Differential Code Biases Research Articles

Triple-Frequency GPS Un-Differenced and Uncombined PPP Ambiguity Resolution Using Observable-Specific Satellite Signal Biases

The new generations of global navigation satellite system (GNSS) space vehicles can transmit three or more frequency signals. Multi-frequency observations bring a significant improvement to precise point positioning ambiguity resolution (PPP AR). However, the multi-frequency satellite code and phase biases need to be properly handled before conducting PPP AR. The traditional satellite bias correction methods, for example, the commonly used differential code biases (DCB), are limited to the dual-frequency ionosphere-free (IF) case and become more and more difficult to extend to multi-GNSS and multi-frequency cases. In this contribution, we propose the observable-specific signal bias (OSB) correction method for un-differenced and uncombined (UDUC) PPP AR. The OSB correction method, which includes observable-specific satellite code and phase bias correction, can directly apply kinds of OSBs to GNSS original observation data, thus, it is more appropriate for multi-GNSS and multi-frequency cases. In order to verify the performance of multi-frequency UDUC-PPP AR based on the OSB correction method, triple-frequency GPS observation data provided by 142 Multi-GNSS Experiment (MGEX) stations were used to estimate observable-specific satellite phase biases at the PPP service end and some of them were also used to conduct AR at the PPP user end. The experiment results showed: the averaged time-to-first-fix (TTFF) of triple-frequency GPS kinematic UDUC-PPP AR with observable-specific satellite code bias (SCB) corrections could reach about 22 min with about 29% improvement, compared with that without observable-specific SCB corrections; TTFF of triple-frequency static UDUC-PPP AR with observable-specific phase-specific time-variant inter-frequency clock bias (IFCB) corrections took about 15.6 min with about 64.3% improvement, compared with that without observable-specific IFCB corrections.

Impact of flex power on GPS Block IIF differential code biases

GPS Block IIF satellites are able to redistribute the transmit power between the signal components. This ability is called flex power, and it has been developed as a remedy against jamming. Since it is operationally not possible to increase the transmit power for all signal components simultaneously, a redistribution between them is necessary under certain operational situations. Flex power has been active on Block IIF satellites since January 2017 over a specific regional area and has an impact on differential code bias estimation as well as the signal-to-noise density ratio. A network of the International GNSS Service stations containing only Septentrio PolaRx5 and PolaRx5TR receivers between August 1 and November 21, 2019 has been used for differential code bias estimation using GPS L1 C/A, L1 P(Y), L2 P(Y), and L2C signals with and without consideration of the flex power in the estimation process for Block IIF satellites. The estimation results are compared with the German Aerospace Center as well as the Chinese Academy of Sciences DCB products to validate the results.

Estimation of the DCB for the BDS-3 new signals based on BDGIM constraints

The hardware delay difference for different navigation signals in satellite or receiver channels is called Differential Code Bias (DCB). DCB is a major error of global navigation satellite system (GNSS), which may lead to tens of Total Electron Content unit (TECu) ionospheric delay errors or several-meter ranging errors. The basic system of BeiDou Navigation Satellite System has opened services and three new navigation signals, namely B1C, B2a at the end of 2018, and B2b at the end of 2019. As the BeiDou satellite broadcast clock difference is based on the phase center of the B3I signal antenna, it is necessary to estimated the DCB values of the new navigation signals relative to B3I. Considering that BDS broadcasts the BeiDou global ionospheric delay correction model (BDGIM) parameters with the highest precision, the method of solving satellite or receiver DCB based on the BDS-3 BDGIM parameter constraint is proposed and verified. Theoretical analysis shows that the mean square error of daily solution is about 0.63–1.96 ns. The measured data of 25 iGMAS stations from January 1 to 30, 2019, are used for verification. The results show that the solution precision of satellite DCB is better than 1.0 ns and that of receiver DCB is slightly worse. This method does not need the support of external ionospheric products and provides a possibility for online real-time estimating of the satellite and receiver DCB.

A new model for total electron content based on ionospheric continuity equation

With the development of the Global Navigation Satellite System (GNSS), the accuracy and reliability requirements on observations are constantly increasing. The effect of ionospheric disturbance is not negligible, especially for accurate GNSS applications. To reduce the influence of ionospheric delay and disturbance, this study uses the Ionospheric Continuity Equation (ICE) to establish a new model for Total Electron Content (TEC), which takes both photochemical and transportation processes into account and named ICE model. Experimental results show that the ICE model achieves accuracies higher than Spherical Harmonic (SH) and polynomial models on geomagnetism quiet and disturbance days, and more prominent advantages appear at nighttime. By the test of Satellite Differential Code Bias (SDCB) estimation, the deviations of the polynomial and the ICE models are nearly the same whatever the geomagnetic field is quiet or disturbance; however, the estimation accuracy of the ICE model is slightly better than the polynomial model. Spatial and temporal adaptability tests indicate that the ICE model can achieve accuracies of 0.5 TECU and 1 TECU on time scales of approximately 25 min and 100 min, respectively. On the weak disturbance or quiet day, these accuracies can be achieved on the spatial scales of approximately 200 km and 600 km. The credibility of the ICE model is evaluated by Nash–Sutcliffe efficiency. The evaluation results assigned that the ICE model has higher credibility than the SH and polynomial models and better immunity to geomagnetic disturbance.

PPP models and performances from single- to quad-frequency BDS observations

Nowadays, China BeiDou Navigation Satellite System (BDS) has been developed well and provided global services with highly precise positioning, navigation and timing (PNT) as well as unique short-message communication, particularly global system (BDS-3) with higher precision multi-frequency signals. The precise point positioning (PPP) can provide the precise position, receiver clock, and zenith tropospheric delay (ZTD) with a stand-alone receiver compared to the traditional double differenced relative positioning mode, which has been widely used in PNT, geodesy, meteorology and so on. However, it has a lot of challenges for multi-frequency BDS PPP with different strategies and more unknown parameters. In this paper, the detailed PPP models using the single-, dual-, triple-, and quad-frequency BDS observations are presented and evaluated. Firstly, BDS system and PPP method are introduced. Secondly, the stochastic models of time delay bias in BDS-2/BDS-3 PPP including the neglection, random constant, random walk and white noise are presented. Then, three single-frequency, four dual-frequency, four triple-frequency and four quad-frequency BDS PPP models are provided. Finally, the BDS PPP models progress and performances including theoretical comparison of the models, positioning performances, precise time and frequency transfer, ZTD, inter-frequency bias (IFB) and differential code bias (DCB) are presented and evaluated as well as future challenges. The results show that the multi-frequency BDS observations will greatly improve the PPP performances.

Performance of ionospheric-free PPP time transfer models with BDS-3 quad-frequency observations

The BeiDou Navigation Satellite System (BDS) has as achieved the status of a global navigation satellite system. Compared to BDS-2 satellites, BDS-3 satellites transmit B1C, B2a, and B2b signals but not B1I and B3I signals. In this study, for the four open-services of BDS-3 satellites, four ionospheric-free combinations, namely (B1I/B3I), (B1I/B2a), (B1C/B2a) and (B1C/B3I) were first proposed. Then, the performance of the BDS-3 carrier-phase time transfer was assessed and investigated. For comparison and evaluation purposes, precise point positioning (PPP) time transfer with BDS-2, GPS, and Galileo was also assessed. Analogous to the BDS-3 signal structure, the performance of BDS-2 based on the B1I/B2I- and B1I/B3I- combined PPP solutions was first validated. Then, with respect to the GPS and Galileo PPP time transfer, the BDS-3 B1I/B3I, B1I/B2a, B1C/B2a, and B1C/B3I; BDS-2 B1I/B2I; and BDS-2 + BDS-3 B1I/B3I PPP time transfers were assessed and analysed. The experimental results demonstrated that the BDS-3 B1I/B3I performance was slightly better than the BDS-3 B1I/B2a, B1C/B3I and B1C/B2a performances due to the differential code bias (DCB) accuracy. Furthermore, the BDS-2 B1I/B3I PPP was worse than the other PPP solutions, while BDS-2 + BDS-3 B1I/B3I PPP outperformed the other PPP solutions. Compared with the BDS-2 B1I/B3I PPP, the mean reductions in the standard deviation (STD) values were approximately (77.2, 21.3, 1.8, 1.3, 13.1) % from (0.241, 0.687, 0.547, 0.218, 0.585) ns to (0.136, 0.566, 0.537, 0.215, 0.517) ns calculated from BDS-3 and (97.5, 23.5, 4.9, 4.8, 16.7)% from (0.241, 0.687, 0.547, 0.218, 0.585) ns to (0.122, 0.556, 0.521, 0.208, 0.501) ns calculated and BDS-2 + BDS-3, respectively. For the frequency stability, BDS-3 demonstrated better results, while BDS-2 + BDS-3 presented significant improvements with respect to BDS-2. The maximum improvement in BDS-2 + BDS-3 reached up to 56.1% from 3.3973E-14 to 1.49E-14. Additionally, the modified Allan deviation (MDEV) of BDS-2 + BDS-3 was analogous to that of GPS and Galileo.

Assessment of multi-frequency global navigation satellite system precise point positioning models using GPS, BeiDou, GLONASS, Galileo and QZSS

The development of a global navigation satellite system (GNSS) brings the benefit of positioning, navigation and timing (PNT) services with three or even more available frequency signals. This paper developed five-system multi-frequency precise point positioning (PPP) models based on mathematical and stochastic models. Static positioning performances were evaluated and analyzed with multi-GNSS experiment (MGEX) network datasets and a vehicle-borne kinematic experiment was conducted to verify the kinematic PPP performances. In addition, the receiver clock, zenith tropospheric delay (ZTD), inter-frequency bias (IFB) and differential code bias (DCB) estimates were discussed. Results show that the triple-frequency PPP performances perform slightly better than the dual-frequency solutions, apart from the GPS-related PPP models based on a single ionosphere-free (IF) combined measurement. By introducing the external ionospheric products, the mean convergence time is reduced. For instance, the mean convergence time of ionosphere-constrained (IC) multi-frequency PPP is reduced by 7.4% from 35.7 to 33.1 min and by 19.0% from 7.8 to 6.3 min, for Galileo-only and five-constellation solutions, respectively, compared with dual-frequency IF PPP models. Similarly, the kinematic PPP can also achieve improved performances with more frequency signals and multi-GNSS observations.

BDS-3 differential code bias estimation with undifferenced uncombined model based on triple-frequency observation

Since 2015, the new generation global BDS system, i.e., BDS-3, has started its development with five experimental satellites demonstration system and has announced its initial global service officially on December 27, 2018. Among the various characteristics to be analyzed for the new generation BDS satellites, the differential code bias (DCB) is of special attention since that it has a direct dependence on the new signals, i.e., B1C and B2a, and it is one of the most intricacy problems in the ionosphere sensing and positioning with multi-GNSS and multi-frequency observations. To take the full capability of the triple-frequency BDS signals, this paper proposed a new method for the DCB estimation in which the undifferenced uncombined observations are processed in PPP mode. In addition, with the intention to estimate all the unknowns, including the DCB, in a single filter, the DESIGN (deterministic plus stochastic ionosphere model for GNSS) method is applied for the ionospheric delay constrains in this method. In the formula derivation, special attention is given to the DCB and clock parameters due to different frequencies for B1I/B1C, etc. Then, the efficiency of the new method is assessed with observations of 23 iGMAS stations capable for BDS triple-frequency tracking and 21 IGS stations capable for GPS triple-frequency tracking during DOY 001 to 090, 2019. Moreover, the traditional DCB estimation method by employing the geometry-free (GF) combination with the ionospheric delay removed by global ionosphere map product is also performed for comparison purpose. The experimental results suggest that by using the undifferenced uncombined solution, rather than the GF combination, the BDS-2 DCB on B1IB2I and B1CB3I can be improved, especially for the MEO satellites. Regarding to the DLR products, the undifferenced uncombined DCB solution presents a RMS of 0.32 ns and 0.27 ns for B1IB2I and B1CB3I, respectively. Concerning the BDS-3 satellites DCB, it is GF combination that performs better by a factor of 12.7% and 15.2% for B1CB2a and B1CB3I, respectively. This is mainly due to the fact that the undifferenced uncombined DCB solution is sensitive to the limited precision of the BDS-3 orbit and clock. This conclusion is further confirmed by the improvement in the GPS DCB solution with the new method. Compared with the GF combination solution, the STD for daily repeatability improves from 0.088 to 0.061 ns and 0.119 to 0.090 ns for satellite on C1WC2W and C1WC5X, respectively, by using the undifferenced uncombined model.

The update of BDS-2 TGD and its impact on positioning

Timing group delay (TGD) is an important parameter that affects the positioning performance of global navigation satellite systems (GNSS). The BeiDou navigation satellite system (BDS) broadcasts TGD corrections from B3I frequency to B1I and B2I frequencies, namely TGD1 and TGD2. On July 21, 2017, BDS updated TGD values with a maximum change of more than 4 ns. In this contribution, we explain the motivation for the BDS TGD update, which is due to the systematic bias between narrowly correlated and widely correlated pseudo-ranges in BDS monitoring receivers. To investigate the impact of the updated TGD, BDS signal-in-space range error (SISRE) and user positioning performance regarding single point positioning (SPP) and precise point positioning (PPP) are analyzed. Results show that after the update of TGD, the difference between the new TGD and multi-GNSS experiment (MGEX) differential code bias (DCB) decreases from 1.38 ns to 0.29 ns on TGD1 and from 0.40 ns to 0.25 ns on TGD2. With the contribution of more accurate TGD, the systematic bias of BDS radial SISRE no longer exists, and the overall BDS SISRE also reduces from 1.33 m to 0.87 m on B1I/B2I frequency, from 1.05 m to 0.89 m on B1I frequency, from 0.92 m to 0.91 m on B2I frequency, respectively, which proves the similar precision of BDS TGD and MGEX DCB. One week of statistical results from 28 globally distributed MGEX stations shows that the SPP performance improves on non-B3I frequencies after the TGD update, with a maximum improvement of more than 22% for the B1I/B2I or B1I/B3I combination. The new TGD mainly reduces SPP positioning bias in the East component. The updated TGD also slightly improves the PPP convergence performance for the B1I/B3I combination, but mostly contributes to a more accurate estimation of the receiver clock and ambiguities.

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.

Estimation of LEO-GPS receiver differential code bias based on inequality constrained least square and multi-layer mapping function

The differential code bias (DCB) of the global navigation satellite system (GNSS) receiver onboard a low earth orbit (LEO) satellite is one of the crucial hardware error sources in ionospheric estimation. In common practice, LEO DCBs are considered as constant within a single day. However, since such receivers are installed on moving platforms, they are subjected to the ever-changing space environment, and thus, LEO DCBs are prone to more frequent fluctuations than receiver DCBs of ground GNSS stations. We estimate the LEO GPS receiver DCBs and plasmaspheric vertical total electron content by using inequality constrained least square and a multi-layer mapping function (MF) approach to minimize mapping errors. The GPS receiver DCBs onboard constellation observing system for meteorology, ionosphere, and climate satellites (COSMIC) is investigated and analyzed during the January 2008 period. The results show that in most cases the multi-layer MF approach performs better than the commonly used single-layer MF. The mapping error from an inaccurately fixed single-layer height can be considerably reduced. Meanwhile, with the help of a multi-layer MF, the mean deviations of DCBs for five receivers compared with those based on a geometric MF at 1400 km are significantly decreased by 59%, 15%, 26%, 47%, and 22%, respectively.

Receiver Differential Code Bias Estimation under Disturbed Ionosphere Status Using Linear Planar Model Based Minimum Standard Deviation Searching Method with Bias Detection

Receiver Differential Code Bias Estimation under Disturbed Ionosphere Status Using Linear Planar Model Based Minimum Standard Deviation Searching Method with Bias Detection

Determination of NavIC differential code biases using GPS and NavIC observations

Delay bias between the pseudo range code is called Differential Code Bias (DCB), this is one of the main error sources to estimate the precise Total Electron Content (TEC) in ionosphere. Differential delay biases between the dual frequency band signals in both satellite and receiver are called satellite differential code biases (SDCBs) and receiver differential code biases (RDCBs). The combined satellite and receiver DCBs lead to measure error TEC. SDCBs and RDCBs should be removed from the Global Navigation Satellite System (GNSS) measurements accurately. The DCB estimations of GNSS signals are important for improving GNSS user positioning accuracy in ionospheric and space weather applications. The Navigation with Indian Constellation (NavIC) system has been developed by India to cater to positioning, navigation, timing, and ionosphere and troposphere remote sensing applications. The proposed DCB estimation algorithm is necessary to calibrate the NavIC TEC measurements for both NavIC satellites and receivers. In this work, a Global Positioning System (GPS)-aided NavIC DCB estimation algorithm has been implemented with the help of single station GPS TEC measurements. TEC data were recorded at the same location by the NavIC as well as the GPS receiver (Koneru Lakshamaiah Education Foundation, Guntur, India, 16.47°N, 80.61°E) for the month of September 2016. The SDCB and RDCB estimations for NavIC satellites and receivers located at Guntur, India. The maximum SDCB value of 9.2 TECU was observed for NavIC satellite pseudo range number (PRN) 3. Standard deviations of NavIC satellites were in the range of 0.53–1.11 TECU. The NavIC RDCB was estimated to be 16 TECU. The DCB results were validated with the Fitted Receiver Bias (FRB) method. The implemented DCB estimation algorithm would be useful for improving positional accuracy of NavIC users.

Advantages of Uncombined Precise Point Positioning with Fixed Ambiguity Resolution for Slant Total Electron Content (STEC) and Differential Code Bias (DCB) Estimation

The determination of slant total electron content (STEC) between satellites and receivers is the first step for establishing an ionospheric model. However, the leveling errors, caused by the smoothed ambiguity solutions in the carrier-to-code leveling (CCL) method, degrade the performance of ionosphere modeling and differential code bias (DCB) estimation. To reduce the leveling errors, an uncombined and undifferenced precise point positioning (PPP) method with ambiguity resolution (AR) was used to directly extract the STEC. Firstly, the ionospheric observables were estimated with CCL, PPP float-ambiguity solutions, and PPP fixed-ambiguity solutions, respectively, to analyze the short-term temporal variation of receiver DCB in zero or short baselines. Then, the global ionospheric map (GIM) was modeled using three types of ionospheric observables based on the single-layer model (SLM) assumption. Compared with the CCL method, the slight variations of receiver DCBs can be obviously distinguished using high precise ionospheric observables, with a 58.4% and 71.2% improvement of the standard deviation (STD) for PPP float-ambiguity and fixed-ambiguity solutions, respectively. For ionosphere modeling, the 24.7% and 27.9% improvements for posteriori residuals were achieved for PPP float-ambiguity and fixed-ambiguity solutions, compared to the CCL method. The corresponding improvement for residuals of the vertical total electron contents (VTECs) compared with the Center for Orbit Determination in Europe (CODE) final GIM products in global accuracy was 9.2% and 13.7% for PPP float-ambiguity and fixed-ambiguity solutions, respectively. The results show that the PPP fixed-ambiguity solution is the best one for the GIM product modeling and satellite DCBs estimation.

Impact of satellite clock offset on differential code biases estimation using undifferenced GPS triple-frequency observations

The uncombined model using undifferenced triple-frequency observations provides an alternative approach for the precise estimation of differential code biases (DCBs) of GPS satellite and receiver. However, the accuracy of DCB estimation is significantly affected by the satellite clock offset due to their strong correlation. We aim to analyze the impact of the satellite clock offset on the estimation of GPS DCB for different frequency pairs. The relationship between satellite clock offset and DCB estimates is rigorously studied based on the derived full-rank triple-frequency uncombined model. Theoretical analysis indicates that the initial satellite clock bias would affect the size of satellite DCB estimates in a linear way. To test our analysis, precise products released by the international GNSS service (IGS), the Center for Orbit Determination in Europe (CODE) and the German Research Centre for Geosciences (GFZ), are used for data processing and comparison. The channel types 1W, 2W and 5X for both GPS triple-frequency code and phase measurements are chosen for DCB estimation. Results show that the average monthly standard deviation (STD) is 0.06 ns and 0.08 ns for GPS satellite C1W–C2W and C1W–C5X DCB estimates. Different satellite clock products have no significant effect on the STDs. When compared with the DCB computed by CODE and Deutsches Zentrum fur Luft-und Raumfahrt (DLR), the mean RMS values of GPS satellite C1W–C2W and C1W–C5X DCB differences are 0.10 ns and 0.42 ns. Different satellite clock products have a significant effect on the RMSs. Through linear regression analysis, we find that 46% and 27% of initial satellite clock biases would be included in the satellite C1W–C2W DCB and C1W–C5X DCB estimates, respectively. In addition, the satellite clock offset has little impact on the receiver DCB estimates though it significantly affects the satellite DCB estimates.

The phase and code biases of Galileo and BDS-3 BOC signals: effect on ambiguity resolution and precise positioning

The modernization of GPS and GLONASS and the development of BDS and Galileo make a variety of new navigation signals available to the users. The wide range of GNSS signals will result in various biases that need to be considered in data processing. In particular, Galileo and BDS-3 binary offset carrier (BOC) signals employ a new approach called dual-frequency constant envelope multiplexing. In this contribution, the phase and code biases of Galileo and BDS-3 BOC signals were estimated and investigated with the observation of iGMAS and MEGX networks during the period of 2015–2018. Initial analyses of BDS-3 BOC signals indicate that satellite-specific pilot-minus-data code biases are close to zero for BDS-3 B2a and B2b signals, while the ones for B1C signal are not. In addition, the satellite differential code biases (DCBs) between Galileo E5a, E5b and E5ab signals, as well as between BDS-3 B2a and B2b signals, are also close to zero. The estimated phase biases for Galileo E5a, E5b and E5ab signals or BDS-3 B2a and B2b signals are the same values, and the resultant phase biases of the extra-wide-lane (EWL) combination are very close to the zero. Besides, no obvious time-varying inter-frequency clock bias could be observed for both Galileo and BDS-3 satellites. Such characteristics of phase/code biases of the new GNSS signals are valuable for ambiguity resolution and precise positioning. Only one set of code or phase bias product is required for Galileo E5a/b/ab (BDS-3 B2a/b) signals. The satellite DCBs between Galileo E5a, E5b and E5ab signals, as well as between BDS-3 B2a and B2b signals, can be ignored in the data processing. And the EWL ambiguities derived from the Galileo E5a/b/ab or BDS-3 B2a/b observations keep their integer feature and can be fixed to integers without phase bias corrections.

Reducing convergence time of precise point positioning with ionospheric constraints and receiver differential code bias modeling

Long convergence time has limited the wide application of traditional precise point positioning (PPP) based on an ionosphere-free combination of dual-frequency observations. Different from the traditional PPP, the uncombined PPP method based on raw observations estimates ionospheric delays. When external ionospheric information is available, it can be applied as a constraint to help shorten the convergence time, as a result of the reduced correlation between the position and the ionospheric parameters. The receiver differential code biases (DCBs) will be a concern, however, when applying the external ionospheric information. For receiver DCBs, it is usually assumed that the biases can be absorbed by the receiver clock parameters. We have demonstrated that the receiver DCBs cannot be fully assimilated by one receiver code clock parameter because the receiver DCBs have different effects on the code and carrier phase measurements at any frequency. Additional parameters are necessary to model the receiver DCBs so that their effects on the positioning solution can be minimized. We developed an ionosphere-constrained PPP model to incorporate ionospheric total electron content (TEC) in the slant (STEC) and vertical (VTEC) when leveraging a regional network and global ionospheric maps (GIMs). Both static and kinematic experimental results show that the convergence time and the positioning accuracy can be improved significantly. Accuracies at the first epoch of 0.4 m for GIM constraints, and 0.2 m for the regional constraints, are achievable. The convergence time to 1 dm horizontal accuracy is reduced to 7.5 min at a 68% confidence level.

A refined carrier-to-code levelling method for retrieving ionospheric measurements from dual-frequency GPS data

The retrieval of ionosphere measurements (IMs) constitutes the first key step in the process of ionospheric monitoring with the global positioning system (GPS). Carrier-to-code levelling (CCL) is one of the most widely used retrieval methods, mainly because of its simplicity and effectiveness. However, CCL-derived IMs have been proven to be susceptible to so-called levelling errors, which is mainly due to the presence of code multipath effects and the occurrence of short-term variability in receiver code biases (RCBs). To address these levelling errors, we propose a refined CCL (rCCL) method, in which the code multipath and RCB variations are jointly estimated as time-varying parameters and thus removed. Compared to the customary CCL method, this new method has two advantages. First, rCCL is able to exclude the effect of code multipath and time-varying RCBs from IMs, thereby greatly reducing the levelling errors of IM estimates. Second, the RCB variations are estimated for each individual code observation using a single receiver, allowing easy and effective calibration of the short-term variability in RCBs and the linear combination of, for instance, receiver differential code biases (rDCBs).

Analysis of the short-term temporal variation of differential code bias in GNSS receiver

The short-term variation of GNSS Receiver Differential Code Biases (RDCB) has become one of the main error sources affecting the retrieval of ionospheric Total Electron Content (TEC). In this paper, the Modified Carrier-to-Code Leveling and its simplified method are used to analyze the RDCB variation characteristics under real and simulation conditions, respectively. The results under simulation conditions show that the receiver types have an obvious impact on the stability of RDCB within 0.5 ns. In contrast, the antenna has little effect. Moreover, a strong correlation between the variation of RDCB and the satellites Doppler shift has been observed. Moreover, the ambient temperature significantly affects the RDCB, and 15 degrees is the most favorable for the stability of RDCB, which is 30–85% higher than other temperature conditions. Additionally, the analysis results under real condition indicate that about 4–10% of the International GNSS Service (IGS) station receivers have significant daily variation in RDCB.

Estimation and Analysis of BDS-3 Differential Code Biases from MGEX Observations

The third generation of China’s BeiDou Navigation Satellite System (BDS-3) began to provide global services on 27 December, 2018. Differential code bias (DCB) is one of the errors in precise BDS positioning and ionospheric modeling, but the impacts on BDS-2 satellites and receiver DCB are unknown after joining BDS-3 observations. In this paper, the BDS-3 DCBs are estimated and analyzed using the Multi-Global Navigation Satellite System (GNSS) Experiment (MGEX) observations during the period of day of year (DOY) 002–031, 2019. The results indicate that the estimated BDS-3 DCBs have a good agreement with the products provided by the Chinese Academy of Sciences (CAS) and Deutsche Zentrum für Luft- und Raumfahrt (DLR). The differences between our results and the other two products are within ±0.2 ns, with Standard Deviations (STDs) of mostly less than 0.2 ns. Furthermore, the effects on satellite and receiver DCB after adding BDS-3 observations are analyzed by BDS-2 + BDS-3 and BDS-2-only solutions. For BDS-2 satellite DCB, the values of effect are close to 0, and the effect on stability of DCB is very small. In terms of receiver DCB, the value of effect on each station is related to the receiver type, but their mean value is also close to 0, and the stability of receiver DCB is better when BDS-3 observations are added. Therefore, there is no evident systematic bias between BDS-2 and BDS-2 + BDS-3 DCB.