Inter-system Bias Research Articles

Clustering and look-up table calibration of receiver inter-system bias for QZSS and GPS integration

The Quasi-Zenith Satellite System (QZSS) from Japan will supplement the global positioning system (GPS) for more accurate and stable positioning, navigation and timing services. However, QZSS and GPS measurements are not free of inter-system bias (ISB) for various receiver brands. After estimating and clustering the receiver ISB values of GPS and QZSS for all three overlapping frequencies L1, L2 and L5, we show that the code ISB of the receivers can have at least two groups and the firmware versions of the groups may overlap with each other, making the look-up-table calibration difficult. Therefore, we propose to discriminate one code ISB value from the other for a specific receiver by the observations of a few epochs so that precise calibration can be implemented within a short time. The phase fractional ISB (F-ISB) calibration of QZSS and GPS is easier as their magnitudes are receiver-type and firmware-version related. Experiments for the code positioning show that the QZSS and GPS integration with the ISB calibrated by the looked-up values has an obviously smaller 3D distance root mean square, i.e. the root mean square of the radial distances from the true position to the calculated positions. The empirical success rate of the ambiguity fixing in precise positioning is also improved with the looked-up phase F-ISB values.

GPS/BDS-2/Galileo Precise Point Positioning Ambiguity Resolution Based on the Uncombined Model

In this study, an uncombined precise point positioning (PPP) model was established and was used for estimating fractional cycle bias (FCB) products and for achieving ambiguity resolution (AR), using GPS, BDS-2, and Galileo raw observations. The uncombined PPP model is flexible and efficient for positioning services and generating FCB. The FCBs for GPS, BDS-2, and Galileo were estimated using the uncombined PPP model with observations from the Multi-GNSS Experiment (MGEX) stations. The root mean squares (RMSs) of the float ambiguity a posteriori residuals associated with all of the three GNSS constellations, i.e., GPS, BDS-2, and Galileo, are less than 0.1 cycles for both narrow-lane (NL) and wide-lane (WL) combinations. The standard deviation (STD) of the WL combination FCB series is 0.015, 0.013, and 0.006 cycles for GPS, BDS-2, and Galileo, respectively, and the counterpart for the NL combination FCB series is 0.030 and 0.0184 cycles for GPS and Galileo, respectively. For the BDS-2 NL combination FCB series, the STD of the inclined geosynchronous orbit (IGSO) satellites is 0.0156 cycles, while the value for the medium Earth orbit (MEO) satellites is 0.073 cycles. The AR solutions produced by the uncombined multi-GNSS PPP model were evaluated from the positioning biases and the success fixing rate of ambiguity. The experimental results demonstrate that the growth of the amount of available satellites significantly improves the PPP performance. The three-dimensional (3D) positioning accuracies associated with the PPP ambiguity-fixed solutions for the respective only-GPS, GPS/BDS-2, GPS/Galileo, and GPS/BDS-2/Galileo models are 1.34, 1.19, 1.21, and 1.14 cm, respectively, and more than a 30% improvement is achieved when compared to the results related to the ambiguity-float solutions. Additionally, the convergence time based on the GPS/BDS-2/Galileo observations is only 7.5 min for the ambiguity-fixed solutions, and the results exhibit a 53% improvement in comparison to the ambiguity-float solutions. The values of convergence time based on the only-GPS observations are estimated as 22 and 10.5 min for the ambiguity-float and ambiguity-fixed solutions, respectively. Lastly, the success fixing rate of ambiguity is also dramatically raised for the multi-GNSS PPP AR. For example, the percentage is approximately 99% for the GPS/BDS-2/Galileo solution over a 10 min processing period. In addition, the inter-system bias (ISB) between GPS, BDS-2, and Galileo, which is carefully considered in the uncombined multi-GNSS PPP method, is modeled as a white noise process. The differences of the ISB series between BDS-2 and Galileo indicate that the clock datum bias of the satellite clock offset estimation accounts for the variation of the ISB series.

Comprehensive analysis of compatibility between QZSS and GPS in Asia-Pacific region: Signal quality, RTK and PPP

The Japanese QZSS, with four operational satellites in orbit, provides positioning, navigation and timing services as a GPS-complementary system. However, the compatibility between GPS and QZSS has not been adequately investigated so far. In this paper, their compatibility, from the aspects of signal quality, GPS + QZSS inter-system RTK and GPS + QZSS combined PPP is comprehensively analyzed. First, the signal quality comparison using 10-day zero-baseline observations shows that although QZSS has slightly lower carrier‑to‑noise‑density ratio (C/N0) than GPS, it has GPS-comparable accuracies of 36.2 and 7.4 cm on L1 and L5 code observations respectively and 2.4, 2.1 and 2.2 mm on L1/L2, L1/L5 and L2/L5 geometry-free phase observations respectively, and an even better one of 22.3 cm on L2 code observation. Second, the short-baseline experiment testifies that GPS + QZSS inter-system double-difference (ISDD) RTK obtains better positioning performance than traditional double-difference (TDD) RTK in terms of ambiguity-fixing success rate (ASR) and positioning accuracy. This indicates a good compatibility between GPS and QZSS which benefits from their same frequency on L1, L2 and L5. And at 10° cut-off elevation, the GPS + QZSS with ISDD RTK model obtains the ASR improvements of 15.3%, 4.0% and 73.1% over GPS-only on L1, L2 and L5, respectively. The significant improvement on L5 is due to only 12 GPS Block IIF satellites transmitting the L5 signal. Third, the theoretical derivation demonstrates that the inter-system bias (ISB) in GPS + QZSS PPP is composed of the constant part (i.e., receiver-dependent hardware delay difference and AC-dependent clock datum difference) and the variable part. The analysis of ISB characteristics shows that ISBWUM and ISBCOM have the same variation trend and behave stably with a daily variation of 0.4 m, while ISBGBM makes a significant daily variation of 1.4 m. Furthermore, the GPS + QZSS dual- and triple-frequency PPP based on the observations of 40 GNSS stations in Asia-Pacific region show that the GPS + QZSS PPP performance using WUM and COM products is superior to that using GBM product. These results denote that WUM and COM products make GPS and QZSS more compatible in PPP. The four-satellite QZSS system is able to effectively improve the performance of RTK and PPP for GPS in Asia-Pacific region and this improvement increases with increasing cut-off elevation.

Particle Filter-Based Inter-System Positioning Model for Non-Overlapping Frequency Code Division Multiple Access Systems

In the process of composing a double-differenced positioning model, it is difficult to separate different frequency signals between code division multiple access (CDMA) systems, the single-difference ambiguity of the pivot satellite and phase differential inter-system biases (PDISBs). Hence it is difficult to calibrate in advance the bias between systems in order to build an inter-system model which only needs one pivot satellite. Based on analysis of the stability of PDISB parameters for non-overlapping frequency CDMA systems, this study adopts a particle filter to estimate the fractional part of the PDISBs (F-PDISBs) between the systems and proposes a particle filter-based inter-system positioning model. Results show that the F-PDISBs and code DISBs for the baselines with the same receiver types and some with different receiver types are rather stable over time and for these baselines it is feasible to use a particle filter to estimate the F-PDISB parameters in the initial stage. Having attained the F-PDISBs, the inter-system model can be constructed to improve positioning accuracy in complex operational environments.

Estimating and Analyzing Long-Term Multi-GNSS Inter-System Bias Based on Uncombined PPP.

With the development of precise positioning with multi-GNSS, the inter-system bias (ISB) has become an issue that cannot be ignored. ISB is introduced from the differences among satellite reference clocks and different receiver hardware delay biases. To analyze the characteristics of multi-GNSS ISB, the precise point positioning (PPP) with full-rank uncombined model was derived for GLONASS, BDS, GALILEO, while the GPS receiver clock was selected as the reference. In addition, a recommended ISB parameter processing model was adopted. Data of 28-days from the Multi-GNSS Experiment (MGEX) station was used to estimate and analyze the ISB parameters. Based on a statistical analysis of the acquired data, results demonstrate that: (a) The rms of multi-GNSS PPP positional bias can reach 4.6 mm, 3.4 mm and 8.5 mm for E, N and U directions, respectively, which guarantees the reliability and accuracy of the ISB parameter solution. (b) The intra-day ISB time series of the three groups is relatively stable with standard deviations less than 0.6 ns. The ISB parameters between the GALILEO and GPS system are the most stable and the standard deviation was the smallest, at about 0.37 ns, which may be related to the good signal quality of the GALILEO system. (c) The mean of the single-day solution of the ISB parameter is not stable and the amplitude of the jump can be up to 60 ns. However, each station shows a similar variation for the same ISB parameter on the same day. The situation is independent of the type of receiver and antenna; however, it may be affected by the satellite reference clock of different systems. (d) There is a clear relationship between the ISB parameters and receiver types.

Detecting and Repairing Inter-system Bias Jumps with Satellite Clock Preprocessing

The key to performing successful multi-GNSS (Global Navigation Satellite System) precise point positioning is calibrating ISB (inter-system bias) in different systems. We can use the method of modeling to eliminate the ISB error. However, the ISB series are commonly discontinuous, as they contain jumps and outliers caused by day boundaries, gaps, or outliers in the precise clock products, which will break the ability of modeling. Thus, before modeling the ISB, we must remove outliers and repair jumps to improve the ISB continuity and achieve a continuous and smooth ISB series. Preprocessing on precise clock products is focused on in this study for the detection of ISB jumps and their repair. From the results, a positive correlation is revealed within the residuals of satellite clock offset and ISB differences between adjacent days. This finding implies ISB continuity can be improved through the preprocessing of precise clock products. It is also found that the exact reason for the occurrence of ISB jumps is the presence of extrema (i.e., maximum or minimum points) in the frequency domain. From the clock data in the frequency domain, larger extrema are identified directly. Meanwhile, for the detection of smaller extrema, a robust estimation method based on the median filter was applied. Then, all smaller extrema were classified into three types. Different preprocessing methods were applied for every type. After that, a new preprocessed precise clock product was obtained. With this preprocessed satellite clock product, the ISB continuity was substantially improved, and the improvement in the ISB continuity can reach 85.1%, on the average. These results indicate that for detecting and repairing ISB jumps, the proposed preprocessing method on satellite clock products is very effective.

Accounting BDS3–BDS2 inter-system biases for precise time transfer

The rapid development of BDS not only brings opportunities but also new problems. This article has been verified that the biases exist between BDS-2 and BDS-3 precise point positioning time transfer. We defined internal biases of BDS as Time Delay Bias (TDB) for a distinguish with inter system biases (ISB) exists in different systems. Motivated by the bias, we make comprehensive analysis for determining the characteristics of TDB referring to all the receivers capable of BDS-2 along with BDS-3 signals, furtherly, three kinds of TDB stochastic model are designed for parameter estimation. Later, the influence on time transfer with different TDB estimation strategy is compared in the end. We have some findings: (1) It is confirmed that TDB indeed exists in JAVAD TRE_3, CETC-54-GMR-4016, GNSS GGR and UB4B0-13478 receivers. Without external frequency source, the characteristics of TDB is related to the receiver itself. With the same external frequency source, TDB tendency of two receivers show the same evolution. (2) It is concluded that the STD (Standard Deviation) and stability of TDB estimated with random walk model and constant model is superior to white noise model because of strong correlation among TDB time series. (3) It is verified that the stability of time transfer estimated as random walk model and constant model is smaller than that of white noise model, what is more, when the two sides were receivers of different brands, the significant deviation exists in the clock offset with TDB estimation and without TDB estimation. When the two sides are receiver of the same brand, marginal deviation exists in the clock offset with TDB estimation and without TDB estimation.

Investigation of Tightly Combined Single-Frequency and Single-Epoch Precise Positioning Using Multi-GNSS Data

The loose combination (LC) and the tight combination (TC) are two different models in the combined processing of four global navigation satellite systems (GNSSs). The former is easy to implement but may be unusable with few satellites, while the latter should cope with the inter-system bias (ISB) and is applicable for few tracked satellites. Furthermore, in both models, the inter-frequency bias (IFB) in the GLObal NAvigation Satellite System (GLONASS) system should also be removed. In this study, we aimed to investigate the performance difference of ambiguity resolution and position estimation between these two models simultaneously using the single-frequency data of all four systems (GPS + GLONASS + Galileo + BeiDou Navigation Satellite System (BDS)) in three different environments, i.e., in an open area, with surrounding high buildings, and under a block of high buildings. For this purpose, we first provide the definition of ISB and IFB from the perspective of the hardware delays, and then propose practical algorithms to estimate the IFB rate and ISB. Thereafter, a comprehensive performance comparison was made between the TC and LC models. Experiments were conducted to simulate the above three observation environments: the typical situation and situations suffering from signal obstruction with high elevation angles and limited azimuths, respectively. The results show that in a typical situation, the TC and LC models achieve a similar performance. However, when the satellite signals are severely obstructed and few satellites are tracked, the float solution and ambiguity fixing rates in the LC model are dramatically decreased, while in the TC model, there are only minor declines and the difference in the ambiguity fixing rates can be as large as 30%. The correctly fixed ambiguity rates in the TC model also had an improvement of around 10%. Once the ambiguity was fixed, both models achieved a similar positioning accuracy.

An Improved Single-Epoch GNSS/INS Positioning Method for Urban Canyon Environment Based on Real-Time DISB Estimation

As one of the commonly used solutions for vehicular dynamic positioning, the stability of Global Navigation Satellite Systems (GNSS)/ Inertial Navigation System (INS) integrated positioning when applied in an urban environment is still struggling. Generally, carrier-phase ambiguity fixing requires continuous several epochs, the stability and continuity of integrated positioning are significantly reduced if the signal is frequently blocked. To reduce the impact of frequent signal blockage, an improved tightly-coupled algorithm is proposed in this paper. Firstly, a step-wise ambiguity processing method is introduced to form instantaneous fixed Wide-Lane (WL) observations for calibrating INS measurement. Secondly, by estimating the differential inter-system bias (DISB) parameter, the pivot satellite can be shared between different constellations, to increase the number of usable satellites under the limited observation conditions and improve the positioning performance. The proposed method is verified with vehicular experiments on semi-simulated and actual urban canyon scenarios. In the artificial GNSS outage experiment, in the situation of 4 satellites that can be observed in BeiDou Navigation Satellite System (BDS) and Global Positioning System (GPS), the positioning accuracy of the proposed algorithm can achieve $4.1cm$ horizontally and $15.2cm$ vertically. An improvement of 26.9% horizontally and 20.4% vertically is gained accordingly compared to the conventional method. In the actual urban environment experiment, in case of insufficient satellite and low signal-to-noise ratio (SNR), when the conventional method can no longer maintain a fixed solution of GNSS/INS integrated positioning, the proposed algorithm can still achieve an accuracy of $49.7cm$ horizontally and $67.0cm$ vertically.

Estimability analysis of differential inter-system biases and differential inter-frequency biases for dual-frequency GPS and BDS combined RTK

Inter-system biases (ISBs) and inter-frequency biases (IFBs) are critical for the data processing of multi-frequency and multi-constellation global navigation satellite systems (GNSS). Calibration of the ISB and IFB makes it possible to enhance the accuracy, integrity, and availability of multi-GNSS positioning, navigation, and timing. In this study, we investigate the magnitude and stability of the code and carrier phase differential inter-system biases (DISBs) between the GPS/BeiDou navigation satellite system (BDS) L1–B1/L2–B2 and differential inter-frequency biases (DIFBs) between L1–L2/B1–B2 using zero and short baseline data collected using both identical and different receiver types in real-time kinematic (RTK) positioning. We verify that the code and carrier phase DISBs of GPS/BDS L1–B1/L2–B2 and DIFBs of L1–L2/B1–B2 are both approximately constant on the timescale of a consecutive five-day observation period from a statistical perspective, and can thus be corrected by a priori DISB and DIFB, respectively. Moreover, the performance of the a priori model correction and real-time estimation strategies for the DISB and DIFB are compared. Both can precisely correct the DISB and DIFB and achieve significantly improved performance for multi-GNSS inter-system and inter-frequency combined RTK positioning compared to the classical intra-system and intra-frequency combined RTK approach. The DISB and DIFB using the a priori model correction and real-time estimation effectively improve the positioning model strength, thereby enabling superior performance for multi-GNSS positioning.

Analysis of the code and phase between-receiver inter-system biases of the overlapping frequencies for GPS/Galileo/BDS

The overlapping-frequency signals from different GNSS constellations are interoperable and can be integrated as one constellation in multi-GNSS positioning when inter-system bias (ISB) is properly disposed. The look-up table method for ISB calibration can enhance the model strength, maximize the number of integer-estimable ambiguities, and thus is preferred. However, the characteristics and magnitudes of the receiver code ISB and phase fractional ISB (F-ISB) are not well known and the wrong values of the biases can seriously degrade the positioning results. In this contribution, we first estimate the between-receiver code ISB and phase F-ISB of hundreds of the baselines up to around 25km in the European Permanent GNSS Network (EPN) and the Multi-GNSS Experiment (MGEX) for the overlapping frequencies L1-E1 (L1), L5-E5a (L5) and E5b-B2b (L7). The data collected from 1st January 2016 to 1st January 2019. Second, the receiver-type and firmware-version combinations for the receivers of Trimble, Leica, Javad, Septentrio and NovAtel are carefully classified. Results show that the Septentrio receivers have consistent code and phase ISB values for the three overlapping frequencies i.e. only one value for each frequency and no receivers are different. The Leica, Trimble and Javad receivers have two or more ISB values for at least one of the three frequencies. A few receivers with biases to the groups are also found and listed. Third, the code ISB and phase F-ISB of the groups are adjusted by the least-squares method. The root mean square errors (RMSE) of the least square adjustment are 0.240 m, 0.250 m and 0.200 m for code of L1, L5 and L7 frequencies, respectively, and are 0.0009 m, 0.0015 m and 0.0031 m for phase of L1, L5 and L7 frequencies, respectively. Finally, the effects of code ISB errors on code positing are investigated with the zero-baseline MAT1_MATZ. The distance root mean square error (DRMS) of L1-E1 code positioning can be reduced by 48.2% with 5 GPS and Galileo satellites and the DRMS degrades quickly when the code ISB error is larger.

Inter-system and inter-frequency code biases: simultaneous estimation, daily stability and applications in multi-GNSS single-frequency precise point positioning

Both inter-system and inter-frequency code biases (ISCB and IFCB) embodied in the multi-GNSS single-frequency precise point positioning (MSFPPP) exhibit mainly as systematic errors. Comprehensive analysis of the ISCB and IFCB characteristics and proper assimilation of their impact on MSFPPP are very important. We study the simultaneous estimation of all ISCBs for GLONASS, BDS and Galileo relative to GPS and the IFCBs of GLONASS by using our newly formulated estimable ionospheric-weighted MSFPPP model. Then, the daily stability of the ISCBs and IFCBs is examined, on which the application strategy is advised to improve the MSFPPP performance. One-month multi-GNSS single-frequency observation at 60 stations with eight types of receivers is carried out. The results indicate that the ISCBs of Galileo are most stable, followed by GLONASS and BDS, and all ISCBs and IFCBs are sufficiently stable on a daily scale. Imposing this stable information on the MSFPPP model can significantly enhance the model strength and improve the positioning performance.

Combined precise orbit determination of GPS and GLONASS with ambiguity resolution

Precise orbit products of Global Navigation Satellite Systems (GNSS) are an essential precondition for precise positioning. Ambiguity resolution (AR) can enhance the orbit accuracy in precise orbit determination (POD). To improve the quality of orbits, we propose a method of combined POD for GPS and GLONASS with AR. Firstly, GLONASS wide-lane and narrow-lane fractional cycle biases (FCBs) are daily estimated. Then, by applying the estimated FCBs, GLONASS and GPS double-differenced wide-lane and narrow-lane ambiguities are successfully resolved, even for the baselines of up to several thousand kilometers. Finally, the ambiguity-resolved solutions are achieved by introducing the constraints of the resolved ambiguities into the real-valued solutions. To prove the contribution of the AR to GPS and GLONASS POD, a network including 141 sites is processed over 2018. The results show that the receiver types and firmware versions seriously affect the stability of the daily wide-lane FCBs. The fluctuation of the inter-system biases between two adjacent days is obviously larger than a half narrow-lane wavelength, causing an irregular change of the daily narrow-lane FCBs. After FCB calibration, the success rate of GLONASS can reach up to 90% over the whole year, which is at the same level compared with that of GPS. The improvements of GLONASS and GPS orbits after AR are confirmed by the orbit comparison with the International GNSS Service final products, the orbit misclosures at day boundaries and satellite laser ranging residuals. Due to some other issues, such as the GLONASS frequency-division multiple access and the high noise of observations, the improvement of GLONASS orbit is still less obvious than that of GPS orbit.

Characteristics of GPS, BDS2, BDS3 and Galileo inter-system biases and their influence on RTK positioning

The Chinese new-generation BeiDou navigation satellite system (BDS3) is under construction and has begun to provide satellite-based positioning, navigation and timing solutions for global users. Inter-system biases (ISBs) are critical for the optimal combination of multiple global navigation satellite systems (GNSSs) observations, and their calibration makes it possible to enhance the interoperability among different GNSSs. We formerly presented a method based on the between-receiver single-differenced (SD) multi-GNSS observation equations for estimating the ISBs in scenarios with both overlapping and non-overlapping frequencies. In this paper, we analyze the ISBs between the new-generation BDS3 and other GNSSs, as well as BDS2, using the previously proposed SD method. For the first time, this article presents the ISBs between BDS3 and GPS, Galileo and BDS2 for identical-receiver baselines. Numerical analysis indicates that (1) the new frequencies of BDS3 that overlap those of GPS and Galileo are, rather unexpectedly, free of ISBs, suggesting the existence of interoperability among these three constellations; (2) there are obvious ISBs between BDS3 and BDS2 at non-overlapping frequencies but not at overlapping frequencies; (3) calibrating the ISBs makes it possible to improve the performance of real-time kinematic (RTK) positioning for multiple GNSSs involving BDS3.

Multi-GNSS time transfer based on the CGGTTS

This paper studies the use of multiple GNSS constellations for time transfer. The performances of each constellation are compared to one another. We show that, with its current constellation, Galileo already provides time transfer results with the same or better performances than GPS, for which some time links are affected by a diurnal variation. The paper also investigates different ways of combining the constellations into one global time transfer solution. We show that in a global AV solution, only one constellation should be calibrated, due to the inherent inter-system biases in the multi-GNSS clock products. Our results show that the best combination is to use the Galileo solution as a reference, and to combine the observations applying one bias (with respect to Galileo) per satellite of the other constellations. This allows reducing the impact of diurnal variations present in some GPS clock solutions. Combining Galileo and GPS satellites in this way improves the time transfer link with respect to the GPS-only or Galileo-only solutions. Adding moreover BeiDou-2 and GLONASS satellites gives, depending on the link, a solution similar or worse than the solution obtained from GPS and Galileo.

Estimating inter-system biases for tightly combined Galileo/BDS/GPS RTK

The overlapping carrier frequencies L1/E1, L5/E5a and B2/E5b from GPS/Galileo/BDS allow inter-system double-differencing of observations, which shows a clear advantage over differencing of the observations of each constellation independently. However, the inter-system biases destroy the integer nature of the inter-system double-differencing ambiguities. Two methods of direct rounding and parameter estimation are used to determine the ISB value. By analyzing data collected from Curtin University from 2015 to 2018, the phase and code inter-system bias (ISB) are related to the receiver type, firmware version and the selected overlapping frequency. Upgrade of receiver firmware version results in changes of ISB values. For example, the upgrade of Javad firmware in Dec, 15, 2017 causes the difference of 0.5 cycles ISB between BDS GEO and non-GEO satellites. By comparing the three dynamic models which include white noise process, random walk process, and random constant in the parameter estimation method, the ISB determined by the random constant model is consistent with the value obtained by the direct rounding method. After the calibration of ISBs, the performances of tightly combined positioning are assessed. The success rate of ambiguity resolution and accuracy of positioning for the tight combination (TC) are significantly improved in comparison with that for the loose combination (LC) over short baselines. For L5/E5a, on which only few satellites can be observed, the maximum increase in success rates of ambiguity resolution can reach 31.7%, i.e., from 54.9% of LC to larger than 86.6% of TC, and the positioning accuracies can even be increased by 0.13 m, i.e., from 0.208 of LC to 0.074 m of TC in East direction for the mix-receiver TRIMBLE NETR9-SEPT POLARX4 in 2018.

An analysis of inter-system biases in BDS/GPS precise point positioning

The ways of estimating the inter-system bias (ISB) have an important influence on BDS/GPS combined precise point positioning (PPP). Ordinarily, in data processing, the precise ephemeris and clock offset from the Center for Orbit Determination in Europe (CODE), Deutsches GeoForschungsZentrum (GFZ), and Wuhan University (WHU), respectively, are applied to obtain the ISB products ( $${\text{ISB}}_{\text{COD}}$$ , $${\text{ISB}}_{\text{GFZ}}$$ , and $${\text{ISB}}_{\text{WHU}}$$ ). Currently, in the case of BDS/GPS PPP, the ISB is generally considered to be a stable value in a given day and estimated as a constant. To better understand the mechanism underlying the generation of ISB in combined PPP, we deduce and establish the ISB estimation formulas and mathematical models and collect and process data from 19 multi-GNSS experimental stations. The results show that the ranges of $${\text{ISB}}_{\text{COD}}$$ and $${\text{ISB}}_{\text{WHU}}$$ at different times are within 0.3 m, while $${\text{ISB}}_{\text{GFZ}}$$ can change up to 6.5 m. An interesting phenomenon is that the $${\text{ISB}}_{\text{GFZ}}$$ for different stations has a similar variation trend in a day, and in some days, the ISBs of all stations show a linear trend which may mainly be influenced by satellite clock offset. In addition, the temporal stability of ISB is independent of receiver software version number and antenna type, while the receiver type has little influence on the stability of ISB. Therefore, it is reasonable to estimate $${\text{ISB}}_{\text{COD}}$$ and $${\text{ISB}}_{\text{WHU}}$$ as constant in a day and use a random walk to obtain $${\text{ISB}}_{\text{GFZ}}$$ . The results show that the convergence speed and accuracy of BDS/GPS PPP obtained by the random walk method are higher than those using the constant method, no matter whether $${\text{ISB}}_{\text{GFZ}}$$ , $${\text{ISB}}_{\text{COD}}$$ , or $${\text{ISB}}_{\text{WHU}}$$ is used.

Multiple GNSS inter-system biases in precise time transfer

Inter-system bias (ISB) is important for the fusion of multiple global navigation satellite system data. They should be estimated or corrected for precise position, navigation, and timing solutions. In this study, ISB estimation models are developed for the global positioning system, GLONASS, BeiDou Navigation Satellite System, and Galileo systems. Six stations, with three different types of receivers, are used to analyze the ISB characteristics of different systems. A pair of time-link stations, which have a short baseline and are connected to a common external time frequency from the laboratory, are used to analyze the performance of three different estimation strategies (epoch-wise variable, 30 min piece-wise constant, and daily constant). These stations are also used to evaluate the influence of ISB constrains on the precise estimation of time transfer, coordinates, and the troposphere. The results illustrate: (a) ISBs variation exists between different systems, and the same types of receivers also exhibit large differences between the different stations due to the receiver hardware delay; (b) the precision of the different measurements and their related errors, the estimated ISBs that exhibit large differences between pseudorange and carrier phase observations, and the different ISB estimation strategies can also cause differences in the final fusion results; and (c) using the estimated ISB as constraints can improve the precise estimation of time transfer, coordinates, and the troposphere in the real-time solution mode.

Regularization and particle filtering estimation of phase inter-system biases (ISB) and the lookup table for Galileo E1-GPS L1 phase ISB calibration

The between-receiver phase inter-system bias (ISB) in multi-GNSS integration needs to be calibrated when fixing the inter-system double difference (DD) ambiguities. As a result, the DD ambiguity fixing performance is affected by the accuracy of the ISB calibration; however, those effects have not been investigated until now. Also, the knowledge about the ISB characteristics is not enough which can lead to failure of ambiguity fixing with ISB calibration value such as obtained by the lookup table method. Our study first integrates the regularization method to the existing particle filtering fractional ISB (F-ISB) estimation to derive more precise F-ISB estimates. Afterward, the effects of the F-ISB accuracy on DD ambiguity fixing in single-epoch GPS L1-Galileo E1 integration are investigated. We show that the errors of F-ISB can degrade the success rate of single-epoch ambiguity fixing, and thus, more precise F-ISB values lead to higher empirical success rates. Finally, the characteristics of the F-ISB are investigated based on baselines shorter than 20 km from the GNSS network of International GNSS Services. According to the 132 F-ISB estimates obtained using data collected from day of year 001 2016 to DOY 001, 2019, the F-ISB values for 4 receiver brands including Leica, Septentrio, Trimble and Javad are analyzed. The receiver-type and software-version combinations from the same manufacturer and with similar F-ISB values are regarded as one group and all the receivers can be classified into 6 groups. With F-ISB of Leica receiver group set to zero value, the other groups have F-ISB values of 0.000 m, + 0.055 m and + 0.095 m and the L1-E1 F-ISBs between different groups are provided in a lookup table for practical use.

Tightly combined GPS + GLONASS positioning with consideration of inter-system code bias and GLONASS inter-frequency code bias

Inter-system code double differencing is an effective method for improving the positioning accuracy of low-cost receivers in complex environments. Due to the adoption of Frequency Division Multiple Access (FDMA), Globalnaya Navigazionnaya Sputnikovaya Sistema (GLONASS) code observations are affected by the Inter-Frequency Code Biases (IFCBs), which makes it difficult to calculate the Differential Inter-System Code Biases (DISCBs) between GLONASS and the Code Division Multiple Access (CDMA) systems directly. In this contribution, the focus is on the performance of tightly combined Global Positioning System (GPS) and GLONASS code Double Difference (DD) positioning. After analysing the relationship between IFCBs and GLONASS channel numbers, an IFCB correction model and an inter-system code differencing model between GLONASS and GPS are proposed. Results show that even if there is no obvious relationship between IFCBs and channel numbers, the long-term stable IFCB values of each satellite can be obtained by using the proposed model. In addition, the GPS + GLONASS DISCB is also stable. Therefore, compared with the intra-system model, the inter-system model can benefit from prior IFCBs and DISCBs parameters and thus can significantly improve the positioning accuracy in obstructed environments.