Ambiguity Fixing Research Articles

Reliability of partial ambiguity fixing with multiple GNSS constellations

Reliable ambiguity resolution (AR) is essential to real-time kinematic (RTK) positioning and its applications, since incorrect ambiguity fixing can lead to largely biased positioning solutions. A partial ambiguity fixing technique is developed to improve the reliability of AR, involving partial ambiguity decorrelation (PAD) and partial ambiguity resolution (PAR). Decorrelation transformation could substantially amplify the biases in the phase measurements. The purpose of PAD is to find the optimum trade-off between decorrelation and worst-case bias amplification. The concept of PAR refers to the case where only a subset of the ambiguities can be fixed correctly to their integers in the integer least squares (ILS) estimation system at high success rates. As a result, RTK solutions can be derived from these integer-fixed phase measurements. This is meaningful provided that the number of reliably resolved phase measurements is sufficiently large for least-square estimation of RTK solutions as well. Considering the GPS constellation alone, partially fixed measurements are often insufficient for positioning. The AR reliability is usually characterised by the AR success rate. In this contribution, an AR validation decision matrix is firstly introduced to understand the impact of success rate. Moreover the AR risk probability is included into a more complete evaluation of the AR reliability. We use 16 ambiguity variance–covariance matrices with different levels of success rate to analyse the relation between success rate and AR risk probability. Next, the paper examines during the PAD process, how a bias in one measurement is propagated and amplified onto many others, leading to more than one wrong integer and to affect the success probability. Furthermore, the paper proposes a partial ambiguity fixing procedure with a predefined success rate criterion and ratio test in the ambiguity validation process. In this paper, the Galileo constellation data is tested with simulated observations. Numerical results from our experiment clearly demonstrate that only when the computed success rate is very high, the AR validation can provide decisions about the correctness of AR which are close to real world, with both low AR risk and false alarm probabilities. The results also indicate that the PAR procedure can automatically chose adequate number of ambiguities to fix at given high-success rate from the multiple constellations instead of fixing all the ambiguities. This is a benefit that multiple GNSS constellations can offer.

An enhanced calibration method of GLONASS inter-channel bias for GNSS RTK

A user of heterogeneous GPS and GLONASS receiver pairs in differential positioning mode will experience ambiguity fixing challenges due to the presence of inter-channel biases. These biases cannot be canceled by differencing GLONASS observations, whether pseudorange or carrier phase. Fortunately, pre-calibration of GLONASS pseudorange and carrier phase observations can make ambiguity fixing for GPS/GLONASS positioning much easier. We propose an effective algorithm that transforms an RTK (real-time kinematic) solution in a mixed receiver baseline from a float to a fixed ambiguity solution. Carrier phase and code inter-channel biases are estimated from a zero baseline. Then, GLONASS both carrier phase and code observations are corrected accordingly. The results show that a mixed baseline can be transformed from a float (~100 %) to a fixed (more than 92 %) solution.

Zero-difference GPS ambiguity resolution at CNES–CLS IGS Analysis Center

CNES (Centre National d’Etudes Spatiales) and CLS (Collecte Localisation Satellites) became an International GNSS Service (IGS) Analysis Center (AC) the 20th of May 2010. Since 2009, we are using the integer ambiguity fixing at the zero-difference level strategy in our software package (GINS/Dynamo) as an alternative to classical differential approaches. This method played a key role among all the improvements in the GPS processing we made during this period. This paper provides to the users the theoretical background, the strategies and the models used to compute the products (GPS orbits and clocks, weekly station coordinate estimates and Earth orientation parameters) that are submitted weekly to the IGS. The practical realization of the two-step, ambiguity-fixing scheme (wide-lane and narrow-lane) is described in detail. The ambiguity fixing improved our orbit overlaps from 6 to 3 cm WRMS in the tangential and normal directions. Since 2008, our products have been also regularly compared to the IGS final solutions by the IGS Analysis Center Coordinator. The joint effects of ambiguity fixing and dynamical model changes (satellite solar radiation pressure and albedo force) improved the consistency with IGS orbits from 35 to 18 mm 3D-WRMS. Our innovative strategy also gives additional powerful properties to the GPS satellite phase clock solutions. Single receiver (zero-difference) ambiguity resolution becomes possible. An overview of the applications is given.

Inter-agency comparison of TanDEM-X baseline solutions

TanDEM-X (TerraSAR-X add-on for Digital Elevation Measurement) is the first Synthetic Aperture Radar (SAR) mission using close formation flying for bistatic SAR interferometry. The primary goal of the mission is to generate a global digital elevation model (DEM) with 2m height precision and 10m ground resolution from the configurable SAR interferometer with space baselines of a few hundred meters. As a key mission requirement for the interferometric SAR processing, the relative position, or baseline vector, of the two satellites must be determined with an accuracy of 1mm (1D RMS) from GPS measurements collected by the onboard receivers. The operational baseline products for the TanDEM-X mission are routinely generated by the German Research Center for Geosciences (GFZ) and the German Space Operations Center (DLR/GSOC) using different software packages (EPOS/BSW, GHOST) and analysis strategies. For a further independent performance assessment, TanDEM-X baseline solutions are generated at the Astronomical Institute of the University of Bern (AIUB) on a best effort basis using the Bernese Software (BSW).Dual-frequency baseline solutions are compared for a 1-month test period in January 2011. Differences of reduced-dynamic baseline solutions exhibit a representative standard deviation (STD) of 1mm outside maneuver periods, while biases are below 1mm in all directions. The achieved baseline determination performance is close to the mission specification, but independent SAR calibration data takes acquired over areas with a well known DEM from previous missions will be required to fully meet the 1mm 1D RMS target. Besides the operational solutions, single-frequency baseline solutions are tested. They benefit from a more robust ambiguity fixing and show a slightly better agreement of below 1mm STD, but are potentially affected by errors caused by an incomplete compensation of differential ionospheric path delays.

Improving the Estimation of Uncalibrated Fractional Phase Offsets for PPP Ambiguity Resolution

Integer ambiguity resolution in Precise Point Positioning (PPP) can shorten convergence time and improve accuracy significantly. Uncalibrated Fractional Offsets (UFOs) originating in the satellites destroy the integer nature of carrier phase ambiguities observed at a single station. Several methods have been developed to estimate UFO information from a reference network for PPP ambiguity resolution. In this paper, we present a new approach for estimating Zero-Differenced (ZD) UFOs via float ZD ambiguities from a reference network. In this new approach, UFOs for receivers and satellites are estimated in an integrated adjustment with integer ambiguities being resolved sequentially, so that UFOs of higher quality can be achieved. The float ZD ambiguities used in the estimation can be from network or PPP solutions. Using those from PPP solutions enables the time-consuming clock estimations and the UFO estimations to be carried out separately, so that UFOs can be estimated more reliably from a much denser network. In this paper, a data processing procedure, from the estimation of UFOs through to PPP ambiguity fixing, is designed and proposed. The approach is validated with several data sets in various scenarios. The results show that satellite UFOs can be estimated precisely and reliably by using the observations from a reference network, in which the station spacing could be up to thousands of kilometres. With 30 minutes of observations, a PPP solution with fixed integer ambiguities is achievable and the positioning accuracy can be improved significantly when compared with its float solution.

Reliable Integer Ambiguity Resolution: Multi-Frequency Code Carrier Linear Combinations and Statistical A Priori Knowledge of Attitude

This paper provides two methods to improve the reliability of carrier phase integer ambiguity resolution: The first one is a group of multi-frequency linear combinations that include both code and carrier phase measurements, and allow an arbitrary scaling of the geometry, an arbitrary scaling of the ionospheric delay, and any preferred wavelength. The maximization of the ambiguity discrimination results in combinations with a wavelength of several meters and a noise level of a few centimeters. These combinations could be beneficial for both Real-Time Kinematics (RTK) and Precise Point Positioning (PPP). The second method incorporates some statistical a priori knowledge of attitude into the actual fixing. The a priori knowledge includes the length and direction of the baseline between two receivers and is given either as a uniform or Gaussian distribution. It enables a substantial reduction of the search space volume but also ensures a large robustness over errors in the a priori information. Both methods improve the accuracy of the float solution, which motivates a simple rounding for ambiguity fixing. A method is described, which enables an efficient computation of its success rate with a few integral transformations. Copyright © 2012 Institute of Navigation.

Vibrations of Mertz Glacier ice tongue, East Antarctica

AbstractAt the time of its calving in February 2010, Mertz Glacier, East Antarctica, was characterized by a 145 km long, 35 km wide floating tongue. In this paper, we use GPS data from the Collaborative Research into Antarctic Calving and Iceberg Evolution (CRAC-ICE) 2007/08 and 2009/10 field seasons to investigate the dynamics of Mertz Glacier. Two months of data were collected at the end of the 2007/08 field season from two kinematic GPS stations situated on each side of the main rift of the glacier tongue and from rock stations located around the ice tongue during 2008/09. Using Precise Point Positioning with integer ambiguity fixing, we observe that the two GPS stations recorded vibrations of the ice tongue with several dominant periods. We compare these results with a simple elastic model of the ice tongue and find that the natural vibration frequencies are similar to those observed. This information provides a better understanding of their possible effects on rift propagation and hence on the glacier calving processes.

Carrier-phase inter-frequency biases of GLONASS receivers

The frequency division multiplexing of the GLONASS signals causes inter-frequency biases in the receiving equipment. These biases vary considerably for receivers from different manufacturers and thus complicate or prevent carrier-phase ambiguity fixing. Complete and reliable ambiguity fixing requires a priori information of the carrier-phase inter-frequency bias differences of the receivers involved. GLONASS carrier-phase inter-frequency biases were estimated for 133 individual receivers from 9 manufacturers. In general, receivers of the same type and even receivers from the same manufacturer show similar biases, whereas the differences among manufacturers can reach up to 0.2 ns (more than 5 cm) for adjacent frequencies and thus up to 2.4 ns (73 cm) for the complete L1 or L2 frequency bands. A few individual receivers were identified whose inter-frequency biases behave differently as compared to other receivers of the same type or whose biases vary with time.

Monument-antenna effects on GPS coordinate time series with application to vertical rates in Antarctica

We examine the electromagnetic coupling of a GPS antenna–monument pair in terms of its simulated affect on long GPS coordinate time series. We focus on the Earth and Polar Observing System (POLENET) monument design widely deployed in Antarctica and Greenland in projects interested particularly in vertical velocities. We base our tests on an absolute robot calibration that included the top ~0.15 m of the monument and use simulations to assess its effect on site coordinate time series at eight representative POLENET sites in Antarctica over the period 2000.0–2011.0. We show that the neglect of this calibration would introduce mean coordinate bias, and most importantly for velocity estimation, coordinate noise which is highly sensitive to observation geometry and hence site location and observation period. Considering only sub-periods longer than 2.5 years, we show vertical site velocities may be biased by up to ±0.4 mm/year, and biases up to 0.2 mm/year may persist for observation spans of 8 years. Changing between uniform and elevation-dependent observation weighting alters the time series but does not remove the velocity biases, nor does ambiguity fixing. The effect on the horizontal coordinates is negligible. The ambiguities fixed series spectra show noise between flicker and random walk with near-white noise at the highest frequencies, with mean spectral indices (frequencies <20 cycles per year) of approximately −1.3 (uniform weighting) and −1.4 (elevation-dependent weighting). While the results are likely highly monument specific, they highlight the importance of accounting for monument effects when analysing vertical coordinate time series and velocities for the highest precision and accuracy geophysical studies.

Ionospheric path delay models for spaceborne GPS receivers flying in formation with large baselines

GPS relative navigation filters could benefit notably from an accurate modeling of the ionospheric delays, especially over large baselines (>100 km) where double difference delays can be higher than several carrier wavelengths. This paper analyzes the capability of ionospheric path delay models proposed for spaceborne GPS receivers in predicting both zero-difference and double difference ionospheric delays. We specifically refer to relatively simple ionospheric models, which are suitable for real-time filtering schemes. Specifically, two ionospheric delay models are evaluated, one assuming an isotropic electron density and the other considering the effect on the electron density of the Sun aspect angle. The prediction capability of these models is investigated by comparing predicted ionospheric delays with measured ones on real flight data from the Gravity Recovery and Climate Experiment mission, in which two satellites fly separated of more than 200 km. Results demonstrate that both models exhibit a correlation in the excess of 80% between predicted and measured double-difference ionospheric delays. Despite its higher simplicity, the isotropic model performs better than the model including the Sun effect, being able to predict double differenced delays with accuracy smaller than the carrier wavelength in most cases. The model is thus fit for supporting integer ambiguity fixing in real-time filters for relative navigation over large baselines. Concerning zero-difference ionospheric delays, results demonstrate that delays predicted by the isotropic model are highly correlated (around 90%) with those estimated using GPS measurements. However, the difference between predicted and measured delays has a root mean square error in the excess of 30 cm. Thus, the zero-difference ionospheric delays model is not likely to be an alternative to methods exploiting carrier-phase observables for cancelling out the ionosphere contribution in single-frequency absolute navigation filters.

チャネル間ハードウェアバイアス較正によるRTK-GPS/GLONASS測位の性能評価

The purpose of this paper is to improve GLONASS positioning performance in RTK-GPS/GLONASS between different types of receivers. Tables of inter-channel code biases in various receivers were obtained with a zero baseline test in order to examine the magnitude of inter-channel code bias between the receivers. We showed the improvement of fix rate and positioning accuracy of RTK-GPS/GLONASS between different types of receivers by using table of the calibrated GLONASS biases and partial ambiguity fixing solution of GPS and GLONASS for the aid of GPS carrier-phase ambiguity resolution. Furthermore, we showed that the bias correction method suggested in this paper is applicable to RTK-GPS/GLONASS with multi-epoch measurements as well as with single-epoch measurements.

Regional reference network augmented precise point positioning for instantaneous ambiguity resolution

Integer ambiguity fixing can significantly shorten the initialization time and improve the accuracy of precise point positioning (PPP), but it still takes approximate 15 min of time to achieve reliable integer ambiguity solutions. In this contribution, we present a new strategy to augment PPP estimation with a regional reference network, so that instantaneous ambiguity fixing is achievable for users within the network coverage. In the proposed method, precise zero-differenced atmospheric delays are derived from the PPP fixed solution of the reference stations, which are disseminated to, and interpolated at user stations to correct for L1, L2 phase observations or their combinations. With the corrected observations, instantaneous ambiguity resolution can be carried out within the user PPP software, thus achieving the position solutions equivalent to the network real-time kinematic positioning (NRTK). The strategy is validated experimentally. The derived atmospheric delays and the interpolated corrections are investigated. The ambiguity fixing performance and the resulted position accuracy are assessed. The validation confirms that the new strategy can provide comparable service with NRTK. Therefore, with this new processing strategy, it is possible to integrate PPP and NRTK into a seamless positioning service, which can provide an accuracy of about 10 cm anywhere, and upgrade to a few centimeters within a regional network.

Quality assessment of GPS rapid static positioning with weighted ionospheric parameters in generalized least squares

Precise GPS positioning requires the processing of carrier-phase observations and fixing integer ambiguities. With increasing distance between receivers, ambiguity fixing becomes more difficult because ionospheric and tropospheric effects do not cancel sufficiently in double differencing. A popular procedure in static positioning is to increase the length of the observing session and/or to apply atmospheric (ionospheric) models and corrections. We investigate the methodology for GPS rapid static positioning that requires just a few minutes of dual-frequency GPS observations for medium-length baselines. Ionospheric corrections are not required, but the ionospheric delays are treated as pseudo-observations having a priori values and respective weights. The tropospheric delays are reduced by using well-established troposphere models, and satellite orbital and clock errors are eliminated by using IGS rapid products. Several numerical tests based on actual GPS data are presented. It is shown that the proposed methodology is suitable for rapid static positioning within 50---70 km from the closest reference network station and that centimeter-level precision in positioning is feasible when using just 1 min of dual-frequency GPS data.

Integer Ambiguity Resolution on Undifferenced GPS Phase Measurements and Its Application to PPP and Satellite Precise Orbit Determination

Integer ambiguity fixing is routinely applied to double-differenced GPS phase measurements to achieve precise positioning. Double-differencing is interesting because it removes most of the common errors between the different signal paths. However, if common errors can be estimated it becomes attractive to fix integer ambiguities on undifferenced measurements. Phase measurements then become pseudorange-like measurements with a noise level of a few millimeters. This paper introduces a new method for fixing dual-frequency GPS ambiguities on undifferenced phase measurements either locally or globally. The clocks for the GPS constellation obtained during this process can be used for precise point positioning of ground based receivers and for precise orbit determination of low Earth orbiting satellites. The resulting positioning precision is comparable to that of standard differential positioning without the need for a reference station. Ambiguity-fixed satellite orbits for the GRACE and Jason satellites are more precise than the most precise solution available today.

Estimates of ocean tide loading displacements and its impact on position time series in Hong Kong using a dense continuous GPS network

Three-dimensional ocean tide loading (OTL) displacements of eight diurnal and semidiurnal constituents at 12 sites in Hong Kong were estimated using 3–7 years of continuous global positioning system (GPS) observations. OTL displacements were estimated using the precise point positioning (PPP) technique on a daily basis and then combined. The OTL displacements obtained by GPS were compared with predictions using seven recent global ocean tide models. The effect of OTL displacements on GPS position time series was also investigated. The study shows that the GPS-derived OTL displacements (excluding K1 and K2 constituents) agree best with those predicted by the GOT4.7 and NAO99b models. The GPS/model agreement is generally at the sub-millimeter level, except for S2, K1, and K2 constituents with relatively large errors. After systematic biases between the GPS and model values are removed, the misfits of all sites for M2, S2, N2, O1, P1, and Q1 are less than 0.5 and 1.0 mm in the horizontal and vertical components, respectively, while larger misfits (within 2.5 mm) are observed for K1 and K2. Integer ambiguity fixing slightly improves the east component of OTL displacement estimates. The study also finds that GPS-derived OTL corrections, instead of model predicts, can be used in daily data processing with the exception of K1 and K2. Including K2 corrections, a secular vertical rate of up to 1 mm/year in position time series can be induced, which needs to be confirmed by further studies.

Advantage of velocity measurements on instantaneous RTK positioning

Instantaneous real-time kinematic (RTK) techniques are one approach to achieving real-time high-accuracy positioning. The object of this paper is to show the advantage of adding velocity information to instantaneous RTK positioning. In this paper, velocity from Doppler frequency measurements is used to help resolve integer ambiguities in the LAMBDA method. In urban areas, pseudorange measurements can suffer from significant errors due to strong multipath, despite the use of advanced multipath-mitigation techniques. However, Doppler frequency measurements do not deteriorate as much as pseudoranges, because the multipath error on Doppler frequency measurements is only in the range of several centimeters. Our proposed method has been tested by both static and moving users in sub-urban environments. Single-epoch ambiguity fixing performance was improved compared to conventional ambiguity resolution without velocity information.

Resolution of GPS carrier-phase ambiguities in Precise Point Positioning (PPP) with daily observations

Precise Point Positioning (PPP) has been demonstrated to be a powerful tool in geodetic and geodynamic applications. Although its accuracy is almost comparable with network solutions, the east component of the PPP results is still to be improved by integer ambiguity fixing, which is, up to now, prevented by the presence of the uncalibrated phase delays (UPD) originating in the receivers and satellites. In this paper, it is shown that UPDs are rather stable in time and space, and can be estimated with high accuracy and reliability through a statistical analysis of the ambiguities estimated from a reference network. An approach is implemented to estimate the fractional parts of the single-difference (SD) UPDs between satellites in wide- and narrow-lane from a global reference network. By applying the obtained SD-UPDs as corrections to the SD-ambiguities at a single station, the corrected SD-ambiguities have a naturally integer feature and can therefore be fixed to integer values as usually done for the double-difference ones in the network mode. With data collected at 450 stations of the International GNSS Service (IGS) through days 106 to 119 in 2006, the efficiency of the presented ambiguity-fixing strategy is validated using IGS Final products. On average, more than 80% of the independent ambiguities could be fixed reliably, which leads to an improvement of about 27% in the repeatability and 30% in the agreement with the IGS weekly solutions for the east component of station coordinates, compared with the real-valued solutions.

Single epoch ambiguity resolution for Galileo with the CAR and LAMBDA methods

This paper investigates single epoch ambiguity resolution performance using Galileo four frequency data. Two commonly used ambiguity resolution methods are used in the tests, including the Cascade Ambiguity Resolution (CAR) and the Least-Squares Ambiguity Decorrelation Adjustment (LAMBDA) methods. For CAR method, four optimal combinations are carefully selected according to their wavelength to noise ratios and success rate for ambiguity fixing. The test results show, in general, the LAMBDA method performs better than the CAR method. The speed of ambiguity resolution is closely related to the carrier phase measurement precision. With carrier phase measurement precision of 3 mm, single epoch ambiguity resolution can be achieved at every epoch with simulated 1-s interval 24-h Galileo data (total epochs 86,400). With the increase of carrier phase noise, ambiguity resolution performances become worse. When the noise level is increased to 12 mm, single epoch ambiguity resolution can only be achieved about 50% of epochs.

Precise orbit determination for GRACE using undifferenced or doubly differenced GPS data

The two GRACE satellites provide the ideal platform to study the performance of different strategies for precise orbit determination using undifferenced or doubly differenced GPS data. We use pseudo-stochastic orbit modeling techniques in a batch least-squares environment for the two GRACE satellites to outline the mutual benefits of processing doubly differenced instead of undifferenced GPS data. We either process the space baseline only, the space-ground baselines only, or both types of baselines together, and show that the fixing of the GPS double difference carrier phase ambiguities has a significant impact on the space baseline, but also on the space-ground baselines. The validation of the relative orbit positions by inter-satellite K-band observations shows precisions of better than 1 mm in the case of fixed space baseline ambiguities, precisions of a few millimeter in the case of fixed space-ground baseline ambiguities, and precisions of about 1 cm in the case of float ambiguities. We discuss the differences between the various GRACE orbit solutions in order to formulate well suited orbit determination strategies tailored to the GRACE configuration. Satellite laser ranging observations indicate that accuracies between 2 cm and 2.5 cm are achieved.

A comparison of GPS, VLBI and model estimates of ocean tide loading displacements

In recent years, ocean tide loading displacements (OTLD) have been measured using the Global Positioning System (GPS) and Very Long Baseline Interferometry (VLBI). This study assesses the accuracy of GPS measurements of OTLD by comparison with VLBI measurements and estimates derived from numerical ocean tide models. A daily precise point positioning (PPP) analysis was carried out on ∼11 years of GPS data for each of 25 sites that have previous OTLD estimates based on data from co-located VLBI sites. Ambiguities were fixed to integer values where possible. The resulting daily estimates of OTLD, at eight principal diurnal and semi-diurnal tidal frequencies, were combined to give GPS measurements of OTLD at each site. The 3D GPS and VLBI measurements of OTLD were compared with estimates computed (by convolution with Green’s functions) from five modern ocean tide models (CSR4.0, FES2004, GOT00.2, NAO99b and TPXO6.2). The GPS/model agreement is shown to be similar to the VLBI/model agreement. In the important radial direction, the GPS/model misfit is shown to be smaller than the VLBI/model misfit for seven of the eight tidal constituents; the exception being the K2 constituent. Fixing of GPS carrier-phase ambiguities to integer values resulted in a marginal improvement to the GPS/model agreement. Statistically, it is shown there is no significance to the difference between the fit of the GPS and VLBI measurements of OTLD to modelled values. Equally, differences in fit of either the complete set of GPS or VLBI estimates to the five sets of model-derived values cannot be identified with statistical significance. It is thus concluded that, overall, we cannot distinguish between GPS and VLBI measurements of OTLD, and that at the global scale, present ocean tide models are accurate to within the current measurement noise of these techniques.