BeiDou Navigation Satellite System Triple-frequency Research Articles

A Cycle Slip Detection and Repair Method Based on Inertial Aiding for BDS Triple-Frequency Signals

Cycle slip detection and repair is a prerequisite to obtain high-precision positioning based on a carrier phase. Traditional triple-frequency pseudorange and phase combination algorithm are highly sensitive to the pseudorange observation accuracy. To solve the problem, a cycle slip detection and repair algorithm based on inertial aiding for a BeiDou navigation satellite system (BDS) triple-frequency signal is proposed. To enhance the robustness, the INS-aided cycle slip detection model with double-differenced observations is derived. Then, the geometry-free phase combination is united to detect the insensitive cycle slip, and the optimal coefficient combination is selected. Furthermore, the L2-norm minimum principle is used to search and confirm the cycle slip repair value. To correct the INS error accumulated over time, the extended Kalman filter based on the BDS/INS tightly coupled system is established. The vehicular experiment is conducted to evaluate the performance of the proposed algorithm from a few aspects. The results indicate that the proposed algorithm can reliably detect and repair all cycle slips that occur in one cycle, including the small and insensitive cycle slips as well as the intensive and continuous cycle slips. Additionally, in signal-challenged environments, the cycle slips occurring 14 s after a satellite signal outage can be correctly detected and repaired.

Evaluation of BDS/GPS Multi-Frequency RTK Positioning Performance under Different Baseline Lengths

The BeiDou Navigation Satellite System (BDS) is fully operational and provides positioning, navigation, and timing services to users worldwide. To comprehensively evaluate the BeiDou-3 Navigation Satellite System (BDS-3) global real-time kinematic (RTK) positioning performance, five sets of IGS/MEGX stations with different baseline lengths are selected in this research, and the visibility of current BDS-3, BDS-2+BDS-3, and Global Positioning System (GPS) system satellites are analyzed. The single frequency, dual-frequency, and triple-frequency positioning accuracy as well as ambiguity fixing rate under short baseline and long baseline are also analyzed. The experimental results show that the positioning accuracies of B1C, BII, L1, and B3I single-frequency bands were about the same, while for band B2a it was lower. For the short baseline dual-frequency RTK positioning mode, the accuracy of BDS-3 (B1C/B2a), BDS-3 (B1I/B3I), triple-frequency BDS-3 (B1C/B2a/B3I), and GPS (L1L2) is comparable and slightly better than that of BDS-3 (B1I/B3I). With the increase in baseline length, the advantages of dual-frequency BDS-3 (B1C/B2a) and triple frequency BDS (B1C/B2a/B3I) are more obvious, with triple-frequency BDS-3 (B1C/B2a/B3I) having the best positioning accuracy. In terms of ambiguity fixing performance, dual-frequency BDS-2+BDS-3 (B1I/B3I) and dual-frequency GPS (L1L2) have the highest ambiguity fixing rate. The ambiguity fixing rate of dual-frequency BDS-3 (B1C/B2a) and triple-frequency BDS-3 (B1C/B2a/B3I) can be higher than 90% within 100 km. In the case of positioning using only the BDS-3 system, the triple-frequency BDS-3 (B1C/B2a/B3I) is superior to both the dual-frequency BDS-3 (B1I/B3I) and dual-frequency BDS-3 (B1C/B2a) in terms of both positioning accuracy and ambiguity fixing rate. The BDS-2+BDS-3 (B1I/B3I) dual-frequency RTK, which has reached a level comparable to GPS, can provide global users with real-time centimeter-level differential positioning services.

A Cycle Slip Detection and Repair Method Using BDS Triple-Frequency Optimization Combination with Wavelet Denoising

The traditional triple-frequency geometry-free pseudorange minus phase (GFPMP) method is very susceptible to the influence of pseudorange observation noise, and it is difficult to detect insensitive small cycle slips. The dual-frequency phase ionospheric residual (PIR) method has the ability to detect sensitive small cycle slips, but its detection results have multivalue problems. In view of this, a GFPMP-PIR combination model method with wavelet denoising (GPW) is proposed here. The idea of this method is as follows: firstly, wavelet transform is used to denoise the pseudorange observations; then, by optimizing and selecting the combination coefficients with the minimum condition, an optimized GFPMP-PIR combination model is constructed. Hence, the source data quality is assured through wavelet denoising. And the accuracy of cycle slip detection and repair is improved by using the dual-frequency PIR combination. Moreover, the multivalue problem of the dual-frequency PIR method is solved by triple-frequency GFPMP combination; finally, the BeiDou navigation satellite system (BDS) triple-frequency measurement data is used for experimental verification. The experiment results show that the GFPMP-PIR optimization combination model with wavelet denoising can detect and repair various cycle slips, especially for insensitive small cycle slips.

A Real-Time Cycle Slip Detection and Repair Method Based on Ionospheric Delay Prediction for Undifferenced Triple-Frequency BDS Signals

The detection and repair of cycle slips are key steps in high-accuracy GNSS (Global Navigation Satellite System) data processing using carrier phase observations. BDS (BeiDou Navigation Satellite System) triple-frequency observations provide better combinations for cycle slip detections and repairs compared to dual-frequency observations. Although a number of algorithms have been developed and may correctly detect cycle slips most of the time, the reliability of empirical thresholds methods cannot be guaranteed. In this study, an adaptive threshold is proposed for three sets of triple-frequency Geometry-Free (GF) pseudorange minus phase combinations to improve the cycle slip detection performance and reduce the false alarm rate of the cycle slip detection by combining the predicted epoch-differenced ionospheric delays under active ionospheric conditions. Moreover, in the cycle slip repair, the integral combined cycle slips are determined by directly rounding the estimated float-combined cycle slips, which will lead to a repair error if the between-epoch ionospheric variation is large. In this study, a new rounding method considering the predicted epoch-differenced ionospheric delays is proposed, and it is proven that the new method has a higher success rate for estimating the integer value of a cycle slip than the traditional method. The performance of the newly proposed method is validated by using static BDS triple-frequency observations that contain simulated cycle slips. BDS triple-frequency observations were collected at 30-s sampling intervals under active ionospheric conditions. The results show that this method can successfully detect and repair all slips of more than one cycle. In addition, dynamic BDS data collected with a vehicle-based receiver at a 1-s sampling intervals are processed, and the results show that the proposed method is also effective in the detection and repair of cycle slips in dynamic data.

Unequal‐weighted TCAR algorithm for long‐baseline static triple‐frequency ambiguity resolution

On account of the different parameters of delay-locked loop, multipath effect and code bandwidth, the Global Navigation Satellite System (GNSS) pseudorange noise of each frequency is different. An unequal-weighted three-carrier ambiguity resolution (TCAR) algorithm for long-baseline static triple-frequency ambiguity resolution is proposed. The algorithm is conducted in cascaded steps. When calculating the coefficients of the extra-wide-lane (EWL), wide-lane (WL) and narrow-lane (NL) combinations, the difference of pseudorange noise is considered. Real triple-frequency BeiDou Navigation Satellite System and Global Positioning System observations from four International GNSS Service stations from 13 January to 19 January 2019 are analysed to evaluate the performance of the proposed algorithm. Results show that compared to the classical equal-weighted algorithm, the proposed algorithm shows a comparable performance for the EWL and WL ambiguity resolution and much better performance for the NL ambiguity resolution.

A Cycle Slip Repair Method Against Ionospheric Effects and Observational Noises for BDS Triple-Frequency Undifferenced Phases.

The cycle slip detection (CSD) and cycle slip repair (CSR) are easily affected by ionospheric delay and observational noise. Aiming at mitigating the above disadvantage, a new BeiDou navigation satellite system (BDS) triple-frequency CSR method (BTCSR) is proposed for the undifferenced phase. BTCSR learns from the classic triple-frequency CSR (CTCSR), with combinations of phases and pseudoranges in correcting ionospheric delay and optimizing observational noise. Different from CTCSR, though, BTCSR has made the following improvements: (1) An optimal model of calculating cycle slip combination is established, which further takes into account the minimization of the effect of residual ionospheric error after the correction. The calculation of cycle slip combination is obtained with the root mean squared errors (0.0646, 0.1261, 0.1069) of cycles, resulting in CSR success rate of 99.9927%, and the wavelengths (4.8842,3.5738,8.1403) of m. (2) A discriminant function is added to guarantee the CSR correctness. This function utilizes epoch-difference value of the ionosphere-free and geometry-free phase to select the correct cycle slip value, which eliminates the interference of large pseudorange errors in determining the final cycle slip. Consequently, the performances of BTCSR and CTCSR have been compared. For the real BDS pseudorange observation with additional 1.5 m errors, which can cover situations of 99.96% pseudorange noise, results of CTCSR show failure, but results of BTCSR keep correct. Moreover, BTCSR has made the following improvements relative to the geometry-free cycle slip detection method (GFCSD) and Melboune–Wubbena cycle slip combination detection method (MWCSD): (1) During a moderate magnetic storm of level 6, CSR testing, with the BDS monitoring station in a low latitude region, showed that some failures occur in GFCSD because of severe ionospheric variation, but BTCSR could correctly identify and fix cycle slips. (2) For the BDS observation data with an additional 1.5 m error on the actual pseudoranges, MWCSD exhibited failures, but the repair results of BTCSR were correct and reliable. (3) For the special slips of (0,59,62) cycles, and equal slips of (1,1,1) cycles on (B1,B2,B3), that are hard to detect by GFCSD and MWCSD, respectively, BTCSR could repair these correctly. Finally, BTCSR obtains reliable repair results under large pseudorange errors and severe ionospheric variations, and the cut-off elevation larger than 10 degrees is the suggested background.

Extra-Wide Lane Ambiguity Resolution and Validation for a Single Epoch Based on the Triple-Frequency BeiDou Navigation Satellite System

The ambiguity resolution (AR) and validation of the global navigation satellite system (GNSS) have been challenging tasks for some decades. Considering the reliability problem of extra-wide-lane (EWL) ambiguity in the triple-carrier ambiguity resolution (TCAR), a method for validating the reliability of the EWL ambiguity using a single epoch was proposed for the BeiDou Navigation Satellite System (BDS). For the initial EWL ambiguity, obtained using a rounding estimator with a geometry-free (GF) model, the double-difference ionospheric delay was first estimated to construct a relative positioning model with an initial fixed ambiguity. Second, based on the theory of gross error detection and the AR characteristics of EWL, the second-best ambiguity candidate was constructed. Finally, among the two sets of ambiguities, the one with the smaller posterior variance was taken as the reliable ambiguity. The study showed that, for a single epoch, when only one or two satellites had incorrect ambiguities, the AR success rate after ambiguity validation and correction could reach 100% for medium baselines. For long baselines, due to the increase of atmospheric error, the validation was affected to some extent. However, the AR success rates for two long baselines increased from 96.82% and 98.44% to 98.80% and 99.67%, respectively.

Calibration of BeiDou Triple-Frequency Receiver-Related Pseudorange Biases and Their Application in BDS Precise Positioning and Ambiguity Resolution.

Global Navigation Satellite System pseudorange biases are of great importance for precise positioning, timing and ionospheric modeling. The existence of BeiDou Navigation Satellite System (BDS) receiver-related pseudorange biases will lead to the loss of precision in the BDS satellite clock, differential code bias estimation, and other precise applications, especially when inhomogeneous receivers are used. In order to improve the performance of BDS precise applications, two ionosphere-free and geometry-free combinations and ionosphere-free pseudorange residuals are proposed to calibrate the raw receiver-related pseudorange biases of BDS on each frequency. Then, the BDS triple-frequency receiver-related pseudorange biases of seven different manufacturers and twelve receiver models are calibrated. Finally, the effects of receiver-related pseudorange bias are analyzed by BDS single-frequency single point positioning (SPP), single- and dual-frequency precise point positioning (PPP), wide-lane uncalibrated phase delay (UPD) estimation, and ambiguity resolution, respectively. The results show that the BDS SPP performance can be significantly improved by correcting the receiver-related pseudorange biases and the accuracy improvement is about 20% on average. Moreover, the accuracy of single- and dual-frequency PPP is improved mainly due to a faster convergence when the receiver-related pseudorange biases are corrected. On the other hand, the consistency of wide-lane UPD among different stations is improved significantly and the standard deviation of wide-lane UPD residuals is decreased from 0.195 to 0.061 cycles. The average success rate of wide-lane ambiguity resolution is improved about 42.10%.

Accuracy Analysis of Ionospheric Prediction Models for Repairing Cycle Slips for BeiDou Triple-Frequency Observations

Both the code–phase combination and the Geometry-Free (GF) phase combination are widely employed to detect and repair cycle slips for BeiDou Navigation Satellite System (BDS) triple-frequency observations. However, the effect of residual ionospheric delay on Narrow-Lane (NL) or GF observations must be considered to avoid incorrect cycle–slip estimation. To improve the accuracy in repairing cycle slips, a corrective ionospheric delay value predicted from the previous ionosphere sequence is used to amend the NL or GF observations at the current epoch. The main purpose of the work reported here is to evaluate the efficacy of a three-step method proposed to detect and repair cycle slip using two extra-wide-lane code–phase and one GF phase combination observations. BDS triple-frequency data were processed in two stages: separate processing of geosynchronous Earth orbit satellites, and the division of inclined geosynchronous satellite orbit and medium Earth orbit satellites into two groups for processing at 30° elevation thresholds. Results revealed that using the prediction models to correct NL or GF observations could ensure a rounding success rate of cycle slip close to 100%, even under high ionospheric activity.

Improving the Performance of Galileo Uncombined Precise Point Positioning Ambiguity Resolution Using Triple-Frequency Observations

Compared with the traditional ionospheric-free linear combination precise point positioning (PPP) model, the un-differenced and uncombined (UDUC) PPP model using original observations can keep all the information of the observations and be easily extended to any number of frequencies. However, the current studies about the multi-frequency UDUC-PPP ambiguity resolution (AR) were mainly based on the triple-frequency BeiDou navigation satellite system (BDS) observations or simulated data. Limited by many factors, for example the accuracy of BDS precise orbit and clock products, the advantages of triple-frequency signals to UDUC-PPP AR were not fully exploited. As Galileo constellations have been upgraded by increasing the number of 19 useable satellites, it makes using Galileo satellites to further study the triple-frequency UDUC-PPP ambiguity resolution (AR) possible. In this contribution, we proposed the method of multi-frequency step-by-step ambiguity resolution based on the UDUC-PPP model and gave the reason why the performance of PPP AR can be improved using triple-frequency observations. We used triple-frequency Galileo observations on day of year (DOY) 201, 2018 provided by 166 Multi-GNSS Experiment (MGEX) stations to estimate original uncalibrated phase delays (UPD) on each frequency and to conduct both dual- and triple-frequency UDUC-PPP AR. The performance of UDUC-PPP AR based on post-processing mode was assessed in terms of the time-to-first-fix (TTFF) as well as positioning accuracy with 2-h observations. It was found that triple-frequency observations were helpful to reduce TTFF and improve the positioning accuracy. The current statistic results showed that triple-frequency PPP-AR reduced the averaged TTFF by 19.6% and also improved the positioning accuracy by 40.9%, 31.2% and 23.6% in the east, north and up directions respectively, compared with dual-frequency PPP-AR. With an increasing number of Galileo satellites, it is expected that the robustness and accuracy of the triple-frequency UCUD-PPP AR can be improved further.

A MEMS-IMU Assisted BDS Triple-Frequency Ambiguity Resolution Method in Complex Environments

Emerging technologies such as smart cities and unmanned vehicles all need Global Navigation Satellite Systems (GNSS) to provide high-precision positioning and navigation services. Fast and reliable carrier phase ambiguity resolution (AR) is a prerequisite for high-precision positioning. The poor satellite geometry and severe multipath effect caused by Beidou Navigation Satellite System (BDS) signal occlusion and reflection in complex environments will degrade the AR performance. In this contribution, a fast triple-frequency AR method combining Microelectromechanical System-Inertial Measurement Unit (MEMS-IMU) and BDS is proposed. First, the Extra-Wide Lane (EWL) ambiguity is fixed with the positioning parameters of MEMS-IMU instead of the pseudorange. Then, the phase noise variance of Narrow Lane (NL) observation is obtained from ambiguity-fixed EWL observation to reduce the total noise level of NL observation, and the NL ambiguity can be reliably fixed, and the BDS positioning result is obtained. Finally, the BDS positioning result is used as the posterior measurement of the extended Kalman filter to update the MEMS-IMU positioning parameters to form the coupling loop of MEMS-IMU and BDS. The data of urban road vehicle experiments were collected to verify the feasibility and effectiveness of the proposed algorithm. Results show that MEMS-IMU can speed up AR, and reduction of total noise level can significantly improve the reliability of AR.

Characteristics and modelling of BDS satellite inter-frequency clock bias for triple-frequency PPP

Aiming at the problem of the inter-frequency clock bias (IFCB) in Beidou Navigation Satellite System (BDS) triple-frequency observations, the variation characteristics with time are analysed in detail. The IFCB models for all the three kinds of satellites, i.e. GEO, IGSO and MEO, are proposed. Then the attenuation and long-term forecasting performance of the models are evaluated. Finally, the validity and benefit of the model are verified by triple-frequency Precise Point Positioning (PPP) experiments. The IFCB results from consecutive two-month BDS triple-frequency observation data of 44 globally distributed Multi-GNSS Experiment (MGEX) stations show that the IFCBs of GEO satellites have prominent periodic variation in general. The correlation coefficient and the determination coefficient of predicted IFCB by the model for GEO satellites are up to 0.957 and 0.915 respectively, which shows the model has a high stability and is suitable for long-term prediction. The IFCBs of IGSO satellites have the same periodicity as those of GEO satellites. Although the model is not as good as that of GEO satellites, it still performs well overall and can be applied to long-term prediction in most instances. The model of MEO satellites performs worse than GEO and IGSO due to the numerical instability and less obvious periodicity of IFCBs. Moreover, the effect of the IFCB models is better than the substitution method of using the IFCB at the corresponding moment in the previous period as the IFCB forecast value of the current period. In the triple-frequency Precise Point Positioning (PPP) experiments, the modelled IFCB correction can improve the positioning accuracy by 21.1%, 9.0% and 9.9% in the north, east and up directions and also shorten the convergence time in these three directions by 26.8%, 10.4% and 24.4% respectively compared with the observation model without IFCB correction.

Rapid initialization method in real-time deformation monitoring of bridges with triple-frequency BDS and GPS measurements

Rapid initialization with ambiguity fixed resolution plays a key role in real-time Global Navigation Satellite Systems (GNSS) displacement monitoring of bridges. In this study, we propose a rapid initialization method with triple-frequency (TF) BeiDou Navigation Satellite System (BDS) and Global Positioning System (GPS) observations. The key step of this method is to form extra-wide-lane (EWL) ambiguity with TF BDS and GPS carrier-phase to reduce the ambiguity search space. In order to improve the speed of ambiguity resolution (AR) further, the strategies of fixing ambiguities in groups sequentially from long wavelength to the short and partial AR (PAR) with elevation and signal-to-noise ratio (SNR) criterions were proposed. Based on the real monitoring data on the Baishazhou Yangtze River Bridge, the AR efficiency was tested. The results illustrate that over 90% EWL ambiguities can be fixed by only single-epoch with TF BDS + GPS. After EWL ambiguity is fixed, the wide-lane (WL) ambiguity success rate could improve from 20% with dual-frequency (DF) BDS + GPS data to 70% with TF BDS + GPS data, 50–80% and 80–90% for the three stations respectively. Since the area of ambiguity search space is reduced by TF observations, the ambiguity search time decrased significantly compared with the DF BDS + GPS. In the Time-to-first-fix (TTFF) test, over 85% of epochs can complete the initialization process by only one epoch with one hour of 10 Hz TF BDS + GPS data. However, DF BDS + GPS has only 25%. As for the initialization time longer than two epochs, the counts of TTFFs of TF BDS + GPS are significantly less than that of DF BDS + GPS. Meanwhile, the integrated BDS and GPS can further enhance the geometry of satellites, which will be better for the precision improvement than the increase of signal frequencies.

Modeling and Assessment of Precise Time Transfer by Using BeiDou Navigation Satellite System Triple-Frequency Signals.

This study proposes two models for precise time transfer using the BeiDou Navigation Satellite System triple-frequency signals: ionosphere-free (IF) combined precise point positioning (PPP) model with two dual-frequency combinations (IF-PPP1) and ionosphere-free combined PPP model with a single triple-frequency combination (IF-PPP2). A dataset with a short baseline (with a common external time frequency) and a long baseline are used for performance assessments. The results show that IF-PPP1 and IF-PPP2 models can both be used for precise time transfer using BeiDou Navigation Satellite System (BDS) triple-frequency signals, and the accuracy and stability of time transfer is the same in both cases, except for a constant system bias caused by the hardware delay of different frequencies, which can be removed by the parameter estimation and prediction with long time datasets or by a priori calibration.

Triple-Frequency Code-Phase Combination Determination: A Comparison with the Hatch-Melbourne-Wübbena Combination Using BDS Signals

Considering the influence of the ionosphere, troposphere, and other systematic errors on double-differenced ambiguity resolution (AR), we present an optimal triple-frequency code-phase combination determination method driven by both the model and the real data. The new method makes full use of triple-frequency code measurements (especially the low-noise of the code on the B3 signal) to minimize the total noise level and achieve the largest AR success rate (model-driven) under different ionosphere residual situations (data-driven), thus speeding up the AR by directly rounding. With the triple-frequency Beidou Navigation Satellite System (BDS) data collected at five stations from a continuously-operating reference station network in Guangdong Province of China, different testing scenarios are defined (a medium baseline, whose distance is between 20 km and 50 km; a medium-long baseline, whose distance is between 50 km and 100 km; and a long baseline, whose distance is larger than 100 km). The efficiency of the optimal code-phase combination on the AR success rate was compared with that of the geometry-free and ionosphere-free (GIF) combination and the Hatch-Melbourne-Wübbena (HMW) combination. Results show that the optimal combinations can always achieve better results than the HMW combination with B2 and B3 signals, especially when the satellite elevation angle is larger than 45°. For the wide-lane AR which aims to obtain decimeter-level kinematic positioning service, the standard deviation (STD) of ambiguity residuals for the suboptimal combination are only about 0.2 cycles, and the AR success rate by directly rounding can be up to 99%. Compared with the HMW combinations using B1 and B2 signals and using B1 and B3 signals, the suboptimal combination achieves the best results in all baselines, with an overall improvement of about 40% and 20%, respectively. Additionally, the STD difference between the optimal and the GIF code-phase combinations decreases as the baseline length increases. This indicates that the GIF combination is more suitable for long baselines. The proposed optimal code-phase combination determination method can be applied to other multi-frequency global navigation satellite systems, such as new-generation BDS, Galileo, and modernized GPS.

Three-frequency BDS precise point positioning ambiguity resolution based on raw observables

All BeiDou navigation satellite system (BDS) satellites are transmitting signals on three frequencies, which brings new opportunity and challenges for high-accuracy precise point positioning (PPP) with ambiguity resolution (AR). This paper proposes an effective uncalibrated phase delay (UPD) estimation and AR strategy which is based on a raw PPP model. First, triple-frequency raw PPP models are developed. The observation model and stochastic model are designed and extended to accommodate the third frequency. Then, the UPD is parameterized in raw frequency form while estimating with the high-precision and low-noise integer linear combination of float ambiguity which are derived by ambiguity decorrelation. Third, with UPD corrected, the LAMBDA method is used for resolving full or partial ambiguities which can be fixed. This method can be easily and flexibly extended for dual-, triple- or even more frequency. To verify the effectiveness and performance of triple-frequency PPP AR, tests with real BDS data from 90 stations lasting for 21 days were performed in static mode. Data were processed with three strategies: BDS triple-frequency ambiguity-float PPP, BDS triple-frequency PPP with dual-frequency (B1/B2) and three-frequency AR, respectively. Numerous experiment results showed that compared with the ambiguity-float solution, the performance in terms of convergence time and positioning biases can be significantly improved by AR. Among three groups of solutions, the triple-frequency PPP AR achieved the best performance. Compared with dual-frequency AR, additional the third frequency could apparently improve the position estimations during the initialization phase and under constraint environments when the dual-frequency PPP AR is limited by few satellite numbers.

Instantaneous BDS + GPS undifferenced NRTK positioning with dynamic atmospheric constraints

As the Chinese BeiDou Navigation Satellite System (BDS) has become operational in the Asia-Pacific region, it is important to better understand and demonstrate the benefits of combining triple-frequency BDS with dual-frequency GPS observations for network-based real-time kinematic (NRTK) services. Undifferenced NRTK is a new NRTK service mode, it extends the concept of NRTK by not requiring reference station and specified reference satellite at the rover processing. In order to realize the undifferenced NRTK service, a strategy for real-time modeling the undifferenced (UD) augmentation information is given, in which the fixed double-differenced ambiguities are transformed into UD ones with the help of datum settings. Since this strategy is insensitive to existing ephemeris products, it is applicable to the services of current BDS regional reference networks. Furthermore, a processing scheme for ambiguity resolution (AR) with arbitrary-frequency observations is also presented in detail. An instantaneous and reliable BDS + GPS positioning service can be provided to the rovers in undifferenced NRTK processing mode. With the data collected at 31 stations from a continuously operating reference station network in Guangdong Province (GDCORS) of China, the efficiency of the proposed approaches using combined BDS and GPS observations is confirmed. For three rover stations during days 327–329, a total of 12,960 1-min tests were performed separately to demonstrate the performance of AR. Thanks to the dynamically refined priori information of residual tropospheric and ionospheric error, and the availability of more satellites and observations, the AR fixing rates of combined BDS and GPS systems improve by 13 to 65%, compared with those of the GPS-only system using the traditional WL-L1-IF scheme. The positioning accuracy has also significantly improved.

Performance Analysis of Optimal Combinations for Triple-Frequency BDS

A theoretical and experimental investigation on the BeiDou Navigation Satellite System triple-frequency combination theory is presented. The optimal combination strategies under different application circumstances are given in consideration of the combination wavelength, noise, and ionosphere amplification factor as the main criteria. On this basis, the performance of BeiDou Navigation Satellite System triple-frequency optimal combinations, in regard to ambiguity resolution and relative positioning accuracy, is verified by four static tests with baseline lengths of 4.8, 12.4, 20, and 36.8 km, respectively. Finally, a few suggestions about the optimal phase combinations used in relative navigation applications are given.

Considering Inter-Frequency Clock Bias for BDS Triple-Frequency Precise Point Positioning

The joint use of multi-frequency signals brings new prospects for precise positioning and has become a trend in Global Navigation Satellite System (GNSS) development. However, a new type of inter-frequency clock bias (IFCB), namely the difference between satellite clocks computed with different ionospheric-free carrier phase combinations, was noticed. Consequently, the B1/B3 precise point positioning (PPP) cannot directly use the current B1/B2 clock products. Datasets from 35 globally distributed stations are employed to investigate the IFCB. For new generation BeiDou Navigation Satellite System (BDS) satellites, namely BDS-3 satellites, the IFCB between B1/B2a and B1/B3 satellite clocks, between B1/B2b and B1/B3 satellite clocks, between B1C/B2a and B1C/B3 satellite clocks, and between B1C/B2b and B1C/B3 satellite clocks is analyzed, and no significant IFCB variations can be observed. The IFCB between B1/B2 and B1/B3 satellite clocks for BDS-2 satellites varies with time, and the IFCB variations are generally confined to peak amplitudes of about 5 cm. The IFCB of BDS-2 satellites exhibits periodic signal, and the accuracy of prediction for IFCB, namely the root mean square (RMS) statistic of the difference between predicted and estimated IFCB values, is 1.2 cm. A triple-frequency PPP model with consideration of IFCB is developed. Compared with B1/B2-based PPP, the positioning accuracy of triple-frequency PPP with BDS-2 satellites can be improved by 12%, 25% and 10% in east, north and vertical directions, respectively.

Real-time cycle-slip detection and repair for BeiDou triple-frequency undifferenced observations

A new method of real-time cycle-slip detection and repair for BeiDou navigation satellite system (BDS) triple-frequency undifferenced observations is proposed. The method, which is based on code–phase and geometry-free phase combinations, composes linearly independent combination observations to uniquely determine the cycle-slip of original frequency phase observations in two independent steps. First, two extra-wide lane (EWL) code–phase combinations are utilised to determine the EWL combination cycle-slips. The sum of the carrier phase combination coefficients is zero. Second, the geometry-free phase combination with a non-zero sum of combination coefficients composes linearly independent combination observations with two code–phase combinations. Ionospheric delay variance systematic deviation is corrected with several previous carrier-phase observations without cycle-slip, which seriously affect the accuracy of the geometry-free phase combination observations. The BDS triple-frequency carrier-phase and code observations of different satellite types at 30 s sampling interval, which involve increased active ionospheric conditions in 1 day, are utilised to test the method. Results show that the proposed method can efficiently detect and accurately repair any cycle-slip combinations in real-time for different satellite types under active ionospheric conditions. No erroneous and leakage judgments are observed in the test.