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

Family language patterns in bilingual families and relationships with children's language outcomes

Past research shows that family language patterns (i.e., which languages are spoken in the family and by whom) are associated with bilingual children's language use. However, it is unclear how input properties such as input quantity, parental proficiency, and language mixing may differ across family language patterns. It is also unclear whether the effects of family language patterns on children's language proficiency remain when differences in input properties are controlled. We investigated (i) which family language patterns occurred in bilingual families in the Netherlands (n = 136), (ii) whether input properties differed across patterns, and (iii) how patterns related to children's proficiency, once input properties were controlled. Home language situations were assessed through a questionnaire, children's proficiency in Dutch and the minority language through vocabulary tests and parent ratings. Three language patterns were found: one-parent-one-language, both parents mixed languages or used the minority language. The results showed differences in input properties across all patterns, as well as effects of these patterns on children's proficiency in Dutch and the minority language that disappeared once input properties were controlled. These findings do not provide robust evidence that family language patterns predict children's proficiency, but rather, that input quantity is crucial.</p