Satellite Differential Code Biases Research Articles

Analysis of factors affecting the estimation of the multi-GNSS satellite differential code biases (SDCBs)

The satellite hardware differential code biases (SDCBs) are one of the important factors affecting the accuracy of GNSS-based ionospheric total electron content (TEC) estimation. The accuracy of SDCB estimates is directly influenced by the data processing methodology. To analyze the impact of various factors on multi-GNSS SDCB estimation, we conducted a comprehensive analysis of the number of contributed stations, the data sampling rate, the contributed satellite systems, and the cut-off elevation angle. Two sets of GNSS data obtained from the International GNSS Service (IGS) multi-GNSS experiment (MGEX) network, covering both high and low solar activities during the ascending phase of solar cycle 25, were processed to estimate multi-GNSS SDCBs. The results indicate that the number of contributed stations is the primary factor that influences multi-GNSS SDCB estimation. To achieve a balance between computational efficiency and accuracy, a data sampling rate of 540 s is recommended, which results in a multi-GNSS SDCB root-mean-square (RMS) error of less than 0.01 ns compared to a 60-second sampling rate. The stability of SDCBs is hardly affected by incorporating observations from multiple GNSS systems, and the differences of the standard deviations of SDCBs for quad-, triple- and dual-system solutions are below 0.002 ns. Additionally, the results indicate that the optimal cutoff elevation angle for multi-GNSS SDCB estimation is between 20° and 30°, which ensures the best stability in the estimated multi-GNSS SDCBs.

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.

A Simple Approach to Determine Single-Receiver Differential Code Bias Using Precise Point Positioning.

In this study, a precise single-receiver differential code bias (DCB) estimation method using the precise point positioning (PPP) model is presented. The first step is to extract the high-precision ionospheric observations, including DCBs, based on the PPP model. Then, the satellite DCBs are corrected using International GNSS Service (IGS) products. Lastly, the algorithm for the minimization of the standard deviation of vertical total electron content (VTECmstd) is employed to determine the value of receiver DCB. To check the method, GNSS data from more than 200 IGS stations around the globe on four days with various geomagnetic and solar activity circumstances are processed. The receiver DCBs are compared to those obtained using previous carried-to-code level (CCL) models. The experimental results show that, compared to the CCL model, the values of VTECmstd for most stations are significantly reduced, the mean number of stations with negative ionospheric measurements is reduced by 40% after correcting the receiver DCBs, and the mean error of estimated receiver DCBs is reduced by approximately 0.6 ns using the PPP model. These results suggest that this method can provide more high-precision receiver DCB estimation.

Estimating BDS-3 Satellite Differential Code Biases with the Single-Frequency Uncombined PPP Model.

Differential Code Bias (DCB) is a crucially systematic error in satellite positioning and ionospheric modeling. This study aims to estimate the BeiDou-3 global navigation satellite system (BDS-3) satellite DCBs by using the single-frequency (SF) uncombined Precise Point Positioning (PPP) model. The experiment utilized BDS-3 B1 observations collected from 25 International GNSS Service (IGS) stations located at various latitudes during March 2023. The results reveal that the accuracy of estimating B1I-B3I DCBs derived from single receiver exhibits latitude dependence. Stations in low-latitude regions show considerable variability in the root mean square (RMS) of absolute offsets for satellite DCBs estimation, covering a wide range of values. In contrast, mid- to high-latitude stations demonstrate a more consistent pattern with relatively stable RMS values. Moreover, it has been observed that the stations situated in the Northern Hemisphere display a higher level of consistency in the RMS values when compared to those in the Southern Hemisphere. When incorporating estimates from all 25 stations, the RMS of the absolute offsets in satellite DCBs estimation consistently remained below 0.8 ns. Notably, after excluding 8 low-latitude stations and utilizing data from the remaining 17 stations, the RMS of absolute offsets in satellite DCBs estimation decreased to below 0.63 ns. These enhancements underscore the importance of incorporating a sufficient number of mid- and high-latitude stations to mitigate the effects of ionospheric variability when utilizing SF observations for satellite DCBs estimation.

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.

A Novel Method to Estimate Multi-GNSS Differential Code Bias without Using Ionospheric Function Model and Global Ionosphere Map

Global navigation satellite system (GNSS) differential code bias (DCB) is one of main errors in ionospheric modeling and applications. Accurate estimation of multiple types of GNSS DCBs is important for GNSS positioning, navigation, and timing, as well as ionospheric modeling. In this study, a novel method of multi-GNSS DCB estimation is proposed without using an ionospheric function model and global ionosphere map (GIM), namely independent GNSS DCB estimation (IGDE). Firstly, ionospheric observations are extracted based on the geometry-free combination of dual-frequency multi-GNSS code observations. Secondly, the VTEC of the station represented by the weighted mean VTEC value of the ionospheric pierce points (IPPs) at each epoch is estimated as a parameter together with the combined receiver and satellite DCBs (RSDCBs). Last, the estimated RSDCBs are used as new observations, whose weight is calculated from estimated covariances, and thus the satellite and receiver DCBs of multi-GNSS are estimated. Nineteen types of multi-GNSS satellite DCBs are estimated based on 200-day observations from more than 300 multi-GNSS experiment (MGEX) stations, and the performance of the proposed method is evaluated by comparing with MGEX products. The results show that the mean RMS value is 0.12, 0.23, 0.21, 0.13, and 0.11 ns for GPS, GLONASS, BDS, Galileo, and QZSS DCBs, respectively, with respect to MGEX products, and the stability of estimated GPS, GLONASS, BDS, Galileo, and QZSS DCBs is 0.07, 0.06, 0.13, 0.11, and 0.11 ns, respectively. The proposed method shows good performance of multi-GNSS DCB estimation in low-solar-activity periods.

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.

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.

A Station-Specific Ionospheric Modeling Method for the Estimation and Analysis of BeiDou-3 Differential Code Bias Parameters

<h3>Abstract</h3> A modified generalized trigonometric series (GTS) function is presented for the joint estimation of local ionospheric activities and differential code biases (DCBs) of the third-generation BeiDou navigation satellite system (BeiDou-3). Using observational data from the iGMAS and IGS-MGEX networks, DCBs between the pilot-, data- and I-components of BeiDou-3 signals are estimated and analyzed for January 2019. The stability of the modified GTS-based satellite DCB estimates an improvement of about 29.7% compared to the original GTS-based results. In comparison with transmitted T_GDB1Cp and T_GDB2ap parameters, the consistency between broadcast and post-processed biases reaches 0.33 and 0.50 ns. The estimated data-pilot biases are observed to be notably smaller than BeiDou-3 transmitted inter-signal correction (ISC) parameters. In the analysis of receiver B1I-B3I, DCBs generated from the independent BeiDou-2 and BeiDou-3 constellations, the receiver-type dependent biases are emphasized, which raises the consideration of using receiver-group specific bias concept in the future estimation of BeiDou DCBs.

Inter-frequency code bias handling and estimation for multi-frequency BeiDou-3/Galileo uncombined precise point positioning

The development of a global navigation satellite system (GNSS) with multi-frequency signals brings new opportunities for providing high-quality positioning, navigation and timing (PNT) services. Proper inter-frequency code bias (IFB) handling is a prerequisite for multi-frequency uncombined precise point positioning (UC-PPP) to ensure reliable and accurate PNT services. This work focuses on analyzing the mathematical representation of estimated parameters, as well as the relationship between different multi-frequency UC-PPP models, caused by whether to correct the inter-frequency satellite differential code bias (DCB) or the external ionosphere. Multi-GNSS experiment (MGEX) network stations tracking Galileo E1/E5a/E5b/E6/E5ab and BeiDou-3 (BDS-3) B1I/B3I/B1C/B2a signals were used to investigate the positioning performance and parameter estimations of three multi-frequency UC-PPP models. The results show that the loosely constrained ionosphere will make the estimated ionosphere and DCB/IFB parameters unable to effectively separate due to their high linear dependence. The 3D positioning accuracy of UC-PPP using Galileo five-frequency, BDS-3 four-frequency and Galileo/BDS-3 multi-frequency signals in static mode is 1.76, 2.36 and 1.39 cm, while the corresponding accuracy in kinematic mode is 6.40, 7.08 and 4.16 cm, respectively. The consistency of Galileo IFBs with respect to the MGEX DCB files is rather good, and the probability of deviations within 0.3 ns is 96.58%. Compared to Galileo, the agreement of the BDS-3 IFBs with respect to the reference values is worse, with 92.69% of them within 1 ns.

GPS satellite differential code bias estimation with current eleven low earth orbit satellites

GPS satellite differential code bias estimation with current eleven low earth orbit satellites

A new method to estimate GPS satellite and receiver differential code biases using a network of LEO satellites

The differential code biases (DCBs) of the global positioning system (GPS) receiver onboard low-Earth orbit (LEO) satellites are commonly estimated by a local spherical symmetry assumption together with the known GPS satellite DCBs from ground-based observations. Nowadays, more and more LEO satellites are equipped with GPS receivers for precise orbit determination, which provides a unique chance to estimate both satellite and receiver DCBs without any ground data. A new method to estimate the GPS satellite and receiver DCBs using a network of LEO receivers is proposed. A multi-layer mapping function (MF) is used to combine multi-LEO satellite data at varying orbit heights. First, model simulations are conducted to compare the vertical total electron content (VTEC) derived from the multi-layer MF and the reference VTEC obtained from the empirical ionosphere model International Reference Ionosphere and Global Core Plasmasphere Model. Second, GPS data are collected from five LEO missions, including ten receivers used to estimate both the satellite and receiver DCBs simultaneously with the multi-layer MF. The results show that the GPS satellite DCB solutions obtained from space-based data are consistent with ground-based solutions provided by the Centre for Orbit Determination in Europe. The proposed normalization procedure combining topside observations from different LEO missions has the potential to improve the accuracies of satellite DCBs of Global Navigation Satellite Systems as well as the receiver DCBs onboard LEO satellites, although the number of LEO missions and spatial–temporal coverage of topside observations are limited.

Ionospheric VTEC and satellite DCB estimated from single-frequency BDS observations with multi-layer mapping function

The ionospheric delays and satellite differential code biases (DCBs) act as the significant error sources in the global navigation satellite system (GNSS) positioning, navigation and timing (PNT) services, and are still challenging to estimate correctly. In this study, the ionospheric vertical total electron content (VTEC) and satellite DCBs are estimated by a refining single-frequency precise point positioning (SFPPP) method based on the multi-layer ionosphere mapping function (MF), as well as the dual-frequency methods, including the carrier-to-code leveling (CCL) and dual-frequency PPP (DFPPP). The solutions isolate the ionospheric VTEC values from the slant ionospheric delay with the generalized trigonometric series function (GTSF) and precisely estimate the satellite DCB with a zero-mean condition. The SFPPP-derived VTEC estimates are validated and evaluated by comparing with the International GNSS Service (IGS) products and using ionosphere-corrected (IC) SFPPP in both static and kinematic scenarios. Using the 74 experimental stations collected from the multi-GNSS experiment (MGEX) network from January to March 2020, the results show that the VTEC estimation precision by applying the multi-layer MF is improved for the SFPPP approach. The positioning performances of the static and kinematic BDS IC SFPPP with the ionospheric correction derived from the multi-layer MF SFPPP are better when compared to the single-layer MF. The estimated BDS DCB with the SFPPP is stable and of high accuracy. The SFPPP approach with multi-layer MF is demonstrated as a promising and reliable method to retrieve the VTEC and satellite DCB with the low-cost property for the GNSS users.

Estimation and Analysis of BDS2 and BDS3 Differential Code Biases and Global Ionospheric Maps Using BDS Observations

Following the continuous and stable regional service of BDS2, the BDS3 officially announced its global service in July 2020. To fully take advantage of the new multi-frequency BDS3 signals in ionosphere sensing and positioning, it is essential to understand the characteristics of the differential code bias (DCB) of new BDS3 signals and BDS performance in global ionospheric maps (GIMs) estimation. This article presents an evaluation of the characteristics of 13 types of BDS DCBs and the accuracy of BDS-based GIM based on the data provided by the International GNSS Service (IGS) and International GNSS Monitoring and Assessment System (iGMAS) for the first time. The GIMs and DCBs are estimated by the APM (Innovation Academy for Precision Measurement Science and Technology) method in a time efficient manner, which can be divided into two main steps. The first step is to produce GIMs based on BDS observations at the B1I, B2I and B3I signals, and the second step is to estimate DCBs among the other frequency bands by removing the ionospheric delay using the precomputed GIMs. Good agreement is found between the APM-based satellite DCB estimates and those from the Chinese Academy of Sciences (CAS) and the German Aerospace Center (DLR) at levels of 0.26 ns and 0.18 ns, respectively. The results, spanning one month, show that the stability of BDS DCB estimates among different frequency bands are related to the contributed observations, and the receiver DCB estimates represent larger STD values than the satellite DCB estimates. The differences in receiver DCB estimates between BDS2 and BDS3 are found to be related to the types of receivers and antennas and firmware version, and the bias of the JAVAD receivers reaches 1.03 ns. The results also indicate that the difference in the single-frequency standpoint positioning (SPP) accuracy using GPS-based and BDS-based GIMs for ionospheric delay corrections is less than 0.03 m in both the horizontal and vertical directions.

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.

Enhancing BDS-3 precise time transfer with DCB modelling

This article is the first report of BDS-3 differential code bias (DCB) correction models. Satellite differential code biases (SDCBs) were regarded as a source of error; if there was no correction, the quality of the GNSS Position & Navigation & Timing service deteriorated. As a newly built global system, the BDS-3 satellite broadcasted B1I, B3I, B1C and B2a signals. To fully exploit all the BDS-3 signals, DCB correction models related to multiple Dual-Frequency (DF) Ionosphere-Free (IF) combinations were developed for BDS-3 application. Two prevailing global navigation satellite system positioning technologies, namely, standard point positioning (SPP) and precise point positioning (PPP), were utilized to validate the efficacy of the DCB parameters provided by the Multi-GNSS Experiment (MGEX). Our numerical analyses clarified how the DCB model performed when it was applied to positioning and time transfer events under the B1C&B3I, B1I&B2a, B1C&B2a cases. With the use of DCB correction in three combinations, the positioning accuracy and frequency stability were significantly improved. The E, N and U position accuracies of the ondcb scheme compared to those of the offdcb scheme were enhanced in the range of 39.51–86.24%, 35.44–78.64% and 41.84–78.64%, respectively. In particular, we observed that the long-term stability of the time link was obviously better than the short-term stability. The stability percentages were improved by at least 3%, and some were improved by up to 66.5%. The frequency stability obtained by B1I&B3I is equivalent to that obtained by B1C&B3I and B1I&B2a. Finally, all the statistics indicate that the B1C&B2a combination recommended as the optimal option is applicable to time transfer communities. The result agrees with our expectation that the appropriate modelling of DCB is vital, which allows for more precise positioning and a stable time link.

Simultaneous estimation of GPS P1-P2 differential code biases using low earth orbit satellites data from two different orbit heights

Global navigation satellite system (GNSS) differential code bias (DCB) is a significant error source in ionosphere modeling that uses GNSS observation data. Given that the orbit altitudes of low earth orbit (LEO) satellites are above the F layer of ionosphere, ionized electrons come from the upper ionosphere or plasmasphere, so observations suffer from small signal delays. DCBs can be estimated using LEO data instead of that from ground stations. In most studies, LEO-based DCB estimation employs onboard data from either a single LEO or from two that are at the same orbit height. In our study, we applied four LEO satellites data from two different orbit heights to estimate GPS satellite DCBs, receiver DCBs and LEO-based vertical total electron content (VTEC) parameters simultaneously. Before DCB estimation, a data preprocessing strategy suitable for LEO onboard data was applied to obtain clear pseudorange data. We developed a processing method for the upper ionosphere using LEO onboard data from different orbit heights, by introducing LEO-based VTEC models in advance to remove its effects; two introduced VTEC models were derived from modeling results using paired data from the same orbital height, respectively. We draw some conclusions as follows. For the LEO satellites at different orbit heights, the more the number of LEO satellites involved, the more stable estimation results achieved. We also noted that the GPS satellite DCBs stability and accuracy with established LEO-based VTEC models, using multi-LEOs data from different orbit heights, were better than the achievable results using onboard data from the same orbital height or using single LEO satellite schemes. Compared with the estimation results of single LEO solutions, the monthly stabilities and accuracies for multi-LEOs solutions from different orbit heights are improved by 25–35% approximately. Among all the tested schemes, superior stability derived from simultaneous estimation using four LEO data at different orbit heights is 0.064 ns; the optimal accuracies for DCB estimation for different GPS satellites are the scheme GRACE-A and JASON-2 (two LEOs) of 0.146 ns, approximately, compared with Center of Orbit Determination of Europe products, respectively. The stability and accuracy of LEO-based DCB estimations were approximately similar to those achieved using ground solutions.

Denoising ionospheric observables based on multipath error modelling with Tikhonov regularization

Multipath errors, as one of the manifest unmodelled error types introduced by code measurements, are absorbed in ionospheric observables, which results in ionospheric pollution. In this work, in order to cancel out random noise by an adjustment algorithm, the modelling of multipath errors derived from code measurement residuals in undifferentiated and uncombined precise point position (PPP) mode is realized by solving a regularization problem, in which Tikhonov regularization (i.e. the first order difference) is taken into account. Furthermore, the Thomas update algorithm is employed to find the solution and a bootstrap method is used to determine the optimal regularization parameter. After the multipath error model was constructed, it was used to extract the ionospheric observables, thereby enabling the correction of the code measurements. In this research, three ionosphere-retrieval schemes were designed, namely the the alignment of the carrier phase to the code (1) before and (2) after the multipath errors were corrected (CCBM and CCAM), and (3) a PPP solution (PPPS). Based on colocation experiments, the effects of removing errors arising from the multipath effect on the code were comparatively assessed. Large differences were found between the three schemes when used in a short-baseline experimental setting, but only small differences were found with a zero-baseline setup. In the short-baseline experiment, CCAM was more reliable than CCBM, with an improvement in accuracy of about 50%, and reached close agreement with PPPS. In addition, 14 continuously operating reference stations in Hong Kong were selected for ionosphere modelling with differential code bias (DCB) estimations. Similar results were seen, as follows: (1) compared with CCBM, CCAM and PPPS had better modelling accuracies, with an improvement of 60%; (2) CCAM and PPPS had better performance at five fitting stations with respect to CCBM, which resulted in improvements in the accuracies ranging from 60% to 80%; (3) taking five-day-average values of the satellite DCBs, the difference between Center for OrbitDetermination in Europe (CODE) and CCAM was around 0.5 ns, which is consistent with the difference between CODE and PPPS, but about a 2 ns difference was verified between CODE and CCBM.

Estimation of QZSS differential code biases using QZSS/GPS combined observations from MGEX

As an important error source in Global Navigation Satellite System (GNSS) positioning and ionospheric modeling, the differential code biases (DCB) need to be estimated accurately, e.g., the regional Quasi-Zenith satellite system (QZSS). In this paper, the DCB of QZSS is estimated by adopting the global ionospheric modeling method based on QZSS/GPS combined observations from Multi-GNSS experiment (MGEX). The performance of QZSS satellite and receiver DCB is analyzed with observations from day of year (DOY) 275–364, 2018. Good agreement between our estimated QZSS satellite DCB and the products from DLR and CAS is obtained. The bias and root mean square (RMS) of DCB are mostly within ±0.3 ns. The day-to-day fluctuation of the DCB time series is less than 0.5 ns with about 96% of the cases for all satellites. However, the receiver DCB is a little less stable than satellite DCB, and their standard deviations (STDs) are within 1.9 ns. The result shows that the stability of the receiver DCBs is not significantly related to the types of receiver or antenna.