Wide-lane Ambiguity Research Articles

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.

GLONASS fractional-cycle bias estimation across inhomogeneous receivers for PPP ambiguity resolution

The key issue to enable precise point positioning with ambiguity resolution (PPP-AR) is to estimate fractional-cycle biases (FCBs), which mainly relate to receiver and satellite hardware biases, over a network of reference stations. While this has been well achieved for GPS, FCB estimation for GLONASS is difficult because (1) satellites do not share the same frequencies as a result of Frequency Division Multiple Access (FDMA) signals; (2) and even worse, pseudorange hardware biases of receivers vary in an irregular manner with manufacturers, antennas, domes, firmware, etc., which especially complicates GLONASS PPP-AR over inhomogeneous receivers. We propose a general approach where external ionosphere products are introduced into GLONASS PPP to estimate precise FCBs that are less impaired by pseudorange hardware biases of diverse receivers to enable PPP-AR. One month of GLONASS data at about 550 European stations were processed. From an exemplary network of 51 inhomogeneous receivers, including four receiver types with various antennas and spanning about 800 km in both longitudinal and latitudinal directions, we found that 92.4 % of all fractional parts of GLONASS wide-lane ambiguities agree well within $$\pm $$ 0.15 cycles with a standard deviation of 0.09 cycles if global ionosphere maps (GIMs) are introduced, compared to only 51.7 % within $$\pm $$ 0.15 cycles and a larger standard deviation of 0.22 cycles otherwise. Hourly static GLONASS PPP-AR at 40 test stations can reach position estimates of about 1 and 2 cm in RMS from ground truth for the horizontal and vertical components, respectively, which is comparable to hourly GPS PPP-AR. Integrated GLONASS and GPS PPP-AR can further achieve an RMS of about 0.5 cm in horizontal and 1–2 cm in vertical components. We stress that the performance of GLONASS PPP-AR across inhomogeneous receivers depends on the accuracy of ionosphere products. GIMs have a modest accuracy of only 2–8 TECU (Total Electron Content Unit) in vertical which confines PPP-AR to an approximately $$800\times 800$$ km area in Europe. We expect that a regional ionosphere map with a better than 1 TECU accuracy is likely to improve the GLONASS PPP-AR efficiency.

Improving GLONASS Precise Orbit Determination through Data Connection.

In order to improve the precision of GLONASS orbits, this paper presents a method to connect the data segments of a single station-satellite pair to increase the observation continuity and, consequently, the strength of the precise orbit determination (POD) solution. In this method, for each GLONASS station-satellite pair, the wide-lane ambiguities derived from the Melbourne–Wübbena combination are statistically tested and corrected for phase integer offsets and then the same is carried out for the narrow-lane ambiguities calculated from the POD solution. An experimental validation was carried out using one-month GNSS data of a global network with 175 IGS stations. The result shows that, on average, 27.1% of the GLONASS station-satellite pairs with multiple data segments could be connected to a single long observation arc and, thus, only one ambiguity parameter was estimated. Using the connected data, the GLONASS orbit overlapping RMS at the day boundaries could be reduced by 19.2% in ideal cases with an averaged reduction of about 6.3%.

Ionospheric effects in uncalibrated phase delay estimation and ambiguity-fixed PPP based on raw observable model

Zero-difference (ZD) ambiguity resolution (AR) reveals the potential to further improve the performance of precise point positioning (PPP). Traditionally, PPP AR is achieved by Melbourne–Wubbena and ionosphere-free combinations in which the ionosphere effect are removed. To exploit the ionosphere characteristics, PPP AR with L1 and L2 raw observable has also been developed recently. In this study, we apply this new approach in uncalibrated phase delay (UPD) generation and ZD AR and compare it with the traditional model. The raw observable processing strategy treats each ionosphere delay as an unknown parameter. In this manner, both a priori ionosphere correction model and its spatio-temporal correlation can be employed as constraints to improve the ambiguity resolution. However, theoretical analysis indicates that for the wide-lane (WL) UPD retrieved from L1/L2 ambiguities to benefit from this raw observable approach, high precision ionosphere correction of better than 0.7 total electron content unit (TECU) is essential. This conclusion is then confirmed with over 1 year data collected at about 360 stations. Firstly, both global and regional ionosphere model were generated and evaluated, the results of which demonstrated that, for large-scale ionosphere modeling, only an accuracy of 3.9 TECU can be achieved on average for the vertical delays, and this accuracy can be improved to about 0.64 TECU when dense network is involved. Based on these ionosphere products, WL/narrow-lane (NL) UPDs are then extracted with the raw observable model. The NL ambiguity reveals a better stability and consistency compared to traditional approach. Nonetheless, the WL ambiguity can be hardly improved even constrained with the high spatio-temporal resolution ionospheric corrections. By applying both these approaches in PPP-RTK, it is interesting to find that the traditional model is more efficient in AR as evidenced by the shorter time to first fix, while the three-dimensional positioning accuracy of the RAW model outperforms the combination model by about \(7.9\,\%\). This reveals that, with the current ionosphere models, there is actually no optimal strategy for the dual-frequency ZD ambiguity resolution, and the combination approach and raw approach each has merits and demerits.

Study on cycle-slip detection and repair methods for a single dual-frequency global positioning system (GPS) receiver

In this work, we assessed the performance of the cycle-slip detection methods: Turbo Edit (TE), Melbourne-Wübbena wide-lane ambiguity (MWWL) and forward and backward moving window averaging (FBMWA). The TE and MWWL methods were combined with ionospheric total electron content rate (TECR), and the FBMWA with second-order time-difference phase ionosphere residual (STPIR) and TECR. Under different scenarios, 10 Global Positioning System (GPS) datasets were used to assess the performance of the methods for cycle-slip detection. The MWWL-TECR delivered the best performance in detecting cycle-slips for 1 s data. The relative comparisons show that the FBMWA-TECR method performed slightly better than its original version, FBMWA-STPIR, detecting 100% and 73%, respectively. For data with a sample rate of 5 s, the FBMWA-TECR performed better than MWWL-TECR. However, the FBMWA is suitable only for post-processing, which refers to applications where the data are processed after the fact.

Divisional ambiguity resolution for long range reference stations in network RTK

Increasing the distance between reference stations of network RTK can improve the flexibility of the reference stations sitting, reduce the number of reference stations and cut down construction and maintenance costs. However, it also decreases the correlation of spatial error and affects the real time performance and effectiveness of the network RTK positioning. Thus, the key techniques of long range network RTK need to be further optimised. Ambiguity resolution between reference stations is one of the most critical techniques. In this paper a divisional ambiguity resolution for long range reference stations of network RTK is proposed, so as to deal with the long fixed time and low success rate, especially for low elevation satellites. Based on conventional two-step ambiguity resolution, firstly, this new method calculates the wide-lane ambiguity by building M-W combination of pseudo-range and carrier observations. Second, the conventional ambiguity resolution as a whole is changed and satellites are divided into high elevation ones and low elevation ones, then the integer ambiguity of high elevation satellites is resolved by means of ionosphere-free combination with the relative zenith wet delay as a parameter. Finally, the Kalman filter model is established by the observation equation of high elevation satellites with calculated ambiguity and low elevation satellites, assisting to obtain the ambiguity of low elevation satellites quickly. The example selects a 196 km long baseline for comparative analysis. Result shows that it takes 733 s to resolve the ambiguity of all satellites by conventional ambiguity resolution in whole, and 252 s by divisional ambiguity resolution proposed in this paper. The new method saves about two thirds of resolution time, and also greatly improves the efficiency of ambiguity resolution of low elevation satellites for long baseline.

Real-time detection and repair of cycle slips in triple-frequency GNSS measurements

Cycle slip detection and repair are prerequisites to the use of the global navigation satellite system (GNSS) carrier phases for precise positioning. Modern GNSS techniques introduce triple- or multi-frequency signals that are beneficial for cycle slip detection and repair. We present a new real-time cycle slip detection and repair method based on the independent linear combinations of undifferenced triple-frequency GNSS observations. The proposed method forms three types of linear combinations based on the original observations. These combinations are called extra-wide lane (EWL), wide lane (WL), and narrow lane (NL). Cycle slips on the combinations are determined sequentially in three cascaded steps. The first step employs the geometry-free and ionosphere-free Hatch–Melbourne–Wübbena combination to determine and repair the EWL cycle slips. The second step subtracts the cycle-slip-repaired EWL combination from the WL combination to eliminate the geometry part of the WL combination. This subtraction results in a new function that contains the WL ambiguity and residual ionospheric delay. This function is differenced at two consecutive epochs to determine the WL cycle slips. The residual ionospheric delay difference is ignored because of its small magnitude relative to WL wavelength. The third step determines the NL cycle slips in the same manner as in the second step. The difference is that the cycle-slip-repaired EWL combination is replaced with the more accurate cycle-slip-repaired WL combination. Moreover, the residual ionospheric delay difference is compensated by the ionospheric delay rate derived from the original carrier phase observations. When the EWL, WL, and NL cycle slips are determined, cycle slips on the original carrier phase observations can be uniquely identified. The proposed approach has been tested on 30-s triple-frequency BeiDou navigation satellite system data under different levels of ionospheric variations, and on 30-s triple-frequency global positioning system and quasi-zenith satellite system data. Results indicate that the approach can effectively detect and correct cycle slips even for one cycle under low sampling rate or active ionospheric conditions on each frequency in real time.

GNSS multi-carrier fast partial ambiguity resolution strategy tested with real BDS/GPS dual- and triple-frequency observations

The regional constellation of BeiDou navigation satellite system (BDS) has been officially in operation since December 27, 2012, and real-time kinematic positioning using BDS and GPS multi-frequency observations is feasible. A heavy computational problem arises when resolving ambiguities in the case of multi-system with multi-frequency observations. A multi-carrier fast partial ambiguity resolution strategy is developed with the property that the extra-wide-lane and wide-lane ambiguities in the multi-frequency case can be resolved reliably in advance. Consequently, the technique resolves ambiguities sequentially instead of the usual batch ambiguity resolution (AR) mode so as to improve the computational efficiency of AR significantly. The strategy is demonstrated with real BDS/GPS dual- and triple-frequency observations. The results have shown that the probability of correct AR by the proposed method is comparable to that of the batch AR. Experimentally, the new method is about 2.5 times as fast as the batch AR in the dual-frequency case, 3 times in the mixed dual- and triple-frequency case and 3.5 times in the triple-frequency case.

GNSS ambiguity resolution with controllable failure rate for long baseline network RTK

Many large-scale GNSS CORS networks have been deployed around the world to support various commercial and scientific applications. To make use of these networks for real-time kinematic positioning services, one of the major challenges is the ambiguity resolution (AR) over long inter-station baselines in the presence of considerable atmosphere biases. Usually, the widelane ambiguities are fixed first, followed by the procedure of determination of the narrowlane ambiguity integers based on the ionosphere-free model in which the widelane integers are introduced as known quantities. This paper seeks to improve the AR performance over long baseline through efficient procedures for improved float solutions and ambiguity fixing. The contribution is threefold: (1) instead of using the ionosphere-free measurements, the absolute and/or relative ionospheric constraints are introduced in the ionosphere-constrained model to enhance the model strength, thus resulting in the better float solutions; (2) the realistic widelane ambiguity precision is estimated by capturing the multipath effects due to the observation complexity, leading to improvement of reliability of widelane AR; (3) for the narrowlane AR, the partial AR for a subset of ambiguities selected according to the successively increased elevation is applied. For fixing the scalar ambiguity, an error probability controllable rounding method is proposed. The established ionosphere-constrained model can be efficiently solved based on the sequential Kalman filter. It can be either reduced to some special models simply by adjusting the variances of ionospheric constraints, or extended with more parameters and constraints. The presented methodology is tested over seven baselines of around 100 km from USA CORS network. The results show that the new widelane AR scheme can obtain the 99.4 % successful fixing rate with 0.6 % failure rate; while the new rounding method of narrowlane AR can obtain the fix rate of 89 % with failure rate of 0.8 %. In summary, the AR reliability can be efficiently improved with rigorous controllable probability of incorrectly fixed ambiguities.

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.

URTK: undifferenced network RTK positioning

Standard network RTK has been widely used since it was proposed in the mid-1990s. Rovers can obtain high-precision estimates of position by resolving double-differenced (DD) ambiguities. The focus of this study is a new undifferenced network RTK method, abbreviated as URTK hereafter, based on undifferenced (UD) observation corrections whose single-differenced (SD) ambiguities between satellites can be resolved in several seconds. The tools for studying the real-time realization of the new method are our developments of logical schemes that have the capability for the real-time modeling of a reference network and the instantaneous resolution of SD ionosphere-free (IF) ambiguities at a single station. This research demonstrates the validity of modeling regional UD-unmodeled errors on the ground and examines the maximum differences when compared to modeling the errors using ionospheric pierce points (IPP). With data collected at 48 stations from a CORS network in Shanxi Province (SXCORS) in China through May 21, 2010, the efficiency of the presented real-time strategies is validated using IGS final products in a postprocessing mode. The results verify that more than 83 % of SD wide-lane (WL) ambiguity can be fixed with 5 s of observation data, and the average resolution time of all the WL tests is 4.96 s. More than 80 % of SD L1 ambiguity can be fixed within 5 s, and the average resolution time is only 6.66 s. Rovers could gain rapidly centimeter-level absolute positioning service, comparable to standard network RTK. In addition, the URTK method transforms the fixed DD-ambiguities of the reference network into UD-ambiguities, and it does not need to set the base station and base satellite. Since the UD-corrections are modeled for each common visible satellite, it breaks down the connections between stations and satellites of the DD-corrections in the current network RTK. The UD-corrections can be broadcast by the base station and automatically selected and optimized by a rover during the real-time kinematic processing, thus avoiding ambiguity in reinitialization due to the change of reference, so it should be very flexible and useful for a wide range of applications.

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.

Towards PPP-RTK: Ambiguity resolution in real-time precise point positioning

Integer ambiguity resolution at a single station can be achieved by introducing predetermined uncalibrated phase delays (UPDs) into the float ambiguity estimates of precise point positioning (PPP). This integer resolution technique has the potential of leading to a PPP-RTK (real-time kinematic) model where PPP provides rapid convergence to a reliable centimeter-level positioning accuracy based on an RTK reference network. Nonetheless, implementing this model is technically subject to how rapidly we can fix wide-lane ambiguities, stabilize narrow-lane UPD estimates, and achieve the first ambiguity-fixed solution. To investigate these issues, we used 7 days of 1-Hz sampling GPS data at 91 stations across Europe. We find that at least 10 min of observations are required for most receiver types to reliably fix about 90% of wide-lane ambiguities corresponding to high elevations, and over 20 min to fix about 90% of those corresponding to low elevations. Moreover, several tens of minutes are usually required for a regional network before a narrow-lane UPD estimate stabilizes to an accuracy of far better than 0.1 cycles. Finally, for hourly data, ambiguity resolution can significantly improve the accuracy of epoch-wise position estimates from 13.7, 7.1 and 11.4 cm to 0.8, 0.9 and 2.5 cm for the East, North and Up components, respectively, but a few tens of minutes is required to achieve the first ambiguity-fixed solution. Therefore, from the timeliness aspect, our PPP-RTK model currently cannot satisfy the critical requirement of instantaneous precise positioning where ambiguity-fixed solutions have to be achieved within at most a few seconds. However, this model can still be potentially applied to some near-real-time remote sensing applications, such as the GPS meteorology.

Estimation of Relative Satellite Position Using Transformed Differential Carrier-Phase GPS Measurements

Techniques for applying differential carrier-phase global positioning systems to satellite formation clusters with large (approximately 100 km or more) baselines are described. Because satellites in the cluster may move relative to each other, it is imperative that the carrier-phase ambiguities be resolved quickly and accurately. We propose a transformation of the m-vector carrier-phase measurement equations that restricts the geometric nonlinearities to a one-dimensional subspace and an almost universal linearization of the position state and integer ambiguities in the remaining m - 1 dimensional subspace. We then show that all of the measurement equations can be processed with an unscented Kalman filter to quickly compute very accurate floating-point valued estimates of the system state and error covariance. By an integer-preserving transformation found in the least-squares ambiguity decorrelation adjustment method, the number of possible hypotheses for the double-differenced wide-lane ambiguity candidates can be reduced. For the hypotheses set applying a multiple hypothesis Wald sequential probability test, using a specially conditioned form of the transformed global positioning system measurements, quickly and almost optimally determines the correct value of the carrier-phase double-differenced ambiguity. Once the double-differenced wide-lane ambiguities are obtained, the L 1 double-differenced ambiguities are resolved by using L 1 and L 2 carrier-phase measurements based on wide-lane integers in the unscented Kalman filter, then using the least-squares ambiguity decorrelation adjustment method for determining the hypotheses, followed by the multiple hypothesis Wald sequential probability test for resolving the L 1 double-differenced ambiguities. Finally, using the L 1 carrier-phase measurements, the unscented Kalman filter produces the relative position estimates. These techniques are then demonstrated on a simulation of a formation of two satellites in low Earth orbits.

Single Epoch Kinematic GPS Positioning Technique in Short Baseline

Resolving for integer phase ambiguities is an important issue of satellite surveying. In general ambiguity resolution techniques, observations made at tens of continuous epochs are required. These techniques usually fail in the case of cycle slip between global positioning system (GPS) satellites and GPS receivers happens intermittently. To overcome this, a novel technique, the single epoch kinematic (SEK) positioning technique, is proposed in this paper to resolve ambiguities based on observations made at only one epoch. In order to improve the computational efficiency, multiple wide lane ambiguities transformations and multiple statistic constraints are proposed and employed in this novel technique. Observations acquired from a 6 km×5 km test area with 25 check points are used for experiments. High accuracy of positioning and ratio of success for the experimental results guarantee the capability of the SEK positioning technique.

Characteristics of Satellite-Satellite Single-Difference Widelane Fractional Carrier-Phase Biases

: To analyze the behavior characteristics of widelane fractional carrier-phase biases, time series of biases were analyzed. Single-differenced widelane fractional phase bias calibration values, involving two GPS satellites and one receiver, were analyzed for long-term characteristics between January 1, 1997, and May 30, 1998. The single differencing removes receiver effects, and differencing of the code and carrier-phase measurements isolates the widelane ambiguity. This ambiguity term includes a fraction that is consistent for each single-difference satellite pair. These fractional terms were averaged over a global tracking network using directional statistics to calculate daily fractional phase bias calibration values. Trends were then computed for individual GPS satellites. Over the test period, satellites 22, 26, 27, 29, 31, and 32 experienced near-zero drift rates in widelane and signals with amplitudes smaller than 0.1 widelane cycle. Satellites 30, 33, and 40 had high drift rates between 0.78 and 1.4 widelane cycles/year and substantial second-order signals. These groupings are correlated with mission launch date.

Research on the estimation of GPS phase ambiguities by triple frequency observations

As a part of modernization of GPS, a new signal with a frequency of 1 176.45 MHz will be made available to the civilian community and can provide more measurements for positioning in combination with the well known L1 and L2 signals. Applying the law of variance-covariance propagation, the correlation of triple-frequency ambiguity is discussed, the L1 ambiguity is week correlate with wide lane and long wide lane ambiguity. Under a comprehensive analysis of linear combination, no matter how long the baseline is, the ambiguities of long wide lane carrier phase combination can be determined using code measurements directly, ambiguity of short baseline can be fixed fast because of very small standard error of ambiguities of many linear combinations, the main efforts of TCAR are ambiguity resolution of long baseline. The ionosphere-free combinations used widely are not suitable for ambiguity resolution because of very short wavelength with high noise and very large standard error of ambiguity. The smoothing procedure reduces the standard errors of code and also can improve ambiguity estimation, the ambiguity of wide-lane, in long baseline case, can be rounded to the nearest integer within six epochs.

AMBIGUITY ONLINE ESTIMATION BY COMBINING DUAL-FREQUENCY PHASE AND CARRIER SMOOTHED CODE MEASUREMENTS: PERFORMANCE ON SHORT BASELINES

For both static and kinematic relative positioning with GPS, the success of baseline resolution depends closely on the reliable fixing of integer ambiguities of carrier phase measurements. Low noise code correlated dual-frequency GPS receivers offer accurate pseudo-range measurements on both carrier frequencies. With the help of a combination of code and carrier phase measurements, wide-lane ambiguities can be easily determined and used for the improvement of ambiguity resolutions on both L1 and L2 frequencies. A carrier smoothing technique is utilized to eliminate the noise of code measurements. Baselines ranging from 1.5 km to 48 km are used to test the proposed method.

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.

Unbiased vs biased estimation of GPS phase ambiguities from dual-frequency code and phase observables

With access to dual-frequency pseudorange and phase Global Positioning System (GPS) data, the wide-lane ambiguity can easily be fixed. Advantage is taken of this information in the linear combination of the above four observables for base ambiguity estimation (i.e. of N 1 and N 2). Starting points for our analysis are the Best Linear Unbiased Estimators BLUE1 and BLUE2. BLUE1 is the best one (with minimum mean square error, MSE) if the ionosphere effect is negligible. If this is not the case, BLUE2 has the smallest variance, but not necessarily the least mean square error. Hence, both estimators may suffer from a non-optimal treatment of the ionosphere bias. BLUE1 ignores possible ionosphere bias, while BLUE2 compensates for this bias in a less favourable way by eliminating it at the price of increased noise. As an alternative, linear estimators are derived, which make a compromise between the ionosphere bias and the random observation errors. This leads to the derivation of the Best Linear Estimator (BLE) and the Restricted Best Linear Estimator (RBLE) with minimum MSE. The former is generally not very useful, while the RBLE is recommended for practical use. It is shown that the MSE of the RBLE is limited by the variances of BLUE1 and BLUE2, i.e. However, as is always the case with a BLE, it cannot be used strictly: some parameter (in this case the ionosphere bias) must be approximately known.