Differential Code Bias Products Research Articles

Reconstructed estimation method of the multi-frequency GNSS inter-frequency clock bias

In consideration of contributions from both carrier phase and pseudorange observations during computation, this study introduces a reconstructed method for estimating the multi-frequency global navigation satellite system (GNSS) inter-frequency clock biases (IFCB). Diverging from conventional approaches that separately calculate the time-varying and constant parts of IFCBs using carrier phase and pseudorange observations, the reconstructed method directly utilizes their combination to estimate satellite IFCBs. To validate the efficacy of the presented approach, 7 d observations from 87 International GNSS Service (IGS) stations are analyzed, with a specific focus on triple-frequency GPS, multi-frequency BDS-3, and Galileo IFCBs, aiming to scrutinize their distinctive characteristics. Furthermore, the performances of satellite IFCB estimation are investigated in both precise point positioning (PPP) and single-point positioning (SPP) using 35 IGS stations over 3 d. The results demonstrate that IFCBs of GPS BLOCK IIIA satellites exhibit centimeter-level variations, distinguishing them from BLOCK IIF counterparts. The Galileo IFCBs vary from millimeter to centimeter level, while those of BDS-3 reach a centimeter level. These variations significantly impact GPS PPP convergence performances but have minimal effects on Galileo and BDS-3 PPP. SPP performances are slightly enhanced when the time-varying IFCB part are taken into account. Additionally, we note disparities in the constant parts of satellite IFCBs computed with differential code bias (DCB) products, reconstructed, and pseudorange observation-based methods, particularly for BDS-3 C2I/C6I-C1P/C5P and C2I/C6I-C1P/C6I combinations. The differences of IFCB estimated with different strategies, SPP and PPP performances show that the reconstructed method is better than others, and the IFCB accuracy decreases when computed with satellite DCB products.

An Analysis of Satellite Multichannel Differential Code Bias for BeiDou SPP and PPP

Differential code bias (DCB) of satellite is an error that cannot be ignored in precise positioning, timing, ionospheric modeling, satellite clock correction, and ambiguity resolution. The completion of the third generation of BeiDou Navigation Satellite System (BDS-3) has extended DCB to multichannel code bias observations and observable-specific signal bias (OSB). In this paper, the DCB and OSB products provided by the Chinese Academy of Sciences (CAS) are analyzed and compared. The DCB parameters for the BDS satellites are applied in both single- and dual-frequency single point positioning (SPP), and the results are intensively investigated. Based on the satellite DCB parameters of the BDS, the performance of precise point positioning (PPP) with different frequency combinations is also analyzed in terms of positioning accuracy and convergence time. The standard deviations (STDs) of DCBs at each signal pair fluctuate from 0.2 ns to 1.5 ns. The DCBs of BDS-2 are slightly more stable than those of BDS-3. The mean values and STDs of C2I and C7I OSBs for BDS-2 are at the same level and are numerically smaller than their counterparts for the C6I OSBs. The mean OSBs for each signal of the BDS-3, excluding C2I, fluctuate between 12.35 ns and 12.94 ns, and the STD fluctuates between 2.11 ns and 3.10 ns. The DCBs and OSBs of the BDS-3 of the IGSO satellites are more stable than those of the MEO satellites. The corrections for TGD and DCB are able to improve the accuracy of single-frequency SPP by 44.09% and 44.07%, respectively, and improve the accuracy of dual-frequency SPP by 6.44% and 12.85%, respectively. The most significant improvements from DCB correction are seen in single-frequency positioning with B1I and dual-frequency positioning with B2a+B3I. DCB correction can improve the horizontal and three-dimensional positioning accuracy of the dual-frequency PPP of different ionosphere-free combinations by 13.53% and 13.84% on average, respectively, although the convergence is slowed.

BDS-3 new signals observable-specific phase biases estimation and PPP ambiguity resolution

Nowadays, the BDS-3 satellites transmit new signals B1c and B2a that are compatible with GPS and Galileo, which is meaningful to fully take advantage of the BDS positioning capability. However, further research and analysis are needed to investigate the precise point positioning ambiguity resolution (PPP-AR) performance of the new BDS-3 signal combination. In this manuscript, the characteristics and quality of differential code bias (DCB), inter frequency clock bias (IFCB) and uncalibrated phase delay (UPD) of BDS-3 new signals are analyzed based on 30 days observation data from 70 multi GNSS experiment (MGEX) stations. The results show that the mean value of IFCB between B1I/B3I and B1C/B2a is 1.18 cm, which indicates that the IFCB can be ignored when performing the clock errors reference conversion of BDS-3 satellites. Besides, the DCB of the new BDS-3 signals varies within 0.2 ns over 30 days, and with a difference of 0.08 ns compared to the DCB product provided by chinese academy of science (CAS). The percentage of the estimated residuals less than 0.25 cycles for new signals WL UPD and less than 0.15 cycles for new signals NL UPD are 99.76% and 97.19%, respectively. The above results demonstrate the reliability of the DCB and UPD products, which can be used to estimate the carrier phase observable specific signal bias (OSB) product for the new signals. The average positioning errors of the BDS-3 new signals static PPP-AR in the N, E and U directions is 0.50 cm, 0.43 cm and 1.62 cm respectively, with an improvement of 18.3%, 28.3% and 11.5% compared to the ambiguity float static PPP. The average positioning errors of the kinematic PPP-AR are 2.09 cm, 2.67 cm and 4.79 cm in three directions, respectively, with an improvement of 17.1%, 23.9% and 15.7% compared to the ambiguity float PPP. The convergence time of the static and kinematic PPP-AR are 26.36 min and 29.12 min, respectively.

BDS and Galileo: Global Ionosphere Modeling and the Comparison to GPS and GLONASS

The ionospheric delay is one of the important error sources in the Global Navigation Satellite System (GNSS) data processing. With the rapid construction and development of GNSS, the abundant satellite resources have brought new opportunities for ionospheric monitoring. To further investigate the performances and abilities of Galileo and BDS in ionosphere modeling, we study the ionosphere modeling based on the 15th order spherical harmonic function, and 364 stations around the world are selected for global ionospheric modeling of GPS, GLONASS, Galileo and BDS systems under ionospheric quiet and active conditions, respectively. The results show that the average biases of the ionospheric models built by GPS, GLONASS and Galileo are relatively small, which are within 2 Total Electron Content Unit (TECU) as compared to the Center for Orbit Determination in Europe (CODE) global ionospheric map (GIM), while the average biases of the models built by BDS are between 6 and 8 TECU during the ionospheric quiet and active days, respectively. In addition, in order to analyze the modeling performances before and after using BDS geostationary earth orbit (GEO) satellites, BDS is divided into two groups, in which one group contains medium earth orbit (MEO), inclined geosynchronous orbit (IGSO) and GEO satellites; and the other group contains only MEO and IGSO satellites. The results show that the influence of GEO satellites on ionospheric modeling is less than 1 TECU. Due to the distribution of the stations, the 0-value region in the ionospheric model is mainly distributed in the mid and high-latitude regions of the southern hemisphere. Since the ionospheric parameters are lumped with the Differential Code Bias (DCB), we also estimate the DCB parameters and analyze their performances. The DCB estimated in ionosphere modeling shows strong stability, with the average biases of GPS, GLONASS, Galileo and BDS under 0.25 ns, 0.25 ns, 0.2 ns and 0.42 ns, respectively. We also estimate other DCB types of the four GNSS systems. The results show that the DCB is stable and shows consistency with Chinese Academy of Sciences (CAS) DCB products.

Accounting for biases between BDS-3 and BDS-2 overlapping B1I/B3I signals in BeiDou global ionospheric modeling and DCB determination

The receiver pseudo-range biases between BDS-2 and BDS-3 in global ionospheric modeling and differential code bias (DCB) determination are investigated by using the overlapping B1I/B3I signals. The BDS receiver DCB estimations from global ionospheric modeling and the results of single-differences of short-baselines reveal that the receiver pseudo-range bias differences between BDS-3 and BDS-2 B1I/B3I observations are related to receiver types, and the maximum of the bias differences can be up to 8.6 ns. It means that the receiver DCB differences between BDS-3 and BDS-2 B1I/B3I observations should be considered in BDS-3/BDS-2 ionospheric modeling and DCB determination. Compared with DCB products from Chinese Academy of Sciences (CAS), the RMS is better than 0.3 ns when the receiver pseudo-range bias differences between BDS-3 and BDS-2 are ignored, while the RMS is about 0.51 ns when the receiver pseudo-range bias differences are considered. Based on the analysis of the BDS-3 and BDS-2 receiver pseudo-range bias differences, it indicates there are significant deviations in existing CAS satellite DCB products for BDS B1I/B3I observations. In terms of the stability of BDS satellite DCB estimations, the monthly STDs of BDS-3 DCB estimations improves by 20.2% on average when the receiver pseudo-range bias differences between BDS-3 and BDS-2 are considered. Taking the final Global Ionospheric Maps (GIMs) from Centre for Orbit Determination in Europe (CODE) as references, the accuracy of BDS-only global ionospheric modeling is 2.4 total electron content unit (TECU) when the receiver DCB differences between BDS-3 and BDS-2 are considered, and improves by 9.3% compared with the results which ignores the receiver pseudo-range bias differences between BDS-3 and BDS-2.

Comprehensive Analyses of PPP-B2b Performance in China and Surrounding Areas

BeiDou Global Navigation Satellite System (BDS-3) provides a regional Precise Point Positioning (PPP) service, called PPP-B2b, for users in China and surrounding areas through B2b signal transmitted from its three geostationary earth orbit (GEO) satellites. The information broadcasted by the B2b signal include satellite orbit corrections, satellite clock offset corrections, and differential code bias (DCB) corrections of BDS-3 satellites. In this study, the accuracies of PPP-B2b corrections along with real-time PPP performance are comprehensively evaluated referenced to precise orbit and clock products from GFZ and the precise DCB products from CAS. The result indicates that the accuracy of the BDS-3 broadcast orbit is similar to that of the PPP-B2b real-time orbit. The PPP-B2b clock offset correction improved the satellite clock offset precision of the BDS-3 broadcast ephemeris. The Signal-in-Space Range Error (SISRE) of broadcast ephemeris and PPP-B2b are calculated, which are 0.536 and 1.24 m, respectively. The large SISRE value of PPP-B2b is caused by the satellite-specified systematic bias to IGS final products. The positioning performance evaluation of real-time PPP with B2b service is carried out and compared with the real-time product provided by Wuhan University (WHU) based on the eight IGS MGEX stations in China and surrounding countries. The positioning accuracy of static positioning mode with PPP-B2b service achieved centimeter-level accuracy in the selected station, and that of kinematic positioning mode achieved decimeter-level accuracy. The availability rate of PPP-B2b corrections in the surrounding area of China, however, degrades from 88.76% to 60.91% in the selected stations. The accuracy of the PPP solution using PPP-B2b correction is better than that of using WHU real-time product within China. The positioning performance of stations located at the boundary of the PPP-B2b service area, however, is affected by the number of PPP-B2b available satellites. The positioning accuracy in kinematic positioning mode is worse than that of using WHU real-time precise product.

Variation Characteristics of Multi-Channel Differential Code Biases from New BDS-3 Signal Observations

A multi-frequency Global Navigation Satellite System (GNSS) provides greater opportunities for positioning and navigation applications, particularly the BeiDou Global Navigation Satellite System (BDS-3) satellites. However, multi-frequency signals import more pseudorange channels, which introduce more multi-channel Differential Code Biases (DCBs). The satellite and receiver DCBs from the new BDS-3 signals are not clear. In this study, 9 DCB types of the new BDS-3 signals from 30-days Multi-GNSS Experiment (MGEX) observations are estimated and investigated. Compared with the DCB values provided by the Chinese Academy of Science (CAS) products, the mean bias and root mean squares (RMS) error of new BDS-3 satellite DCBs are within ±0.20 and 0.30 ns, respectively. The satellite DCBs are mostly within ±0.40 ns with respect to the product of the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The four sets of constructed closure errors and their mean values are within ±0.30 ns and ±0.15 ns, respectively. The mean standard deviation (STD) of the estimated satellite DCBs is less than 0.10 ns. In particular, our estimated satellite DCBs are more stable than DCB products provided by CAS and DLR. Unlike satellite DCBs, the receiver DCBs have poor compliance and show an obvious relationship with the geographic latitude when compared to the CAS products. The STDs of our estimated receiver DCBs are less than 1.00 ns. According to different types of receiver DCBs, the distribution of STDs indicates that the coefficient of the ionospheric correction has an influence on the stability of the receiver DCBs under the ionosphere with the same accuracy level. In addition, the type of receiver shows no regular effects on the stability of receiver DCBs.

Estimation and Analysis of the Observable-Specific Code Biases Estimated Using Multi-GNSS Observations and Global Ionospheric Maps

Observable-specific bias (OSB) parameterization allows observation biases belonging to various signal types to be flexibly addressed in the estimation of ionosphere and global navigation satellite system (GNSS) clock products. In this contribution, multi-GNSS OSBs are generated by two different methods. With regard to the first method, geometry-free (GF) linear combinations of the pseudorange and carrier-phase observations of a global multi-GNSS receiver network are formed for the extraction of OSB observables, and global ionospheric maps (GIMs) are employed to correct ionospheric path delays. Concerning the second method, satellite and receiver OSBs are converted directly from external differential code bias (DCB) products. Two assumptions are employed in the two methods to distinguish satellite- and receiver-specific OSB parameters. The first assumption is a zero-mean condition for each satellite OSB type and GNSS signal. The second assumption involves ionosphere-free (IF) linear combination signal constraints for satellites and receivers between two signals, which are compatible with the International GNSS Service (IGS) clock product. Agreement between the multi-GNSS satellite OSBs estimated by the two methods and those from the Chinese Academy of Sciences (CAS) is shown at levels of 0.15 ns and 0.1 ns, respectively. The results from observations spanning 6 months show that the multi-GNSS OSB estimates for signals in the same frequency bands may have very similar code bias characteristics, and the receiver OSB estimates present larger standard deviations (STDs) than the satellite OSB estimates. Additionally, the variations in the receiver OSB estimates are shown to be related to the types of receivers and antennas and the firmware version. The results also indicate that the root mean square (RMS) of the differences between the OSBs estimated based on the CAS- and German Aerospace Center (DLR)-provided DCB products are 0.32 ns for the global positioning system (GPS), 0.45 ns for the BeiDou navigation satellite system (BDS), 0.39 ns for GLONASS and 0.22 ns for Galileo.

Estimation and analysis of GNSS receiver differential code bias in Southeast Asia using a new method

The receiver differential code bias (DCB) is one of the main errors in GNSS navigation and positioning as well as ionospheric monitoring. In this paper, we present a new method to estimate the receiver differential code bias (DCB) using Multi-GNSS observations in Southeast Asia. Different from the traditional method by using ionosphere modeling or Global ionospheric map (GIM), the total electron content (TEC) of station in the new method is estimated directly together with the receiver DCB. The data of one year with 34 stations were used to evaluate the performance of the presented method. The results show a good agreement between our estimated receiver DCBs and the MGEX DCB products and the RMS of eight types of GNSS receiver DCB are mostly less than 1ns with respect to the MGEX products. Finally, the stability of GNSS receiver DCB was analysed for eight stations located in Southeast Asia as examples. The result indicates that those stations were relatively stable with mostly less than 1ns of STD of receiver DCB. Moreover, no evidence of latitudinal and receiver type dependencies of the stability of receiver DCB for those selected stations was found.

Estimation and Analysis of GNSS Differential Code Biases (DCBs) Using a Multi-Spacing Software Receiver

In the use of global navigation satellite systems (GNSS) to monitor ionosphere variations by estimating total electron content (TEC), differential code biases (DCBs) in GNSS measurements are a primary source of errors. Satellite DCBs are currently estimated and broadcast to users by International GNSS Service (IGS) using a network of GNSS hardware receivers which are inside structure fixed. We propose an approach for satellite DCB estimation using a multi-spacing GNSS software receiver to analyze the influence of the correlator spacing on satellite DCB estimates and estimate satellite DCBs based on different correlator spacing observations from the software receiver. This software receiver-based approach is called multi-spacing DCB (MSDCB) estimation. In the software receiver approach, GNSS observations with different correlator spacings from intermediate frequency datasets can be generated. Since each correlator spacing allows the software receiver to output observations like a local GNSS receiver station, GNSS observations from different correlator spacings constitute a network of GNSS receivers, which makes it possible to use a single software receiver to estimate satellite DCBs. By comparing the MSDCBs to the IGS DCB products, the results show that the proposed correlator spacing flexible software receiver is able to predict satellite DCBs with increased flexibility and cost-effectiveness than the current hardware receiver-based DCB estimation approach.

Improving GPS and Galileo precise data processing based on calibration of signal distortion biases

The quality of pseudorange observations plays an important role in Global Navigation Satellite System (GNSS) data processing, especially for satellite clock estimation, differential code bias (DCB) estimation and wide-lane ambiguity resolution. It is validated that receivers with different correlator spaces and front-end designs will bring signal distortion bias (SDB) in pseudorange observations, which will affect GNSS data processing based on inhomogenous receivers, such as receivers from Multi-GNSS EXperiment (MGEX) network. We firstly evaluate the consistency of GPS and Galileo satellite clock and DCB products among different International GNSS Service (IGS) Analysis Centers (ACs). The results show that inconsistency exists in both satellite clocks and DCB products among different ACs. For example, the difference of GPS satellite DCB (C1C-C1W) between two ACs, which is free of ionospheric delay, can reach up to 0.31 ns. To improve GPS and Galileo precise data processing based on inhomogenous receivers, pseudorange biases caused by receiver signal distortion are analyzed and calibrated for 5 receiver brands and 19 receiver models. The maximum pseudorange biases of ionospheric-free combination caused by receiver SDBs can reach up to ±3 ns and ±1 ns for GPS and Galileo, respectively. Then, bias corrections of GPS and Galileo raw pseudorange observations are divided into 11 groups and estimated by each receiver group. Finally, several experiments are carried out for the validation of GPS and Galileo SDB correction models. For example, the GPS and Galileo satellite clock offset differences estimated by different receiver brands are decreased by 86.59% and 50.0% for specific receiver type, respectively. The improvement ratio of satellite DCB accuracy is related to satellite systems and signal types, and the maximum improvement can reach up to 95%. Moreover, the maximum improvement of GPS undifferenced wide-lane ambiguity resolution success rate can reach up to 22% for specific receiver types. All these results prove that the bias corrections proposed could greatly eliminate the effect of receiver SDBs on GPS and Galileo precise data processing. Hence, we suggest the corrections of SDBs should be taken into consideration for IGS data reprocessing.

Estimation of differential code biases with Jason-2/3 onboard GPS observations

With the continuous deployment of Low Earth Orbit (LEO) satellites, the estimation of differential code biases (DCBs) based on GNSS observations from LEO has gained increasing attention. Previous studies on LEO-based DCB estimation are usually using the spherical symmetry ionosphere assumption (SSIA), in which a uniform electron density is assumed in a thick shell. In this study, we propose an approach (named the SHLEO method) to simultaneously estimate the satellite and LEO onboard receiver DCBs by modeling the distribution of the global plasmaspheric total electron content (PTEC) above the satellite orbit with a spherical harmonic (SH) function. Compared to the commonly used SSIA method, the SHLEO model improves the GPS satellite DCB estimation accuracy by 13.46% and the stability by 22.34%, respectively. Compared to the GPS satellite DCBs estimated based on the Jason-3-only observations, the accuracy and monthly stability of the satellite DCBs can be improved by 14.42% and 26.8% when both Jason-2 and Jason-3 onboard observations are jointly processed. Compared with the Jason-2 solutions, the GPS satellite DCB estimates based on the fusion of Jason-2 and Jason-3 observations have an improved consistency of better than 18.26% and 9.71% with the products provided by the Center for Orbit Determination in Europe (CODE) and Chinese Academy of Sciences (CAS). Taking the DCB products provided by the German Aerospace Center (DLR) as references, there is no improvement in accuracy of the GPS satellite DCB estimates based on the fusion of Jason-2 and Jason-3 observations than the Jason-2 solutions alone. A periodic variation is found in the time series of both the Jason-3 and Jason-2 onboard receiver DCB estimates. Preliminary analysis of the PTEC distribution based on the estimated SH coefficients are also presented.

Understanding long-term variations in GPS differential code biases

Precise estimation of satellite differential code biases (DCBs) plays a crucial role in precise ionospheric modeling, positioning, and timing. Due to the rank deficiency, a constraint or a datum is required in order to separate the satellite DCBs from the receiver DCBs. A common practice is to impose a zero-mean constraint on all the visible satellites. However, datum selection is affected by satellite replacement and variation of the DCBs. As a result, the long-term variations of current DCB products vary significantly. Taking the DCBs of SVN 44 (PRN 28) as a reference, we analyzed the long-term variations of DCBs over a period of 20 years, between 2000 and 2019. Based on this reference, the results indicate that the change of the zero-mean datum is responsible for the variation of current DCB products. The datum change is attributed to the satellite replacement as well as the discontinuities and their variations. We found that discontinuities for the same satellite vehicle reach 1.8 ns, which is related to satellite changes announced in the Notice Advisory to Navstar Users message and to flex power. The magnitude of the DCBs depends on the satellite type. DCBs for Block IIR-A and IIR-M satellites are close to each other, while DCBs of Block IIR-B satellites are approximately 5 ns larger and DCBs for the Block IIF are 8 ns smaller. In addition, the satellite biases between GPS P1 and C1 are also briefly examined, and the results show that they are also affected by the satellite replacements and discontinuities. However, the satellite bias differences between P1 and C1 for different satellite types are minor.

Estimation of GPS Differential Code Biases Based on Independent Reference Station and Recursive Filter

The differential code bias (DCB) of the Global Navigation Satellite Systems (GNSS) receiver should be precisely corrected when conducting ionospheric remote sensing and precise point positioning. The DCBs can usually be estimated by the ground GNSS network based on the parameterization of the global ionosphere together with the global ionospheric map (GIM). In order to reduce the spatial-temporal complexities, various algorithms based on GIM and local ionospheric modeling are conducted, but rely on station selection. In this paper, we present a recursive method to estimate the DCBs of Global Positioning System (GPS) satellites based on a recursive filter and independent reference station selection procedure. The satellite and receiver DCBs are estimated once per local day and aligned with the DCB product provided by the Center for Orbit Determination in Europe (CODE). From the statistical analysis with CODE DCB products, the results show that the accuracy of GPS satellite DCB estimates obtained by the recursive method can reach about 0.10 ns under solar quiet condition. The influence of stations with bad performances on DCB estimation can be reduced through the independent iterative reference selection. The accuracy of local ionospheric modeling based on recursive filter is less than 2 Total Electron Content Unit (TECU) in the monthly median sense. The performance of the recursive method is also evaluated under different solar conditions and the results show that the local ionospheric modeling is sensitive to solar conditions. Moreover, the recursive method has the potential to be implemented in the near real-time DCB estimation and GNSS data quality check.

GPS satellite inter-frequency clock bias estimation using triple-frequency raw observations

This study proposes a unified uncombined model to estimate GPS satellite inter-frequency clock bias (IFCB) in both triple-frequency code and carrier-phase observations. In the proposed model, the formulae of both phase-based and code-based IFCBs are rigorously derived. Specifically, satellite phase-based IFCB refers to its time-variant part and it is modeled as a periodic function related to the sun–spacecraft–earth angle. A zero-mean condition of all available GPS satellites that support triple-frequency data is introduced to render satellite code-based IFCB estimable. Three months of data from 40 globally distributed stations of the International GNSS Service Multi-GNSS Experiment are used to test our method. The results show that the four-order periodic function is suitable for eliminating the 12-h, 6-h, 4-h, and 3-h periods that exist in the a posteriori phase residuals when no periodic function is used. For comparison, the geometry-free and ionosphere-free (GFIF) phase combination and differential code bias (DCB) products released by DLR (German Aerospace Center) and IGG (Institute of Geodesy and Geophysics, China) are also used to calculate the satellite phase-based and code-based IFCBs, respectively. The results show that (1) the average root mean square (RMS) of the phase-based IFCB difference between the proposed method and the GFIF phase combination is 4.3 mm; (2) the average RMS in the eclipse period increased by 50% compared with the average RMS in the eclipse-free period; (3) the mean monthly STD for code-based IFCB from the proposed method is 0.09 ns; and (4) the average RMS values of code-based IFCB differences between the proposed method and the DCB products released by DLR and IGG are 0.32 and 0.38 ns. This proposed model also provides a general approach for multi-frequency GNSS applications such as precise orbit and clock determination.

Combined use of single-frequency data and global ionosphere maps to estimate BDS and Galileo satellite differential code biases

Differential code biases (DCBs) are the most significant systematic errors in global navigation satellite system (GNSS) ionospheric total electronic content (TEC) monitoring and other code-related applications. Currently, the rapidly developing GNSS benefits from the Chinese BeiDou (BDS) and European Galileo systems that continuously offer position, navigation and timing services. In this way, the determination of satellite DCBs (SDCBs), which generally relies on GNSS data collected from dual- or multi-frequency geodetic-grade receivers, is of great importance, especially for the BDS and Galileo systems. However, the use of these geodetic-grade receivers implies a substantial increase in hardware costs. In this paper, we propose a single-frequency (SF) method for the determination of the BDS (C2I–C7I) and Galileo (C1X–C5X) satellite DCBs with the combined use of global ionospheric maps (GIMs) and low-cost SF multi-GNSS receivers, thereby greatly reducing the hardware costs. The mean precision of the satellite DCBs estimated from multi-GNSS data of a standalone SF receiver is approximately 0.35 ns for Galileo and 0.51 ns for BDS, when taking the DCB products published by the Deutsches zentrum für Luft-und Raumfahrt (DLR) as ground truth.

Differential code bias estimation based on uncombined PPP with LEO onboard GPS observations

Differential Code Bias (DCB) is an essential correction that must be provided to the Global Navigation Satellite System (GNSS) users for precise position determination. With the continuous deployment of Low Earth Orbit (LEO) satellites, DCB estimation using observations from GNSS receivers onboard the LEO satellites is drawing increasing interests in order to meet the growing demands on high-quality DCB products from LEO-based applications, such as LEO-based GNSS signal augmentation and space weather research. Previous studies on LEO-based DCB estimation are usually using the geometry-free combination of GNSS observations, and it may suffer from significant leveling errors due to non-zero mean of multipath errors and short-term variations of receiver code and phase biases. In this study, we utilize the uncombined Precise Point Positioning (PPP) model for LEO DCB estimation. The models for uncombined PPP-based LEO DCB estimation are presented and GPS observations acquired from receivers onboard three identical Swarm satellites from February 1 to 28, 2019 are used for the validation. The results show that the average Root Mean Square errors (RMS) of the GPS satellite DCBs estimated with onboard data from each of the three Swarm satellites using the uncombined PPP model are less than 0.18 ns when compared to the GPS satellite DCBs obtained from IGS final daily Global Ionospheric Map (GIM) products. Meanwhile, the corresponding average RMS of GPS satellite DCBs estimated with the conventional geometry-free model are 0.290, 0.210, 0.281 ns, respectively, which are significantly larger than those obtained with the uncombined PPP model. It is also noted that the estimated GPS satellite DCBs by Swarm A and C satellites are highly correlated, likely attributed to their similar orbit type and space environment. On the other hand, the Swarm receiver DCBs estimated with uncombined PPP model, with Standard Deviation (STD) of 0.065, 0.037 and 0.071 ns, are more stable than those obtained from the official Swarm Level 2 products with corresponding STD values of 0.115, 0.101, and 0.109 ns, respectively. The above indicates that high-quality DCB products can be estimated based on uncombined PPP with LEO onboard observations.

FY-3D and FY-3C onboard observations for differential code biases estimation

With the development of Low Earth Orbit satellites, differential code biases (DCBs) estimation based on onboard observations has been widely studied. In this study, onboard observations of BDS and GPS satellites by the Chinese Fengyun-3D (FY-3D) and Fengyun-3C (FY-3C) satellites are applied to estimate BDS and GPS DCBs. Since only the code observations of C1C and C2W for GPS, and C2I and C7I for BDS are tracked by FY-3D and FY-3C, the DCB types of GPS C1C-C2W and BDS C2I-C7I are estimated with code multipath considered. First, the DCB estimates based on FY-3D onboard observations are analyzed. When jointly processing BDS + GPS onboard observations, the stability of satellite and receiver DCBs for both BDS and GPS has better consistency with the DCB products of the German Aerospace Center (DLR) and the Chinese Academy of Science than that for the single-system solutions (BDS-only solution and GPS-only solution). This is reasonable because more onboard observations are used in BDS + GPS solution, which can improve the strength of the DCB estimation. The variations of receiver DCB are analyzed as a function of geomagnetism and solar activity, but little relationship between them has been found. Compared with the FY-3C solution, the FY-3D solution can achieve a more stable satellite DCB with a stability improvement of 33%, 48%, 62% and 56% for GPS, BDS GEO, IGSO, and MEO satellites, respectively. Meanwhile, the receiver DCB of FY-3D is more stable than that of FY-3C as well. These improvements of satellite and receiver DCBs can be due to the enhancement of FY-3D GNSS Occultation Sounder (GNOS) instrument, which provides more observations with higher quality. Furthermore, both FY-3D and FY-3C onboard observations are processed together to estimate BDS and GPS DCBs. Compared with the FY-3D solution, the stability of satellite DCB can be improved by 16%, 9% and 7% for GPS, BDS GEO and IGSO satellites DCB, respectively, when both FY-3D and FY-3C onboard observations are jointly processed. The impact of DCB estimation on estimating the vertical total electron content (VTEC) is also investigated. Compared with FY-3D GPS-only and BDS + GPS solutions, the VTEC estimates along the FY-3D orbit can achieve more realistic results for FY-3D + FY-3C solution.

Investigation and Validation of the Time-Varying Characteristic for the GPS Differential Code Bias

The time-varying characteristic of the bias in the GPS code observation is investigated using triple-frequency observations. The method for estimating the combined code bias is presented and the twelve-month (1 January–31 December 2016) triple-frequency GPS data set from 114 International GNSS Service (IGS) stations is processed to analyze the characteristic of the combined code bias. The results show that the main periods of the combined code bias are 12, 8, 6, 4, 4.8 and 2.67 h. The time-varying characteristic of the combined code bias, which is the combination of differential code bias (DCB) (P1–P5) and DCB (P1–P2), shows that the real satellite DCBs are also time-varying. The difference between the two sets of the computed constant parts of the combined code bias, with the IGS DCB products of DCB (P1–P2) and DCB (P1–P2) and the mean of the estimated 24-h combined code bias series, further show that the combined code bias cannot be replaced by the DCB (P1–P2) and DCB (P1–P5) products. The time-varying part of inter-frequency clock bias (IFCB) can be estimated by the phase and code observations and the phase based IFCB is the combinations of the triple-frequency satellite uncalibrated phase delays (UPDs) and the code-based IFCB is the function of the DCBs. The performances of the computed the IFCB with different methods in single point positioning indicate that the accuracy for the constant part of the combined code bias is reduced, when the IGS DCB products are used to compute. These performances also show that the time-varying part of IFCB estimated with phase observation is better than that of code observation. The predicted results show that 98% of the predicted constant part of the combined code bias can be corrected and the attenuation of the predicted accuracy is much less evident. However, the accuracy of the predicted time-varying part decreases significantly with the predicted time.

Modeling and performance analysis of precise time transfer based on BDS triple-frequency un-combined observations

In this study, a model of precise time transfer is developed based on the triple-frequency un-combined observations of the BeiDou navigation satellite system, known as UC-PPP. In this model, except for the traditional position, troposphere delay and receiver clock parameters, ionosphere delays are estimated as unknown parameters by adding the prior, spatial and temporal constraints. In addition, receiver differential code biases (DCB) are also estimated as unknown parameters. The standard triple-frequency ionosphere-free model is also introduced, named as IF-PPP. To assess the performance of the model, datasets with short baseline and common external time frequency are used. The results show that the triple-frequency UC-PPP model can be used for precise time transfer, with accuracy and stability identical to those of the IF-PPP model. The model can also provide the receiver DCB and ionosphere total electron content products.