Narrow-lane Ambiguity Resolution Research Articles

Research on single-epoch ambiguity resolution method based on BDS-3 ionosphere-reduced combinations

In conventional ambiguity resolution method for single-epoch, narrow-lane (NL) observations are often inaccurately fixed due to improper treatment of error sources such as ionospheric and tropospheric delays. Therefore, based on the multi-frequency observation data of the BeiDou Global Navigation Satellite System (BDS-3), this paper adopts a single-epoch resolution method based on the ionosphere-reduced combinations. Firstly, select the appropriate combination observations of extra wide-lane (EWL) and wide-lane (WL) and get its fixed ambiguity. Then the ambiguity of the WL combination with high observation accuracy is obtained by linear combination, which goes to constrain the fixation of NL ambiguity. Secondly, different from the conventional method, the ionosphere-reduced combinations observations were selected as NL, thus significantly weakening the influence of first-order ionospheric delay error on the fixation of narrow-lane ambiguity. The experimental results show that compared with the conventional methods, the method adopted in this paper can significantly improve the fixing success rate of the wide-lane ambiguity. In the NL ambiguity resolution part, using the ionosphere-reduced combinations as NL for solving improved the average fixing success rate from 68.6%/44.4%/35.7%/27.6% to 97.4%/95.4%/95.6%/87.3% compared to the conventional method of using the base frequency as NL for solving. Furthermore, the N/E/U positioning accuracy improved by an average of 63.05%/64.37%/52.27%. To sum up, this method effectively improves the success rate of NL ambiguity fixing and positioning accuracy.

Global single-epoch narrow-lane ambiguity resolution with multi-constellation and multi-frequency precise point positioning

Global single-epoch narrow-lane ambiguity resolution with multi-constellation and multi-frequency precise point positioning

Assessment of Multi-Frequency PPP Ambiguity Resolution Using Galileo and BeiDou-3 Signals

From network RTK to PPP-RTK, it is highly expected that high-precision positioning within a few minutes can be achieved with a sparse reference network. In this study, we investigate a rapid multi-frequency PPP convergence strategy based on Galileo E1/E5a/E6 and BeiDou-3 B1C/B2a/B3I signals, whose unambiguous wide-lane observables can efficiently assist in speeding up narrow-lane ambiguity resolution. Furthermore, frequency-specific biases existing on the third-frequency observables have been observed to slow down multi-frequency PPP-AR convergence. In this study, we partially mitigated their effects by estimating a second satellite clock for the third frequency of signals. We validated this approach with one month of data collected from 22 stations. On average, it took about 18 min for PPP wide-lane ambiguity resolution (PPP-WAR) to converge, while 32 min were required for ambiguity-float PPP. Compared with dual-frequency PPP-AR, which needed nearly 12 min to converge, multi-frequency PPP-AR required 6 min only. Once there were more than 10 satellites involved in PPP, the convergence could be achieved within 3 min on average. Meanwhile, 81% and 62% of multi-frequency PPP-AR solutions converged successfully within 5 and 1 min, respectively. Finally, we carried out a vehicle-borne experiment to validate this approach in a kinematic environment. Owing to frequent cycle slips during the movement of vehicle, it took 14 min for B1C/B2a/B3I and E1/E5a/E6 PPP-AR to obtain reliable positions, and 19 min for those using the other signal combinations B1C/B2a/B2b and E1/E5a/E5b, owning to higher noise. Overall, these results are promising for achieving high-precision PPP positioning globally within a few minutes if multi-frequency biases can be handled well in the data processing.

Performance of Single-Epoch EWL/WL/NL Ambiguity-Fixed Precise Point Positioning with Regional Atmosphere Modelling

Precise point positioning (PPP) with ambiguity resolution (AR) can improve positioning accuracy and reliability. The narrow-lane (NL) AR solution can reach centimeter-level accuracy but there is a certain initialization time. In contrast, extra-wide-lane (EWL) or wide-lane (WL) ambiguity can be fixed instantaneously. However, due to the limited correction accuracy of the empirical atmospheric model, the positioning accuracy is only a few decimeters. In order to further improve the real-time performance of PPP while ensuring accuracy, we developed a multi-system multi-frequency uncombined PPP single-epoch EWL/WL/NL AR method with regional atmosphere modelling. In the proposed method, the precise atmosphere, including zenith wet-troposphere delay (ZWD) and the slant ionosphere, is extracted through multi-frequency stepwise AR, which then is both interpolated and broadcast to users. By adding regional atmosphere constraints, users can achieve single-epoch PPP AR with centimeter-level accuracy. To verify the algorithm, four sets of reference networks with different inter-station distances are used for experiments. With atmosphere constraints, the accuracy of the single-epoch WL solution can be improved from the decimeter level to a few centimeters, with an improvement of more than 90%, and the epoch fix rate can also be improved to varying degrees, especially for the dual-frequency case. Due to the enlarged noise of the EWL combination, its accuracy is at the decimeter level, while the accuracy of the WL/NL solution can reach several centimeters. However, reliable NL ambiguity-fixing tightly relies on atmosphere constraints with sufficiently high accuracy. When the modelling of the atmosphere correction is not accurate enough, the NL AR performance is degraded, although this situation can be improved to a certain extent through the multi-GNSS combination. In contrast, in this case, the WL ambiguity can be successfully fixed and can support the precise positioning with an accuracy of several centimeters.

Performance investigation of LAMBDA and bootstrapping methods for PPP narrow-lane ambiguity resolution

ABSTRACT Precise point positioning with ambiguity resolution (PPP-AR) is a powerful tool for geodetic and time-constrained applications that require high precision. The performance of PPP-AR highly depends on the reliability of the correct integer carrier-phase ambiguity estimation. In this study, the performance of narrow-lane ambiguity resolution of PPP using the Least-squares AMBiguity Decorrelation (LAMBDA) and bootstrapping methods is extensively investigated using real data from 55 IGS stations over one-month in 2020. Static PPP with 24-, 12-, 8-, 4-, 2-, 1- and ½-h sessions using two different cutoff angles (7° and 30°) was conducted with three PPP modes: i.e. ambiguity-float and two kinds of ambiguity-fixed PPP using the LAMBDA and bootstrapping methods for narrow-lane AR, respectively. The results show that the LAMBDA method can produce more reliable results for 2 hour and shorter observation sessions compared with the bootstrapping method using a 7° cutoff angle. For a 30° cutoff angle, the LAMBDA method outperforms the bootstrapping method for observation sessions of 4 h and less. For long observation times, the bootstrapping method produced much more accurate coordinates compared with the LAMBDA method without considering the wrong fixes cases. The results also show that occurrences of fixing the wrong integer ambiguities using the bootstrapping method are higher than that of the LAMBDA method.

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.

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.

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.

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.

A new method for GLONASS inter-frequency bias estimation based on long baselines

Frequency division multiplexing of GLONASS signals causes inter-frequency bias (IFB) in receiving equipment. IFB significantly increases the difficulties of fixing GLONASS ambiguities and limits the accuracy and reliability of GLONASS positioning and orbit determination. Accurately estimating and calibrating IFB can effectively solve such a problem. However, at present, most methods of IFB estimation are based on zero and short baselines, in which case it is not only difficult to realize fast and efficient IFB estimation but also one cannot fully utilize publicly available IGS and CORS data. Therefore, we present a new method for GLONASS IFB estimation based on long baselines. First, to weaken the influence of inter-frequency code bias, the wide-lane ambiguities are calculated directly based on the wide-lane combinations of observations. Then, according to the range of inter-frequency phase bias (IFPB) rates, a IFPB defined as the difference in IFPBs between adjacent frequencies, a step-by-step search schedule is designed to remove the impacts of IFPB on wide-lane and narrow-lane ambiguity resolution. Finally, after fixing integer wide-lane and narrow-lane ambiguities, the IFPB rate can be estimated. An experimental network is set up to verify the validity of this method; the experiment includes the data observed for 31 days at 542 stations in Europe and North America. The IFPB rates of 38 receiver types from nine manufacturers are successfully determined. Experimental results show that the estimated IFPB rates for the same receiver type stabilize within a month with a standard deviation of less than 1.4 mm/∆f (millimeters per frequency number increment, ∆f denotes the frequency difference of adjacent GLONASS frequencies with frequency number increment of 1). Generally, the difference in IFPB rates of receiver types from the same manufacturer does not exceed 2.5 mm/∆f. However, the estimated IFPB rates of Septentrio’s newly produced receivers, as compared with that of the old receiver types, show a rate difference of up to 50 mm/∆f. This significant difference should be considered for practical applications.

ERTK: extra-wide-lane RTK of triple-frequency GNSS signals

Triple-frequency signals have thus far been available for all satellites of BeiDou and Galileo systems and for some GPS satellites. The main benefit of triple-frequency signals is their formation of extra-wide-lane (EWL) combinations whose ambiguities can be instantaneously fixed for several 10–100 km baselines. Yet, this benefit has not been fully exploited and only used as a constraint for narrow-lane (NL) ambiguity resolution (AR) in most previous studies. In this study, we comprehensively investigate the real-time kinematic (RTK) capabilities of EWL observations, also referred to as EWL RTK (ERTK). We begin by mathematically expressing the ease of EWL AR and the difficulty of NL AR, respectively, using a numerical demonstration. We then present the mathematical models for ERTK including the ionosphere-ignored, ionosphere-float and ionosphere-smoothed types. The experiments are conducted using a four-station network of real triple-frequency BeiDou data with baseline lengths from 33 to 75 km. The results show that the ionosphere-ignored ERTK achieves real-time solutions with a horizontal accuracy of about 10 cm. Although the ionosphere-float ERTK solutions are very noisy, they can be quickly improved at the centimetre level by further applying the ionosphere-smoothed model. Note that such accurate results are very promising and already satisfy many applications without complicated NL AR. To the best of our knowledge, this is the first comprehensive study to make full use of EWL observations of triple-frequency signals on RTK.

Method and assessment of BDS triple-frequency ambiguity resolution for long-baseline network RTK

Fast and reliable ambiguity resolution (AR) over long baselines is one of the major challenges for large-scale GNSS network real-time kinematic (RTK) positioning. In the dual-frequency case, wide-lane (WL) ambiguities are usually fixed firstly, followed by the procedure of determination of the narrow-lane (NL) ambiguity integers based on the ionosphere-free model. For long-baseline network RTK, there are still some problems faced, for instance the WL AR still needs to take over tens of epochs due to the influence of pseudorange noises and multipath effects; the NL AR model is weak since more parameters are introduced, thus much time is needed to guarantee the precision of float ambiguity solutions. Along with the deployment of modernized GPS, BDS and other GNSS, the triple-frequency AR with real data gradually becomes feasible. Many literatures have proven the third signal will make AR faster and more reliable. In this paper based on the characteristics of network RTK, a modified ionosphere-free model was proposed to improve EWL/WL AR performance for long baselines, followed by the comparison with several commonly-used EWL/WL AR models. The NL AR performance with triple-frequency and dual-frequency observations was also tested. In NL AR, a partial AR strategy is adopted to weaken the negative influence of new-rising or low-elevation satellites. Experiments were conducted using real BDS triple-frequency data on 74km and 241km baselines. Results show that the modified ionosphere-free model can balance the ambiguity precision and ambiguity bias better for the EWL/WL AR. The NL AR with triple-frequency observations also performs significantly better than that in dual-frequency mode, with 29.9% and 32.9% improvement in time-to-first-fix (TTFF) for the 74km and 241km baselines respectively.

BDS ambiguity resolution with the modified TCAR method for medium-long baseline

As the BeiDou Navigation Satellite System (BDS) attained initial regional operational status at the end of October 2012, real BDS data are available to investigate triple-frequency ambiguity resolution (AR) performance. In this contribution, analysis of the classical TCAR models and limiting factors is given. It is concluded that the residual double-differencing (DD) ionospheric delay is the key limitation to the medium and long range narrow-lane (NL) ambiguity resolution in step 3 of the classical TCAR method. To improve this algorithm, the third step of the classical TCAR method is modified accordingly. The modified TCAR comprises three major steps. Step 1 and step 2 are the same as the classical TCAR, which is the geometry-free determination of the extra-wide-lane (EWL) and wide-lane (WL) ambiguities. Then we can derive the DD first-order ionospheric delay estimated from the ambiguity-fixed EWL. As the noise term of the estimated DD ionospheric delay is very large, the smooth method is employed to correct the estimated DD ionospheric delay. In step 3, the smooth DD ionospheric delay is used to completely correct the single-epoch DD float NL ambiguity resolution. It is also noted that there exist cycle-slips which influence the process of the ionosphere-smooth method. This is followed by a procedure to repair the cycle-slip. As a result, the modified-TCAR method shows a much better performance than the classical TCAR method over medium and long baseline.

Assessment of code bias variations of BDS triple-frequency signals and their impacts on ambiguity resolution for long baselines

Carrier phase ambiguity resolution over long baselines is challenging in BDS data processing. This is partially due to the variations of the hardware biases in BDS code signals and its dependence on elevation angles. We present an assessment of satellite-induced code bias variations in BDS triple-frequency signals and the ambiguity resolutions procedures involving both geometry-free and geometry-based models. First, since the elevation of a GEO satellite remains unchanged, we propose to model the single-differenced fractional cycle bias with widespread ground stations. Second, the effects of code bias variations induced by GEO, IGSO and MEO satellites on ambiguity resolution of extra-wide-lane, wide-lane and narrow-lane combinations are analyzed. Third, together with the IGSO and MEO code bias variations models, the effects of code bias variations on ambiguity resolution are examined using 30-day data collected over the baselines ranging from 500 to 2600 km in 2014. The results suggest that although the effect of code bias variations on the extra-wide-lane integer solution is almost ignorable due to its long wavelength, the wide-lane integer solutions are rather sensitive to the code bias variations. Wide-lane ambiguity resolution success rates are evidently improved when code bias variations are corrected. However, the improvement of narrow-lane ambiguity resolution is not obvious since it is based on geometry-based model and there is only an indirect impact on the narrow-lane ambiguity solutions.

A New Method of Ambiguity Resolution for Triple-Frequency GPS PPP

At present, reliable ambiguity resolution in GPS precise point positioning (PPP) can be achieved through the traditional model called “EWL-WL-NL”. In this paper, we proposed a new model of ambiguity resolution,“WL-WL-WL”, where making use of linear independence of coefficient vector of wide-lane combination. Firstly, using the Melbourne-Wubbena combination observable on L2 and L5, we could resolve extra-wide-lane ambiguity instantaneously. Then, the resolved unambiguous extra-wide-lane carrier-phase assists wide-lane ambiguity resolution (AR). Three wide-lane combinations whose coefficient vectors are linearly independent are chosen to compose one full-rank matrix so that the three narrow-lane ambiguity resolution can be achieved. As a result, with the triple-frequency signals, the correctness rate of narrow-lane ambiguity resolution achieves 90% within 60s, in contrast to only 63% within 180s in dual-frequency PPP. Therefore, we demonstrate that triple-frequency PPP has the potential to achieve ambiguity-fixed solutions within a few minutes and the efficiency of ambiguity resolution in triple-frequency PPP is higher.

Real-time kinematic positioning over long baselines using triple-frequency BeiDou signals

The BeiDou system is the first global navigation satellite system in which all satellites transmit triple-frequency signals that can provide the positioning, navigation, and timing independently. A benefit of triple-frequency signals is that more useful combinations can be formed, including some extrawide-lane combinations whose ambiguities can generally be instantaneously fixed without distance restriction, although the narrow-lane ambiguity resolution (NL AR) still depends on the interreceiver distance or requires a long time to achieve. In this paper, we synthetically study decimeter and centimeter kinematic positioning using BeiDou triple-frequency signals. It starts with AR of two extrawide-lane signals based on the ionosphere-free or ionosphere-reduced geometry-free model. For decimeter positioning, one can immediately use two ambiguity-fixed extrawide-lane observations without pursuing NL AR. To achieve higher accuracy, NL AR is the necessary next step. Despite the fact that long-baseline NL AR is still challenging, some NL ambiguities can indeed be fixed with high reliability. Partial AR for NL signals is acceptable, because as long as some ambiguities for NL signals are fixed, positioning accuracy will be certainly improved.With accumulation of observations, more and more NL ambiguities are fixed and the positioning accuracy continues to improve. An efficient Kalman-filtering system is established to implement the whole process. The formulated system is flexible, since the additional constraints can be easily applied to enhance the model's strength. Numerical results from a set of real triple-frequency BeiDou data on a 50 km baseline show that decimeter positioning is achievable instantaneously.With only five data epochs, 84% of NL ambiguities can be fixed so that the real-time kinematic accuracies are 4.5, 2.5, and 16 cm for north, east, and height components (respectively), while with 10 data epochs more than 90% of NL ambiguities are fixed, and the real-time kinematic solutions are improved to centimeter level for all three coordinate components.

Three-carrier ambiguity resolution using the modified TCAR method

Multi-frequency technique is expected to be widely adopted with the new generations of global navigation satellite system, which is anticipated to benefit ambiguity resolution (AR). Three-carrier AR (TCAR) is a classical AR method based on triple-frequency observations, which is efficient for AR of short baseline. However, this method ignores the residual ionospheric delay, which degrades the reliability in active ionosphere situations and reduces the success of AR for medium and long baselines. We investigate the classical TCAR method and identify the major deficiency that hampers its application, especially to medium and long baselines. To improve this algorithm, the second and third steps of the classical TCAR are modified accordingly. In step 2, the ambiguity-resolved extra-wide-lane (EWL) combination and three pseudorange observations are employed to eliminate or reduce the residual ionospheric delay, in addition to the geometry term. In step 3, besides the EWL combination and pseudoranges, the ambiguity-resolved wide-lane (WL) combination is used to completely eliminate the ionosphere and geometry terms. The combination coefficients of these pseudoranges and combinations are optimized to minimize the noise of the ambiguity estimates. In order to assess the performances, real triple-frequency observations of BeiDou navigation system of baselines with different lengths are processed by the two methods. Results show that, the classical TCAR method is very sensitive to ionospheric delay and limited to short baseline application, while the modified TCAR method is free from ionospheric delay and can be applied to AR of median and long baselines. For WL AR, the modified TCAR method shows a comparable performance with the classical TCAR method, and a better performance can be expected when the baseline becomes longer, e.g., from 100s to 1,000s kilometers. For narrow-lane AR, the modified TCAR method performs much better than the classical TCAR method for median and long baselines.

Triple-frequency GPS precise point positioning with rapid ambiguity resolution

At present, reliable ambiguity resolution in real-time GPS precise point positioning (PPP) can only be achieved after an initial observation period of a few tens of minutes. In this study, we propose a method where the incoming triple-frequency GPS signals are exploited to enable rapid convergences to ambiguity-fixed solutions in real-time PPP. Specifically, extra-wide-lane ambiguity resolution can be first achieved almost instantaneously with the Melbourne-Wubbena combination observable on L2 and L5. Then the resultant unambiguous extra-wide-lane carrier-phase is combined with the wide-lane carrier-phase on L1 and L2 to form an ionosphere-free observable with a wavelength of about 3.4 m. Although the noise of this observable is around 100 times the raw carrier-phase noise, its wide-lane ambiguity can still be resolved very efficiently, and the resultant ambiguity-fixed observable can assist much better than pseudorange in speeding up succeeding narrow-lane ambiguity resolution. To validate this method, we use an advanced hardware simulator to generate triple-frequency signals and a high-grade receiver to collect 1-Hz data. When the carrier-phase precisions on L1, L2 and L5 are as poor as 1.5, 6.3 and 1.5 mm, respectively, wide-lane ambiguity resolution can still reach a correctness rate of over 99 % within 20 s. As a result, the correctness rate of narrow-lane ambiguity resolution achieves 99 % within 65 s, in contrast to only 64 % within 150 s in dual-frequency PPP. In addition, we also simulate a multipath-contaminated data set and introduce new ambiguities for all satellites every 120 s. We find that when multipath effects are strong, ambiguity-fixed solutions are achieved at 78 % of all epochs in triple-frequency PPP whilst almost no ambiguities are resolved in dual-frequency PPP. Therefore, we demonstrate that triple-frequency PPP has the potential to achieve ambiguity-fixed solutions within a few minutes, or even shorter if raw carrier-phase precisions are around 1 mm. In either case, we conclude that the efficiency of ambiguity resolution in triple-frequency PPP is much higher than that in dual-frequency PPP.

Rapid re-convergences to ambiguity-fixed solutions in precise point positioning

Integer ambiguity resolution at a single receiver can be achieved if the fractional-cycle biases are separated from the ambiguity estimates in precise point positioning (PPP). Despite the improved positioning accuracy by such integer resolution, the convergence to an ambiguity-fixed solution normally requires a few tens of minutes. Even worse, these convergences can repeatedly occur on the occasion of loss of tracking locks for many satellites if an open sky-view is not constantly available, consequently totally destroying the practicability of real-time PPP. In this study, in case of such re-convergences, we develop a method in which ionospheric delays are precisely predicted to significantly accelerate the integer ambiguity resolution. The effectiveness of this method consists in two aspects: first, wide-lane ambiguities can be rapidly resolved using the ionosphere-corrected wide-lane measurements, instead of the noisy Melbourne–Wübbena combination measurements; second, narrow-lane ambiguity resolution can be accelerated under the tight constraints derived from the ionosphere-corrected unambiguous wide-lane measurements. In the test at 90 static stations suffering from simulated total loss of tracking locks, 93.3 and 95.0% of re-convergences to wide-lane and narrow-lane ambiguity resolutions can be achieved within five epochs of 1-Hz measurements, respectively, even though the time latency for the predicted ionospheric delays is up to 180 s. In the test at a mobile van moving in a GPS-adverse environment where satellite number significantly decreases and cycle slips frequently occur, only when the predicted ionospheric delays are applied can the rate of ambiguity-fixed epochs be dramatically improved from 7.7 to 93.6% of all epochs. Therefore, this method can potentially relieve the unrealistic requirement of a continuous open sky-view by most PPP applications and improve the practicability of real-time PPP.

Cycle Slip Detection and Ambiguity Resolution Algorithms for Dual-Frequency GPS Data Processing

This article investigates the problem of cycle slip detection and ambiguity resolution using dual-frequency GPS data. Several algorithms are proposed and described. F or cycle slip detection, three L1/L2 observable combinations have been integrated to formulate a new algorithm for cycle slip detection. For ambiguity resolution, both widelane and narrowlane ambiguity resolution algorithms are presented, but the focus is on the narrowlane ambiguity resolution. Numerical results are included to evaluate the performance of the proposed algorithms, which have shown that cycle slips can be effectively detected and the narrowlane ambiguities can be resolved almost instantaneously after successful determination of the widelane ambiguities.