Wide-lane Ambiguity Research Articles

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.

Beyond three frequencies: an extendable model for single-epoch decimeter-level point positioning by exploiting Galileo and BeiDou-3 signals

GNSS is indispensable to self-driving vehicles by delivering decimeter-level or better absolute positioning solutions. Such a high precision normally requires a convergence time spanning seconds to minutes, which is, however, unrealistic in extremely difficult driving conditions where GNSS signals are obstructed frequently. Such convergences, no matter how short, will greatly risk and discredit autonomous driving in satisfying stringent life safety standards. In this study, we therefore developed an extendable GNSS precise point positioning (PPP) model to exploit the advanced Galileo/BeiDou-3 more-than-three-frequency signals with the goal of achieving instant or single-epoch 10–30 cm positioning accuracy and over 99% availability for the horizontal components over wide areas. In particular, uncombined Galileo/BeiDou-3 signals on all available frequencies were injected simultaneously into PPP to perform single-epoch wide-lane ambiguity resolution (PPP-WAR) after phase bias calibrations on raw observations. Experimenting on the Galileo five-frequency data from 36 stations in Australia, we found that instant PPP-WAR was accomplished at more than 99.5% of all epochs; we achieved an instant positioning accuracy of 0.10 and 0.11 m (1σ) for the east and north components, respectively, using Galileo E1/E5a/E5/E5b/E6 signals from less than 10 satellites, while 0.16 and 0.23 m using BeiDou-3 B1C/B1I/B2a/B2b/B3I signals from only 5–6 satellites per epoch observed by 10 stations within China. Moreover, we carried out vehicle-borne experiments collecting multi-frequency Galileo/BeiDou-3 signals in the case of overpass and tunnel adversities. With 7 Galileo/BeiDou-3 satellites per epoch on average, instant PPP-WAR reached a mean positioning accuracy of 0.23 and 0.24 m for the horizontal components, which can be further improved to 0.14 and 0.12 m when multi-epoch filtering is preferably enabled. More encouragingly, though this positioning accuracy can also be ensured with triple-frequency data, the data redundancy favored by even more frequencies can reduce the high-precision recovery time from up to 4 s to 2 s in the case of total signal blockages. With the rapidly ongoing deployment of Galileo, BeiDou-3 and other GNSS constellations, we can envision an instant global positioning service characterized by around 20 cm horizontal accuracy and over 99% availability for self-driving vehicles.

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.

Analysis of BDS Fractional Cycle Biases and PPP Ambiguity Resolution.

It is difficult to enable traditional precise point positioning (PPP) with ambiguity resolution (AR) due to fractional cycle biases (FCBs), which limit the accuracy and reliability of positioning results. The BeiDou Navigation Satellite System (BDS) has been providing continuous positioning, navigation, and timing (PNT) services in the global region since the end of 2018. The BDS constellation includes geostationary earth orbit (GEO), inclined geostationary orbit (IGSO), and medium earth orbit (MEO) satellites. However, its hybrid constellation structure and the satellite-side multipath effect have hindered the BDS PPP AR for two main reasons: (1) some receivers have half-cycle biases between GEO and non-GEO satellites, which result in the inconsistency of hardware delays for each satellite type; (2) the correction model for elevation-dependent satellite-side multipath effect is only applicable to IGSO and MEO, while in the case of GEO the effect cannot be effectively weakened or eliminated. To rectify these problems, a suitable strategy is proposed for estimating BDS FCBs, whereby the GEO FCBs and non-GEO FCBs are estimated independently. Results show that the FCBs estimated by the new strategy for GEO and non-GEO are more stable compared to the traditional strategy. The GEO wide-lane (WL) FCBs fluctuate less than 0.3 cycle in one month, except for C05, while the variation of non-GEO WL FCBs is about 0.1 cycle. In addition, compared to the traditional strategy, the fractions of GEO WL ambiguities after the removal of FCBs estimated by the new strategy can be improved noticeably from 53.5% to 78.5%, and from 71.8% to 92.3% for <0.15 cycle and <0.25 cycle respectively, which could be comparable with non-GEO. Simultaneously, the improvement of GEO narrow-lane (NL) ambiguities is from 28.9% to 40.2%, and from 40.4% to 53.3% for <0.10 cycle and <0.15 cycle respectively, are less noticeable. This is mainly due to the low precision IGS products for GEO. After PPP AR, the mean convergence time is shorted from 56.0 min to 43.6 min, and from 71.6 min to 62.7 min for static PPP and kinematic PPP, 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%.

FCB estimation with three different PPP models: equivalence analysis and experiment tests

Precise and reliable fractional cycle bias (FCB) products are critical for ambiguity resolution of precise point positioning (PPP). Three PPP models are usually adopted to generate the FCB products, i.e., the traditional ionospheric-free combined PPP (IF-PPP) model, the uncombined and unconstrained PPP model (UU-PPP) as well as the ionospheric-constrained PPP (IC-PPP) model. Considering that different observation models and ionospheric delay constraints are used, the applicability and interoperability of the obtained FCB products are assessed. We presented the equivalent conversion formulas of different FCB products obtained with different PPP models, which are then converted and compared. The root mean square (RMS) of the wide-lane (WL) FCB differences for IF-UU, IF-IC and UU-IC is 0.021, 0.024 and 0.010 cycles, while the RMS of the narrow-lane FCB differences is 0.028, 0.018 and 0.021 cycles. The precision of the WL FCBs based on the uncombined PPP models is higher than that based on the IF-PPP model since the new WL ambiguities derived from the uncombined ambiguities are free of the pseudorange noise. The equivalence of the FCB products estimated from the three different PPP models is confirmed in theory and by experiment results. To further evaluate the performance of the three PPP models, the positioning accuracy, the convergence time and the ambiguity fixing success rate are calculated using IGS data. The maximum positioning difference is less than 0.9 mm among the three PPP models. Compared to the float solutions of the IF-PPP, UU-PPP, IC-PPP with GIM model or with re-injected ionospheric delay corrections, the convergence time is shortened by 38.5%, 46.2%, 50.0% and 87.8% and the positioning accuracy improved by 27.1%, 38.9%, 41.1% and 25.7%, respectively. The ambiguity fixing success rate of the uncombined PPP models is slightly higher than that of the combined model, and fast ambiguity-fixed solution can proceed with high-precise ionospheric delay corrections.

An assessment of wide-lane ambiguity resolution methods for multi-frequency multi-GNSS precise point positioning

We assess the time-to-first-fix (TTFF) and the ambiguity fixing rate of two PPP wide-lane ambiguity resolution (WL-AR) methods, namely the geometry-based and ionospheric-free (GB-IF) method, and the geometry-free and ionospheric-free (GF-IF) method. First, an optimal GF-IF WL linear combination is selected based on the ratio between the code and carrier phase measurement noise (). Then, the relation between ambiguity variance and satellite geometry in the GB-IF WL-AR is investigated. Both simulated and real data from 31 GNSS stations over 37 consecutive days in 2017 were used. Numerical results show that the GF-IF WL-AR method has shorter TTFF and higher ambiguity fixing rate compared to the GB-IF method when . However, when , the GB-IF method outperforms the GF-IF method. Depending on RT values used, 2–10 min would be required to resolve the WL ambiguities when using GNSS measurements with one second sampling rate.

Improving real-time clock estimation with undifferenced ambiguity fixing

Real-time clock products are essential and a prerequisite for global navigation satellite system (GNSS) real-time applications, which are usually estimated from real-time observations from reference stations of a ground network without fixing the phase ambiguities. To improve the precision of real-time clock products further, an undifferenced ambiguity fixing algorithm is proposed for the classical real-time clock estimation method without requiring modifications to the current data processing strategy and product consistency. Combined with the wide-lane (WL) ambiguities, the ionospheric-free (IF) ambiguity estimates generated by the traditional clock estimation method are fed to an independent ambiguity fixing process to estimate the WL and narrow-lane (NL) uncalibrated phase delays (UPDs) and fix the WL/NL undifferenced ambiguities at each epoch. The fixed IF ambiguities are recovered from these WL/NL UPDs and integer ambiguities and then used to constrain the float-ambiguity clock solution. The proposed strategy is tested on 30 days of observations of 85 globally distributed reference stations of the international GNSS service (IGS) and multi-GNSS experiment (MGEX) networks. In the experiment, 99.96% of the post-fit residuals of WL ambiguities and 99.31% of the post-fit residuals of NL ambiguities fall in the range of (??0.3, 0.3), with standard deviations (STD) of 0.045 and 0.074 cycles, respectively, which demonstrates the high precision and consistency of UPDs and fixed ambiguities. With the constraint of fixed ambiguities, the traditional float-ambiguity clock solution is further refined, resulting in ambiguity-fixed clock solution. Comparison with the IGS 30?s final clock products shows that ambiguity-fixing brings as much as 50---87% precision improvement to the float-ambiguity clock solution, and with an average improvement over 30?days of 24---50% for each satellite. When used in the float-ambiguity kinematic PPP test, the ambiguity-fixed clock brings at least 5% improvement to the north component and at least 10% improvement to the east and vertical components.

Regional BDS satellite clock estimation with triple-frequency ambiguity resolution based on undifferenced observation

Real-time precise satellite clock estimation is fundamental for real-time precise point positioning. To take full advantage of the triple-frequency (TF) signal of the BeiDou Navigation Satellite System (BDS), we used undifferenced observations directly in the TF data processing. In addition, TF ambiguity resolution (AR) was performed in a sequential manner, in which the wide-lane (WL) and extra wide-lane (EWL) ambiguities were fixed based on the Hatch---Melbourne---W?bbena (HMW) combination of B1/B2 and B1/B3, and then we followed with narrow-lane (NL) ambiguity fixing. By applying the ambiguity integer constraints, an ambiguity-fixed BDS satellite clock was obtained. To evaluate the contribution of TF AR in a satellite clock solution when a regional station network is involved, experiments were carried out based on the observations collected by 11 stations from the crustal movement observation network of China (CMONOC) during day of year (DOY) 153---188, 2016. First, the temporal behavior of the BDS EWL and WL uncalibrated phase delays (UPDs) were analyzed. The results suggested that they were quite stable over several days with standard deviations (STDs) of 0.004 cycles and 0.015 cycles for EWL and WL, respectively. Then, the ambiguity-fixed solution of the satellite clock was obtained with average success rates of about 99%, 90%, and 63% for EWL, WL, and NL AR, respectively. Attributed to the efficient AR, not only was the initialization time shortened but also the precision was improved. The overall averaged improvements of STD for geostationary orbits (GEO), inclined geostationary orbits (IGSO), and medium-altitude Earth orbit (MEO) satellites were 48.7%, 40.8%, and 34.4%, respectively, for the EWL and WL ambiguity-fixed solution and 72.6%, 71.0%, and 62.8%, respectively, for the NL ambiguity-fixed solution. As a comparison, a dual-frequency (DF) ambiguity-fixed satellite clock was also generated. Although the precision of the NL ambiguity-fixed clock was roughly the same, a larger root mean square (RMS) by a factor of 52% suggested that the DF ambiguity-fixed solution may have been biased with incorrect integer ambiguity fixing, whereas the TF ambiguity-fixed solution was more reliable because of the third frequency observation. In addition, precise point positioning (PPP) was carried out to assess the performance of these clock products. Compared with the PPP based on the ambiguity-float clock solution, improvements of 32.0% and 42.9% for horizontal and vertical, respectively, were obtained with the TF NL ambiguity-fixed clock solution.

A Simulation Study on Triple Frequency Ambiguity Resolution for Reference Stations Using Different Strategy Regarding Elevation angles

This paper proposes an ambiguity resolution method using triple frequency for reference stations. Using the reference coordinate information, geometry based ambiguity resolution performance is analysed. Although orbit errors and tropospheric model errors still remain, wide lane ambiguity could be fixed in several epochs. However, the narrow lane wave length of about 10cm is too short to overcome error sources by simply combining the measurement. Therefore, we have divided the elevation angle into 5 degree intervals and investigated the measurement errors and the time to fix of each section. For high elevation satellites, it is possible to determine in several epochs by integer rounding. On the other hand, if the elevation is lower than 30 degrees, the tropospheric zenith delay must be estimated with ambiguities. The proposed algorithm estimates ambiguities and tropospheric zenith delay simultaneously utilizing ambiguity free observations of high elevation satellites. Ambiguities for high elevation satellites are resolved by integer rounding in several epochs. The algorithm has been verified by generating the simulated observation data for the ‘Cheon-an’ and ‘Boen’ reference stations in the Korea.

Toward global instantaneous decimeter-level positioning using tightly coupled multi-constellation and multi-frequency GNSS

Autonomous driving represents one of the emerging applications that require both high-precision positions and highly critical timeliness to reach stringent safety standards. We develop a method to potentially achieve global instantaneous decimeter-level positioning by virtue of tightly coupled multi-GNSS triple-frequency observations contributing to precise point positioning (PPP). Inter-system phase biases (ISPBs) for two wide-lane observables are first computed for each station to form inter-GNSS resolvable ambiguities, and then correspondingly two wide-lane fractional-cycle biases (FCBs) are computed for each satellite to recover the integer property of single-station ambiguities. With both ISPB and FCB products, we can accomplish tightly coupled multi-GNSS PPP wide-lane ambiguity resolution (PPP-WAR) using only a single epoch of triple-frequency observations on a global scale. To verify this method, we used 1 month of GPS/BeiDou/Galileo/QZSS data from 107 globally distributed stations and 1 h of such multi-GNSS data collected on a vehicle moving in an urban area. We found that both ISPB and FCB products could be estimated every 24 h with high precisions of around or below 0.1 cycles; 83–98% of their day-to-day variations fell within 0.1 cycles, facilitating their precise predictions for real-time applications. Using these corrections, we achieved both instantly and reliably ambiguity-fixed solutions at 91.2% of all epochs at the 107 stations on average; the resultant single-epoch positions reached a mean accuracy of 0.22 m, 0.18 m and 0.63 m for the east, north and up components, respectively, in case of abundant triple-frequency observations from over 15 satellites. Similarly, for the vehicle-borne test, we obtained instantaneous PPP-WAR solutions at 99.31% of all epochs and achieved a positioning accuracy of 0.29, 0.35 and 0.77 m for the east, north and up components, respectively, which improved significantly the identification of road lanes as opposed to other single-epoch solutions. Finally, we expect that the prospect of instantaneous PPP-WAR in aiding driverless vehicles can be more promising if, whenever possible, integrated with inertial sensors and/or smoothed through multi-epoch data.

Characteristics of BeiDou Navigation Satellite System (BDS) Code Observations for Different Receiver Types and Their Influence on Wide-Lane Ambiguity Resolution.

The Chinese BeiDou Navigation Satellite System (BDS) has been an important constitute of the Global Navigation Satellite System (GNSS), and the combination of GPS and BDS shows significant improvements when compared with single GPS system for real-time kinematic (RTK) positioning, and improves on availability and fixing rates, especially in the East Asian area. While network RTK might have different types of receivers, both for global and regional networks, different types of receiver may adopt different internal multipath mitigation methods and other techniques that result in different pseudorange characteristics, especially for a multipath. Then, the performance of wide-lane ambiguity resolution (WL AR) is affected. In this study, we first analyze and compare the characteristics of BDS dual-frequency observations for different types of receivers, including Trimble, Leica, Javad, and Septentrio, based on multipath (MP) observables, and then we assess their influence on double-differenced (DD) WL AR. The numerical results show that an obvious low-frequency component exists in MP observables of BDS geostationary earth-orbit satellites (GEOs) for Leica receivers, while its high-frequency measurement noise is very small. For geosynchronous orbit satellites (IGSOs) and medium earth-orbit satellites (MEOs), a slight fluctuation can also be observed that is similar to that of GPS satellites, except for the satellite-included code bias. In Trimble, Javad, and Septentrio receivers, the MP series are dominated by high-frequency measurement noise, both for GEOs and non-GEOs, except for satellite-included code bias. Furthermore, the characteristic of Leica receivers for BDS GEOs seriously affects WL AR and, even for a short baseline, it takes a long time for WL ambiguities to converge, or not converge for many GEO-related DD WL ambiguities, while Trimble, Javad, and Septentrio receivers perform well for short and medium baselines. Then, a time-difference method is proposed to mitigate the multipath of BDS GEOs for a Leica receiver. After applying the proposed method, WL ambiguity fixing rates of GEO-related satellite pairs are improved significantly and the convergence time is shortened from several hours to ten minutes.

Improved Method for GLONASS Long Baseline Ambiguity Resolution without Inter-Frequency Code Bias Calibration

Use of a frequency-division multiple access strategy causes Globalnaya Navigatsionnaya Sputnikovaya Sistema (GLONASS) receiving equipment to experience both inter-frequency phase bias (IFPB) and inter-frequency code bias (IFCB). While IFPB can be calibrated using a linear model, there is no general model for IFCB calibration, which causes great difficulty in GLONASS ambiguity resolution over long baselines; most current GLONASS ambiguity resolution research is confined to short baselines. In this paper, based on a single-differencing between-receivers (SDBR) model, a wide-lane phase combination-based approach is proposed to fix the GLONASS ambiguities over long baselines. External precise ionospheric products are introduced to eliminate the ionospheric delay. To mitigate the effect of the residual ionospheric delays, we fix the relative wide-lane ambiguity using the Hatch–Melbourne–Wubbena (HMW) combination. The results show that 96% and 55% of the wide-lane round-off residuals are within 0.2 cycles for the Global Positioning System (GPS) and GLONASS, respectively, if the traditional HMW method is used. The method proposed here for GLONASS can improve these percentages significantly, reaching up to 95.5%. The root mean square (RMS) position errors are 1.43, 1.06 and 4.32 mm for GPS in the north, east and up directions, respectively. When GLONASS with ambiguity fixing is added, the corresponding RMS values are reduced significantly to 1.26, 1.02 and 3.87 mm, respectively.

GLONASS real-time wide-lane ambiguity resolution with an enhanced geometry-based model for medium-range baselines

Double-difference wide-lane integer ambiguities are determined first to assist narrow-lane ambiguity resolution in real-time medium-range baseline resolution. The Hatch-Melbourne-Wübbena (HMW) is the conventional strategy for wide-lane ambiguity resolution, which works well for GPS/Galileo/BDS. However, the un-canceled inter frequency bias (IFB) on double-difference measurement causes that the HMW combination is invalid on GLONASS wide-lane ambiguity resolution. The residual IFB in double-difference measurement may be several meters, especially between inhomogeneous stations. There is no effective method to model or tabulate ranging-codes IFB. In this paper, we propose a device-independent geometry-based model to achieve GLONASS real-time wide-lane ambiguity resolution for medium-range baselines. Since zenith tropospheric delay and slant ionospheric delay of satellites are estimated as unknown parameters, we utilize atmosphere-weighted algorithm to enhance the model strength and decrease the dependence on pseudorange measurement. For medium-range baseline, GLONASS wide-lane ambiguity float solutions can converge quickly by the geometry-based model. The cumulative frequency of WL AR can exceed 99% during 10 epochs. Benefiting from rapid and reliable wide-lane ambiguity resolution, the positioning accuracy of GPS/GLONASS RTK fixed solution are greater than 3 cm in level direction and 5 cm in upward direction for 40–100 km baselines. An approximately 35% improvement rate is observed in each direction compared with that of single GPS RTK. Since less available satellites for single-GLONASS mode, the convergence of wide-lane ambiguity becomes slower. For base stations with known coordinates, the enhanced geometry-based model can be developed to the enhanced geometry-fixed model and are applied to longer baselines. Assisting with the geometry-fixed model, more than 90% of GLONASS narrow-lane ambiguities can be fixed quickly for 80–160 km baselines. It should be noted that the accuracy of external atmospheric delay is critical to the performance of the geometry-based method. Its benefit may weaken with the increasing of baseline length, when external atmospheric delays are set as 0 directly.

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.

Analysis of Wide-Lane Ambiguities Derived from Geometry-Free and Geometry-Based Precise Point Positioning Models and Their Implication for Orbit and Clock Quality.

Orbit and clock products are used in real-time global navigation satellite systems (GNSS) precise point positioning (PPP) without knowing their quality. This study develops a new approach to detect orbit and clock errors through comparing geometry-free and geometry-based wide-lane ambiguities in a PPP model. The reparameterization and estimation procedures of the geometry-free and geometry-based ambiguities are described in detail. The effects of orbit and clock errors on ambiguities are given in analytical expressions. The numerical similarity and differences of geometry-free and geometry-based wide-lane ambiguities are analyzed using different orbit and clock products. Furthermore, two types of typical errors in orbit and clock are simulated and their effects on wide-lane ambiguities are numerically produced and analyzed. The contribution discloses that the geometry-free and geometry-based wide-lane ambiguities are equivalent in terms of their formal errors. Although they are very close in terms of their estimates when the used orbit and clock for geometry-based ambiguities are precise enough, they are not the same, in particular, in the case that the used orbit and clock, as a combination, contain significant errors. It is discovered that the discrepancies of geometry-free and geometry-based wide-lane ambiguities coincide with the actual time-variant errors in the used orbit and clock at the line-of-sight direction. This provides a quality index for real-time users to detect the errors in real-time orbit and clock products, which potentially improves the accuracy of positioning.

A dynamic single epoch ambiguity resolution algorithm with straight trajectory constraints

The key problem of dynamic single epoch GPS positioning is the rapid decomposition of ambiguity. In some special dynamic positioning, the coordinates of the GPS receiver placed on the rover station often have the constraint condition, if this useful information is taken into account, it will help to determine the ambiguity accurately. For this reason, the straight motion trajectory of GPS receiver is used as the constraint condition for single epoch ambiguity resolution, in this algorithm, the wide lane ambiguities are calculated by using the MW combination, and the corresponding alternative combinations of wide lane ambiguity are established, which are determined according to the minimum residual sum of squares criterion firstly, it is used to lay the foundation for the subsequent L1 and L2 frequency ambiguity solution. And then the straight trajectory constraint condition and the phase observation equation are used to solve the integer ambiguity N1. When the constraint information of the trajectory is unknown, the space linear equation is established as the constraint condition according to the coordinate solution by wide lane relative positioning to solve the unknown problem. By designing the corresponding experiments, the results of this algorithm are proved to be feasible by using different experimental schemes, such as without constraint condition, the constraint condition with known prior information, introducing dynamic constraint condition and so on, to solve the ambiguity, and to compare and analyze the effect of linear constraint condition on single epoch ambiguity resolution.

Network-based triple-frequency carrier phase ambiguity resolution between reference stations using BDS data for long baselines

Network-based ambiguity resolution (AR) between reference stations is the prerequisite to realize a precise real-time kinematic positioning service. With the help of BDS triple-frequency signals, we can efficiently deal with the ionospheric delay and tropospheric delay, and achieve rapid and reliable AR. To overcome the inaccurate ionospheric delay estimated by the geometry-free three carrier ambiguity resolution (GF TCAR) technique, which leads to failure in the original ambiguity resolution, we propose an ionospheric-free (IF) TCAR method to resolve the ambiguity between the reference stations over long baselines. Taking full advantage of the known positions of the reference stations, the easily resolved extra-wide-lane (EWL) ambiguity, and the IF phase combinations, we can reliably fix the wide-lane (WL) ambiguity. A Kalman filter is applied to estimate precise IF ambiguities and the original ambiguity is resolved with the fixed WL ambiguity. A numerical analysis with triple-frequency BDS data from three long baselines of a CORS network is provided to compare the AR performance of GF TCAR with that of IF TCAR. The results show that both methods can reliably resolve the WL ambiguity with a remarkable correctly-fixed rate of higher than 99%, and the reliably-fixed rates of the IF TCAR slightly increase from 92.19, 94.67 and 94.61–98.26, 99.54 and 97.51% for the three baselines. Herein “correctly-fixed” and “reliably-fixed” mean the difference between the float ambiguity and the true one are less than ± 0.5 and ± 0.25 cycles, respectively. On the other hand, the AR performance of the original signals with the IF TCAR method is much better than that with the GF TCAR method attaining a 100% correctly-fixed rate, while the GF TCAR method can hardly fix the original ambiguity with the largest bias being as much as 4 cycles because of the amplified systematic bias.

An Improved BeiDou-2 Satellite-Induced Code Bias Estimation Method.

Different from GPS, GLONASS, GALILEO and BeiDou-3, it is confirmed that the code multipath bias (CMB), which originate from the satellite end and can be over 1 m, are commonly found in the code observations of BeiDou-2 (BDS) IGSO and MEO satellites. In order to mitigate their adverse effects on absolute precise applications which use the code measurements, we propose in this paper an improved correction model to estimate the CMB. Different from the traditional model which considering the correction values are orbit-type dependent (estimating two sets of values for IGSO and MEO, respectively) and modeling the CMB as a piecewise linear function with a elevation node separation of 10°, we estimate the corrections for each BDS IGSO + MEO satellite on one hand, and a denser elevation node separation of 5° is used to model the CMB variations on the other hand. Currently, the institutions such as IGS-MGEX operate over 120 stations which providing the daily BDS observations. These large amounts of data provide adequate support to refine the CMB estimation satellite by satellite in our improved model. One month BDS observations from MGEX are used for assessing the performance of the improved CMB model by means of precise point positioning (PPP). Experimental results show that for the satellites on the same orbit type, obvious differences can be found in the CMB at the same node and frequency. Results show that the new correction model can improve the wide-lane (WL) ambiguity usage rate for WL fractional cycle bias estimation, shorten the WL and narrow-lane (NL) time to first fix (TTFF) in PPP ambiguity resolution (AR) as well as improve the PPP positioning accuracy. With our improved correction model, the usage of WL ambiguity is increased from 94.1% to 96.0%, the WL and NL TTFF of PPP AR is shorten from 10.6 to 9.3 min, 67.9 to 63.3 min, respectively, compared with the traditional correction model. In addition, both the traditional and improved CMB model have a better performance in these aspects compared with the model which does not account for the CMB correction.

A new single-epoch method for resolving GNSS ambiguity with segmented search in short baseline

Correct ambiguity resolution (AR) is the key for high-precision positioning. In order to reliably resolve ambiguity, an improved single-epoch segmented ambiguity algorithm is proposed in this paper. First, the double-difference wide-lane ambiguities resolved by the Melbourne-Wubbena (MW) method are divided into two categories based on signal-to-noise ratio (SNR). One is the primary ambiguity group that can be resolved easily, and the other is the secondary ambiguity group that is difficult to resolve. Second, the primary ambiguity group is divided into different levels in a (4,1,1…1) pattern based on SNR, and the different levels’ ambiguities are resolved successively, and the method being that the integer ambiguity is selected from two candidate values built by rounding up and down. Third, the resolved primary ambiguity group is used as a constraint to solve the secondary ambiguity group. Finally, the wide-lane observation equation and the basic observation equation are combined to solve the ambiguity of the L1 carrier phase, and the resolved ambiguity can be obtained by directly rounding. The candidate values of the primary ambiguity group is limited to two cycles through this step, and the (4,1,1…1) pattern can reduce ambiguity search times. Since an appropriate selection enables the correct resolution of primary wide-lane ambiguities, the selection of the threshold for SNR is the key to group wide-lane ambiguity, and we use the data from Novatel receiver as an example to discuss how to confirm the SNR threshold in this paper. Static short baseline data and kinematic short baseline that were less than 20 km were collected to validate the proposed algorithm. These results were compared with the commercial software IE8.60. These experiments show that the successful rate reaches 100%, 99.78% and 99.42%, respectively, for the new single-epoch segmented ambiguity method in the three data sets, and the accuracy is similar to that of IE8.60 for the combined double difference.