Three Carrier Ambiguity Resolutions : Generalised problems, models and solutions

In this paper, the problems of three carrier phase ambiguity resolution (TCAR) and position estimation (PE) are generalized as real time GNSS data processing problems for a continuously observing network on large scale. In order to describe these problems, a general linear equation system is presented to uniform various geometry-free, geometry-based and geometry-constrained TCAR models, along with state transition questions between observation times. With this general formulation, generalized TCAR solutions are given to cover different real time GNSS data processing scenarios, and various simplified integer solutions, such as geometry-free rounding and geometry-based LAMBDA solutions with single and multiple-epoch measurements. In fact, various ambiguity resolution (AR) solutions differ in the floating ambiguity estimation and integer ambiguity search processes, but their theoretical equivalence remains under the same observational systems models and statistical assumptions. TCAR performance benefits as outlined from the data analyses in some recent literatures are reviewed, showing profound implications for the future GNSS development from both technology and application perspectives.

Computed success rates of various carrier phase integer estimation solutions and their comparison with statistical success rates

The success rate of carrier phase ambiguity resolution (AR) is the probability that the ambiguities are successfully fixed to their correct integer values. In existing works, an exact success rate formula for integer bootstrapping estimator has been used as a sharp lower bound for the integer least squares (ILS) success rate. Rigorous computation of success rate for the more general ILS solutions has been considered difficult, because of complexity of the ILS ambiguity pull-in region and computational load of the integration of the multivariate probability density function. Contributions of this work are twofold. First, the pull-in region mathematically expressed as the vertices of a polyhedron is represented by a multi-dimensional grid, at which the cumulative probability can be integrated with the multivariate normal cumulative density function (mvncdf) available in Matlab. The bivariate case is studied where the pull-region is usually defined as a hexagon and the probability is easily obtained using mvncdf at all the grid points within the convex polygon. Second, the paper compares the computed integer rounding and integer bootstrapping success rates, lower and upper bounds of the ILS success rates to the actual ILS AR success rates obtained from a 24 h GPS data set for a 21 km baseline. The results demonstrate that the upper bound probability of the ILS AR probability given in the existing literatures agrees with the actual ILS success rate well, although the success rate computed with integer bootstrapping method is a quite sharp approximation to the actual ILS success rate. The results also show that variations or uncertainty of the unit–weight variance estimates from epoch to epoch will affect the computed success rates from different methods significantly, thus deserving more attentions in order to obtain useful success probability predictions.

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.

Estimating the probability of an invalid match for matching using the rank transform

A fundamental problem faced by stereo matching algorithms is the matching or correspondence problem. A wide range of algorithms have been proposed for the correspondence problem. For all matching algorithms, it would be useful to be able to compute a measure of the probability of correctness, or reliability of a match. This paper focuses in particular on one class for matching algorithms, which are based on the rank transform. The interest in these algorithms for stereo matching stems from their invariance to radiometric distortion, and their amenability to fast hardware implementation. This work differs from previous work in that it derives, from first principles, an expression for the probability of a correct match. This method was based on an enumeration of all possible symbols for matching. The theoretical results for disparity error prediction, obtained using this method, were found to agree well with experimental results. However, disadvantages of the technique developed in this chapter are that it is not easily applicable to real images, and also that it is too computationally expensive for practical window sizes. Nevertheless, the exercise provides an interesting and novel analysis of match reliability.

A fiber Bragg grating temperature sensor with adjustable sensitivity [一种灵敏度系数可调的光纤光栅温度传感器]

As a novel sensing element, fiber Bragg grating (FBG) is sensitive to both temperature and strain. Basing on this character, high sensitivity FBG temperature sensor can be made. However, as a result of the strain limit of the fiber, the temperature range it can endure is quite narrow. This drawback limits its application and complicates its storage and transport. We design and manufacture a FBG temperature sensor with tunable sensitivity. By tuning its sensitivity, its temperature range is changed, which enlarges its application field, solves the problem of storage and transport, and brighten the future of FBG in temperature measurement. In experiment, by changing the fixing position of the bimetal we tuned the sensitivity of the high sensitivity FBG sensor to different values (-47 pm/℃，-97.7 pm/℃，-153.3 pm/℃).

A reference station-based GNSS computing mode to support unified precise point positioning and real-time kinematic services

Currently, the GNSS computing modes are of two classes: network-based data processing and user receiver-based processing. A GNSS reference receiver station essentially contributes raw measurement data in either the RINEX file format or as real-time data streams in the RTCM format. Very little computation is carried out by the reference station. The existing network-based processing modes, regardless of whether they are executed in real-time or post-processed modes, are centralised or sequential. This paper describes a distributed GNSS computing framework that incorporates three GNSS modes: reference station-based, user receiver-based and network-based data processing. Raw data streams from each GNSS reference receiver station are processed in a distributed manner, i.e., either at the station itself or at a hosting data server/processor, to generate station-based solutions, or reference receiver-specific parameters. These may include precise receiver clock, zenith tropospheric delay, differential code biases, ambiguity parameters, ionospheric delays, as well as line-of-sight information such as azimuth and elevation angles. Covariance information for estimated parameters may also be optionally provided. In such a mode the nearby precise point positioning (PPP) or real-time kinematic (RTK) users can directly use the corrections from all or some of the stations for real-time precise positioning via a data server. At the user receiver, PPP and RTK techniques are unified under the same observation models, and the distinction is how the user receiver software deals with corrections from the reference station solutions and the ambiguity estimation in the observation equations. Numerical tests demonstrate good convergence behaviour for differential code bias and ambiguity estimates derived individually with single reference stations. With station-based solutions from three reference stations within distances of 22–103 km the user receiver positioning results, with various schemes, show an accuracy improvement of the proposed station-augmented PPP and ambiguity-fixed PPP solutions with respect to the standard float PPP solutions without station augmentation and ambiguity resolutions. Overall, the proposed reference station-based GNSS computing mode can support PPP and RTK positioning services as a simpler alternative to the existing network-based RTK or regionally augmented PPP systems.

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.

A new ambiguity acceptance test threshold determination method with controllable failure rate

The ambiguity acceptance test is an important quality control procedure in high precision GNSS data processing. Although the ambiguity acceptance test methods have been extensively investigated, its threshold determine method is still not well understood. Currently, the threshold is determined with the empirical approach or the fixed failure rate (FF-) approach. The empirical approach is simple but lacking in theoretical basis, while the FF-approach is theoretical rigorous but computationally demanding. Hence, the key of the threshold determination problem is how to efficiently determine the threshold in a reasonable way. In this study, a new threshold determination method named threshold function method is proposed to reduce the complexity of the FF-approach. The threshold function method simplifies the FF-approach by a modeling procedure and an approximation procedure. The modeling procedure uses a rational function model to describe the relationship between the FF-difference test threshold and the integer least-squares (ILS) success rate. The approximation procedure replaces the ILS success rate with the easy-to-calculate integer bootstrapping (IB) success rate. Corresponding modeling error and approximation error are analysed with simulation data to avoid nuisance biases and unrealistic stochastic model impact. The results indicate the proposed method can greatly simplify the FF-approach without introducing significant modeling error. The threshold function method makes the fixed failure rate threshold determination method feasible for real-time applications.

A novel ambiguity acceptance test threshold determination method with controllable failure rate

Ambiguity validation as an important procedure of integer ambiguity resolution is to test the correctness of the fixed integer ambiguity of phase measurements before being used for positioning computation. Most existing investigations on ambiguity validation focus on test statistic. How to determine the threshold more reasonably is less understood, although it is one of the most important topics in ambiguity validation. Currently, there are two threshold determination methods in the ambiguity validation procedure: the empirical approach and the fixed failure rate (FF-) approach. The empirical approach is simple but lacks of theoretical basis. The fixed failure rate approach has a rigorous probability theory basis, but it employs a more complicated procedure. This paper focuses on how to determine the threshold easily and reasonably. Both FF-ratio test and FF-difference test are investigated in this research and the extensive simulation results show that the FF-difference test can achieve comparable or even better performance than the well-known FF-ratio test. Another benefit of adopting the FF-difference test is that its threshold can be expressed as a function of integer least-squares (ILS) success rate with specified failure rate tolerance. Thus, a new threshold determination method named threshold function for the FF-difference test is proposed. The threshold function method preserves the fixed failure rate characteristic and is also easy-to-apply. The performance of the threshold function is validated with simulated data. The validation results show that with the threshold function method, the impact of the modelling error on the failure rate is less than 0.08%. Overall, the threshold function for the FF-difference test is a very promising threshold validation method and it makes the FF-approach applicable for the real-time GNSS positioning applications.

Multi-GNSS precise point positioning with raw single-frequency and dual-frequency measurement models

The emergence of multiple satellite navigation systems, including BDS, Galileo, modernized GPS, and GLONASS, brings great opportunities and challenges for precise point positioning (PPP). We study the contributions of various GNSS combinations to PPP performance based on undifferenced or raw observations, in which the signal delays and ionospheric delays must be considered. A priori ionospheric knowledge, such as regional or global corrections, strengthens the estimation of ionospheric delay parameters. The undifferenced models are generally more suitable for single-, dual-, or multi-frequency data processing for single or combined GNSS constellations. Another advantage over ionospheric-free PPP models is that undifferenced models avoid noise amplification by linear combinations. Extensive performance evaluations are conducted with multi-GNSS data sets collected from 105 MGEX stations in July 2014. Dual-frequency PPP results from each single constellation show that the convergence time of undifferenced PPP solution is usually shorter than that of ionospheric-free PPP solutions, while the positioning accuracy of undifferenced PPP shows more improvement for the GLONASS system. In addition, the GLONASS undifferenced PPP results demonstrate performance advantages in high latitude areas, while this impact is less obvious in the GPS/GLONASS combined configuration. The results have also indicated that the BDS GEO satellites have negative impacts on the undifferenced PPP performance given the current “poor” orbit and clock knowledge of GEO satellites. More generally, the multi-GNSS undifferenced PPP results have shown improvements in the convergence time by more than 60 % in both the single- and dual-frequency PPP results, while the positioning accuracy after convergence indicates no significant improvements for the dual-frequency PPP solutions, but an improvement of about 25 % on average for the single-frequency PPP solutions.

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.

RTK-based GNSS drifters for monitoring in estuaries and rivers: Field experimental results

This paper presents a novel RTK-based GNSS Lagrangian drifter system that is capable of monitoring water velocity, turbulence and dispersion coefficients of river and estuarine. The Lagrangian drifters use the dual-frequency real time kinematic (RTK) technique for both position and velocity estimations. The capsule is designed to meet the requirements such as minimizing height, diameter, minimizing the direct wind drag, positive buoyancy for satellite signal reception and stability, and waterproof housing for electronic components, such as GNSS receiver and computing board. The collected GNSS data are processed with post-processing RTK software. Several experiments have been carried out in two rivers in Brisbane and Sunshine Coast in Queensland. Results show that the high accuracy GNSS-drifters can be used to measure dispersion coefficient resulting from sub-tidal velocity fluctuations in shallow tidal water. In addition, the RTK-GNSS drifters respond well to vertical motion and thus could be applicable to flood monitoring.

Geometry-free stochastic analysis of BDS triple frequency signals

The paper presents a geometry-free approach to assess the variation of covariance matrices of undifferenced triple frequency GNSS measurements and its impact on positioning solutions. Four independent geometryfree/ ionosphere-free (GFIF) models formed from original triple-frequency code and phase signals allow for effective computation of variance-covariance matrices using real data. Variance Component Estimation (VCE) algorithms are implemented to obtain the covariance matrices for three pseudorange and three carrier-phase signals epoch-by-epoch. Covariance results from the triple frequency Beidou System (BDS) and GPS data sets demonstrate that the estimated standard deviation varies in consistence with the amplitude of actual GFIF error time series. The single point positioning (SPP) results from BDS ionosphere-free measurements at four MGEX stations demonstrate an improvement of up to about 50% in Up direction relative to the results based on a mean square statistics. Additionally, a more extensive SPP analysis at 95 global MGEX stations based on GPS ionosphere-free measurements shows an average improvement of about 10% relative to the traditional results. This finding provides a preliminary confirmation that adequate consideration of the variation of covariance leads to the improvement of GNSS state solutions.