Triple-frequency Precise Point Positioning Research Articles

The benefit of B1C/B2a signals for BDS-3 wide-area decimeter-level and centimeter-level point positioning with observable-specific signal bias

In addition to the legacy of B1I and B3I signals, the BeiDou Global Navigation Satellite System (BDS-3) also broadcasts the new navigation signals B1C and B2a, which brings new challenges and opportunities for multi-frequency precise point positioning (PPP) ambiguity resolution. The observable-specific signal biases (OSBs) provide flexible frequency choice for multi-frequency ambiguity resolution. This work implemented an approach of BDS-3 quad-frequency OSBs estimation, making the generated OSBs suitable for ambiguity resolution using any frequency combinations. Besides, the benefits of B1C and B2a signals for wide-area decimeter-level and centimeter-level point positioning with ambiguity resolution are also investigated in depth. The BDS-3 B1C/B1I/B2a/B3I measurement data of 110 stations over one month were used to estimate the quad-frequency phase OSBs, and 20 different stations were selected for PPP verification. The results indicate that the BDS-3 quad-frequency phase OSBs are relatively stable, with a standard deviation of less than 0.06 cycle. The B1C/B2a/B3I combination achieved the highest positioning accuracy for instantaneous triple-frequency PPP wide-lane ambiguity resolution, with root-mean-square of 0.15 m, 0.17 m and 0.51 m in the east, north and up directions, respectively. Moreover, the use of B1C/B2a instead of the traditional frequency B1I/B3I as the WL combination improved the single-epoch PPP accuracy. Furthermore, after solving the narrow-lane ambiguity, the median convergence time of B1C/B2a for the kinematic and static mode was 26.5 min and 14.5 min, which was shortened by 29.3 % and 27.5 % compared to the results of the B1I/B3I combination, respectively.

Analysis of Galileo five-frequency precise satellite clock estimation models and its effect on multi-frequency PPP

The Global Navigation Satellite System (GNSS) multi-frequency observations are widely used in positioning applications, while precise orbit determination and precise clock estimation (PCE) techniques typically employ dual-frequency undifferenced (UD) ionospheric-free (IF) observations from global networks. To fully utilize the multi-frequency GNSS observations for generating satellite products at the server-end, we develop some new five-frequency PCE models based on Galileo data, which are five-frequency uncombined FFUC model, FFIF0 model combining the E1/E5a, E1/E5b, E1/E5 and E1/E6 IF observables, FFIF1 model combining E1/E5a and E1/E5a/E5b/E5/E6 IF observables, respectively. The traditional dual-frequency UD IF and triple-frequency PCE models are also introduced for comparison. The new multi-frequency PCE models can not only make full use of the modern GNSS multi-frequency observations, but also obtain satellite clock offsets and inter-frequency clock bias (IFCB) at the same time, which can better support multi-frequency precise point positioning (PPP) applications. The multi-frequency models can improve the stability of GNSS satellite clock estimation, the precision of satellite clock offsets can be improved by 8–12% for triple-frequency models, and 19–27% for five-frequency PCE models compared with the traditional dual-frequency IF model. The PPP positioning accuracy using only multi-frequency satellite clock offsets can be improved by 7–18% for dual-frequency PPP, 8–15% for triple-frequency PPP, 4–16% for five-frequency PPP. The positioning accuracy can be further improved by 11–28% for triple-frequency PPP and 7–20% for five-frequency PPP. Therefore, the new multi-frequency PCE models are demonstrated to support PPP applications with better performance.

GPS, BDS-3, and Galileo Inter-Frequency Clock Bias Deviation Time-Varying Characteristics and Positioning Performance Analysis

Multi-frequency observations are now available from GNSSs, thereby bringing new opportunities for precise point positioning (PPP). However, they also introduce new challenges, such as inter-frequency clock bias (IFCB) between the new frequencies and the original dual-frequency observations due to triple-frequency observations, which severely impact the PPP. In this paper, we studied the estimation and correction methods of uncombined inter-frequency clock bias of GPS, BDS-3, and Galileo, analyzed the time-varying characteristics and short-term stability of IFCB, and analyzed the influence of IFCB on the positioning of the GPS, BDS-3, and Galileo, based on a triple-frequency un-differential non-combined PPP model. The obtained results show that the amplitude of Block IIF satellites of the GPS can reach up to 10–20 cm, and the IFCB in BDS-3, Galileo, and GPS Block III satellites can be neglected. After correction by IFCB, the 3D positioning accuracy of the GPS triple-frequency PPP was 1.73 cm and 4.75 cm in the static and kinematic modes, respectively, while the convergence time was 21.64 min and 39.61 min. Compared with the triple-frequency GPS PPP without any correction with IFCB, the static and kinematic 3D positioning accuracy in this work was improved by 27.39% and 17.34%, and the corresponding convergence time was improved by 10.55% and 15.22%, respectively. Furthermore, the delayed IFCB was also used for positioning processing, and it was found that a positioning performance comparable to that of the same day can be obtained. The standard deviation of IFCB for a single satellite was found to be no more than 1 cm, when the IFCB value of a neighboring day was subtracted from the IFCB value of same day, which proves the short-term stability of IFCB.

Toward BDS/Galileo/GPS/QZSS triple-frequency PPP instantaneous integer ambiguity resolutions without atmosphere corrections

Multi-frequency precise point positioning (PPP) has drawn attention along with the modernization of the Global Navigation Satellite Systems. There are now nearly 90 satellites providing multi-frequency signals. This contribution aims to achieve fast convergence of a few seconds for BDS/Galileo/GPS/QZSS integrated triple-frequency PPP with integer ambiguity resolution (IAR) without atmosphere corrections. A unified model of an uncombined and undifferenced manner for PPP-IAR with dual- and triple-frequency observations is presented. The uncalibrated phase delays (UPD) of extra wide-lane (EWL), wide-lane (WL), and N1 ambiguities for triple-frequency PPP are estimated with standard deviations of 0.02, 0.05, and 0.10 cycles achieved, respectively. The PPP-IAR validation based on 20 stations evenly distributed in China is conducted using UPD products generated from a regional network covering a large part of China. The EWL, WL, and N1 ambiguities are sequentially fixed utilizing the least-squares ambiguity decorrelation adjustment (LAMBDA) technique. In terms of convergence time, PPP instantaneous IAR is achievable without using atmosphere corrections, thanks to the contribution of the multi-frequency and multi-constellation observations. This has been proved by performing PPP-IAR restart every 10-min over 2520 times in our case study. For PPP-IAR solutions produced with BDS/Galileo/GPS/QZSS triple-frequency observations with an interval of 1 s, the convergence is fulfilled within 1 s for the horizontal components with an accuracy of better than 5 cm, while 2 s for the vertical component with better than 10 cm accuracy, and both are at 95% confidence level.

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.

Modeling and assessment of multi-frequency GPS/BDS-2/BDS-3 kinematic precise point positioning based on vehicle-borne data

BeiDou Navigation Satellite System (BDS) has got the global service capability after the launch of the 55th BDS-3 satellite on June 23, 2020. Currently, BDS is operated based on both the second generation BDS (BDS-2, with 15 satellites transmitting triple frequency signals) and the third generation BDS (BDS-3, with 30 satellites providing services from five frequency signals). To make full use of BDS-2 + BDS-3 signals (B1I, B2I, B3I, B1C, and B2a), this paper introduces an undifferenced and uncombined Precise Point Positioning (PPP) model by using single/dual/triple frequencies BDS-2/BDS-3 data. Results illustrate that the Root Mean Square (RMS) values of BDS-3 PPP based on new signals are better than BDS-2 (B1I + B3I), but worse than BDS-3 (B1I + B3I). Generally, triple-frequency PPP is close to that of dual-frequency PPP, but it is improved significantly in the north, east and vertical components comparing with single-frequency PPP. Compared to the positioning accuracy of GPS + BDS-2 PPP, those of GPS + BDS-3 PPP are improved obviously, especially in the horizontal components. While using BDS-2 and BDS-3 together, the positioning performance of GPS + BDS-2 + BDS-3 is upgraded significantly in both horizontal and vertical directions. Moreover, the convergence time of BDS-3 (especially B1C + B2a) and BDS-2 + BDS-3 PPP are accelerated visibly comparing with that of BDS-2 PPP.

Estimating the Fractional Cycle Biases for GPS Triple-Frequency Precise Point Positioning with Ambiguity Resolution Based on IGS Ultra-Rapid Predicted Orbits

We investigate the estimation of the fractional cycle biases (FCBs) for GPS triple-frequency uncombined precise point positioning (PPP) with ambiguity resolution (AR) based on the IGS ultra-rapid predicted (IGU) orbits. The impact of the IGU orbit errors on the performance of GPS triple-frequency PPP AR is also assessed. The extra-wide-lane (EWL), wide-lane (WL) and narrow-lane (NL) FCBs are generated with the single difference (SD) between satellites model using the global reference stations based on the IGU orbits. For comparison purposes, the EWL, WL and NL FCBs based on the IGS final precise (IGF) orbits are estimated. Each of the EWL, WL and NL FCBs based on IGF and IGU orbits are converted to the uncombined FCBs to implement the static and kinematic triple-frequency PPP AR. Due to the short wavelengths of NL ambiguities, the IGU orbit errors significantly impact the precision and stability of NL FCBs. An average STD of 0.033 cycles is achieved for the NL FCBs based on IGF orbits, while the value of the NL FCBs based on IGU orbits is 0.133 cycles. In contrast, the EWL and WL FCBs generated based on IGU orbits have comparable precision and stability to those generated based on IGF orbits. The use of IGU orbits results in an increased time-to-first-fix (TTFF) and lower fixing rates compared to the use of IGF orbits. Average TTFFs of 23.3 min (static) and 31.1 min (kinematic) and fixing rates of 98.1% (static) and 97.4% (kinematic) are achieved for the triple-frequency PPP AR based on IGF orbits. The average TTFFs increase to 27.0 min (static) and 37.9 min (kinematic) with fixing rates of 97.0% (static) and 96.3% (kinematic) based on the IGU orbits. The convergence times and positioning accuracy of PPP and PPP AR based on IGU orbits are slightly worse than those based on IGF orbits. Additionally, limited by the number of satellites transmitting three frequency signals, the introduction of the third frequency, L5, has a marginal impact on the performance of PPP and PPP AR. The GPS triple-frequency PPP AR performance is expected to improve with the deployment of new-generation satellites capable of transmitting the L5 signal.

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.

Performance Evaluation of Triple-Frequency GPS/Galileo Techniques for Precise Static and Kinematic Applications.

The objective of this research was to develop new precise point positioning (PPP) processing models using triple-frequency GPS/Galileo observations. Different triple-frequency PPP models were developed including undifferenced, between-satellite single-difference (BSSD) and semi-decoupled PPP models. Additionally, a dual-frequency ionosphere-free undifferenced PPP model was developed. The performance of our developed PPP models was evaluated for both static and kinematic applications. To validate the proposed PPP models for static applications, triple-frequency GPS/Galileo observations spanning three successive days from eight globally distributed reference stations were acquired. Then, the observations were processed using the four static PPP solutions. It is found that the 3D positioning accuracy of the triple-frequency semi-decoupled, BSSD and undifferenced PPP models is enhanced after 10 min by about 50, 41 and 29%, respectively, compared with the dual-frequency undifferenced PPP model. After 20 min of processing, improvements in the 3D positioning accuracy by 40, 31 and 21% are obtained for the triple-frequency semi-decoupled, BSSD and undifferenced PPP models, respectively, with respect to the dual-frequency PPP model. The 3D positioning accuracy is also improved after 60 min, compared with the dual-frequency solution, by 40, 40 and 35% for the triple-frequency semi-decoupled, BSSD and undifferenced PPP solutions, respectively. For kinematic application validation, a vehicle trajectory was carried out. The collected triple-frequency GPS/Galileo observations were processed using the four kinematic PPP solutions. It is shown that the triple-frequency semi-decupled, BSSD and undifferenced PPP solutions enhance the 3D positioning accuracy by 31, 23 and 10%, respectively, in comparison with the dual-frequency undifferenced PPP solutions.

Investigation of real-time kinematic multi-GNSS precise point positioning with the CNES products

The performance of precise point positioning (PPP) is easily affected by the observation environment, satellite number, spatial geometry and the accuracy of the external data products being used. In this study, we formulate the mathematical model of multi-GNSS PPP and assess the performance of dual- and triple-frequency PPP modes using the real-time products from the Centre National d’Etudes Spatiales (CNES). For real-time dual-frequency PPP performance, the results show that, no significant differences can be observed from different combined PPP modes at a low cut-off angle (e.g. 10°). With the increase of the elevation cut-off angle, the multi-GNSS PPP shows a better performance in the convergence speed and positioning accuracy, especially in the early stage of convergence. Of all six PPP modes, the convergence time of GPS/BDS/GLONASS/GALILEO PPP is the shortest one under the elevation cut-off angle of 30°. Compared with GPS-only, the convergence time is reduced by 47.6% when converging to 0.2 m, and by 30.4% when converging to 0.1 m. For the position accuracy, under the elevation cut-off angle of 30°, 80.4% of positioning biases are within 0–10 cm in horizontal direction, and 91.0% within 0–20 cm in vertical direction for the GPS/BDS/GLONASS/GALILEO combined PPP. In addition, the convergence performance of triple-frequency PPP is improved by at least 36.4% in two schemes compared with dual-frequency PPP. The accuracy also be improved, with a mean RMS of about 2.9, 3.2 and 6.0 cm in East, North and Up directions.

GPS\u2009+\u2009Galileo\u2009+\u2009BeiDou precise point positioning with triple-frequency ambiguity resolution

Along with the rapid development of GNSS, not only BeiDou, but also Galileo, and the newly launched GPS satellites can provide signals on three frequencies at present. To fully take advantage of the multi-frequency multi-system GNSS observations on precise point positioning (PPP) technology, this study aims to implement the triple-frequency ambiguity resolution (AR) for GPS, Galileo, and BeiDou-2 combined PPP using the raw observation model. The processing of inter-frequency clock bias (IFCB) estimation and correction in the context of triple-frequency PPP AR has been addressed, with which the triple-frequency uncalibrated phase delay (UPD) estimation is realized for real GPS observations for the first time. In addition, the GPS extra-wide-line UPD quality is significantly improved with the IFCB correction. Because of not being contaminated by the IFCB, the raw UPD estimation method is directly employed for Galileo which currently has 24 satellites in operation. An interesting phenomenon is found that all Galileo satellites except E24 have a zero extra-wide-lane UPD value. With the multi-GNSS observations provided by MGEX covering 15 days, the positioning solutions of GPS + Galileo + BeiDou triple-frequency PPP AR have been conducted and analyzed. The triple-frequency kinematic GNSS PPP AR can achieve an averaged 3D positioning error of 2.2 cm, and an averaged convergence time of 10.8 min. The average convergence time can be reduced by triple-frequency GNSS PPP AR by 15.6% compared with dual-frequency GNSS PPP AR, respectively. However, the additional third frequency has only a marginal contribution to positioning accuracy after convergence.

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

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

Research on empirical correction models of GPS Block IIF and BDS satellite inter-frequency clock bias

Triple-frequency observations will introduce an inter-frequency clock bias (IFCB) between the new frequency and the original dual-frequency observations. It has been verified that satellite IFCB can reach dozens of centimeters and several centimeters for GPS Block IIF satellite and BDS satellite, respectively. The existence of satellite IFCB will significantly affect undifferenced triple-frequency data processing. Based on 4-year data collected from 80 globally distributed stations, the long-term characteristics of IFCB coefficients obtained by using harmonic analysis have been studied. The results demonstrate that the coefficients of IFCB periodic model cannot be well fitted only by using sun elevation angle. Also, coefficients have obvious periodic characteristics and their periods differ among different satellites. Thus, a new linear-plus-periodic model is proposed to fit the long-term coefficients. Then, IFCB empirical correction models for 12 GPS Block IIF satellites and BDS GEO and IGSO satellites are built. In order to validate the correction model, IFCB standard deviation (STD), triple-frequency precise point positioning (PPP) and undifferenced extra-wide-lane (EWL) ambiguity resolution are employed. The results based on more than 4-year observations show that, with correction model applied, the average IFCB STD decreases by about 65.5% and 45.5% for GPS and BDS satellites, respectively. Compared to triple-frequency PPP without IFCB correction, triple-frequency PPP results with IFCB correction show that Up, North and East components accuracy are improved by 12.3%, 16.0% and 13.2%, respectively. Besides, IFCB correction will greatly improve the consistence of EWL fractional cycle bias among different stations and improve the success rate of EWL ambiguity resolution.

Speeding up PPP ambiguity resolution using triple-frequency GPS/BeiDou/Galileo/QZSS data

Precise point positioning (PPP) has been suffering from slow convergences to ambiguity-fixed solutions. It is expected that this situation can be relieved or even resolved using triple-frequency GNSS data. We therefore attempt an approach where uncombined triple-frequency GPS/BeiDou/Galileo/QZSS (Quasi-zenith satellite system) data are injected into PPP, whereas their raw ambiguities are mapped into the extra-wide-lane, wide-lane and narrow-lane combinations for integer-cycle resolution at a single station (i.e., PPP-AR). Once both extra-wide-lane and wide-lane ambiguities are fixed to integers, the resulting unambiguous (extra-)wide-lane carrier-phase can usually outweigh the raw pseudorange to improve the convergence of positions and narrow-lane ambiguities. We used 31 days of triple-frequency multi-GNSS data from 76 stations over the Asia Oceania regions and divided them into hourly pieces for real-time PPP-AR. We found that the positioning accuracy for the first 10 min of epochs could be improved by about 50% from 0.23, 0.18 and 0.43 m to 0.12, 0.08 and 0.27 m for the east, north and up components, respectively, once wide-lane ambiguity fixing was achieved for triple-frequency PPP. Consequently, 48% of PPP solutions could be initialized successfully with narrow-lane ambiguities resolved within 2 min, in contrast to only 26% for dual-frequency PPP. On average, 6 min of epochs were required to achieve triple-frequency PPP-AR, whereas 9 min for its dual-frequency counterpart. Of particular note, the more satellites contribute to triple-frequency PPP-AR, the faster the initializations will be; as a typical example, the mean initialization time declined to 3 min in case of 20–21 satellites. We therefore envision that only a few minutes of epochs will suffice to reliably initialize real-time PPP once all GPS, BeiDou, Galileo and QZSS constellations emitting triple-frequency signals are complete in the near future.

Galileo PPP rapid ambiguity resolution with five-frequency observations

Galileo, transmitting signals on the five frequencies E1, E5a, E5b, E5 and E6, has completed the fundamental constellation with 26 satellites and now can provide the global positioning service independently. Multi-frequency (triple-frequency or above) signals allow a variety of combinations on different frequencies, which has the potential to improve the performance of the precise point positioning (PPP) ambiguity resolution (AR). We developed a multi-frequency PPP AR method to make use of the Galileo five-frequency observations. The stable multi-frequency uncalibrated phase delay (UPD) products of Galileo were estimated first. It is interesting to find that the extra-wide-lane (EWL) UPDs on the E5a/E5b, E5a/E5 and E5/E5b combined frequencies are very close to zero. With the obtained UPD products, the Galileo triple-, quad- and five-frequency PPP AR was conducted. Triple-frequency PPP AR with different frequency combinations can improve the positioning accuracy of 30 min by 36.6–86.8% compared with float solutions, and 2.3–62.5% compared with dual-frequency PPP AR. Among the five types of frequency combinations, the triple-frequency PPP AR on E1/E5/E6 frequencies shows the best positioning performance with the averaged convergence time shortened to 16.9 min. Furthermore, the averaged convergence time is 15.3 min and 15.0 min for quad- and five-frequency PPP AR, respectively. Compared with the time to first fix (TTFF) of 19.9 min for narrow-lane ambiguity resolution with dual-frequency observations, the TTFF is only shortened by about 1 min with multi-frequency observations. It is beneficial that the EWL and wide-lane (WL) ambiguities can be fixed to integers instantaneously, and the decimeter-level positioning accuracy can be achieved within 0.5 min by utilizing triple-/quad-/five-frequency PPP wide-lane AR (WAR). Moreover, the positioning accuracy of the first epoch derived from Galileo five-frequency PPP WAR is (0.112, 0.144, 0.641) m in the east, north and up components, which has an improvement of 2.1–42.0% compared to triple-/quad-frequency PPP WAR.

A comparison of three widely used GPS triple-frequency precise point positioning models

Triple-frequency signals can be transmitted by the GPS Block IIF satellites. With triple-frequency integration, more choices are available for the implementation of precise point positioning (PPP), namely the PPP using L1/L2 and L1/L5 dual-frequency ionospheric-free (IF) combinations (IF-PPP1), the PPP using the triple-frequency IF combination (IF-PPP2), and the PPP with uncombined (UC) measurements (UC-PPP). A comprehensive comparison of the three GPS triple-frequency PPP models is carried out in this study. A total of 197 3-h datasets with pure triple-frequency observables are used for analysis in the static situation. The mathematical equivalence among the three PPP models in the converged stage is proven. Only an accuracy difference of smaller than 6 mm exists among the three triple-frequency PPP models. For all the three models, the positioning accuracies are 37.2–42.9, 11.6–13.0 and 36.2–40.6 mm in east, north and vertical directions, respectively. As to the average convergence time, IF-PPP2 shows the best performance, which is 29.7, 11.0 and 29.9 min in the three directions, respectively. The improvement for IF-PPP2 on the average convergence time is 6%, 3% and 1% over IF-PPP1, and 17%, 13% and 20% over UC-PPP in the three directions, respectively. These numerical differences may be attributed to the different parameterization and noise levels for the three triple-frequency PPP models.

Modeling of multi-sensor tightly aided BDS triple-frequency precise point positioning and initial assessments

The BeiDou global navigation satellite system (BDS), which is the first satellite navigation system that provides triple-frequency signals on B1, B2, and B3 for civil applications, has been applied widely around the Asian-Pacifica region. The current BDS precise point positioning (PPP) approaches are mainly based on the B1&B2 dual-frequency observations. To make full use of BDS’ triple-frequency observations, the motion sensors measurements, and the platform motion information, this paper proposes an inertial sensor, odometer, and heading measurement tightly aided BDS triple-frequency PPP model. In this model, inertial sensor biases, odometer scale factor, residuals of slant ionospheric delays, and inter-frequency code biases are estimated simultaneously in a unique extended Kalman filter. The Rauch-Tung-Striebel (RTS) smoother is further adopted to reduce the solution's noises and enhance the stability and relative measuring accuracy. To evaluate the capability of this method, a set of triple-frequency BDS raw observations, inertial measurements, odometer data, and heading measurements collected by a customized hardware system on Lanzhou-Urumqi high speed railway track in China, are processed and analyzed. Results illustrated that both positioning accuracy and cycle slip detection capability are upgraded significantly by applying B3 frequency observations in BDS PPP. About 13–55% position accuracy enhancements from B3 observations, inertial sensors, and RTS smoother, and over 70% heading improvements from the aids of heading measurements can be obtained. Moreover, such multi-sensor tight integration system can directly provide millimeter-level positioning accuracy in term of repeatability and provide sub-millimeter-level accuracy indirectly by transforming attitude solutions into distance solutions. Such accuracy is much higher than the state-of-art GNSS and such method presents potential capability in 3D geometry measuring.

Triple-frequency PPP ambiguity resolution with multi-constellation GNSS: BDS and Galileo

Multi-constellation GNSS (multi-GNSS) and multi-frequency signals open new prospects for fast ambiguity resolution (AR) of precise point positioning (PPP). Currently, all the BDS and Galileo satellites are capable of transmitting signals on three or more frequencies. In this contribution, we investigate the triple-frequency PPP ambiguity resolution with B1, B2 and B3 observations from BDS satellites and E1, E5a and E5b observations from Galileo satellites and evaluate the contribution of BDS + Galileo combination to triple-frequency PPP AR. The uncalibrated phase delay (UPD) products are estimated based on triple-frequency observations, and the temporal characteristic as well as the residual distributions are analyzed. Our results show that the extra-wide-lane (EWL) and wide-lane (WL) UPDs for BDS and Galileo satellites are both stable during the 30 days and the daily narrow-lane (NL) UPD series are also steady with no obvious fluctuation. The Galileo UPDs exhibit better performance than BDS UPDs due to the high-quality observations. It is also interesting to find that the EWL UPD corrections for all Galileo satellites are very close to the zero. With the precise UPD products, the triple-frequency PPP AR with BDS and Galileo observations was implemented in both static and kinematic modes. Compared to the ambiguity-float solution, the performance can be significantly improved by triple-frequency PPP AR with the positioning accuracy improved by 30–70% in both static and kinematic modes. Moreover, the triple-frequency PPP fixed solutions also present better performance than the dual-frequency PPP fixed solutions in terms of time to the first fix and positioning accuracy, especially for the Galileo-only and BDS + Galileo solutions. And the fusion of multi-GNSS (BDS and Galileo) can further improve the position estimations compared to the single system with more satellites and better spatial geometry.

Improving GNSS PPP Convergence: The Case of Atmospheric-Constrained, Multi-GNSS PPP-AR.

GNSS positioning performance has been shown to improve with the ingestion of data from Global Ionospheric Maps (GIMs) and tropospheric zenith path delays, which are produced by, e.g., the International GNSS Service (IGS). For both dual- and triple-frequency Precise Point Positioning (PPP) processing, the significance of GIM and tropospheric products in processing is not obvious in the quality of the solution after a few hours. However, constraining the atmosphere improves PPP initialization and solution convergence in the first few minutes of processing. The general research question to be answered is whether there is any significant benefit in constraining the atmosphere in multi-frequency PPP? A key related question is: regarding time and position accuracy, how close are we to RTK performance in the age of multi-GNSS PPP-AR? To address these questions, this paper provides insight into the conceptual analyses of atmospheric GNSS PPP constraints. Dual- and triple-frequency scenarios were investigated. Over 60% improvement in convergence time was observed when atmospheric constraints are applied to a dual-frequency multi-GNSS PPP-AR solution. Future work would involve employing the constraints to improve low-cost PPP solutions.

A Unified Model for Multi-Frequency PPP Ambiguity Resolution and Test Results with Galileo and BeiDou Triple-Frequency Observations

With the modernization of Global Navigation Satellite System (GNSS), triple- or multi-frequency signals have become available from more and more GNSS satellites. The additional signals are expected to enhance the performance of precise point positioning (PPP) with ambiguity resolution (AR). To deal with the additional signals, we propose a unified modeling strategy for multi-frequency PPP AR based on raw uncombined observations. Based on the unified model, the fractional cycle biases (FCBs) generated from multi-frequency observations can be flexibly used, such as for dual- or triple- frequency PPP AR. Its efficiency is verified with Galileo and BeiDou triple-frequency observations collected from globally distributed MGEX stations. The estimated FCB are assessed with respect to residual distributions and standard deviations. The obtained results indicate good consistency between the input float ambiguities and the generated FCBs. To assess the performance of the triple-frequency PPP AR, 11 days of MGEX data are processed in three-hour sessions. The positional biases in the ambiguity-fixed solutions are significantly reduced compared with the float solutions. The improvements are 49.2%, 38.3%, and 29.6%, respectively, in east/north/up components for positioning with BDS, while the corresponding improvements are 60.0%, 29.0%, and 21.1% for positioning with Galileo. These results confirm the efficiency of the proposed approach, and that the triple-frequency PPP AR can bring an obvious benefit to the ambiguity-float PPP solution.