Inter-system Bias Research Articles

Multi-GNSS clock combination with consideration of inconsistent nonlinear variation and satellite-specific bias

As part of the International GNSS Service (IGS) multi-GNSS Pilot Project (MGEX), precise orbit and clock products for multi-GNSS constellations have been submitted by several analysis centers (ACs) since 2012. Based on the 30 s satellite clocks from 6 MGEX ACs, the multi-GNSS clocks are combined and the consistency of the AC clocks is assessed in this study. Usually, a linear transformation between the combined and AC solution is used for clock combination, and the clock residuals of the AC solution w.r.t the combined solution are used to determine the weights of the AC. However, any inconsistent satellite-specific bias or nonlinear variations in the clocks induced by the AC’s processing strategy can contaminate the linear transformation as well as the determination of the weight. In this study, the analysis center and satellite-specific bias (ASB) of the MGEX AC clock solutions is first identified and estimated by using observations from globally distributed stations. Moreover, the clock solutions with nonlinear variations induced by the reference clock or nonconstant intersystem bias (ISB) are corrected by aligning the clocks to the selected reference solution before clock combination. With the correction of the ASB, the root-mean-square of the clock residuals decreases significantly and reaches 14-26, 37-91, 33-48 and 12-44 ps for GPS, GLONASS, BDS-2 and Galileo, respectively. In general, the consistency of AC solutions w.r.t the combination reaches 8-16, 27-58, 13-27 and 9-36 ps for GPS, GLONASS, BDS-2 and Galileo in terms of the standard deviation. Finally, the individual AC and combined orbit as well as clock solutions of different constellations are assessed by precise point positioning, and the combined multi-GNSS solutions show competitive performance with the best AC solution in terms of both the positioning accuracy and stability of the reference frame parameters. However, inconsistent scale parameters of both the AC and combined solutions are identified and require more investigation.Graphical

Preliminary Analysis of Intersystem Biases in BDS-2/BDS-3 Precise Time and Frequency Transfer

The Chinese BeiDou global satellite system (BDS-3) and regional system (BDS-2) are predicted to coexist over the next decade. Intersystem biases (ISBs) in BDS-2/BDS-3 play a key role in maintaining the consistency and continuity from the BDS-2 to BDS-3 time transfer. Here, we discuss the temporal characteristics, parameter composition, generation mechanism, and the effect of ISBs in BDS-2/BDS-3 on time and frequency transfer. The satellite orbits and clock products from three international GNSS service analysis centers, namely Wuhan University (WUM, China), GeoForschungsZentrum Potsdam (GFZ, Germany), and the Center for Orbit Determination in Europe (CODE), were employed to investigate the time-transfer stability of ISBs when BDS-2 and BDS-3 were used in combination. We analyzed the intrinsic characteristics of ISBs, the receiver types, antennas, and frequency standards. Our first results showed that ISBs are stable for different analysis center products, although the mean values of daily results differed markedly for the three analysis centers. With respect to the relationship between station attribution and ISB difference for a time link, the receiver type, antenna, and frequency standard influence the ISB differences in time and frequency transfer. The effect of three ISB stochastic models was evaluated with respect to time and frequency transfer. The “walk” and “constant” schemes were slightly superior to “noise”, with the improvement in their frequency stability being approximately 5% compared with that of “noise”.

Modeling and performance assessment of precise point positioning with multi-frequency GNSS signals

The development of global navigation satellite system (GNSS), especially BeiDou-3 Navigation Satellite System (BDS-3) and Galileo system, promotes the progress of the precise point positioning (PPP) technology. Considering deep integration and full utilization of GNSS signals, this study dedicates to developing three triple-frequency PPP models apply to multi-constellation including BDS-3. Besides, some biases such as differential code bias (DCB), inter-frequency bias (IFB) and inter-system bias (ISB) are also addressed. Through the ten-day datasets collected from 160 stations, the multi-frequency PPP models for single- and multi-constellation solutions are analyzed and compared from static positioning performance, Zenith Tropospheric Delay (ZTD) estimates, ISB estimates, and IFB estimates. The results indicate that the triple-frequency PPP models, especially two dual-frequency ionosphere-free (IF) model namely TIF with triple-constellation which can provide the positioning accuracy of 4.89, 7.31, 12.1 mm after the convergence time of 10.75 min in east, north and up directions, perform better than the dual-frequency model, apart from the a single IF model with GPS. Interestingly, the positioning performance is not effectively improved with the addition of GLONASS constellation. But the performance of GAL/BDS combined constellation is comparable to GPS single constellation in TIF model.

Precise point positioning with BDS-2 and BDS-3 constellations: ambiguity resolution and positioning comparison

The full operational capability of the Chinese BeiDou Satellite Navigation System (BDS) has injected additional energy into the GNSS field. Along with the visibility of more BDS satellites, as well as the precise products generated by analysis centers, ambguity float and fixed Precise Point Positioning (PPP) performance with the complete BDS constellation are evaluated in this study. Inter-system biases (ISBs) between BDS-2 and BDS-3 are first evaluated, and Fractional Cycle Bias (FCB) with and without consideration of ISBs are assessed. Results show that when considering ISBs, the standard deviations of ambiguity residuals are smaller, thus the ISB parameter should be taken into consideration in BDS PPP. BDS-3 FCBs are more stable than those of BDS-2, at the same time, the ambiguity residuals are smaller for BDS-3 satellites. In terms of PPP performance, BDS-3 outperforms BDS-2 in positioning accuracy, convergence time and time to first fix. For the test data, the convergence time of BDS-2 static solution is 48.8 min, while for BDS-2 + 3 it is 19.7 min, representing an improvement of 59.6% when BDS-3 satellites are included. The positioning accuracy of static BDS-2, BDS-3 and BDS-2 + 3 PPP can reach the same millimeter-level after a long time of convergence, while kinematic solution with BDS-2 + 3 has the highest accuracy compared to BDS-2 and BDS-3, which could reach 1.7, 1.3 and 4.3 cm in East, North and Up components, respectively, in fixed solution. The performance of BDS PPP with the whole constellation is comparable with that of GPS and Galileo, and the PPP performance is promising with receiver updates for MGEX stations in the near future.

Long-term and short-term stability characteristics of receiver inter system bias for BDS3/BDS2

The study of inter-system bias (ISB) is important for multi-system fusion and the performance of different signal compatibility. In this paper, the stability of ISB at the BDS3/BDS2 receiver end is calculated and analyzed for different time spans (DOY 060∼090 in 2021) from a total of 31 MGEX and iGMAS stations. We adopted two estimation strategies, random walk and constant approach, using the precision products of orbit and clock bias provided by WUM, the influence of which on ISB was also analyzed. Our results showed that the ISB value varied little within a day, and the mean of daily ISB standard deviation was only 0.037 m when the observation condition was good. The signal reception was continuous, indicating a high ISB stability for one day. If extending the time series to one month, however, the ISB standard deviation calculated by constant approach, in which a constant ISB is estimated on a daily basis was about 0.1 m, and the results of adjacent days were not continuous, with no apparent pattern. Concerning the random walk approach, the obtained ISB time series also had a jump, and the conclusion was the same as that of the constant strategy. Besides, receiver types showed a strong regularity in ISB numerical situation, and the distribution of ISB values corresponding to the same receiver type was relatively close. Therefore, we conclude that the ISB parameters remain stable in the short term (one day) and less stable in the long-term period. It is recommended that the ISB term should be set as a constant estimate every day in BDS3/BDS2 solutions, regardless of receiver type consistency.

Progress of global ARAIM availability of BDS-2/BDS-3 with TGD and ISB

Advanced RAIM (ARAIM) is an extension of Receiver Autonomous Integrity Monitoring (RAIM) by enhancing the vertical navigation capabilities, so it can be applied to higher-level precision applications. But when the observation data of the second-generation BDS (BDS-2) and the third-generation BDS (BDS-3) are combined to evaluate the ARAIM availability, the evaluation be affected by the Timing Group Delay (TGD) of different satellite signals and the Inter-system Bias (ISB) between BDS-2 and BDS-3, which is ignored by the existing literature. Therefore, in this article, we deduced the mathematical model of ARAIM and the calculation formula of Vertical Protection Level (VPL) by considering the combined influence of TGD and ISB. We used the BDS observation data and ephemeris files of 374 International GNSS Service (IGS) stations around world on July 29, 2021 to calculate the VPL of these stations and evaluate the global ARAIM availability. The results show that the number of visible BDS satellites is 8.033 and the Geometric Dilution of Precision (GDOP) is 1.721, which can meet the basic needs of global ARAIM calculations. Furthermore, when considering the combined influence of TGD and ISB, the evaluation of ARAIM availability by Multiple Hypothesis Solution Separation (MHSS) is significantly better than that without considering the combined TGD and ISB influence or their individual influence. The average VPL values at these IGS stations in four cases are 124.925 m, 13.882 m, 32.610 m and 9.962 m, and the corresponding ARAIM availability percentages are 90.839%, 95.241%, 96.751% and 99.660%, respectively. Finally, the minimum coverage of ARAIM availability based on BDS can reach 90.839%, which can meet the aviation demand for ARAIM availability in most regions and times.

Research on the Impact of BDS-2/3 Receiver ISB on LEO Satellite POD

In recent years, the multi-GNSS positioning application is becoming more and more popular, same to the low Earth orbit (LEO) satellite precise orbit determination (POD) based on the onboard multi-GNSS measurements. The third-generation Beidou navigation satellite system (BDS-3) provides a new option to obtain the LEO satellite orbit solutions. However, the receiver intersystem bias (ISB) of different GNSS is unavoidable in multi-GNSS data processing. This paper’s main goal is absorption of the impact of the ISB between BDS-3 and BDS-2 on the LEO satellite POD. Taking GPS-based POD solutions for the reference orbit, this paper evaluates the orbit accuracy of BDS-2-based POD, BDS-3-based POD, BDS-2 and BDS-3 combined PODs with/without ISB. The BDS-3-based POD accuracy is 6.57 cm in the 3D direction, a 56% improvement over BDS-2-based POD. When the ISB between BDS-3 and BDS-2 is estimated, the BDS-2/3 combined POD accuracy of 5.37 cm in the 3D direction is better than that without ISB, which is a 64% improvement over BDS-2-based POD and 18% improvement over BDS-3-based POD. For GPS and BDS-2/3 combined POD, the GPS and BDS-3 joint POD solutions have the smallest RMS differences in overlapping consistency and smallest RMS differences compared to GPS-based POD. This study indicates that estimating the BDS-2/3 receiver ISB in BDS-2/3 joint POD could improve the orbit accuracy, and the GPS and BDS-3 joint POD solution is better than another combined POD. This paper will provide meaningful references for the LEO satellite multi-GNSS-based POD.

Accounting for BDS-2/BDS-3 inter-system biases in PPP and RTK models

Inter-system Biases (ISBs) between BDS-2 and BDS-3 is of great importance for BDS-2/BDS-3 joint processing, which is meaningful to make full use of BDS observations and improves PNT (positioning, navigation, timing) performance. In this contribution, we derive ISBs models with undifferenced ionospheric-free (IF) combination and double-differenced model, corresponding to the IF-PPP and RTK technologies respectively. To analyze code ISB in PPP, we processed data from 55 multi-GNSS experimental (MGEX) stations including 7 receiver types and conduct a detailed analysis on the effect of frequency, receiver types and precise products from 4 IGS Analysis Centers (ACs). It is concluded that both the overlapping frequency and the non-overlapping frequency of BDS-2/BDS-3 exist code ISB. The code ISB of overlapping frequency is within 10 ns, and that of the non-overlapping frequency can reach 100 ns. Moreover, code ISB is influenced by receiver types as well as satellite geometry. The average ISB of stations with the same manufactures tend to be the same while that of stations from different manufactures are different. Affected by the satellite geometry, the ISB characteristics exhibit periodicity and correlation with the location of stations. By comparing ISB calculated by different precise clock products from the GeoForschungsZentrum (GFZ), European Space Agency (ESA), Center for Orbit Determination in Europe (CODE) and Wuhan University (WHU), the results show that the daily STD of ISBCOD, ISBWHU,ISBESA and ISBGFZ is about 0.2 ns, which means that ISB can be calibrated by constant estimation when these products are used. As for Differential Inter-system Biases (DISBs) in RTK, it is discovered that the BDS-2/BDS-3 joint differenced system has a Differential Inter-system Code Bias (DISCB) that exceeds 3.4 ns when the rover and base receivers are made by different manufactures and the hardware delay characteristics for BDS-2 and BDS-3 are different. However, the Differential Inter-system Phase Bias (DISPB) doesn’t exist and can be neglected in RTK.

Assessment of IRNSS-Only Data Processing: Availability, Single-Frequency SPP and Short-Baseline RTK

The Indian Regional Navigation Satellite System (IRNSS) currently can provide independent positioning services with eight in-orbit satellites. This study provides a comprehensive assessment of IRNSS-only data processing, including the availability of satellite constellation, the performance of single-frequency single point positioning (SPP), and the performance of single-frequency short-baseline real-time kinematic (RTK) positioning. Regarding the availability of IRNSS-only case in its primary service areas, the average number of visible satellites is 6–8, and the average Position Dilution of Precision (PDOP) value falls between 3.3 and 6.2, under a service rate of nearly 100.0%. The datasets from 14 stations located in the IRNSS service areas spanning a week are used for position determination. The results show that under the IRNSS single-system case, the positioning accuracy of the SPP is 6.031, 6.015, and 9.668 m in the east, north, and up directions, respectively, and the mean positioning bias of short-baseline RTK is 5.4, −21.1, and −0.2 mm with a standard deviation (STD) error of 7.8, 19.2, and 29.0 mm in the three directions, respectively. For comparative analysis, the results of the GPS single-system and GPS/IRNSS dual-system combination cases are also presented. The positioning performance of IRNSS is inferior to that of GPS, and the performance improvement of GPS/IRNSS dual-system integrated solutions over GPS single-system solutions is not significant. Furthermore, based on the GPS/IRNSS dual-system solutions, the inter-system bias estimates from SPP, the code observation residuals from SPP, and the carrier phase observation residuals from short-baseline RTK are characterized.

Multi-antenna GNSS tight combination attitude determination in the urban environment

It is a challenging task to determine dynamic vehicle attitude using a single-frequency single-epoch multi-antenna global navigation satellite system (GNSS). In the urban environment, the number of visible satellites drops sharply due to the occlusion of trees and tall buildings, hence it is difficult to obtain the high-precision attitude of vehicles using only a multi-antenna GNSS. The GNSS tight combination algorithm selects the same reference satellite between different systems, and can effectively increase the number of observation equations after eliminating the inter-system bias, to improve the attitude accuracy of vehicles in complex environments. Compared with the loose combination algorithm, which selects reference satellites separately between different systems, the tight combination algorithm can further improve the locatable performance when there are fewer satellites. Dynamic vehicle experiments were carried out in an open environment and a complex environment, respectively, using the GPS/BDS/GALILEO three-system single-frequency and single-epoch positioning mode. The results show that the tight combination algorithm and the loose combination algorithm have the same accuracy when there are enough visible satellites in the open environment. In a complex environment, with a cut-off elevation angle of 40°, the percentage of pitch angle error, yaw angle error and roll angle error within 2° increased by 6.1%, 8.07% and 13.43%, respectively, and the ambiguity fixed rate was increased by 14.78%.

Modeling and Prediction of Inter-System Bias for GPS/BDS-2/BDS-3 Combined Precision Point Positioning

The combination of Precision Point Positioning (PPP) with Multi-Global Navigation Satellite System (MultiGNSS), called MGPPP, can improve the positioning precision and shorten the convergence time more effectively than the combination of PPP with only the BeiDou Navigation Satellite System (BDS). However, the Inter-System Bias (ISB) measurement of Multi-GNSS, including the time system offset, the coordinate system difference, and the inter-system hardware delay bias, must be considered for Multi-GNSS data fusion processing. The detected ISB can be well modeled and predicted by using a quadratic model (QM), an autoregressive integrated moving average model (ARIMA), as well as the sliding window strategy (SW). In this study, the experimental results indicate that there is no apparent difference in the ISB between BDS-2 and BDS-3 observations if B1I/B3I signals are used. However, an obvious difference in ISB can be found between BDS-2 and BDS-3 observations if B1I/B3I and B1C/B2a signals are used. Meanwhile, the precision of the Predicted ISB (PISB) on the next day of all stations is about 0.1−0.6 ns. Besides, to effectively utilize the PISB, a new strategy for predicting the PISB for MGPPP is proposed. In the proposed strategy, the PISB is used by adding two virtual observation equations, and an adaptive factor is adopted to balance the contribution of the Observed ISB (OISB) and the PISB to the final estimations of ISB. To validate the effectiveness of the proposed method, some experimental schemes are designed and tested under different satellite availability conditions. The results indicate that in open sky environment, the selective utilization of the PISB achieves almost the same positioning precision of MGPPP as the direct utilization of the PISB, but the convergence time of MGPPP is reduced by 7.1% at most in the north (N), east (E), and up (U) components. In the blocked sky environment, the selective utilization of the PISB contributes to more significant improvement of the positioning precision and convergence time than that in the open sky environment. Compared with the direct utilization of the PISB, the selective utilization of the PISB improves the positioning precision and convergence time by 6.7% and 12.7% at most in the N, E, and U components, respectively.

Evaluation of BDS-2 and BDS-3 Satellite Atomic Clock Products and Their Effects on Positioning

The BeiDou Navigation Satellite System (BDS) has completed third phase construction and currently provides global services, with a mixed constellation of BDS-2 and BDS-3. The newly launched BDS-3 satellites are equipped with rubidium and passive hydrogen maser (PHM) atomic clocks. The performance of atomic clocks is one of the cores of satellite navigation system, which will affect the performance of positioning, navigation and timing (PNT). In this paper, we systematically analyze the characteristics of BDS-2 and BDS-3 atomic clocks, based on more than one year of precise satellite clock products and broadcast ephemeris. Firstly, the results of overlapping Allan variations demonstrate that BDS-3 Rb and PHM clocks improve better in stability than BDS-2 Rb clock and are comparable to GPS IIF Rb and Galileo PHM clocks. Accordingly, the STDs of BDS-3 broadcast satellite clock are better than GPS and BDS-2, which are at the same level with that of Galileo. Secondly, the inter-system bias (ISB) between BDS-2 and BDS-3 is analyzed by satellite clock datum comparison and precise point positioning (PPP). Surprisingly, the discrepancy between BDS-2 and BDS-3 satellite clock datum has a great difference between products that could reach up to about 10 ns for WHU satellite clock products and broadcast ephemeris. Moreover, the ISBs between BDS-2 and BDS-3 satellite clocks are quite stable over one-year periods. Thirdly, due to the improved stability of BDS-3 atomic clock, the 68% positioning accuracy is better than 0.65 m at 10 min for BDS-3 PPP, based on broadcast ephemeris. Besides, the non-negligible bias between BDS-2 and BDS-3 will greatly affect the BDS precise data processing. The accuracy of positioning is greatly improved when considering the ISB.

Combined BDS-2/BDS-3 real-time satellite clock estimation with the overlapping B1I/B3I signals

With the deployment of the global BeiDou navigation satellite system (BDS-3) constellation completed, the positioning service performance is significantly improved relative to the regional system (BDS-2). As one of the key prerequisite products for precise point positioning (PPP), the combined BDS-2/BDS-3 real-time satellite clock estimation method is investigated in this study. Since the code hardware delays of BDS-2/BDS-3 satellites might be inconsistent for some receiver types, the inter-system bias (ISB) is considered in the combination although the overlapping B1I/B3I signal is employed. The combination processing is comprehensively evaluated in terms of satellite tracking coverage, clock precision, and convergence time. The experiment results indicate that the standard deviation (STD) precision of the estimated clocks for BDS-2 medium Earth orbit (MEO) satellites is improved by 52.0% from 0.150 ns to 0.072 ns due to the improvement of satellite tracking coverage. Introducing the ISB parameter as piece-wise constants improves the STD of all clocks from 0.076 ns to 0.067 ns and BDS-3 clocks achieve a significant improvement from 0.065 ns to 0.053 ns. The real-time clock estimation is demonstrated to be robust for the clock datum transformation, and it converges after one hour to a stable precision but needs a long time to achieve the best performance. The BDS real-time clocks are estimated with the proposed strategy to support real-time PPP based on the predicted satellite orbits. Results show that the clock STD is about 0.1 ns for most BDS satellites and the positioning accuracy of about 10 cm is achieved after the convergence time of about half an hour. The clock precision for BDS-2 geostationary Earth orbit (GEO) satellites is degraded to 0.303 ns due to the impact of poor orbit prediction errors. The exclusion of GEO satellites improves the positioning accuracy by about 15% and reduces the convergence time by 22–34%.

Intersystem Bias in GPS, GLONASS, Galileo, BDS-3, and BDS-2 Integrated SPP: Characteristics and Performance Enhancement as a Priori Constraints

Global navigation satellite systems (GNSSs) have been booming in recent years, and the space segment of all four of the GNSSs, including BDS (BDS-3/BDS-2), Galileo, GPS, and GLONASS, has almost been fully deployed at present. The single point positioning (SPP) technology, which is widely used in satellite navigation and low-accuracy positioning, can benefit from the multi-GNSS integration, but the additional intersystem bias (ISB) parameters should be introduced to ensure the compatibility among different GNSSs. In this study, the ISB estimates derived from four-system integrated SPP are carefully characterized, and the performance enhancement attributed to a priori ISB constraints by prediction for position solutions under open sky and constrained visibility environments is rigorously evaluated. The results indicate that the ISB between BDS-3 and BDS-2 cannot be ignored. The daily ISBs show step changes when encountering the replacement of receiver types, while it is not the case for the receiver firmware versions. The daily ISBs are roughly consistent for the stations equipped with the same type of receivers. The short-term stability of epochwise ISBs for GLONASS, Galileo, BDS-2, and BDS-3 with respect to GPS can be 2.335, 1.262, 1.741, and 1.532 ns, respectively, whereas the corresponding long-term stability for daily ISBs can be 1.258, 1.288, 2.713, and 2.566 ns, respectively. The single-day prediction accuracy of daily ISBs for GLONASS, Galileo, BDS-2, and BDS-3 with respect to GPS can be 1.055, 0.640, 1.242, and 0.849 ns, respectively. The improvements on positioning accuracy after introducing a priori ISB constraints can be over 20% at an elevation mask of 40° and 50° with a time span of ISB prediction of a day. As to the availability, it is only 64.0% for traditional four-system SPP under a cutoff elevation of 50°, while the corresponding availability is increased to approximately 90.0% after considering a priori ISB constraints. For completeness, the characteristics of ISBs estimated with the low-cost u-blox M8T receiver and the Xiaomi Mi8 smartphone as well as the contribution of a priori ISB constraints to the multisystem SPP solutions with these devices are also investigated.

BDS-2/BDS-3 combined precise time-frequency transfer with different inter-system bias estimation strategies and different analysis centres products

BeiDou-3 global navigation satellite system (BDS-3) has officially provided positioning, navigation, and timing (PNT) services on 31 July 2020. The difference between BDS-2 and BDS-3 in signal modulation mode, signal characteristics, and satellite clock datum leads to the possible existence of inter-system bias (ISB) between BDS-2 and BDS-3, which becomes the key to affect the time-frequency transfer of BDS. In this contribution, taking the observations of BRUX, CEBR, USUD, WTZZ, and YEL2 stations as an example, five analysis centres (ACs) precise products (for instance, CNT, COM, ESA, GBM, and WUM) are used to evaluate the BDS-2 BII/B3I combined BDS-3 B1I/B3I, B1I/B2a, B1C/B2a, and B1C/B2a time-frequency transfer performance with ignore ISB (ISBNO), estimating ISB as constant (ISBCT), estimating ISB as a random walk (ISBRW) and estimating ISB as white noise (ISBWN) four kinds of estimation strategies. The results show that the ISBCT estimation strategy has the better performance than ISBNO, ISBRW, and ISBWN estimation strategies in BDS-2/BDS-3 combined time-frequency transfer; compared with ignoring ISB strategy, the standard deviation and frequency instability of ISBCT estimation strategy is decreased by 9.16% and 12.83% on average, respectively. Meanwhile, an ISBCT estimation strategy is recommended when using CNT, COM, ESA, and WUM precision products for BDS-2/BDS-3 combined time-frequency transfer.

Estimation of Inter-System Biases between BDS-3/GPS/Galileo and Its Application in RTK Positioning

For the development of a global navigation satellite system (GNSS), the third generation of BeiDou Navigation Satellite System (BDS-3) achieved full constellation for worldwide service on 23 June 2020. The new signals, B1C and B2a of BDS-3, further enhance the compatibility and interoperability between different GNSSs. In this study, we first assessed the quality of all the signals in BDS-3/GPS/Galileo. Then, to achieve the interoperability among BDS-3/GPS/Galileo, the inter-system bias (ISB), which appears if an inter-system difference exists between two GNSSs, was estimated at overlapping frequencies. Finally, we used the estimated ISBs in real-time kinematic (RTK) positioning. The results show the higher quality of the overlapping frequency B2a/L5/E5a than B1C/L1/E1 in terms of pseudo range multipath. The ISBs are stable both in the short term for one day and in the long term for over a year, which fit a zero-mean normal distribution well when the identical type of receiver is applied. Thus, it is reasonable to ignore the ISBs in the inter-system differences. With the estimated ISBs, the inter-system double-difference RTK can be achieved, which is called a tightly combined model (TCM) RTK. Compared with the traditional intra-system double-difference RTK, which is called a loosely combined model (LCM) RTK, the TCM RTK can achieve a higher success rate (SR) in terms of ambiguity resolution and higher positioning accuracy. In addition, the higher the cutoff elevation angle set, the greater the promotion can be obtained in SR. Even with a cutoff elevation angle of 50°, the SR of TCM is over 80%. Thus, it is important to apply TCM RTK when the observation conditions are limited, such as in dense jungles or the urban canyons.

Real-Time Estimation of GPS-BDS Inter-System Biases: An Improved Particle Swarm Optimization Algorithm

The restart of the receiver will lead to the change in the non-overlapping frequency inter-system biases (ISB), which will make it difficult to apply the tightly combined RTK method of pre-calibrating ISB to the actual scene. Particle swarm optimization (PSO) algorithm can be used to estimate the fractional part of the inter-system phase bias (F-ISPB) in real time, which is not affected by the receiver restart. However, the standard PSO can easily fall into local optimum and cannot accurately estimate the value of F-ISPB. In this contribution, based on the characteristics of F-ISPB, we propose an improved PSO with adaptive search space and elite reservation strategy to estimate the F-ISPB in real time. When the value of F-ISPB is close to the boundary of the search space, the improved PSO will transform the search space so that F-ISPB will be located near the central region of the new search space, which will greatly reduce the situation of the standard PSO easily falling into local optimum. Since F-ISPB is very stable, an elite retention strategy will help us to estimate F-ISPB faster and more accurately. Three sets of short baseline static data were selected for testing. The results show that the inter-system differenced model based on the improved PSO has a higher ambiguity fixed rate and positioning accuracy than the inter-system differenced model based on the standard PSO and the classical intra-system differenced model, and the fewer the number of satellites, the more obvious the effect.

Feasibility of easy-to-implement methods to analyze systematic errors of multipath, differential code bias, and inter-system bias for low-cost receivers

Feasibility of easy-to-implement methods to analyze systematic errors of multipath, differential code bias, and inter-system bias for low-cost receivers

Characteristics of receiver-related biases between BDS-3 and BDS-2 for five frequencies including inter-system biases, differential code biases, and differential phase biases

It is foreseeable that the BeiDou navigation satellite system with global coverage (BDS-3) and the BeiDou navigation satellite (regional) system (BDS-2) will coexist in the next decade. Care should be taken to minimize the adverse impact of the receiver-related biases, including inter-system biases (ISBs), differential code biases (DCB), and differential phase biases (DPB) on the positioning, navigation, and timing (PNT) provided by global navigation satellite systems (GNSS). Therefore, it is important to ascertain the intrinsic characteristics of receiver-related biases, especially in the context of the combination of BDS-3 and BDS-2, which have some differences in their signal level. We present a method that enables time-wise retrieval of between-receiver ISBs, DCB, and DPB from multi-frequency multi-GNSS observations. With this method, the time-wise estimates of the receiver-related biases between BDS-3 and BDS-2 are determined using all five frequencies available in different receiver pairs. Three major findings are suggested based on our test results. First, code ISBs are significant on the two overlapping frequencies B1II and B2b/B2I between BDS-3 and BDS-2 for a baseline with non-identical receiver pairs, which disrupts the compatibility of the two constellations. Second, epoch-wise DCB estimates of the same type in BDS-3 and BDS-2 can show noticeable differences. Thus, it is unreasonable to treat them as one constellation in PNT applications. Third, the DPB of BDS-3 and BDS-2 may have significant short-term variations, which can be attributed to, on the one hand, receivers composing baselines, and on the other hand, frequencies.

Estimation and analysis of multi-GNSS observable-specific code biases

The proper handling of code biases is essential for realizing precise ionospheric modeling, positioning and timing. It is common to treat code biases in a differential mode as a differential code bias (DCB). With the modernization of GPS and GLONASS and the implementation of Galileo and BDS, the traditional DCB calibrations are complex and not easily extendable to different frequencies and modulations. An alternative treatment of code biases is to use observable-specific signal biases (OSBs). In this contribution, all possible OSBs are estimated for the latest GNSS and analyzed from the perspectives of precision, consistency and stability. The precisions of the GPS and Galileo OSBs are significantly better than those of the GLONASS and BDS OSBs. Considering the inter-frequency bias (IFB) of GLONASS and the inter-system bias (ISB) of BDS can improve the precisions of their OSB estimates. OSB comparisons among different agencies reveal that GPS and Galileo show good agreement at the level of 0.2–0.3 ns, while the differences of the GLONASS and BDS OSBs reach 0.5–1.0 ns. In addition, agreement of 0.4–0.5 ns is demonstrated for IGSO and MEO OSBs, while the consistency of GEO OSBs is worse by a factor of 2–3. The stability of the OSB estimates is at the level of 0.03–0.09 ns for GPS, 0.10–0.25 ns for Galileo, 0.14–0.48 ns for GLONASS, and 0.16–0.44 ns for BDS. In general, the BDS-3 OSB estimates show better stability than the BDS-2 OSBs. Moreover, the code biases at the same or at a close central frequency show similar performance. This is particularly obvious for Galileo and BDS, which adopt the dual-frequency constant envelope multiplexing (DCEM) technique. For instance, the code bias estimates of C5Q, C5X, C7Q, and C8Q are close to each other for individual Galileo satellites, and the BDS code biases of C5P, C7Z, and C8X are comparable to each other.