Inter-frequency Bias Research Articles

Carrier Phase Common-View Single-Differenced Time Transfer via BDS Penta-Frequency Signals

The BeiDou Navigation Satellite System (BDS-3) has officially provided services worldwide since July 2020. BDS-3 has added new signals for B1C, B2a and B2b based on old BDS-2 B1I and B3I signals, which brings opportunities for achieving high-precision time transfer. In this research, the BDS-3/BDS-2 combined penta-frequency common-view (CV) single-differenced (SD) precise time transfer model is established with B1I, B3I, B2I, B1C, B2a and B2b signals, including dual-, triple-, quad- and penta-frequency (abbreviated as DF, TF, QF and PF) ionosphere-free (IF) combination CV SD models. Taking four long baseline time links (from 637.6 km to 1331.6 km) as examples, the accuracy and frequency stability of the BDS-3/BDS-2 combined DF, TF, QF and PF SD time transfer models were evaluated. The experimental results show that the frequency stability of the TF, QF and PF SD models were improved by 2.5%, 5.3% and 8.5%, on average, over the DF SD model. Compared with the traditional DF (B1I/B3I IF combination) SD model, the standard deviation (STD) of the multi-frequency SD model was reduced by 5.9%, on average, and the frequency stability was improved by 4.0% on average, which had the most apparent effect on the improvement of short-term frequency stability. Specifically, the DF1 (B1C and B2a DF IF combination), TF1 (B1C, B2a and B2b TF IF combination), QF1 (B1C, B1I, B2a and B2b QF IF combination) and PF4 (B1C, B1I, B2a, B2b and B3I PF IF combination) SD models had better performance in timing, and the PF4 SD model had the best performance. Considering that the PF4 (one PF signal IF combination) SD model does not require an estimated inter-frequency bias and that its noise factor is minor compared with the PF1 (four DF signal IF combination), PF2 (three TF signal IF combination) and PF3 (two QF signal IF combination) SD models, we recommend the PF4 SD model for multi-frequency time transfer and the use of the PF2, PF2 or PF3 SD model to supplement the PF4 SD model in cases of penta-frequency observation loss.

BDS-3 full-frequency precise point positioning: models and performance comparison

With the BDS-3 six-frequency integration, there are more choices to implement the precise point positioning (PPP) technology. To take full advantage of the multi-frequency combination, six typical BDS-3 six-frequency PPP models using uncombined observation (UC), five dual-frequency ionospheric-free (IF) combinations (IF2), four triple-frequency IF combinations (IF3), three four-frequency IF combinations (IF4), two five-frequency IF combinations (IF5), and a single six-frequency IF combination (IF6) were constructed and compared. The results indicate that the positioning accuracies of the six models are similar. The static positioning accuracies reach 4, 3–5, and 12–14 mm in the east, north, and up directions, respectively, while the kinematic positioning accuracies are 24–26, 16–17, and 42–43 mm, respectively. Regarding the convergence times, the UC model is slightly worse than the five IF combined models, except for some cases in the up direction. In the static mode, benefiting from the smallest noise amplification factor, the average convergence times of the IF6 model are the shortest, reaching 12.1, 5.5, and 13.4 min in the three directions, respectively, and are 7%, 15%, and 6% shorter than those of the UC model. In the kinematic mode, with the combination of more signals into a single IF combination, the noise level of the IF combined observables gradually decreases, but the convergence times gradually increase. The kinematic average convergence times of the IF2 model are 15.3, 3.6, and 18.0 min in the three directions, which are 6%, 22%, and 10% and 1%, 16%, and 20% shorter than those of the UC and IF6 models, respectively. In addition, the observation residuals, and the estimates of inter-frequency bias, tropospheric zenith wet delay, and receiver clock offsets from different models were compared and analyzed. The specific selection of the model should be based on actual situations.

BDS-3 Triple-Frequency Timing Group Delay/Differential Code Bias and Its Effect on Positioning

BeiDou Global Navigation Satellite System (BDS-3) broadcasts multifrequency signals that offer more choices of frequencies and more signal combinations for positioning. This paper analyzes the effect of timing group delay (TGD) and differential code bias (DCB) of BDS-3 on the corresponding triple-frequency positioning. The triple-frequency observation models of BDS-3 are summarized and the DCB correction models are derived for the four different frequency combinations of triple-frequency ionospheric-free (IF) combination (IF123), two dual-frequency IF combinations (IF1213) and triple-frequency uncombined (UC123) positioning modes. Standard point positioning (SPP) and precise point positioning (PPP) experiments were conducted using 30 days of observations from 25 multi-GNSS experiment (MGEX) stations. The results show that the TGD/DCB correction has a significant impact on the accuracy of SPP. The positioning accuracy using IF123 and IF1213 models improved by about 73~90% after TGD correction, in comparison to a 27~30% improvement achieved using the UC123 model. In addition, the correction effect of DCB is slightly better than TGD. The DCB correction significantly improves accuracy in the initial epoch of the PPP, which helps the convergence of the filtering and reduces the convergence time. The average convergence times of IF123, IF1213 and UC123 are 26.1, 26.9 and 38.3 min, respectively, which are reduced by 6.79, 2.54 and 8.59% with DCB correction. The pseudorange residuals are closer to zero-mean random noise after DCB correction. Furthermore, the DCB affects the evaluation of the inter-frequency bias (IFB), ionospheric delay and floating ambiguity parameters. However, the tropospheric delay is almost unaffected by DCB.

Analysis of inter-frequency bias in multi-mode multi-frequency GNSS-IR water level retrieval and correction method

Analysis of inter-frequency bias in multi-mode multi-frequency GNSS-IR water level retrieval and correction method

Modelling and Assessment of a New Triple-Frequency IF1213 PPP with BDS/GPS

The currently available triple-frequency signals give rise to new prospects for precise point positioning (PPP). However, they also bring new bias, such as time-varying parts of the phase bias in the hardware of receivers and satellites due to the fact that dual-frequency precise clock products cannot be directly applied to triple-frequency observation. These parameters generate phase-based inter-frequency clock bias (PIFCB), which impacts the PPP. However, the PIFCBs of satellites are not present in all GNSSs. In this paper, various IF1213 PPP models are constructed for these parts, namely, the triple-frequency PIFCB (TF-C) model with PIFCB estimation, the TF inter-frequency bias (IFB) (TF-F) model ignoring the PIFCB, and the TF-PIFCB-IFB (TF-CF) model with one system PIFCB estimation. Additionally, this study compares these IF1213 PPP models with the dual-frequency ionosphere-free (DF) model. We conducted single system static PPP, dual-system static and kinematic PPP experiments based on BDS/GPS observation data. The GPS static PPP experiment demonstrates the reliability of the TF-C model, as well as the non-negligibility of the GPS PIFCB. The BDS static PPP experiment demonstrates the reliability of the TF-F and TF-CF models, and that the influence of the BDS-2 PIFCB can be neglected in BDS. The BDS/GPS PPP experimental results show that the third frequency does not significantly improve the positioning accuracy but shortens the convergence time. The positioning accuracy of TF-C and TF-CF for static PPP is better than 1.0 cm, while that for kinematic PPP is better than 2.0 cm and 4.0 cm in the horizontal and vertical components, respectively. Compared with the DF model, the convergence time of the TF-C and TF-CF models for static PPP is improved by approximately 23.5%/18.1%, 13.6%/9.7%, and 19.8%/12.1%, while that for kinematic PPP is improved by approximately 46.2%/49.6%, 33.5%/32.4%, and 35.1%/36.1% in the E, N and U directions, respectively. For dual-system PPP based on BDS/GPS observations, the TF-C model is recommended.

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

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

Effect of Stochastic Modeling for Inter-Frequency Biases of Receiver on BDS-3 Five-Frequency Undifferenced and Uncombined Precise Point Positioning

The third generation of the Beidou navigation satellite system (BDS-3) broadcasts navigation signals of five frequencies. Focusing on the deep integration of five-frequency signals, we applied the joint BDS-3 five-frequency undifferenced and uncombined precise point positioning (UC-PPP) to analyze the receiver inter-frequency biases (IFB). Firstly, 12 Multi-GNSS Experiment tracking (MGEX) stations are selected to investigate the time-varying characteristics of receiver IFB and, according to random characteristics, three random modeling schemes are proposed. Secondly, the effects of three stochastic modeling methods on zenith tropospheric delay, ionospheric delay, floating ambiguity, and quality control are analyzed. Finally, the effects of three IFB stochastic modeling methods on positioning performance are evaluated. The results showed that the amplitude in the IFB for B2b is 5.139 m, B2a is 1.964 m, and B1C is 0.950 m by measuring one week’s observation data. The IFB stochastic modeling method based on random walks can shorten the PPP convergence time by 4~12%, diminish the false alarm of quality control, and improve the positioning accuracy. The random walk model is recommended to simulate the variation of IFB, which can not only overcome the disadvantage of the time constant model being unable to accurately describe the time-varying characteristics of the IFB, but also avoid reducing the strength of the kinematic PPP positioning model due to the large process noise of the white noise model.

Galileo Time Transfer with Five-Frequency Uncombined PPP: A Posteriori Weighting, Inter-Frequency Bias, Precise Products and Multi-Frequency Contribution

Galileo satellites can broadcast signals on five frequencies, namely E1, E5A, E5B, E5 (A+B), and E6. The multi-frequency integration has become an emerging trend in Global Navigation Satellite System (GNSS) data processing. This study focused on the precise time transfer based on Galileo five-frequency uncombined precise point positioning (PPP), including the performance comparison of PPP time transfer with a priori and a posteriori weighting strategies, with different inter-frequency bias (IFB) dynamic models, and with the precise satellite products from different analysis centers, as well as the contribution of multi-frequency observations for time transfer. Compared with the a priori weighting strategy, the short-term frequency stability of time transfer adopting the Helmert variance component estimation method can be improved by 28.9–37.6% when the average time is shorter than 100 s. The effect of IFB dynamic models on Galileo five-frequency PPP time transfer is not obvious. When employing the post-processed precise satellite products from seven analysis centers, the accuracy of time transfer can be better than 0.1 ns, while an accuracy of 0.253 ns can be obtained in the real-time mode. At an average time of approximately 10,000 s, the post-processed time transfer with Galileo five-frequency PPP can provide a frequency stability of 3.283 × 10−14 to 3.459 × 10−14, while that in real-time mode can be 3.541 × 10−14. Compared with dual-frequency PPP results, the contribution of multi-frequency combination to the accuracy and frequency stability of time transfer is not significant, but multi-frequency PPP can achieve more reliable time transfer results when the signal quality is poor.

Construction of nominal ionospheric gradient using satellite pair based on GNSS CORS observation in Indonesia

Ground-Based Augmentation System (GBAS) is a GNSS augmentation system that meets International Civil Aviation Organization (ICAO) requirements to support precision approach and landing. GBAS is based on local differential GNSS technique with reference stations located around an airport to provide necessary integrity and accuracy. The performance of the GBAS system can be affected by gradient in the ionospheric delay between aircraft and reference stations. A nominal ionospheric gradient, which is bounded by a conservative error bound, is represented by a parameter σvig. The parameter σvig is commonly determined using station pair to GNSS Continuous Operating Reference Station (CORS) data. The station-pair method is susceptible to doubling of the estimation error of receiver inter-frequency bias (IFB) and is not suitable with the CORS conditions in Indonesia. We propose a satellite-pair method that is found to be more suitable for the CORS network over Indonesia which is centered in Java and Sumatra islands. An overall value of σvig (5.21 mm/km) was obtained using this method along with preliminary results of a comparison of σvig from Java and Sumatra islands.Graphical

Modeling and assessment of five-frequency BDS precise point positioning

Since its full operation in 2020, BeiDou Satellite Navigation System (BDS) has provided global services with highly precise Positioning, Navigation, and Timing (PNT) as well as unique short-message communication. More and more academics focus on multi-frequency Precise Point Positioning (PPP) models, but few on BDS five-frequency PPP models. Therefore, this study using the uncombined and Ionospheric-Free (IF) observations develops five BDS five-frequency PPP models and compares them with the traditional dual-frequency model, known as Dual-frequency IF (DF) model. Some biases such as Inter-Frequency Biases (IFB) and Differential Code Bias (DCB) are also addressed. With the data collected from 20 stations, the BDS dual- and five-frequency PPP models are comprehensively evaluated in terms of the static and simulated kinematic positioning performances. Besides, the study also analyzes some by-product estimated parameters in five-frequency PPP models such as Zenith Troposphere Delay (ZTD). The results of experiment show that five-frequency PPP models have different levels of improvement compared with the DF model. In the static mode, the one single Five-Frequency IF combination (FF5) model has the best positioning consequent, especially in the up direction, and in the simulated kinematic mode, the Three Dual-frequency IF combinations (FF3) model has the largest improvement in convergence time.

Impact of receiver inter-frequency bias residual uncertainty on dual-frequency GBAS integrity

Dual-Frequency Ground-Based Augmentation Systems (GBAS) can be affected by receiver Inter-Frequency Bias (IFB) when Ionosphere-Free (Ifree) smoothing is applied. In the framework of the proposed GBAS Approach Service Type F (GAST-F), the IFB in the Ifree smoothed pseudorange can be corrected. However, IFB residual uncertainty still exists, which may threaten the integrity of the system. This paper presents an improved algorithm for the airborne protection level considering the residual uncertainty of IFBs to protect the integrity of dual-frequency GBAS. The IFB residual uncertainty multiplied by a frequency factor is included in the Ifree protection level together with the uncertainty of other error sources. To verify the proposed protection level algorithm, we calculate the IFB residual uncertainties of ground reference receivers and user receiver based on BDS B1I and B3I dual-frequency observation data and carry out a test at the Dongying Airport GBAS station. The results show that the proposed Ifree protection level with IFB residual uncertainty is 1.48 times the current protection level on average. The probability of Misleading Information (MI) during the test is reduced from 3.2 × 10−4 to the required value. It is proven that the proposed protection level can significantly reduce the integrity risk brought by IFB residual uncertainty and protect the integrity of dual-frequency GBAS.

Water Level Retrieval Using a posteriori Residual of GNSS Pseudorange and Carrier-Phase Observations

In global navigation satellite system reflectometry (GNSS-R) applications, water level variations could be determined by analyzing one of three types of data: signal-to-noise ratios (SNRs), combinations of carrier-phase observations and combinations of pseudorange and carrier-phase observations. In this study, a new GNSS-R method based on the analysis of a posteriori residual of undifferenced and uncombined (UC) Precise Point Positioning (PPP) and ionosphere free (IF) PPP was proposed. Double peaks were found in the Lomb-Scargle periodogram (LSP) estimates of dual-frequency carrier-phase residuals generated with UC PPP, the pseudorange and carrier-phase residuals generated with IF PPP and the detrended S2 time series. To solve this problem, we present a strict quality control strategy to screen the correct retrievals. Two experimental datasets from a dam and a bridge-deformation monitoring system, respectively, were used to evaluate the performance of this new method in water-level retrieval. The results show that the water level retrievals from the pseudorange and carrier-phase residuals generated by the UC PPP and IF PPP positioning method show a good agreement with the in-situ references, with correlation coefficient exceeding 0.95 and 0.97 in the two experiments, which is comparable to the SNR method. The retrievals showed a root mean square error (RMSE) of 6-7cm in the reservoir experiment and 13-16cm in the Ganjiang river experiment. Meanwhile, the inter-frequency bias for about 0.2m was found in the Xilongchi reservoir water level retrieval experiment, however, it is not the case for the Ganjiang River water level retrieval experiment. The new method enables the simultaneous determination of displacements and water-level variation and is thus applicable for in-situ displacement monitoring of civil engineering structures, such as dam, bridge etc.

Ionospheric corrections tailored to the Galileo High Accuracy Service

The Galileo High Accuracy Service (HAS) is a new capability of the European Global Navigation Satellite System that is currently under development. The Galileo HAS will start providing satellite orbit and clock corrections (i.e. non-dispersive effects) and soon it will also correct dispersive effects such as inter-frequency biases and, in its full capability, ionospheric delay. We analyse here an ionospheric correction system based on the fast precise point positioning (Fast-PPP) and its potential application to the Galileo HAS. The aim of this contribution is to present some recent upgrades to the Fast-PPP model, with the emphasis on the model geometry and the data used. The results show the benefits of integer ambiguity resolution to obtain unambiguous carrier phase measurements as input to compute the Fast-PPP model. Seven permanent stations are used to assess the errors of the Fast-PPP ionospheric corrections, with baseline distances ranging from 100 to 1000 km from the reference receivers used to compute the Fast-PPP corrections. The 99% of the GPS and Galileo errors in well-sounded areas and in mid-latitude stations are below one total electron content unit. In addition, large errors are bounded by the error prediction of the Fast-PPP model, in the form of the variance of the estimation of the ionospheric corrections. Therefore, we conclude that Fast-PPP is able to provide ionospheric corrections with the required ionospheric accuracy, and realistic confidence bounds, for the Galileo HAS.

Modeling and performance assessment of BDS-2/BDS-3 triple-frequency ionosphere-free and uncombined precise point positioning

The global BeiDou Navigation Satellite System (BDS-3) and regional system (BDS-2) are transmitting signals on three or more frequencies, which brings more choices and practical potentials. To deeply integrate and fully utilize the BDS-2 and BDS-3 signals, this research is dedicated to the modeling and assessment of BDS-2/BDS-3 combined triple-frequency precise point positioning (PPP). Using ionosphere-free (IF) or uncombined (UC) observables, the three BDS-2/BDS-3 combined triple-frequency PPP models, named IF1213, IF123 and UC123 are developed. Particularly, new biases such as time delay bias (TDB) and inter-frequency bias (IFB) as well as the parameterizations are addressed. One-month observations from 16 Multi-GNSS Experiment (MGEX) stations located in and around the Asia-Pacific region are utilized to validate and compare the models. The results indicate that the three BDS-2/BDS-3 combined triple-frequency PPP models have the comparable performance and the third frequency makes a marginal contribution to positioning and tropospheric delay retrieval. The TDB should be carefully considered in BDS-2/BDS-3 combined PPP, especially for the IF123 model due to the larger absolute values of TDB. Meanwhile, the IFB derived from receiver uncalibrated code delay difference shows good stability in both short-term (intra-day) and long-term (inter-day) periods. In addition, demonstrated by triple-frequency PPP tests with one-month data from 47 global MGEX stations, the BDS-2/BDS-3 combined constellation allows for significantly improved convergence performance and positioning accuracy over the BDS-2-only constellation. The BDS-3 system has the ability to provide the precise positioning accuracy of 12.1/9.8/15.8 mm on a global scale after the convergence time of 23.0/13.2/24.7 min in east/north/up components.

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

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

Optimization of Kalman Filtering in Estimating Ionospheric Delay

IFB (Inter-Frequency Bias) is the difference between the hardware delays of two frequencies in the GPS (Global Positioning System) satellite transmitter and the user receiver. It is also called the Instrumental Bias, which will introduce errors into the solution of the ionospheric delay. The current method of eliminating the IFB from the ionospheric delay is to establish a vertical ionospheric model based on GPS dual-frequency observation data and estimate the ionospheric model coefficients and IFBs in real time using Kalman filtering. However, the measurement noise covariance matrix in the filtering process does not consider the correlation between the system observations, which leads to inaccurate filtering models. Finally, it will affect the accuracy of the solved ionospheric delay. In this paper, the GPS dual-frequency observation data of 19 reference stations in the United States are selected, and the ionospheric model coefficients and the IFBs are estimated in real time by Kalman filter. In the filtering process, the estimation noise variance matrix is optimized by introducing the estimated variance of a priori IFB into the measurement noise variance. The calculation results show that the IFBs of satellites after optimization is closer to the related IFB of CODE (the Center for Orbit Determination in Europe). Substituting ionospheric delay after optimization into pseudo-range resolution, the standard deviation of the position error obtained by substituting the pseudo-range solution decreases by 12.5% and 15.4% respectively in the eastward direction and the zenith direction. The average error in the zenith direction decreased by 17.6%, thus the positioning accuracy was improved.

ON GLONASS pseudo-range inter-frequency bias solution with ionospheric delay modeling and the undifferenced uncombined PPP

With the development of multi-GNSS, the differential code bias (DCB) has been an increasing interest in the multi-frequency multi-GNSS community. Unlike code division multiple access (CDMA) mode used by GPS, BDS and Galileo etc., the GLONASS signals are modulated with frequency division multiple access (FDMA) mode. Up to now, the FDMA-aware GLONASS bias products are provided by two individual IGS analysis center (AC), i.e., CODE and GFZ. However, only the ionosphere-free (IF) combination IFB of P1 and P2 is available, while it is founded that the GLONASS IFB of GFZ on both frequencies are identical for the same receiver-satellite pair. In this contribution, the GLONASS IFB (inter-frequency bias) solution based on the spherical-harmonic (SH) ionospheric delay modeling as well as the undifferenced and uncombined PPP were carried out and evaluated. Based on the theoretical analysis, observations from 236 CMONOC stations and 172 IGS stations were collected for 2014 March and 2017 March for the numerical verification. The results suggested that the precision of IFB estimates was mainly subjected to the ionospheric status. Concerning the SH ionospheric delay modeling solution, the STD was 0.85 ns and 0.51 ns for 2014 and 2017, respectively. Concerning the undifferenced and uncombined PPP solution, the IFB was further dependent on the signal frequencies, and the STD was 1.43 ns and 1.94 ns for $${\mathrm{IFB}}_{1}$$ and $${\mathrm{IFB}}_{2}$$ in 2014, and the STD was 0.97 ns and 1.17 ns for $${\mathrm{IFB}}_{1}$$ and $${\mathrm{IFB}}_{2}$$ in 2017. When converted to the GF IFB from the individual IFB on each frequency, and compared to that of GF IFB of SH solution, it is revealed that the undifferenced and uncombined PPP solution has its advantages for IFB estimation on each individual frequency, and more efficient in data processing, while the solution based on the SH ionospheric delay modeling has its advantage in the precision of the GF IFB estimates. Thus, it is suggested that the SH model should be preferred for non-time-critical GF IFB concerned-only applications. Otherwise, the undifferenced and uncombined PPP solution is preferred. These IFB on each frequency was further converted to the ionosphere-free IFB and compared with the products of CODE analysis center.

Impact of the third frequency GNSS pseudorange and carrier phase observations on rapid PPP convergences

New GNSS signals have significantly augmented positioning service and promoted algorithmic innovations such as rapid PPP convergence. With the emerging of multifrequency signals, it becomes essential to thoroughly explore the contribution of third frequency pseudorange and carrier phase toward PPP. In this study, we research the role of the third frequency observations on accelerating PPP convergence, commencing from both stochastic and functional models. We first constructed the stochastic model depending on the observation noise and then introduced two uncombined functional models with respect to different inter-frequency bias (IFB) estimation strategies. The double-differenced residuals based on a zero baseline were used to evaluate the signal noises, which were 0.09, 0.07, 0.11, 0.01 and 0.09 m for Galileo E1/E5a/E5b/E5/E6 pseudorange and 0.24, 0.31 and 0.05 m for BeiDou B1/B2/B3. Besides, carrier phase observations E5a/E5/E6/B1I/B3I shared a comparable signal noise of 0.002 m, while the signal noises of E1/E5b/B2I were 0.003 m. Both BeiDou-2/Galileo and Galileo-only float PPP were implemented based on the dataset collected from 25 stations, spanning 30 days. Triple-frequency Galileo PPP achieved convergence successfully in 19.9 min if observations were weighted according to observation precision, showing a comparable performance of dual-frequency PPP. Meanwhile, the convergence time of triple-frequency float PPP was further shortened to 19.2 min when satellite pair IFBs were eliminated by estimating a second satellite clock. While the improvement of triple-frequency float PPP was marginal, triple-frequency PPP-AR using signals E1/E5a/E6 shortened the initialization time of the dual-frequency counterpart by 38%. Moreover, the performance of triple-frequency PPP-AR kept almost unchanged after we excluded the third frequency pseudorange observations. We thus suggest that the contribution of the third frequency to PPP mainly rests on ambiguity resolution, favored by the additional carrier phase observations.

A modified estimation method of GLONASS inter-frequency bias for RTK based on short baseline

Different from other satellite systems, GLONASS adopts frequency-division multiple access (FCMA) technology to distinguish satellites, which directly leads to frequency deviation during signal receiving. Additionally, carrier phase inter-frequency bias (IFB) between different types of receivers are usually not zero, hindering the ambiguity resolution and thus limits the positioning accuracy. To solve this problem, a new estimating method of GLONASS ambiguity is proposed in this paper. According to the variation range of frequency deviation rate, the influence of phase IFB on ambiguity was removed by using searching method and quadratic linear model, thus reducing phase IFB estimating problem to an optimizing one. Feasibility of the method was verified using zero baseline data. Results show that the method can guarantee the reliability of RTK solution effectively.

Analytical performance and validations of the Galileo five-frequency precise point positioning models

Galileo navigation satellite system provides global services with five-frequency signals. The contribution of this study is to develop four Galileo five-frequency precise point positioning (PPP) models with the ionospheric-free (IF) and uncombined observables, namely FF1, FF2, FF3 and FF4 models, respectively. Galileo dual- and triple-frequency IF models, known as DF and TF models, are also investigated for comparisons. The Galileo dual- and multi-frequency PPP models are comprehensively evaluated with thirty consecutive days period observations collected from 26 multi-GNSS experiment (MGEX) network stations, together with a dynamic experiment dataset, in terms of the static and kinematic performance. The by-product estimated parameters in five-frequency PPP models including the receiver clock, tropospheric delay, receiver inter-frequency biases (IFBs) and differential code bias (DCB) are also analyzed. The experimental results show that the FF1, FF2, and FF3 models perform basic consistent and the FF4 model exhibits inconsistency due to the external ionospheric constraint. The Galileo kinematic PPP performance is significantly improved with the multi-frequency observations under the limited observed satellites circumstance. The significance and potency of the Galileo five-frequency PPP is demonstrated for future Galileo applications.