Penalized GNSS Ambiguity Resolution

Global Navigation Satellite System (GNSS) carrier phase ambiguity resolution is the process of resolving the carrier phase ambiguities as integers. It is the key to fast and high precision GNSS positioning and it applies to a great variety of GNSS models which are currently in use in navigation, surveying, geodesy and geophysics. A new principle of carrier phase ambiguity resolution is introduced. The idea is to give the user the possibility to assign penalties to the possible outcomes of the ambiguity resolution process: a high penalty for an incorrect integer outcome, a low penalty for a correct integer outcome and a medium penalty for the real valued float solution. As a result of the penalty assignment, each ambiguity resolution process has its own overall penalty. Using this penalty as the objective function which needs to be minimized, it is shown which ambiguity mapping has the smallest possible penalty. The theory presented is formulated using the class of integer aperture estimators as a framework. This class of estimators was introduced elsewhere as a larger class than the class of integer estimators. Integer aperture estimators, being of a hybrid nature, can have integer outcomes as well as non-integer outcomes. The minimal penalty ambiguity estimator is an example of an integer aperture estimator. The computational steps involved for determining the outcome of the minimal penalty estimator are given. The additional complexity in comparison with current practice is minor, since the optimal integer estimator still plays a major role in the solution of the minimal penalty ambiguity estimator

An Integrity and Quality Control Procedure for use in Multi Sensor Integration

Real-time estimation of parameters in dynamic systems becomes increasingly important in thefield of high precision navigation. The real-time estimation inevitably requires real-time integrity monitoring of the models underlying the navigation system. This paper presents a real-time recursive detection, identification, and adaptation (DIA) procedure for use in integrated navigation systems. It is based on the concept of multi-sensor integration and makes use of the redundancy information stemming from both the measurement model and dynamic model. The tests proposed are optimal in the uniformly-most-powerful invariant sense. Their inverted power function is used to introduce the concept of minimal detectable biases (MDB). The MDB is a diagnostic tool for inferring the detectability of particular model errors. It can be used for the design of a navigation filter that allows for a sufficient control on the presence of bias

On InSAR ambiguity resolution for deformation monitoring

Integer carrier phase ambiguity resolution is the key to fast and high-precision satellite positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. It also applies to stacked radar interferometry for deformation monitoring, see e.g. [Hanssen, et al., 2001]. In this contribution we apply the integer least-squares' principle to the rank defect model of stacked InSAR carrier phase data. We discuss two ways of dealing with the rank defect for ambiguity resolution. One is based on the use of a priori data, the other is based on the use of an interval constraint on the deformation rate

An invariant upper bound for the GNSS bootstrapped ambiguity success-rate

Abstract. Carrier phase ambiguity resolution is the key to 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

The Lambda Method for the GNSS Compass

Global Navigation Satellite System carrier phase ambiguity resolution is the key to high precision positioning and attitude determination. In this contribution we consider the GNSS compass model. We derive the integer least-squares estimators and discuss the various steps involved in the ambiguity resolution process. This includes the method that has successfully been used in (Park and Teunissen, 2003). We emphasize the unaided, single frequency, single epoch case, since this is considered the most challenging mode of GNSS attitude determination

A General Multivariate Formulation of the Multi-antenna GNSS Attitude Determination Problem

Global Navigation Satellite System (GNSS) carrier phase ambiguity resolution is the key to high precision positioning, navigation and attitude determination. In this contribution we present a general formulation for the multi-antenna GNSS attitude determination problem. This multivariate formulation provides a general framework for solving various GNSS attitude determination problems. With the use of this formulation we show how the constrained integer least-squares carrier phase ambiguities and corresponding attitude matrix can be solved

On the probability density function of the GNSS ambiguity residuals

Integer GNSS ambiguity resolution involves estimation and validation of the unknown integer carrier phase ambiguities. A problem then is that the classical theory of linear estimation does not apply to the integer GPS model, and hence rigorous validation is not possible when use is made of the classical results. As with the classical theory, a first step for being able to validate the integer GPS model is to make use of the residuals and their probabilistic properties. The residuals quantify the inconsistency between data and model, while their probabilistic properties can be used to measure the significance of the inconsistency. Existing validation methods are often based on incorrect assumptions with respect to the probabilistic properties of the parameters involved. In this contribution we will present and evaluate the joint probability density function (PDF) of the multivariate integer GPS carrier phase ambiguity residuals. The residuals and their properties depend on the integer estimation principle used. Since it is known that the integer least-squares estimator is the optimal choice from the class of admissible integer estimators, we will only focus on the PDF of the ambiguity residuals for this estimator. Unfortunately the PDF cannot be evaluated exactly. It will therefore be shown how to obtain a good approximation. The evaluation will be completed by some examples

Array-Aided Multifrequency GNSS Ionospheric Sensing: Estimability and Precision Analysis

The dual-frequency Global Positioning System has proven to be an effective means of measuring the Earth's ionosphere and its total electron content (TEC). With the advent of multifrequency signals from more Global Navigation Satellite Systems (GNSSs), the opportunity arises to construct many more ionosphere-sensing combinations of GNSS data. With such diversity, various estimable ionospheric delays with differing interpretations (and of different precision) can be formed. How such estimable ionospheric delays should be interpreted, and the extent to which they contribute to the precision with which the unbiased TEC can be estimated, are the topics of this paper. Based on multifrequency GNSS code-only, phase-only, and phase-and-code data, we derive the closed-form solutions of different types of ionospheric observables that each can serve as input of an externally provided ionospheric model for TEC determination. Within such a general least-squares framework, we generalize the widely used phase-to-code levelling technique to its multifrequency version. We also show that only certain specific linear combinations of the observables contribute to the TEC solutions. As a further improvement of the multifrequency GNSS-derived TEC solution, we propose and study the usage of an array of GNSS antennas. Analytical solutions, supported by numerical examples, of this array-based concept are presented, together with a discussion on its relevance for TEC determination. This concerns the roles of time averaging and time differencing, of integer ambiguity resolution, and of the number of frequencies and number of array antennas in determining TEC

On the foundation of the popular ratio test for GNSS ambiguity resolution

Integer carrier phase ambiguity resolution is the key to fast and high-precision global navigation satellite system (GNSS) positioning and navigation. It is the process of resolving the unknown cycle ambiguities of the double-differenced carrier phase data as integers. For the problem of estimating the ambiguities as integers a rigorous theory is available. The user can choose from a whole class of integer estimators, from which integer least-squares is known to perform best in the sense that no other integer estimator exists which will have a higher success rate. Next to the integer estimation step, also the integer validation plays a crucial role in the process of ambiguity resolution. Various validation procedures have been proposed in the literature. One of the earliest and most popular ways of validating the integer ambiguity solution is to make use of the so-called Ratio Test. In this contribution we will study the properties and underlying concept of the popular Ratio Test. This will be done in two parts. First we will criticize some of the properties and underlying principles which have been assigned in the literature to the Ratio Test. Despite this criticism however, we will show that the Ratio Test itself is still an important, albeit not optimal, candidate for validating the integer solution. That is, we will also show that the procedure underlying the Ratio Test can indeed be given a firm theoretical footing. This is made possible by the recently introduced theory of Integer Aperture Inference. The necessary ingredients of this theory will be briefly described. It will also be shown that one can do better than the Ratio Test. The optimal test will be given and the difference between the optimal test and the Ratio Test will be discussed and illustrated