and 14% after SDB calibrations with improvements of WL ambiguity fixing rates, the corresponding NL ambiguity fixing rates can further increase greatly, which can reach approximately 16%, 27% and 22%, respectively. Additionally, after the calibration, both WL and NL phase bias series become more stable. The standard deviations (STDs) of WL phase bias series for GPS and BDS can improve by more than 46%, while those of NL phase bias series can yield improvements of more than 13%. Ultimately, the calibration can make more WL and NL ambiguity residuals concentrated in ranges within ±0.02 cycles. All these results demonstrate that SDBs for phase bias estimation cannot be ignored and must be considered when inhomogeneous receivers are used.

Assessment of Multi-Frequency PPP Ambiguity Resolution Using Galileo and BeiDou-3 Signals

From network RTK to PPP-RTK, it is highly expected that high-precision positioning within a few minutes can be achieved with a sparse reference network. In this study, we investigate a rapid multi-frequency PPP convergence strategy based on Galileo E1/E5a/E6 and BeiDou-3 B1C/B2a/B3I signals, whose unambiguous wide-lane observables can efficiently assist in speeding up narrow-lane ambiguity resolution. Furthermore, frequency-specific biases existing on the third-frequency observables have been observed to slow down multi-frequency PPP-AR convergence. In this study, we partially mitigated their effects by estimating a second satellite clock for the third frequency of signals. We validated this approach with one month of data collected from 22 stations. On average, it took about 18 min for PPP wide-lane ambiguity resolution (PPP-WAR) to converge, while 32 min were required for ambiguity-float PPP. Compared with dual-frequency PPP-AR, which needed nearly 12 min to converge, multi-frequency PPP-AR required 6 min only. Once there were more than 10 satellites involved in PPP, the convergence could be achieved within 3 min on average. Meanwhile, 81% and 62% of multi-frequency PPP-AR solutions converged successfully within 5 and 1 min, respectively. Finally, we carried out a vehicle-borne experiment to validate this approach in a kinematic environment. Owing to frequent cycle slips during the movement of vehicle, it took 14 min for B1C/B2a/B3I and E1/E5a/E6 PPP-AR to obtain reliable positions, and 19 min for those using the other signal combinations B1C/B2a/B2b and E1/E5a/E5b, owning to higher noise. Overall, these results are promising for achieving high-precision PPP positioning globally within a few minutes if multi-frequency biases can be handled well in the data processing.

Performance of GPS Precise Time and Frequency Transfer with Integer Ambiguity Resolution

The global positioning system (GPS) carrier-phase (CP) technique is a widely used spatial tool for remote precise time and frequency transfer. However, the performance of traditional GPS time and frequency transfer has been limeted because the ambiguity paramter is still the float solution. This study focuses on the performance of GPS precise time and frequency transfer with integer ambiguity resolution and discusses the corresponding mathematical model. Fractional-cycle bias (FCB) products were estimated by using an ionosphere-free combination. The results show that the satellite wide-lane (WL) FCB products are stable, with a standard deviation (STD) of 0.006 cycles. The narrow-lane (NL) FCB products were estimated over 15 min with the STD of 0.020 cycles. More than 98% of the WL and NL residuals are smaller than 0.25 cycles, which helps to fix the ambiguity into integers during the time and frequency transfer. Subsequently, the performances of the time transfers with integer ambiguity resolution at two time links between international laboratories were assessed in real-time and post-processing modes and compared. The results show that fixing the ambiguity into an integer in the real-time mode significantly decreases the convergence time compared with the traditional float approach. The improvement is ~49.5%. The frequency stability of the fixed solution is notably better than that of the float solution. Improvements of 48.15% and 27.9% were determined for the IENG–USN8 and WAB2–USN8 time links, respectively.

Performance of Single-Epoch EWL/WL/NL Ambiguity-Fixed Precise Point Positioning with Regional Atmosphere Modelling

Precise point positioning (PPP) with ambiguity resolution (AR) can improve positioning accuracy and reliability. The narrow-lane (NL) AR solution can reach centimeter-level accuracy but there is a certain initialization time. In contrast, extra-wide-lane (EWL) or wide-lane (WL) ambiguity can be fixed instantaneously. However, due to the limited correction accuracy of the empirical atmospheric model, the positioning accuracy is only a few decimeters. In order to further improve the real-time performance of PPP while ensuring accuracy, we developed a multi-system multi-frequency uncombined PPP single-epoch EWL/WL/NL AR method with regional atmosphere modelling. In the proposed method, the precise atmosphere, including zenith wet-troposphere delay (ZWD) and the slant ionosphere, is extracted through multi-frequency stepwise AR, which then is both interpolated and broadcast to users. By adding regional atmosphere constraints, users can achieve single-epoch PPP AR with centimeter-level accuracy. To verify the algorithm, four sets of reference networks with different inter-station distances are used for experiments. With atmosphere constraints, the accuracy of the single-epoch WL solution can be improved from the decimeter level to a few centimeters, with an improvement of more than 90%, and the epoch fix rate can also be improved to varying degrees, especially for the dual-frequency case. Due to the enlarged noise of the EWL combination, its accuracy is at the decimeter level, while the accuracy of the WL/NL solution can reach several centimeters. However, reliable NL ambiguity-fixing tightly relies on atmosphere constraints with sufficiently high accuracy. When the modelling of the atmosphere correction is not accurate enough, the NL AR performance is degraded, although this situation can be improved to a certain extent through the multi-GNSS combination. In contrast, in this case, the WL ambiguity can be successfully fixed and can support the precise positioning with an accuracy of several centimeters.

Estimating the Fractional Cycle Biases for GPS Triple-Frequency Precise Point Positioning with Ambiguity Resolution Based on IGS Ultra-Rapid Predicted Orbits

We investigate the estimation of the fractional cycle biases (FCBs) for GPS triple-frequency uncombined precise point positioning (PPP) with ambiguity resolution (AR) based on the IGS ultra-rapid predicted (IGU) orbits. The impact of the IGU orbit errors on the performance of GPS triple-frequency PPP AR is also assessed. The extra-wide-lane (EWL), wide-lane (WL) and narrow-lane (NL) FCBs are generated with the single difference (SD) between satellites model using the global reference stations based on the IGU orbits. For comparison purposes, the EWL, WL and NL FCBs based on the IGS final precise (IGF) orbits are estimated. Each of the EWL, WL and NL FCBs based on IGF and IGU orbits are converted to the uncombined FCBs to implement the static and kinematic triple-frequency PPP AR. Due to the short wavelengths of NL ambiguities, the IGU orbit errors significantly impact the precision and stability of NL FCBs. An average STD of 0.033 cycles is achieved for the NL FCBs based on IGF orbits, while the value of the NL FCBs based on IGU orbits is 0.133 cycles. In contrast, the EWL and WL FCBs generated based on IGU orbits have comparable precision and stability to those generated based on IGF orbits. The use of IGU orbits results in an increased time-to-first-fix (TTFF) and lower fixing rates compared to the use of IGF orbits. Average TTFFs of 23.3 min (static) and 31.1 min (kinematic) and fixing rates of 98.1% (static) and 97.4% (kinematic) are achieved for the triple-frequency PPP AR based on IGF orbits. The average TTFFs increase to 27.0 min (static) and 37.9 min (kinematic) with fixing rates of 97.0% (static) and 96.3% (kinematic) based on the IGU orbits. The convergence times and positioning accuracy of PPP and PPP AR based on IGU orbits are slightly worse than those based on IGF orbits. Additionally, limited by the number of satellites transmitting three frequency signals, the introduction of the third frequency, L5, has a marginal impact on the performance of PPP and PPP AR. The GPS triple-frequency PPP AR performance is expected to improve with the deployment of new-generation satellites capable of transmitting the L5 signal.

Comparisons of CODE and CNES/CLS GPS satellite bias products and applications in Sentinel-3 satellite precise orbit determination

To resolve undifferenced GNSS phase ambiguities, dedicated satellite products are needed, such as satellite orbits, clock offsets and biases. The International GNSS Service CNES/CLS analysis center provides satellite (HMW) Hatch-Melbourne-Wübbena bias and dedicated satellite clock products (including satellite phase bias), while the CODE analysis center provides satellite OSB (observable-specific-bias) and integer clock products. The CNES/CLS GPS satellite HMW bias products are determined by the Hatch-Melbourne-Wübbena (HMW) linear combination and aggregate both code (C1W, C2W) and phase (L1W, L2W) biases. By forming the HMW linear combination of CODE OSB corrections on the same signals, we compare CODE satellite HMW biases to those from CNES/CLS. The fractional part of GPS satellite HMW biases from both analysis centers are very close to each other, with a mean Root-Mean-Square (RMS) of differences of 0.01 wide-lane cycles. A direct comparison of satellite narrow-lane biases is not easily possible since satellite narrow-lane biases are correlated with satellite orbit and clock products, as well as with integer wide-lane ambiguities. Moreover, CNES/CLS provides no satellite narrow-lane biases but incorporates them into satellite clock offsets. Therefore, we compute differences of GPS satellite orbits, clock offsets, integer wide-lane ambiguities and narrow-lane biases (only for CODE products) between CODE and CNES/CLS products. The total difference of these terms for each satellite represents the difference of the narrow-lane bias by subtracting certain integer narrow-lane cycles. We call this total difference “narrow-lane” bias difference. We find that 3% of the narrow-lane biases from these two analysis centers during the experimental time period have differences larger than 0.05 narrow-lane cycles. In fact, this is mainly caused by one Block IIA satellite since satellite clock offsets of the IIA satellite cannot be well determined during eclipsing seasons. To show the application of both types of GPS products, we apply them for Sentinel-3 satellite orbit determination. The wide-lane fixing rates using both products are more than 98%, while the narrow-lane fixing rates are more than 95%. Ambiguity-fixed Sentinel-3 satellite orbits show clear improvement over float solutions. RMS of 6-h orbit overlaps improves by about a factor of two. Also, we observe similar improvements by comparing our Sentinel-3 orbit solutions to the external combined products. Standard deviation value of Satellite Laser Ranging residuals is reduced by more than 10% for Sentinel-3A and more than 15% for Sentinel-3B satellite by fixing ambiguities to integer values.

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.

Using Galileo and BDS-3 Quad-Frequency Signals for Long-Baseline Instantaneous Decimeter-Level Positioning

As the signals of Galileo and the global BDS-3 navigation satellite system have been accessible, positioning users can use quad-frequency even five-frequency signals nowadays. With multifrequency signals, one can form some useful combinations to improve the positioning performance, e.g., the widely used extra-wide-lane (EWL)/wide-lane (WL) in triple-frequency cases. For quad-frequency or five-frequency cases, better positioning performance can be expected since additional frequencies are introduced. In this study, we systematically analyse the benefits of Galileo and BDS-3 quad-frequency signals on long-baseline instantaneous positioning. First, the theoretical analysis of EWL/WL ambiguity resolution (AR) and satellite-station range estimation with a single-satellite geometry-free and ionosphere-free model is studied, along with the comparison with triple-frequency cases. Second, using the quad-frequency advantages, an instantaneous decimeter-level positioning model is proposed, where the geometry-free model is adopted for the first two EWL AR and the geometry-based model is adopted for the third WL AR. In the end, the AR and positioning performance are evaluated using real long-baseline date containing Galileo and BDS-3 quad-frequency observations. The results indicate that, with quad-frequency observations, both Galileo and BDS-3 EWL/WL ambiguities can be fixed reliably with a single epoch. Contributed by the resolved EWL/WL ambiguities, instantaneous decimeter-level positioning can be obtained, with the accuracies of 0.116 m/0.126 m/0.351 m in north, east, and up directions, respectively.

Comparison of Single-Site Ionospheric Delays Derived from Geometry-Free and Geometry-Based Approaches with an Analysis of Orbit and Clock Error Effects

The reparameterization of the geometry-free and geometry-based approaches to derive single-site ionospheric delays using Global Navigation Satellite System (GNSS) measurements is described. Kalman filtering is used to compute the geometry-free and geometry-based ionospheric delays in a forward computation procedure, aiming for a real-time application case. The numerical similarity and differences between the geometry-free and geometry-based ionospheric delays are assessed in terms of both formal and experimental errors (precision). The differences between geometry-free and geometry-based ionospheric delays are derived using two types of precise orbit and clock products. The effects of the precise orbit and clock residual errors are analyzed. The correlation coefficients between the L1 and L2 wide-lane ambiguities with the ionospheric delay are derived and analyzed. It is discovered that the geometry-based ionospheric delay is negatively correlated with geometry-based wide-lane ambiguities, while the geometry-free ionospheric delay and wide-lane ambiguities are much less correlated. A simulation analysis indicates that the impacts on geometry-based ionospheric delay estimates are partly coincided with the actual time-variant errors of the used orbit and clock in the line-of-sight direction.

Algorithm for Real-Time Cycle Slip Detection and Repair for Low Elevation GPS Undifferenced Data in Different Environments

Real-time cycle slip detection and repair is one of the key issues in global positioning system (GPS) high precision data processing and application. In particular, when GPS stations are in special environments, such as strong ionospheric disturbance, sea, and high-voltage transmission line interference, cycle slip detection and repair in low elevation GPS observation data are more complicated than those in normal environments. For low elevation GPS undifferenced carrier phase data in different environments, a combined cycle slip detection algorithm is proposed. This method uses the first-order Gauss–Markov stochastic process to model the pseudorange multipath in the wide-lane phase minus narrow-lane pseudorange observation equation, and establishes the state equation of the wide-lane ambiguity with the pseudorange multipath as a parameter, and it uses the Kalman filter for real-time estimation and detects cycle slips based on statistical hypothesis testing with a predicted residual sequence. Meanwhile, considering there are certain correlations among low elevation, observation epoch interval, and ionospheric delay error, a second-order difference geometry-free combination cycle slip test is constructed that takes into account the elevation. By combining the two methods, real-time cycle slip detection for GPS low elevation satellite undifferenced data is achieved. A cycle slip repair method based on spatial search and objective function minimization criterion is further proposed to determine the correct solution of the cycle slips after they are detected. The whole algorithm is experimentally verified using the static and kinematic measured data of low elevation satellites under four different environments: normal condition, high-voltage transmission lines, dynamic condition in the sea, and ionospheric disturbances. The experimental results show that the algorithm can detect and repair cycle slips accurately for low elevation GPS undifferenced data, the difference between the float solution and the true value for the cycle slip does not exceed 0.5 cycle, and the differences obey the normal distribution overall. At the same time, the wide-lane ambiguity and second-order difference GF combination sequence calculated by the algorithm is smoother, which give further evidence that the algorithm for cycle slip detection and repair is feasible and effective, and has the advantage of being immune to the special observation environments.