Satellite Clock Estimation Research Articles

Adopting the empirical CODE orbit model to Galileo satellites

In 2012 the Center for Orbit Determination in Europe (CODE) joined the Multi-GNSS-EXtension project (MGEX) of the International GNSS Service (IGS). Since the end of 2013 the CODE MGEX contributions were based on combined solutions of five already established and emerging GNSS: GPS, GLONASS, Galileo, BeiDou and QZSS. This undertaking was made possible thanks to the continuous development of new models and approaches and their introduction in our processing schemes in order to ensure the delivery of products of the highest quality.The European Galileo system with a total number of active satellites reaching the level of a nominal constellation is currently complete. Because of their relatively high area-to-mass ratio the Galileo spacecraft are more sensitive to non-gravitational forces than other GNSS satellites. The introduction of the extended empirical CODE orbit model (ECOM2) to the CODE MGEX solutions in early 2015 resulted in a significant improvement of the Galileo products. The use of the Galileo satellites metadata, which were made publicly available in the course of 2016 and 2017, has further enhanced the quality of the produced solutions. However, they still show significant degradations during eclipse seasons in particular for long-arc solutions (e.g., over three days), which are similarly observed in solutions of other IGS MGEX analysis centers to different extents. In particular this is reflected in elevated orbit misclosures, deterioration of the estimated satellite clock corrections and excessive satellite laser ranging (SLR) residuals during these periods. Since the ECOM2 parameters are designed to absorb the effect of solar radiation pressure, they are switched off during eclipses. Hence, there is no empirical force parameter left that can absorb any unmodelled perturbations (e.g., due to thermal radiation (TR)) during an eclipse period.In this study we take advantage of the satellites’ metadata to address the Galileo TR-induced accelerations and, therefore, to advance our orbit model further. By adjusting existing and introducing additional empirical parameters, as well as adopting a priori accelerations to account for unmodelled perturbations we achieve a significant improvement of our solutions. In particular, the use of a once-per-revolution sine term in the ECOM E3 direction (satellite-Sun), activation of the constant term in the ECOM E2 direction (along the solar panels) during the Earth’s shadow transitions as well as the use of a priori acceleration in the body-fixed +X direction substantially improve our solutions. The introduced modifications to the orbit model allow for an efficient adjustment of satellite velocity along the orbit that is necessary due to the apparent presence of unmodelled thermal radiation. The refinements lead to a substantial reduction of orbit misclosures and improvement of the radial orbital component accuracy during eclipse seasons up to 14% w.r.t. the solutions using ECOM2.

Refining GPS/GLONASS Satellite Clock Offset Estimation in the Presence of Pseudo-Range Inter-Channel Biases

Because of the frequency division multiple access (FDMA) technique, Russian global navigation satellite system (GLONASS) observations suffer from pseudo-range inter-channel biases (ICBs), which adversely affect satellite clock offset estimation. In this study, the GLONASS pseudo-range ICB is treated in four different ways: as ignorable parameters (ICB-NONE), polynomial functions of frequency (ICB-FPOL), frequency-specific parameters (ICB-RF), and satellite-specific parameters (ICB-RS). Data from 110 international global navigation satellite system (GNSS) service stations were chosen to obtain the ICBs and were used for satellite clock offset estimation. The ICBs from the different schemes varied from −20 ns to 80 ns. The ICB-RS model yielded the best results, improving the clock offset accuracy from 300 ps to about 100 ps; it could improve the GLONASS precise point positioning (PPP) accuracy and the converging time by approximately 50% and 30%, respectively. Along similar lines, we introduced the GPS-ICB parameters in the process of GPS satellite clock estimation and GPS/GLONASS PPP, as ICBs may exist for GPS because of different chip shape distortions among GPS satellites. This possibility was found to be the case. Further, the GPS-ICB magnitude ranged from −2 ns to 2 ns, and the estimated satellite clock offsets could improve the accuracy of the GPS and combined GPS/GLONASS PPP by 10%; it also accelerated the converging time by more than 15% thanks to the GPS-ICB calibration.

Performance of real-time undifferenced precise positioning assisted by remote IGS multi-GNSS stations

The heavy reliance of real-time precise point positioning (RTPPP) on external satellite clock products may lead to discontinuity or even failure in time-critical applications. We present an alternative approach of real-time undifferenced precise positioning (RUP) that, by combining satellite clock estimation and precise point positioning based on the extended Kalman filter, is independent of external satellite clock corrections. The approach is evaluated in simulated real time with the assistance of a variable number of IGS multi-GNSS stations located between 1359.7 and 4852.5 km from the users. The results show that even with a single auxiliary IGS station, RUP is still feasible and able to retain centimeter-level positioning accuracy. Typically, with three auxiliary IGS stations about 2000–3000 km away, an accuracy of about 2 cm in the horizontal and 5 cm in the vertical can be achieved. The performance of RUP is comparable to that of PPP using 5-s satellite clock products and notably exhibits superior short-term precision in dealing with high-rate (1 Hz) GPS/GLONASS observations. The addition of GLONASS observations reduces the convergence time by 56.9% and improves the 3-D position accuracy by 31.8% while increasing the processing latency by a factor of about 1.6. Employing three IGS stations over 2400 km away from the epicenter, RUP is applied for the rapid determination of coseismic displacements and waveforms for the 2016 Kaikoura earthquake, yielding highly consistent results compared to those obtained from post-processed PPP in the global reference frame. We also explore its potential in facilitating real-time online services in terms of real-time precise positioning, zenith tropospheric delay retrieving, and satellite clock estimation.

A flexible strategy for handling the datum and initial bias in real-time GNSS satellite clock estimation

Investigations have been focused on improving the precision of real-time GNSS satellite clock corrections. There is a little literature analyzing the characteristics and impacts of the clock datum and initial clock biases (ICBs). We illustrated the clock anomaly caused by the two factors and proposed a new strategy flexible for global and regional clock estimation based on the mixed-difference observation method. First, we analyzed the clock datum of the final and real-time clocks. Results show that the variation in the clock datum can be up to 500 ns per day depending on the datum definition strategy. The variations can lead to large inconsistencies in the datum for the re-convergence of estimated satellite clocks when using the mixed-difference clock estimation method. To resolve the problem, we propose a new method for the epoch-by-epoch datum definition, which can screen out any outliers in broadcast ephemeris in real time, and particularly suitable for the real-time clock estimation. Second, we analyzed ICB jumps in real-time clock corrections, which can degrade the accuracy of precise point positioning. We propose a short-term prediction of clock corrections for bridging ICBs with the precision better than 0.1 ns for stable satellite clocks within a gap of 1–3 min. The prediction can decrease the magnitude of ICB jumps and thus improve a user solution. Considering the prediction of clocks, high precision cannot be guaranteed for all conditions, we thus recommend broadcasting a discontinuity flag for real-time service, so the re-convergence of ICB can be identified by users for resetting carrier-phase ambiguities.

Assessment of L5 frequency on real-time GPS orbit and clock estimation

Real-time precise Global Navigation Satellite System (GNSS) satellite orbit and clock products are the prerequisite of real-time GNSS-based applications. With the modernization of Global Positioning System (GPS) constellation and construction of other GNSS constellations, the number of GNSS satellites transmitting multi-frequency signals is increasing rapidly. While the benefit of multi-frequency to Precise Point Positioning (PPP) and Precise Clock Estimation (PCE) has been evaluated by many researchers, the impact of multi-frequency on orbit and clock estimation is still limited, especially in real-time mode. Different from the dual-frequency model, the Inter-Frequency Clock Bias (IFCB) must be treated carefully in the triple-frequency model. In this contribution, the triple-frequency model for real-time GPS satellite orbit and clock estimation is derived and the influence of L5 frequency observations on real-time GPS satellite orbit and clock are analyzed. With observation data from globally distributed stations spanning January 1 to 14 of 2018, real-time GPS orbit and clock are estimated with both dual-frequency and triple-frequency observations. Numerical experiments indicate that, compared to the International GNSS Service (IGS) final orbit, the L5 frequency observations improve the consistency of the normal component with about 0.2 cm, however lead to a decrease of consistency in the radial component with about 0.6 cm; the influence on the tangential direction varies with different number of stations. The potential reason for this phenomenon may partly be attributed to the absence of accurate antenna information for L5 frequency. However, due to the tight constraint of dynamic force model on orbit, the overall differences for each component is less than 1.0 cm, which means the contribution of the third frequency observation on the orbit is limited.

Corrections of BDS Code Multipath Error in Geostationary Orbit Satellite and Their Application in Precise Data Processing.

Multipath error is a main error source in Global Navigation Satellite System (GNSS) data processing, which cannot be removed by a differential technique because of the strong relationship with the environment around the station. The multipath effect of the code observables is more complex than that of the carrier-phase observables, especially for BeiDou Navigation Satellite System (BDS) geostationary orbit (GEO) satellites. In this contribution, we deeply analyzed the characteristic and effect on the precise data processing of GEO satellite multipath errors based on a large number of permanent GNSS stations. A linear combination of code and carrier-phase observables was used to analyze the characteristics of repeatability for BDS GEO’s multipath. Then, a correction method was proposed to eliminate the multipath error of the GEO code observables, based on wavelet transform. The experiment data were collected at 83 globally distributed stations, from multi-GNSS experiments and national BDS augmentation systems, from days 32 to 66 in 2017. The results show that the systematic multipath variation component of the GEO code observables can be obtained with wavelet transform, which can significantly contribute to correcting the multipath error of GEO satellites. The average root mean square error (RMSE) of the multipath series is decreased by approximately 19.5%, 20.2%, and 7.5% for B1, B2, and B3, respectively. In addition, some experiments, including ionospheric delay extraction and satellite clock estimation, were conducted in simulated real-time mode in order to validate the effect of the correction methods. For the ionospheric delay estimation, the average RMSE of the slant ionospheric delay is reduced by approximately 15.5%. Moreover, the multipath correction can contribute greatly to shortening the convergence time of the satellite clock estimation of the BDS GEO satellites.

Multi-GNSS precise orbit positioning for airborne gravimetry over Antarctica

Accurate position, velocity, and acceleration information are critical for airborne gravimetry. In Antarctica, there is a sparse distribution of IGS stations, and the rough terrain makes it impractical to set up nearby reference stations. Therefore, traditional differential GPS techniques may be difficult to implement. Precise point positioning (PPP) is independent of reference stations and provides an unlimited operating distance. However, it is highly dependent on precise satellite orbit and clock information, and may be vulnerable to discontinuities of orbit or clock offsets. An extended PPP method, called precise orbit positioning (POP), is implemented towards multi-GNSS. This approach introduces a widely spaced network of stations and is independent of precise clock information, as it estimates satellite clock offsets and drifts "on-the-fly" and only relies on precise orbit information. The advantage of being independent of clock information is that the IGS ultra-rapid (predicted) products can be applied to real-time POP since the orbits can achieve an accuracy of about 5 cm. This becomes significant when applied to airborne gravimetry, as gravity results calculated from gravity measurements and GNSS solutions can be investigated in real time. By means of processing of 5 IGS stations over Antarctica, it turns out that the PPP solution is greatly affected by discontinuities of the IGS orbit and clock offsets at the day boundaries, accompanied with some biases as large as several decimeters in the vertical component. However, the POP solution remains robust and almost no large positioning errors appear, and the accuracy improves by about 50% in the north, east, and up coordinate components. The aircraft positions derived from relative positioning, PPP, and POP during the kinematic flight period generally agree at the decimeter level because of the lack of observed satellites with elevation angles higher than 60°. Nevertheless, the potential for POP to generate centimeter-level kinematic vertical positions over long baselines is illustrated. Moreover, the POP and double-difference derived velocities and accelerations agree with each other well in both static and kinematic flight periods. After a low-pass filter, the GNSS-based accelerations are of the order of 1 mgal and are considered useful for separating the disturbing kinematic accelerations from the gravity measurements.

Regional BDS satellite clock estimation with triple-frequency ambiguity resolution based on undifferenced observation

Real-time precise satellite clock estimation is fundamental for real-time precise point positioning. To take full advantage of the triple-frequency (TF) signal of the BeiDou Navigation Satellite System (BDS), we used undifferenced observations directly in the TF data processing. In addition, TF ambiguity resolution (AR) was performed in a sequential manner, in which the wide-lane (WL) and extra wide-lane (EWL) ambiguities were fixed based on the Hatch---Melbourne---W?bbena (HMW) combination of B1/B2 and B1/B3, and then we followed with narrow-lane (NL) ambiguity fixing. By applying the ambiguity integer constraints, an ambiguity-fixed BDS satellite clock was obtained. To evaluate the contribution of TF AR in a satellite clock solution when a regional station network is involved, experiments were carried out based on the observations collected by 11 stations from the crustal movement observation network of China (CMONOC) during day of year (DOY) 153---188, 2016. First, the temporal behavior of the BDS EWL and WL uncalibrated phase delays (UPDs) were analyzed. The results suggested that they were quite stable over several days with standard deviations (STDs) of 0.004 cycles and 0.015 cycles for EWL and WL, respectively. Then, the ambiguity-fixed solution of the satellite clock was obtained with average success rates of about 99%, 90%, and 63% for EWL, WL, and NL AR, respectively. Attributed to the efficient AR, not only was the initialization time shortened but also the precision was improved. The overall averaged improvements of STD for geostationary orbits (GEO), inclined geostationary orbits (IGSO), and medium-altitude Earth orbit (MEO) satellites were 48.7%, 40.8%, and 34.4%, respectively, for the EWL and WL ambiguity-fixed solution and 72.6%, 71.0%, and 62.8%, respectively, for the NL ambiguity-fixed solution. As a comparison, a dual-frequency (DF) ambiguity-fixed satellite clock was also generated. Although the precision of the NL ambiguity-fixed clock was roughly the same, a larger root mean square (RMS) by a factor of 52% suggested that the DF ambiguity-fixed solution may have been biased with incorrect integer ambiguity fixing, whereas the TF ambiguity-fixed solution was more reliable because of the third frequency observation. In addition, precise point positioning (PPP) was carried out to assess the performance of these clock products. Compared with the PPP based on the ambiguity-float clock solution, improvements of 32.0% and 42.9% for horizontal and vertical, respectively, were obtained with the TF NL ambiguity-fixed clock solution.

Real-time GPS satellite orbit and clock estimation based on OpenMP

Real-time precise GNSS satellite orbit and clock products are the prerequisite of real-time GNSS-based applications. To obtain real-time GNSS satellite orbit and clock, three approaches exist currently, namely, the prediction-estimation approach, the prediction-correction approach and the estimation approach. Different from the former two approaches, which are based on the predicted orbit, the last approach estimates orbit and clock in an integrated way, thus it is the most rigorous one. However, the simultaneously estimation of both orbit and clock parameters makes it very time-consuming. In this contribution, the extended Kalman filter with parallel computation proposed for real-time GPS satellite clock estimation (Gao et al., 2017) is introduced to improve the computational efficiency. In the introduced method, the epoch observations are processed sequentially and the covariance update process is accelerated with the Open Multi-Processing. With observation data from about 70 globally distributed stations spanning days 001–003 of 2018, the real-time GPS orbit and clock are estimated for validation. The epoch average processing time of the introduced method achieves around 2.9 s on average with 16 CPU cores, while that of the traditional method without Open Multi-Processing is about 4.1 s. When compare the estimated orbit and clock to the IGS final products, the daily constellation-mean RMS of orbit achieve 2.7, 5.7, 4.9 cm for the radial, along-track and cross-track respectively, while the daily constellation-mean STD of the clock is about 0.10 ns. The numerical experiments indicate that the introduced method is able to provide real-time sub-decimeter GPS orbit and clock within 10.0 s considering the time for data collection and corrections broadcast.

Yaw attitude modeling for BeiDou I06 and BeiDou-3 satellites

The estimated yaw angles of the BeiDou I06 satellite demonstrated that the satellite experienced midnight- or noon-turn maneuvers when the sun elevation angle above the orbital plane (β angle) was in the range of [− 3°, + 3°] and the orbital angle was in the range of approximately [− 6°, 6°] or [174°, 186°]. The behavior of yaw attitude maneuvers in the vicinity of the midnight and noon points was identical. An alternative yaw attitude model similar to that used for the Galileo Full-Operation-Capacity (FOC) satellites was developed on the basis of the estimated BeiDou I06 yaw angles with an accuracy of approximately 3.4° to reproduce the yaw attitude behaviors. However, a discrepancy in the form of a reversal in yaw direction during the midnight-turn maneuver was observed for BeiDou I06 when the β angle was extremely small (< 0.1°). The derived yaw attitude model was proved to model the yaw attitude of the BeiDou-3 experimental satellites, and reduces the observation residuals in the vicinity of the midnight and noon points to normal levels, and facilitates continuous satellite clock estimation during eclipse periods. Compared to the yaw attitude model developed by the European Space Operations Centre (ESOC), a similar performance has been achieved with maximum yaw differences up to 9.2° when the β angle is close to 0°. The average agreement between the models is about 1°. However, the ESOC model was developed based on a patented eclipsing model, the developed model in this study is open access.

Consideration of GLONASS Inter-Frequency Code Biases in Precise Point Positioning (PPP) International Time Transfer

International time transfer based on Global Navigation Satellite System (GLONASS) precise point positioning (PPP) is influenced by inter-frequency code biases (IFCBs) because of the application of frequency division multiple access technique. This work seeks to gain insight into the influence of GLONASS IFCBs on international time transfer based on GLONASS-only PPP. With a re-parameterization process, three IFCB handling schemes are proposed: neglecting IFCBs, estimating IFCB for each GLONASS frequency number, and estimating IFCB for each GLONASS satellite. Observation data collected from 39 globally distributed stations in a 71-day period (DOY 227–297, 2017) was exclusively processed. For the comparison reason, Global Positioning System (GPS)-only PPP solutions were regarded as reference values. The clock differences derived from GPS- and GLONASS-only PPP solutions were then analyzed. The experimental results demonstrated that considering GLONASS IFCBs could reduce standard deviation (STD) of the clock differences for both identical receiver types and mixed receiver types, of which reduction was from 3.3% to 62.6%. Furthermore, compared with neglecting IFCBs, STD of the clock differences with estimating IFCB for each GLONASS satellite in coordinate-fixed mode was reduced by more than 30% from 0.30 to 0.20 ns, and by 10% from 0.40 to 0.35 ns, for 1-day arc solutions and 10-day arc solutions, respectively. Moreover, different precise products from three International GNSS Service (IGS) analysis centers were also evaluated. Even though different IFCB handling schemes were adopted in GLONASS satellite clock estimation, our numerical results showed that international time transfer on the basis of estimating IFCB for each GLONASS satellite better than the other two processing schemes. To achieve high-precision GLONASS-only PPP-based international time transfer, it is highly recommended to estimate IFCB for each GLONASS satellite.

Multi-GNSS satellite clock estimation constrained with oscillator noise model in the existence of data discontinuity

During the past years, real-time precise point positioning has been proven to be an efficient tool in the applications of navigation, precise orbit determination of LEO as well as earthquake and tsunami early warning, etc. One of the most crucial issues of these applications is the high-precision real-time GNSS satellite clock. Though the performance and character of the GNSS onboard atomic frequency standard have been widely studied, the white noise model is still the most popular hypothesis that employed in the real-time GNSS satellite clock estimation. However, concerning the real-time applications, significant data discontinuity may arise either due to the fact that only regional stations involved, or the failure in the stations, satellites and network connections. These data discontinuity would result in an arbitrary clock jump between adjacent arcs when the clock offsets are modeled as white noise. In addition, it is also expected that the detection and identification of outliers would be benefited from the constrains of the satellite oscillator noise model. Thus in this contribution, based on the statistic analysis of almost 2-year multi-GNSS precise clock products, we developed the oscillator noise model for the satellites of GPS, GLONASS, BDS and Galileo according to the oscillator type as well as the block type. Then, the efficiency of this oscillator noise model in multi-GNSS satellite clock estimation is demonstrated with 2-months data for both regional and global networks in simultaneous real-time mode. For the regional network, the results suggest that compared with the traditional solution based on white noise model, the improvement is 44.4 and 12.1% on average for STD and RMS, respectively, and the improvement is mainly attributed to the efficiency of the oscillator noise model during the convergence period and the gross error resistance. Concerning the global experiment, since the stations guarantee the continuous tracking of the satellites with redundant observable, the improvement is not as evident as that of regional experiment for GPS, GLONASS and BDS. The STD of Galileo clock improves from 0.28 to 0.19 ns due to that, the satellites E14 and E18 still suffer significant data discontinuity during our experimental period.

Evaluation and calibration of BeiDou receiver-related pseudorange biases

The pseudorange observation of the BeiDou satellite navigation system (BDS) reveals some special errors. Except for the elevation-dependent code bias variations, we present receiver-related code biases of BDS. We have analyzed the BDS receiver-related code biases based on observations from 257 stations whose receivers are from seven different manufacturers. The results demonstrate that BDS code biases are related to receiver manufacturers and receiver models and that they can reach up to about 3.0 ns among different receiver types. Moreover, BDS receiver-related code biases are quite stable over a long period of time. There is no doubt that when mixed receiver types are used, the BDS receiver-related code biases will affect the accuracy of data processing, e.g., satellite differential code bias and satellite clock bias estimation. Thus, to correct BDS receiver-related code biases, correction models for different receiver types are established. Three experiments, including satellite clock estimation, single point positioning (SPP) and pseudorange residual analysis are carried out to validate the corrections proposed. The results prove that, with BDS receiver-related code biases corrected, the biases of satellite clock estimated by mixed receiver types are more consistent with those estimated by the same receiver types. Compared with results without code bias corrections, the average satellite clock bias decreases from 1.04 to 0.22 ns which is an improvement of about 78.8%. Moreover, the accuracy of the ionospheric-free SPP is improved by 7.3 and 10.0% on average in vertical and horizontal components, respectively. The percentage of pseudorange residuals distributed between − 0.5 and 0.5 m is improved from 49.8 to 53.2%.

Real-time estimation of BDS/GPS high-rate satellite clock offsets using sequential least squares

The real-time precise satellite clock product is one of key prerequisites for real-time Precise Point Positioning (PPP). The accuracy of the 24-hour predicted satellite clock product with 15 min sampling interval and an update of 6 h provided by the International GNSS Service (IGS) is only 3 ns, which could not meet the needs of all real-time PPP applications. The real-time estimation of high-rate satellite clock offsets is an efficient method for improving the accuracy. In this paper, the sequential least squares method to estimate real-time satellite clock offsets with high sample rate is proposed to improve the computational speed by applying an optimized sparse matrix operation to compute the normal equation and using special measures to take full advantage of modern computer power. The method is first applied to BeiDou Navigation Satellite System (BDS) and provides real-time estimation with a 1 s sample rate. The results show that the amount of time taken to process a single epoch is about 0.12 s using 28 stations. The Standard Deviation (STD) and Root Mean Square (RMS) of the real-time estimated BDS satellite clock offsets are 0.17 ns and 0.44 ns respectively when compared to German Research Center for Geosciences (GFZ) final clock products. The positioning performance of the real-time estimated satellite clock offsets is evaluated. The RMSs of the real-time BDS kinematic PPP in east, north, and vertical components are 7.6 cm, 6.4 cm and 19.6 cm respectively. The method is also applied to Global Positioning System (GPS) with a 10 s sample rate and the computational time of most epochs is less than 1.5 s with 75 stations. The STD and RMS of the real-time estimated GPS satellite clocks are 0.11 ns and 0.27 ns, respectively. The accuracies of 5.6 cm, 2.6 cm and 7.9 cm in east, north, and vertical components are achieved for the real-time GPS kinematic PPP.

An efficient solution of real-time data processing for multi-GNSS network

Global navigation satellite systems (GNSS) are acting as an indispensable tool for geodetic research and global monitoring of the Earth, and they have been rapidly developed over the past few years with abundant GNSS networks, modern constellations, and significant improvement in mathematic models of data processing. However, due to the increasing number of satellites and stations, the computational efficiency becomes a key issue and it could hamper the further development of GNSS applications. In this contribution, this problem is overcome from the aspects of both dense linear algebra algorithms and GNSS processing strategy. First, in order to fully explore the power of modern microprocessors, the square root information filter solution based on the blocked QR factorization employing as many matrix–matrix operations as possible is introduced. In addition, the algorithm complexity of GNSS data processing is further decreased by centralizing the carrier-phase observations and ambiguity parameters, as well as performing the real-time ambiguity resolution and elimination. Based on the QR factorization of the simulated matrix, we can conclude that compared to unblocked QR factorization, the blocked QR factorization can greatly improve processing efficiency with a magnitude of nearly two orders on a personal computer with four 3.30 GHz cores. Then, with 82 globally distributed stations, the processing efficiency is further validated in multi-GNSS (GPS/BDS/Galileo) satellite clock estimation. The results suggest that it will take about 31.38 s per epoch for the unblocked method. While, without any loss of accuracy, it only takes 0.50 and 0.31 s for our new algorithm per epoch for float and fixed clock solutions, respectively.

Analysis of estimated satellite clock biases and their effects on precise point positioning

This study provides a systemic analysis to identify the biases in estimated satellite clocks and illustrates their effects in precise point positioning (PPP). First, the precise satellite clock estimation method considering pseudorange and carrier phase hardware delays is derived. Two methods for satellite clock estimation are compared, and their equivalency is discussed. The results show that apart from the well-known constant code hardware biases, the time-variant phase hardware biases are also absorbed by the estimated clocks. Also, the satellite clocks contain biases caused by modeling errors. To analyze the effects of these biases, they are grouped into initial clock biases (ICBs) and time-dependent biases (TDBs). Then, a detailed analysis of the impact of the biases on PPP-based troposphere and coordinate estimates is conducted. The experimental analysis demonstrates that TDBs affect positioning and tropospheric estimates, and their impacts are more significant in the static mode. The ICBs affect coordinate accuracy, zenith total delay mean bias, and its standard deviations only at the millimeter level for kinematic and static PPP, which is negligible. However, the ICBs affect the convergence period for both static and real kinematic PPP, and the magnitude of their impact largely depends on data quality. Note that satellites clocks are generally estimated with the P1/P2 and L1/L2 ionospheric-free combinations, and that hardware-specific parts of ICBs and TDBs cancel if users employ the same type of observables as the clock providers. Otherwise, the effects of biases cannot be ignored, especially for triple-frequency applications. Also, modeling-specific parts of ICBs and TDBs are significant in real-time clocks, which also affect user applications. Our conclusion is applicable for understanding the effects of these biases.

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

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

Benefits of satellite clock modeling in BDS and Galileo orbit determination

In the process of Global Navigation Satellite Systems (GNSS) satellite orbit determination, satellite clocks are typically estimated as epoch-wise white noise in addition to the satellite orbit parameters. This results in high correlation between the satellite clock estimates and the satellite orbits, especially the radial component. In this paper, an appropriate clock model is proposed to constrain the satellite clock estimates in order to improve the accuracy of satellite orbits. SLR residuals and Day Boundary Discontinuities (DBDs) were used to assess the orbit quality. The benefits of satellite clock modeling in BDS and Galileo orbit determination is validated by experimental data sets. The results show that for satellites equipped with highly stable clocks, employing an appropriate linear model constraint for the clock estimates in orbit determination can result in an improved orbit consistency as well as accuracy. In detail, for Medium Earth Orbit (MEO) satellites, C12, C14, E11, E12 and E19, the improvements of DBDs range from 10% to 20%. Furthermore, the Standard Deviation (STD) of Satellite Laser Ranging (SLR) residuals were improved up to 30% for the Galileo satellites (using a passive hydrogen maser clock).

Method for evaluating real-time GNSS satellite clock offset products

Real-time satellite clock offset products are frequently utilized in navigation and positioning service fields. The precision of such products is a key issue for their application. The evaluation methods existed for satellite clock offset products are mostly based on post-processed satellite clock offset solutions, which will encounter problems in real-time product evaluation, especially for real-time satellite clock offset products estimated from data with regional stations only. We propose an improved evaluation method for global navigation satellite system (GNSS) satellite clock offset products. In the proposed method, we use all-satellite reference method instead of single-satellite reference method to eliminate the timescale in satellite clock offset products. Moreover, a preprocessing step is suggested to detect gross errors and initial clock bias before evaluating the precision of the satellite clock offsets. We conduct two examples to verify our method, and the experimental results show that the proposed method is more reasonable in assessing the GNSS satellite clock offset precision, and it also provides a reliable approach to analyzing the estimated satellite clock offset in both real-time and post-processed, or globally and regionally.

Handling the satellite inter-frequency biases in triple-frequency observations

The new generation of GNSS satellites, including BDS, Galileo, modernized GPS, and GLONASS, transmit navigation sdata at more frequencies. Multi-frequency signals open new prospects for precise positioning, but satellite code and phase inter-frequency biases (IFB) induced by the third frequency need to be handled. Satellite code IFB can be corrected using products estimated by different strategies, the theoretical and numerical compatibility of these methods need to be proved. Furthermore, a new type of phase IFB, which changes with the relative sun–spacecraft–earth geometry, has been observed. It is necessary to investigate the cause and possible impacts of phase Time-variant IFB (TIFB). Therefore, we present systematic analysis to illustrate the relevancy between satellite clocks and phase TIFB, and compare the handling strategies of the code and phase IFB in triple-frequency positioning. First, the un-differenced L1/L2 satellite clock corrections considering the hardware delays are derived. And IFB induced by the dual-frequency satellite clocks to triple-frequency PPP model is detailed. The analysis shows that estimated satellite clocks actually contain the time-variant phase hardware delays, which can be compensated in L1/L2 ionosphere-free combinations. However, the time-variant hardware delays will lead to TIFB if the third frequency is used. Then, the methods used to correct the code and phase IFB are discussed. Standard point positioning (SPP) and precise point positioning (PPP) using BDS observations are carried out to validate the improvement of different IFB correction strategies. Experiments show that code IFB derived from DCB or geometry-free and ionosphere-free combination show an agreement of 0.3ns for all satellites. Positioning results and error distribution with two different code IFB correcting strategies achieve similar tendency, which shows their substitutability. The original and wavelet filtered phase TIFB long-term series show significant periodical characteristic for most GEO and IGSO satellites, with the magnitude varies between – 5cm and 5cm. Finally, BDS L1/L3 kinematic PPP is conducted with code IFB corrected with DCB combinations, and TIFB corrected with filtered series. Results show that the IFB corrected L1/L3 PPP can achieve comparable convergence and positioning accuracy as L1/L2 combinations in static and kinematic mode.