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

Ambiguity resolution performance with GPS and BeiDou for LEO formation flying

The evolving BeiDou Navigation Satellite System constellation brings new opportunities for high-precision applications. In this contribution the focus will be on one such application, namely precise and instantaneous relative navigation of a formation of LEO satellites. The aim is to assess the ambiguity resolution performance with the future GPS and BeiDou constellations depending on system choice (GPS, BeiDou, or GPS+BeiDou), single- or dual-frequency observations, receiver noise, and uncertainties in ionosphere modelling. In addition, for the GPS+BeiDou constellation it will be shown how the growing BeiDou constellation in the years to come can already bring an important performance improvement compared to the GPS-only case. The performance will be assessed based on the percentage of time that the required precision can be obtained with a partial ambiguity resolution strategy

GNSS carrier phase ambiguity resolution: challenges and open problems

Integer carrier phase ambiguity resolution is the key to fast and high-precision global navigation satellite system (GNSS) positioning and application. Although considerable progress has been made over the years in developing a proper theory for ambiguity resolution, the necessary theory is far from complete. In this contribution we address three topics for which further developments are needed. They are: (1) Ambiguity acceptance testing; (2) Ambiguity subset selection; and (3) Integer-based GNSS model validation. We will address the shortcommings of the present theory and practices, and discuss directions for possible solution

DIA-datasnooping and identifiability

In this contribution, we present and analyze datasnooping in the context of the DIA method. As the DIA method for the detection, identification and adaptation of mismodelling errors is concerned with estimation and testing, it is the combination of both that needs to be considered. This combination is rigorously captured by the DIA estimator. We discuss and analyze the DIA-datasnooping decision probabilities and the construction of the corresponding partitioning of misclosure space. We also investigate the circumstances under which two or more hypotheses are nonseparable in the identification step. By means of a theorem on the equivalence between the nonseparability of hypotheses and the inestimability of parameters, we demonstrate that one can forget about adapting the parameter vector for hypotheses that are nonseparable. However, as this concerns the complete vector and not necessarily functions of it, we also show that parameter functions may exist for which adaptation is still possible. It is shown how this adaptation looks like and how it changes the structure of the DIA estimator. To demonstrate the performance of the various elements of DIA-datasnooping, we apply the theory to some selected examples. We analyze how geometry changes in the measurement setup affect the testing procedure, by studying their partitioning of misclosure space, the decision probabilities and the minimal detectable and identifiable biases. The difference between these two minimal biases is highlighted by showing the difference between their corresponding contributing factors. We also show that if two alternative hypotheses, say (Formula presented.) and (Formula presented.), are nonseparable, the testing procedure may have different levels of sensitivity to (Formula presented.)-biases compared to the same (Formula presented.)-biases

Review and principles of PPP-RTK methods

PPP-RTK is integer ambiguity resolution-enabled precise point positioning. In this contribution, we present the principles of PPP-RTK, together with a review of different mechanizations that have been proposed in the literature. By application of S-system theory, the estimable parameters of the different methods are identified and compared. Their interpretation is essential for gaining a proper insight into PPP-RTK in general, and into the role of the PPP-RTK corrections in particular. We show that PPP-RTK is a relative technique for which the ‘single-receiver user’ integer ambiguities are in fact double-differenced ambiguities. We determine the transformational links between the different methods and their PPP-RTK corrections, thereby showing how different PPP-RTK methods can be mixed between network and users. We also present and discuss four different estimators of the PPP-RTK corrections. It is shown how they apply to the different PPP-RTK models, as well as why some of the proposed estimation methods cannot be accepted as PPP-RTK proper. We determine analytical expressions for the variance matrices of the ambiguity-fixed and ambiguity-float PPP-RTK corrections. This gives important insight into their precision, as well as allows us to discuss which parts of the PPP-RTK correction variance matrix are essential for the user and which are not

GLONASS CDMA L3 ambiguity resolution and positioning

A first assessment of GLONASS CDMA L3 ambiguity resolution and positioning performance is provided. Our analyses are based on GLONASS L3 data from the satellite pair SVNs 755-801, received by two JAVAD receivers at Curtin University, Perth, Australia. In our analyses, four different versions of the two-satellite model are applied: the geometry-free model, the geometry-based model , the height-constrained geometry-based model, and the geometry-fixed model. We study the noise characteristics (carrier-to-noise density, measurement precision), the integer ambiguity resolution performance (success rates and distribution of the ambiguity residuals), and the positioning performance (ambiguity float and ambiguity fixed). The results show that our empirical outcomes are consistent with their formal counterparts and that the GLONASS data have a lower noise level than that of GPS, particularly in case of the code data. This difference is not only seen in the noise levels but also in their onward propagation to the ambiguity time series and ambiguity residuals distribution

IRNSS/NavIC and GPS: a single- and dual-system L5 analysis

The Indian Regional Navigation Satellite System (IRNSS) has recently (May 2016) become fully operational. In this contribution, for the fully operational IRNSS as a stand-alone system and also in combination with GPS, we provide a first assessment of L5 integer ambiguity resolution and positioning performance. While our empirical analyses are based on the data collected by two JAVAD receivers at Curtin University, Perth, Australia, our formal analyses are carried out for various onshore locations within the IRNSS service area. We study the noise characteristics (carrier-to-noise density, measurement precision, time correlation), the integer ambiguity resolution performance (success rates and ambiguity dilution of precision), and the positioning performance (ambiguity float and ambiguity fixed). The results show that our empirical outcomes are consistent with their formal counterparts and that the GPS L5-data have a lower noise level than that of IRNSS L5-data, particularly in case of the code data. The underlying model in our assessments varies from stand-alone IRNSS (L5) to IRNSS (Formula presented.) GPS (L5), from unconstrained to height-constrained and from kinematic to static. Significant improvements in ambiguity resolution and positioning performance are achievable upon integrating L5-data of IRNSS with GPS

IRNSS stand-alone positioning: first results in Australia

The Indian Regional Navigation Satellite System (IRNSS) currently under development is expected to reach full operational capability before 2017. As a large part of the Australian continent lies in IRNSS’s service area, it is important to gain an understanding of its navigational potential and actual positioning capabilities for Australian users. The goals of this contribution are therefore to provide insight into IRNSS, to demonstrate its current positioning performance using actual L5 pseudorange tracking data, and to analyse its expected positioning performance for when the system is fully operational. As such this contribution provides the very first results of the IRNSS stand-alone positioning capabilities over Australia