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     

An alternative approach to calculate the posterior probability of GNSS integer ambiguity resolution

When precise positioning is carried out via GNSS carrier phases, it is important to make use of the property that every ambiguity should be an integer. With the known float solution, any integer vector, which has the same degree of freedom as the ambiguity vector, is the ambiguity vector in probability. For both integer aperture estimation and integer equivariant estimation, it is of great significance to know the posterior probabilities. However, to calculate the posterior probability, we have to face the thorny problem that the equation involves an infinite number of integer vectors. In this paper, using the float solution of ambiguity and its variance matrix, a new approach to rapidly and accurately calculate the posterior probability is proposed. The proposed approach consists of four steps. First, the ambiguity vector is transformed via decorrelation. Second, the range of the adopted integer of every component is directly obtained via formulas, and a finite number of integer vectors are obtained via combination. Third, using the integer vectors, the principal value of posterior probability and the correction factor are worked out. Finally, the posterior probability of every integer vector and its error upper bound can be obtained. In the paper, the detailed process to calculate the posterior probability and the derivations of the formulas are presented. The theory and numerical examples indicate that the proposed approach has the advantages of small amount of computations, high calculation accuracy and strong adaptability.     

On the probability distribution of GNSS carrier phase observations

When processing observational data from global navigation satellite systems (GNSS), the carrier phase measurements are generally assumed to follow a normal distribution. Although full knowledge of the probability distribution of the observables is not required for parameter estimation, for example when using the least-squares method, the distributional properties of GNSS observations play a key role in quality control procedures, such as outlier and cycle-slip detection, in ambiguity resolution, as well as in the reliability assessment of estimation results. In addition, when applying GNSS positioning under critical observation conditions with respect to multipath and atmospheric effects, the validity of the normal distribution assumption of GNSS observables certainly comes into doubt. This paper illustrates the discrepancies between the normal distribution assumption and reality, based on a large and representative data set of GPS phase measurements covering a range of factors, including multipath impact, baseline length, and atmospheric conditions. The statistical inferences are made using the first through fourth sample moments, hypothesis tests, and graphical tools such as histograms and quantile–quantile plots. The results show clearly that multipath effects, in particular the near-field component, produce the dominant influence on the distributional characteristics of GNSS observables. Additionally, using surface meteorological data, considerable correlations between distributional deviations from normality on the one hand and atmospheric relative humidity on the other are detected.     

Influence of ambiguity precision on the success rate of GNSS integer ambiguity bootstrapping

In this contribution, we study the dependence of the bootstrapped success rate on the precision of the GNSS carrier phase ambiguities. Integer bootstrapping is, because of its ease of computation, a popular method for resolving the integer ambiguities. The method is however known to be suboptimal, because it only takes part of the information from the ambiguity variance matrix into account. This raises the question in what way the bootstrapped success rate is sensitive to changes in precision of the ambiguities. We consider two different cases. (1) The effect of improving the ambiguity precision, and (2) the effect of using an approximate ambiguity variance matrix. As a by-product, we also prove that integer bootstrapping is optimal within the restricted class of sequential integer estimators.     

GNSS antenna array-aided CORS ambiguity resolution

Array-aided precise point positioning is a measurement concept that uses GNSS data, from multiple antennas in an array of known geometry, to realize improved GNSS parameter estimation proposed by Teunissen (IEEE Trans Signal Process 60:2870–2881, 2012). In this contribution, the benefits of array-aided CORS ambiguity resolution are explored. The mathematical model is formulated to show how the platform-array data can be reduced and how the variance matrix of the between-platform ambiguities can profit from the increased precision of the reduced platform data. The ambiguity resolution performance will be demonstrated for varying scenarios using simulation. We consider single-, dual- and triple-frequency scenarios of geometry-based and geometry-free models for different number of antennas and different standard deviations of the ionosphere-weighted constraints. The performances of both full and partial ambiguity resolution (PAR) are presented for these different scenarios. As the study shows, when full advantage is taken of the array antennas, both full and partial ambiguity resolution can be significantly improved, in some important cases even enabling instantaneous ambiguity resolution. PAR widelaning and its suboptimal character are hereby also illustrated.     

On the impact of GNSS ambiguity resolution: geometry,ionosphere, time and biases

Integer ambiguity resolution (IAR) is the key to fast and precise GNSS positioning and navigation. Next to the positioning parameters, however, there are several other types of GNSS parameters that are of importance for a range of different applications like atmospheric sounding, instrumental calibrations or time transfer. As some of these parameters may still require pseudo-range data for their estimation, their response to IAR may differ significantly. To infer the impact of ambiguity resolution on the parameters, we show how the ambiguity-resolved double-differenced phase data propagate into the GNSS parameter solutions. For that purpose, we introduce a canonical decomposition of the GNSS network model that, through its decoupled and decorrelated nature, provides direct insight into which parameters, or functions thereof, gain from IAR and which do not. Next to this qualitative analysis, we present for the GNSS estimable parameters of geometry, ionosphere, timing and instrumental biases closed-form expressions of their IAR precision gains together with supporting numerical examples.     

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.     

Integer ambiguity validation in high accuracy GNSS positioning

GPS Solutions - Carrier phase observations are required for high-accuracy positioning with Global Navigation Satellite Systems. This requires that the correct number of whole carrier cycles in each...     

Fast GNSS ambiguity resolution as an ill-posed problem

A linear observational equation system for real-time GNSS carrier phase ambiguity resolution (AR) is often severely ill-posed in the case of poor satellite geometry. An ill-posed system may result in unreliable or unsuccessful AR if no care is taken to mitigate this situation. In this paper, the GNSS AR model as an ill-posed problem is solved by regularizing its baseline and ambiguity parameters, respectively, with the threefold contributions: (i) The regularization parameter is reliably determined in context of minimizing mean square error of regularized solution where the covariance matrix of initial values of unknowns is used as an approximate smoothness term instead of the quadratic matrix of the true values of unknowns; (ii) The different models for computing initial values of unknowns are systematically discussed in order to address the potential schemes in real world applications; (iii) The superior performance of the regularized AR are demonstrated through the numerically random simulations as well as the real GPS experiments. The results show that the proposed regularization strategies can effectively mitigate the model’s ill-condition and improve the success AR probability of the observational system with a severely ill-posed problem.     

The ratio test for future GNSS ambiguity resolution

The performance of the popular ambiguity ratio test is analyzed. Based on experimental and simulated data, it is demonstrated that the current usage of the ratio test with fixed critical value is not sustainable in light of the enhanced variability that future global navigation satellite system (GNSS) ambiguity resolution will bring. As its replacement, the model-driven ratio test with fixed failure rate is proposed. The characteristics of this fixed-failure rate ratio test are described, and a performance analysis is given. The relation between its critical value and various GNSS model parameters is also studied. Finally, a procedure is presented for the creation of fixed failure rate look-up tables for the critical values of the ratio test.     

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.     

A new practical approach to GNSS high-dimensional ambiguity decorrelation

Based on both the lower and the upper triangular Cholesky decomposition algorithms, the (inverse) lower triangular Cholesky integer transformation and the (inverse) upper triangular Cholesky integer transformation are defined, and the (inverse) paired Cholesky integer transformation is proposed. Then, for the case of high-correlation ambiguity, a multi-time (inverse) paired Cholesky integer transformation is given. In addition, a simple and practical criterion is presented to solve the uniqueness problem of the integer transformation. It is verified by an example that (1) the (inverse) paired Cholesky integer transformation is very convenient and very efficient in practical computation; (2) the (inverse) paired Cholesky integer transformation is better than both the (inverse) lower triangular Cholesky integer transformation and the (inverse) upper triangular Cholesky integer transformation; and that (3) the inverse paired Cholesky integer transformation outperforms the paired Cholesky integer transformation slightly in the most cases.     

Generalized integer aperture estimation for partial GNSS ambiguity fixing

In satellite navigation, the key to high precision is to make use of the carrier-phase measurements. The periodicity of the carrier-phase, however, leads to integer ambiguities. Often, resolving the full set of ambiguities cannot be accomplished for a given reliability constraint. In that case, it can be useful to resolve a subset of ambiguities. The selection of the subset should be based not only on the stochastic system model but also on the actual measurements from the tracking loops. This paper presents a solution to the problem of joint subset selection and ambiguity resolution. The proposed method can be interpreted as a generalized version of the class of integer aperture estimators. Two specific realizations of this new class of estimators are presented, based on different acceptance tests. Their computation requires only a single tree search, and can be efficiently implemented, e.g., in the framework of the well-known LAMBDA method. Numerical simulations with double difference measurements based on Galileo E1 signals are used to evaluate the performance of the introduced estimation schemes under a given reliability constraint. The results show a clear gain of partial fixing in terms of the probability of correct ambiguity resolution, leading to improved baseline estimates.     

Integrity monitoring-based ratio test for GNSS integer ambiguity validation

GPS Solutions - The combination of multiple global navigation satellite systems (GNSSs) is able to improve the accuracy and reliability, which is beneficial for navigation in safety–critical...     

Robustness of GNSS integer ambiguity resolution in the presence of atmospheric biases

Both the underlying model strength and biases are two crucial factors for successful integer GNSS ambiguity resolution (AR) in real applications. In some cases, the biases can be adequately parameterized and an unbiased model can be formulated. However, such parameterization will, as trade-off, reduce the model strength as compared to the model in which the biases are ignored. The AR performance with the biased model may therefore be better than with the unbiased model, if the biases are sufficiently small. This would allow for faster AR using the biased model, after which the unbiased model can be used to estimate the remaining unknown parameters. We assess the bias-affected AR performance in the presence of tropospheric and ionospheric biases and compare it with the unbiased case. As a result, the maximum allowable biases are identified for different situations where CORS, static and kinematic baseline models are considered with different model settings. Depending on the size of the maximum allowable bias, a user may decide to use the biased model for AR or to use the unbiased model both for AR and estimating the other unknown parameters.     

GNSS模糊度降相关性能的条件方差平稳度评价法

GNSS模糊度降相关通过整数变换优化条件方差的排列顺序,提高搜索效率。降相关和条件方差的关系及其评价是关键问题之一。针对这一问题,本文从理论上分析了排序后模糊度降相关与条件方差之间的数值关系,发现降相关性能与条件方差数值序列的平稳性有关,降相关性能越强,条件方差数值序列越平稳。基于这一理论关系,给出了"条件方差平稳度"定义,并将其作为评价降相关性能的指标。通过模拟和实测数据验证,并采用条件方差变化趋势图和搜索时间来定性和定量评价降相关性能,用以判定条件方差平稳度的合理性。试验结果表明,条件方差平稳度可以较精确直观地衡量模糊度的降相关性能。本文定义的指标揭示了模糊度降相关的本质。     

GNSS three carrier ambiguity resolution using ionosphere-reduced virtual signals

This paper presents a general modeling strategy for ambiguity resolution (AR) and position estimation (PE) using three or more phase-based ranging signals from a global navigation satellite system (GNSS). The proposed strategy will identify three best “virtual” signals to allow for more reliable AR under certain observational conditions characterized by ionospheric and tropospheric delay variability, level of phase noise and orbit accuracy. The selected virtual signals suffer from minimal or relatively low ionospheric effects, and thus are known as ionosphere-reduced virtual signals. As a result, the ionospheric parameters in the geometry-based observational models can be eliminated for long baselines, typically those of length tens to hundreds of kilometres. The proposed modeling comprises three major steps. Step 1 is the geometry-free determination of the extra-widelane (EWL) formed between the two closest L-band carrier measurements, directly from the two corresponding code measurements. Step 2 forms the second EWL signal and resolves the integer ambiguity with a geometry-based estimator alone or together with the first EWL. This is followed by a procedure to correct for the first-order ionospheric delay using the two ambiguity-fixed widelane (WL) signals derived from the integer-fixed EWL signals. Step 3 finds an independent narrow-lane (NL) signal, which is used together with a refined WL to resolve NL ambiguity with geometry-based integer estimation and search algorithms. As a result, the above two AR processes performed with WL/NL and EWL/WL signals respectively, either in sequence or in parallel, can support real time kinematic (RTK) positioning over baselines of tens to hundreds of kilometres, thus enabling centimetre-to-decimentre positioning at the local, regional and even global scales in the future.     

Cycled efficient V-BLAST GNSS ambiguity decorrelation and search complexity estimation

GPS Solutions - Decorrelation reduction and search techniques play important roles in solving integer ambiguities of global navigation satellite system (GNSS) positioning. This contribution focuses...