Probability Of Correct Integer Estimation Research Articles

Partial Ambiguity Resolution for Ground and Space-Based Applications in a GPS+Galileo scenario: A simulation study

Integer Ambiguity Resolution (IAR) is the key to fast and precise GNSS positioning. The proper diagnostic metric for successful IAR is provided by the ambiguity success rate being the probability of correct integer estimation. In this contribution we analyse the performance of different GPS+Galileo models in terms of number of epochs needed to reach a pre-determined success rate, for various ground and space-based applications. The simulation-based controlled model environment enables us to gain insight into the factors contributing to the ambiguity resolution strength of the different GPS+Galileo models. Different scenarios of modernized GPS+Galileo are studied, encompassing the long baseline ground case as well as the medium dynamics case (airplane) and the space-based Low Earth Orbiter (LEO) case. In our analyses of these models the capabilities of partial ambiguity resolution (PAR) are demonstrated and compared to the limitations of full ambiguity resolution (FAR). The results show that PAR is generally a more efficient way than FAR to reduce the time needed to achieve centimetre-level positioning precision. For long single baselines, PAR can achieve time reductions of fifty percent to achieve such precision levels, while for multiple baselines it even becomes more effective, reaching reductions up to eighty percent for four station networks. For a LEO, the rapidly changing observation geometry does not even allow FAR, while PAR is then still possible for both dual- and triple-frequency scenarios. With the triple-frequency GPS+Galileo model the availability of precise positioning improves by fifteen percent with respect to the dual-frequency scenario.

An integer ambiguity resolution method for the global positioning system (GPS)-based land vehicle attitude determination

During attitude determination using a global positioning system (GPS), cycle slips occur due to the loss of lock and noise disturbance. Therefore, the integer ambiguity needs re-computation to isolate the error in carrier phase. This paper presents a fast method for integer ambiguity resolution for land vehicle application. After the cycle slips are detected, the velocity vector is utilized to obtain the rough baseline vector. The obtained baseline vector is substituted into carrier phase observation equations to solve the float ambiguity solution which can be used as a constraint to accelerate the integer ambiguity search procedure at next epochs. The probability of correct integer estimation in the expanded search space is analyzed. Experimental results demonstrate that the proposed method gives a fast approach to obtain new fixed ambiguities while the regular method takes longer time and sometimes results in incorrect solutions.

On InSAR Ambiguity Resolution For Deformation Monitoring

On InSAR Ambiguity Resolution For Deformation Monitoring

Ambiguity resolution for permanent scatterer interferometry

In the permanent scatterer technique of synthetic aperture radar interferometry, there is a need for an efficient and reliable nonlinear parameter inversion algorithm that includes estimation of the phase cycle ambiguities. Present techniques make use of a direct search of the solution space, treating the observations as deterministic and equally weighted, and which do not yield an exact solution. Moreover, they do not describe the quality of the estimated parameters. Here, we use the integer least squares estimator, which has the highest probability of correct integer estimation for problems with a multivariate normal distribution. With this estimator, the propagated variance-covariance matrix of the estimated parameters can be obtained. We have adapted the LAMBDA method, part of an integer least squares estimator developed for the ambiguity resolution of carrier phase observations in global positioning systems, to the problem of permanent scatterers. Key elements of the proposed method are the introduction of pseudo-observations to regularize the system of equations, decorrelation of the ambiguities for an efficient estimation, and the combination of a bootstrap estimator with an integer least squares search to obtain the final integer estimates. The performance of the proposed algorithm is demonstrated using simulated and real data.

On the approximation of the integer least-sqaures success rate: which lower or upper bound to use?

The probability of correct integer estimation, the success rate, is an important measure when the goal is fast and high precision positioning with a Global Navigation Satellite System. Integer ambiguity estimation is the process of mapping the least-squares ambiguity estimates, referred to as the float ambiguities, to an integer value. It is namely known that the carrier phase ambiguities are integer-valued, and it is only after resolution of these parameters that the carrier phase observations start to behave as very precise pseudorange measurements. The success rate equals the integral of the probability density function of the float ambiguities over the pull-in region centered at the true integer, which is the region in which all real values are mapped to this integer. The success rate can thus be computed without actual data and is very valuable as an a priori decision parameter whether successful ambiguity resolution is feasible or not. The pull-in region is determined by the integer estimator that is used and therefore the success rate also depends on the choice of the integer estimator. It is known that the integer least-squares estimator results in the maximum success rate. Unfortunately, it is very complex to evaluate the integral when integer least-squares is applied. Therefore, approximations have to be used. In practice, for example, the success rate of integer bootstrapping is often used as a lower bound. But more approximations have been proposed which are known to be either a lower or upper bound of the actual integer least-squares success rate. In this contribution an overview of the most important lower and upper bounds will be given. These bounds are compared theoretically as well as based on their performance. The performance is evaluated using simulations, since it is then possible to compute the ’actual’ success rate. Simulations are carried out for the two-dimensional case, since its simplicity makes evaluation easy, but also for the higher-dimensional geometry-based case, since this gives an insight to the performance that can be expected in practice.

An Invariant Upperbound for the GNSS Bootstrapped Ambiguity Success Rate

Carrier phase ambiguity resolution is the keyto fast and high precision GPS positioning. Critical in the application of ambiguity resolution is the quality of the computed integer ambiguities. Unsuccessful ambiguity resolution, when passed unnoticed, will too often lead to unacceptable errors in the positioning results. The success or failure of carrier phase ambiguity resolution can be predicted by means of the probability of correct integer estimation, also referred to as the ambiguity success-rate. Upperbounds of the success-rate can be used to decide that ambiguity resolution has become unreliable. In this contribution we prove an upperbound for the bootstrapped success-rate. The upperbound is easy to compute and it is invariant for the class of admissible ambiguity transformations.

Theory of carrier phase ambiguity resolution

Carrier phase ambiguity resolution is the key to high precision Global Navigation Satellite System (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. A proper handling of carrier phase ambiguity resolution requires a proper understanding of the underlying theory of integer inference. In this contribution a brief review is given of the probabilistic theory of integer ambiguity estimation. We describe the concept of ambiguity pull-in regions, introduce the class of admissible integer estimators, determine their probability mass functions and show how their variability affect the uncertainty in the so-called ‘fixed’ baseline solution. The theory is worked out in more detail for integer least-squares and integer bootstrapping. It is shown that the integer least-squares principle maximizes the probability of correct integer estimation. Sharp and easy-to-compute bounds are given for both the ambiguity success rate and the baseline’s probability of concentration. Finally the probability density function of the ambiguity residuals is determined. This allows one for the first time to formulate rigorous tests for the integerness of the parameters.

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.

An optimality property of the integer least-squares estimator

A probabilistic justification is given for using the integer least-squares (LS) estimator. The class of admissible integer estimators is introduced and classical adjustment theory is extended by proving that the integer LS estimator is best in the sense of maximizing the probability of correct integer estimation. For global positioning system ambiguity resolution, this implies that the success rate of any other integer estimator of the carrier phase ambiguities will be smaller than or at the most equal to the ambiguity success rate of the integer LS estimator. The success rates of any one of these estimators may therefore be used to provide lower bounds for the LS success rate. This is particularly useful in case of the bootstrapped estimator.