An approach to GLONASS ambiguity resolution

 When processing global navigation satellite system (GLONASS) carrier phases, the standard double-differencing (DD) procedure cannot cancel receiver clock terms in the DD phase measurement equations due to the multiple frequencies of the carrier phases. Consequently, a receiver clock parameter has to be set up in the measurement equations in addition to baseline components and DD ambiguities. The resulting normal matrix unfortunately becomes singular. Methods to deal with this problem have been proposed in the literature. However, these methods rely on the use of pseudo-ranges. As pseudo-ranges are contaminated by multi-path and hardware delays, biases in these pseudo-ranges are significant, which may result in unreliable ambiguity resolution. A new approach is addressed that is not sensitive to the biases in the pseudo-ranges. The proposed approach includes such steps as converting the carrier phases to their distances to cancel the receiver clock errors, and searching for the most likely single-differenced (SD) ambiguity. Based on the results from the theoretical investigation, a practical procedure for GLONASS ambiguity resolution is presented. The initial experimental results demonstrate that the proposed approach is useable in cases of GLONASS and combined global positioning system (GPS) and GLONASS positioning. Received: 19 August 1998 / Accepted: 12 November 1999     

The least-squares ambiguity decorrelation adjustment: its performance on short GPS baselines and short observation spans

The least-squares ambiguity decorrelation adjustment is a method for fast GPS double-difference (DD) integer ambiguity estimation. The performance of the method will be discussed, and although it is stressed that the method is generally applicable, attention is restricted to short-baseline applications in the present contribution. With reference to the size and shape of the ambiguity search space, the volume of the search space will be introduced as a measure for the number of candidate grid points, and the signature of the spectrum of conditional variances will be used to identify the difficulty one has in computing the integer DD ambiguities. It is shown that the search for the integer least-squares ambiguities performs poorly when it takes place in the space of original DD ambiguities. This poor performance is explained by means of the discontinuity in the spectrum of conditional variances. It is shown that through a decorrelation of the ambiguities, transformed ambiguities are obtained which generally have a flat and lower spectrum, thereby enabling a fast and efficient search. It is also shown how the high precision and low correlation of the transformed ambiguities can be used to scale the search space so as to avoid an abundance of unnecessary candidate grid points. Numerical results are presented on the spectra of conditional variances and on the statistics of both the original and transformed ambiguities. Apart from presenting numerical results which can typically be achieved, the contribution also emphasizes and explains the impact on the method's performance of different measurement scenarios, such as satellite redundancy, single vs dual-frequency data, the inclusion of code data and the length of the observation time span. Received: 31 October 1995 / Accepted: 21 March 1997     

The Bayesian approach applied to GPS ambiguity resolution. A mixture model for the discrete–real ambiguities alternative

 The problem of phase ambiguity resolution in global positioning system (GPS) theory is considered. The Bayesian approach is applied to this problem and, using Monte Carlo simulation to search over the integer candidates, a practical expression for the Bayesian estimator is obtained. The analysis of the integer grid points inside the search ellipsoid and their evolution with time, while measurements are accumulated, leads to the development of a Bayesian theory based on a mathematical mixture model for the ambiguity. Received: 29 March 2001 / Accepted: 3 September 2001     

Confidence regions for GPS baselines by Bayesian statistics

 The global positioning system (GPS) model is distinctive in the way that the unknown parameters are not only real-valued, the baseline coordinates, but also integers, the phase ambiguities. The GPS model therefore leads to a mixed integer–real-valued estimation problem. Common solutions are the float solution, which ignores the ambiguities being integers, or the fixed solution, where the ambiguities are estimated as integers and then are fixed. Confidence regions, so-called HPD (highest posterior density) regions, for the GPS baselines are derived by Bayesian statistics. They take care of the integer character of the phase ambiguities but still consider them as unknown parameters. Estimating these confidence regions leads to a numerical integration problem which is solved by Monte Carlo methods. This is computationally expensive so that approximations of the confidence regions are also developed. In an example it is shown that for a high confidence level the confidence region consists of more than one region. Received: 1 February 2001 / Accepted: 18 July 2001     

The success rate and precision of GPS ambiguities

 An application of a theorem on the optimality of integer least-squares (LS) is described. This theorem states that the integer LS estimator maximizes the ambiguity success rate within the class of admissible integer estimators. This theorem is used to show how the probability of correct integer estimation depends on changes in the second moment of the ambiguity `float' solution. The distribution of the `float' solution is considered to be a member of the broad family of elliptically contoured distributions. Eigenvalue-based bounds for the ambiguity success rate are obtained. Received: 11 January 1999 / Accepted: 2 November 1999     

An integer ambiguity resolution procedure for GPS/pseudolite/INS integration

In order to achieve a precise positioning solution from GPS, the carrier-phase measurements with correctly resolved integer ambiguities must be used. Based on the integration of GPS with pseudolites and Inertial Navigation Systems (INS), this paper proposes an effective procedure for single-frequency carrier-phase integer ambiguity resolution. With the inclusion of pseudolites and INS measurements, the proposed procedure can speed up the ambiguity resolution process and increase the reliability of the resolved ambiguities. In addition, a recently developed ambiguity validation test, and a stochastic modelling scheme (based on-line covariance matrix estimation) are adapted to enhance the quality of ambiguity resolution. The results of simulation studies and field experiments indicate that the proposed procedure indeed improves the performance of single-frequency ambiguity resolution in terms of both reliability and time-to-fix-ambiguity.     

The parameter distributions of the integer GPS model

 A parameter estimation theory is incomplete if no rigorous measures are available for describing the uncertainty of the parameter estimators. Since the classical theory of linear estimation does not apply to the integer GPS model, rigorous probabilistic statements cannot be made with reference to the classical results. The fact that integer parameters are involved in the estimation process forces a reappraisal of the propagation of uncertainty. It is with this purpose in mind that the joint and marginal distributional properties of both the integer and non-integer parameters of the GPS model are determined. These joint distributions can also be used to determine the distribution of functions of the parameters. As an important example, the distribution of the vector of ambiguity residuals is determined. Received: 30 January 2001 / Accepted: 31 July 2001     

Quality-control issues relating to instantaneous ambiguity resolution for real-time GPS kinematic positioning

An integrated method for the instantaneous ambiguity resolution using dual-frequency precise pseudo-range and carrier-phase observations is suggested in this paper. The algorithm combines the search procedures in the coordinate domain, the observation domain and the estimated ambiguity domain (and therefore benefits from the integration of their most positive elements). A three-step procedure is then proposed to enhance the reliability of the ambiguity resolution by: (1) improving the stochastic model for the double-differenced functional model in real time; (2) refining the criteria which distinguish the integer ambiguity set that generates the minimum quadratic form of residuals from that corresponding to the second minimum one; and (3) developing a fault detection and adaptation procedure. Three test scenarios were considered, one static baseline (11.3 km) and two kinematic experiments (baseline lengths from 5.2 to 13.7 km). These showed that the mean computation time for one epoch is less than 0.1 s, and that the success rate reaches 98.4% (compared to just 68.4% using standard ratio tests). Received: 5 June 1996; Accepted: 16 January 1997     

A new method for three-carrier GNSS ambiguity resolution

A new method for resolving the carrier-phase integer ambiguity in Global Navigation Satellite Systems (GNSS) is presented: the MOdified Cholesky factorization for Ambiguity (MOCA) resolution. The characteristics and features of this method are described and results obtained using a software simulator and an emulator are presented to validate its efficiency. The results are then compared to those obtained using another existing method and good performance of the MOCA method in new GNSS systems is shown. Furthermore, the proposed method yields accurate results even when short time spans are used or when there are poor estimations of measurement error, making it immune to non-ideal conditions and ultimately a practical solution for real applications.     

A discrimination test procedure for ambiguity resolution on-the-fly

Ambiguity validation tests include the acceptance test and discrimination test, which are both important steps in the Global Positioning System ambiguity resolution process. An ambiguity discrimination test procedure based on a test statistic which is constructed by the difference (not the ratio as used in current procedures) between the minimum and second minimum quadratic form of the residuals in ambiguity identification, and its standard deviation, is proposed. The distribution function of the proposed test statistic is theoretically identified as a standard normal distribution when the known a priori variance factor is used, or as a Student's t distribution when the estimated variance factor is used. With this procedure, the ambiguity discrimination test is based on a more rigorous test statistic whose critical value can be calculated with any chosen level of significance. Test results indicate that the proposed ambiguity discrimination test procedure is reliable for use in ambiguity resolution on-the-fly. Received: 17 December 1997 / Accepted: 30 June 1998     

Random simulation and GPS decorrelation

 (i) A random simulation approach is proposed, which is at the centre of a numerical comparison of the performances of different GPS decorrelation methods. The most significant advantage of the approach is that it does not depend on nor favour any particular satellite–receiver geometry and weighting system. (ii) An inverse integer Cholesky decorrelation method is proposed, which will be shown to out-perform the integer Gaussian decorrelation and the Lenstra, Lenstra and Lovász (LLL) algorithm, and thus indicates that the integer Gaussian decorrelation is not the best decorrelation technique and that further improvement is possible. (iii) The performance study of the LLL algorithm is the first of its kind and the results have shown that the algorithm can indeed be used for decorrelation, but that it performs worse than the integer Gaussian decorrelation and the inverse integer Cholesky decorrelation. (iv) Simulations have also shown that no decorrelation techniques available to date can guarantee a smaller condition number, especially in the case of high dimension, although reducing the condition number is the goal of decorrelation. Received: 26 April 2000 / Accepted: 5 March 2001     

The probability distribution of the ambiguity bootstrapped GNSS baseline

 The purpose of carrier phase ambiguity resolution is to improve upon the quality of the estimated global navigation satellite system baseline by means of the integer ambiguity constraints. However, in order to evaluate the quality of the ambiguity resolved baseline rigorously, its probability distribution is required. This baseline distribution depends on the random characteristics of the estimated integer ambiguities, which in turn depend on the chosen integer estimator. In this contribution is presented an exact and closed-form expression for the baseline distribution in the case that use is made of integer bootstrapping. Also presented are the bootstrapped probability mass function and easy-to-compute measures for the bootstrapped baseline's probability of concentration. Received: 28 September 2000 / Accepted: 11 January 2001     

On determination of heights by using terrestrial clocks and GPS signals

 Considering a GPS satellite and two terrestrial stations, two types of equations are derived relating the heights of the two stations to the measured data (frequency ratio or clock rate differences) and the coordinates and velocity components of all three participating objects. The potential possibilities of using such relations for the determination of heights (in terms of geopotential numbers or orthometric heights) are discussed. Received: 6 December 2000 / Accepted: 9 July 2001     

GPS network monitors the Western Alps' deformation over a five-year period: 1993–1998

  The Western Alps are among the best studied collisional belts with both detailed structural mapping and also crustal geophysical investigations such as the ECORS and EGT seismic profile. By contrast, the present-day kinematics of the belt is still largely unknown due to small relative motions and the insufficient accuracy of the triangulation data. As a consequence, several tectonic problems still remain to be solved, such as the amount of N–S convergence in the Occidental Alps, the repartition of the deformation between the Alpine tectonic units, and the relation between deformation and rotation across the Alpine arc. In order to address these problems, the GPS ALPES group, made up of French, Swiss and Italian research organizations, has achieved the first large-scale GPS surveys of the Western Alps. More than 60 sites were surveyed in 1993 and 1998 with a minimum observation of 3 days at each site. GPS data processing has been done by three independent teams using different software. The different solutions have horizontal repeatabilities (N–E) of 4–7 mm in 1993 and 2–3 mm in 1998 and compare at the 3–5-mm level in position and 2-mm/yr level in velocity. A comparison of 1993 and 1998 coordinates shows that residual velocities of the GPS marks are generally smaller than 2 mm/yr, precluding a detailed tectonic interpretation of the differential motions. However, these data seem to suggest that the N–S compression of the Western Alps is quite mild (less than 2 mm/yr) compared to the global convergence between the African and Eurasian plate (6 mm/yr). This implies that the shortening must be accomodated elsewhere by the deformation of the Maghrebids and/or by rotations of Mediterranean microplates. Also, E–W velocity components analysis supports the idea that E–W extension exists, as already suggested by recent structural and seismotectonic data interpretation. Received: 27 November 2000 / Accepted: 17 September 2001     

Stochastic assessment of GPS carrier phase measurements for precise static relative positioning

 Global positioning system (GPS) carrier phase measurements are used in all precise static relative positioning applications. The GPS carrier phase measurements are generally processed using the least-squares method, for which both functional and stochastic models need to be carefully defined. Whilst the functional model for precise GPS positioning is well documented in the literature, realistic stochastic modelling for the GPS carrier phase measurements is still both a controversial topic and a difficult task to accomplish in practice. The common practice of assuming that the raw GPS measurements are statistically independent in space and time, and have the same accuracy, is certainly not realistic. Any mis-specification in the stochastic model will inevitably lead to unreliable positioning results. A stochastic assessment procedure has been developed to take into account the heteroscedastic, space- and time-correlated error structure of the GPS measurements. Test results indicate that the reliability of the estimated positioning results is improved by applying the developed stochastic assessment procedure. In addition, the quality of ambiguity resolution can be more realistically evaluated. Received: 13 February 2001 / Accepted: 3 September 2001     

Time transfer using GPS carrier phase: error propagation and results

 A joint time-transfer project between the Astronomical Institute of the University of Berne (AIUB) and the Swiss Federal Office of Metrology and Accreditation (METAS) was initiated to investigate the power of the time transfer using GPS carrier phase observations. Studies carried out in the context of this project are presented. The error propagation for the time-transfer solution using GPS carrier phase observations was investigated. To this purpose a simulation study was performed. Special interest was focussed on errors in the vertical component of the station position, antenna phase-center variations and orbit errors. A constant error in the vertical component introduces a drift in the time-transfer results for long baselines in east–west directions. The simulation study was completed by investigating the profit for time transfer when introducing the integer carrier phase ambiguities from a double-difference solution. This may reduce the drift in the time-transfer results caused by constant vertical error sources. The results from the present time-transfer solution are shown in comparison to results obtained with independent time-transfer techniques. The interpretation of the comparison benefits from the investigations of the error propagation study. Two types of solutions are produced on a regular basis at AIUB: one based on the rapid orbits from CODE, the other on the CODE final orbits. The rapid solution is available the day after the observations and has nearly the same quality as the final solution, which has a latency of about one week. The differences between these two solutions are below the nanosecond level. The differences from independent time-transfer techniques such as TWSTFT (two-way satellite time and frequency transfer) are a few nanoseconds for both products. Received: 15 November 2001 / Accepted: 6 September 2002 Correspondence to:R. Dach     

An improvement to ionospheric delay correction for single-frequency GPS users – the APR-I scheme

 For the commonly used GPS wide-area augmentation systems (WAAS) with a grid ionospheric model, the efficient modelling of ionospheric delays in real time, for single-frequency GPS users, is still a crucial issue which needs further research. This is particularly necessary when differential ionospheric delay corrections cannot be broadcast, when users cannot receive them, or when there are ionospheric anomalies. Ionospheric delays have a severe effect on navigation performance of single-frequency receivers. A new scheme is proposed which can efficiently address the above problems. The robust recurrence technique is based on the efficient combination of single-frequency GPS observations by users and the high-precision differential ionospheric delay corrections from WAAS. Its effectiveness is verified with examples. Received: 24 December 1999 / Accepted 21 February 2001     

Dual algebraic formulation of differential GPS

 A new approach to differential GPS is presented. The corresponding theoretical framework calls on elementary concepts of algebraic graph theory. The notion of double difference, which is related to that of closure in the sense of Kirchhoff, is revisited in this context. The Moore–Penrose pseudo-inverse of the closure operator plays a key role in the corresponding dual formulation. This approach, which is very attractive from a conceptual point of view, sheds a new light on the Teunissen formulation. Received: 21 May 2002 / Accepted: 18 November 2002 Correspondence to: A. Lannes     

Auto-covariance estimation of variable samples (ACEVS) and its application for monitoring random ionospheric disturbances using GPS

 Ionospheric variation may be considered as a stationary time series under quiet conditions. However, the disturbance of a stationary random process from stationarity results in the bias of corresponding samples from the stationary observations, and in the change of statistical model parameters of the process. From a general mathematical aspect, a new method is presented for monitoring ionospheric variations, based on the characteristic of time-series observation of GPS, and an investigation of the statistical properties of the estimated auto-covariance of the random ionospheric delay when changing the number of samples in the time series is carried out. A preliminary scheme for monitoring ionospheric delays is proposed. Received: 18 August 2000 / Accepted: 12 April 2001     

Reliable single-epoch ambiguity resolution for short baselines using combined GPS/BeiDou system

GNSS single-epoch real-time kinematic (RTK) positioning depends on correct ambiguity resolution. If the number of observed satellites in a single epoch is insufficient, which often happens with a standalone GNSS system, the ambiguity resolution is difficult to achieve. China’s BeiDou Navigation Satellite System has been providing continuous passive positioning, navigation and timing services since December 27, 2012, covering China and the surrounding area. This new system will increase the number of satellites in view and will have a significant effect on successful ambiguity resolution. Since the BeiDou system is similar to GPS, the procedure of data processing is easier than that for the Russian GLONASS system. We briefly introduce the time and the coordinate system of BeiDou and also the BeiDou satellite visibility in China, followed by the discussion on the combined GPS/BeiDou single-epoch algorithm. Experiments were conducted and are presented here, in which the GPS/BeiDou dual-frequency static data were collected in Wuhan with the baseline distance varying from 5 to 13 km, and processed in separate and combined modes. The results indicate that, compared to a standalone GPS or BeiDou system, the combined GNSS system can increase the successful ambiguity fixing rate for single epochs and can also improve the precision of short baselines determination.