Carrier-phase ambiguity validation is essential to ensure the reliability of integer ambiguity resolution in high-precision GNSS positioning. Although integer equivariant (IE) estimators provide optimal integer candidates within their class, noise and model limitations may lead to incorrect fixing. Validation procedures are therefore crucial for safeguarding the transition from float to fixed solutions, particularly in high-precision and safety-critical applications. In this contribution we introduce the concept of Fourier ambiguity validation and show how it is rooted in the principles of integer aperture (IA) estimation and its periodic representation. Unlike classical integer estimators that always return an integer solution, IA estimators introduce adjustable acceptance regions in the float ambiguity domain and fix ambiguities only when sufficient statistical evidence is present. As a result we present a general Fourier representation of IA estimators and provide an analytical description of the probabilistic properties of integer-aperture bootstrapping. We also present a hybrid description and show how the spatial and frequency representations can be mixed so as to do justice to the practical situation when carrier-phase ambiguities have a wide range of varying precision.

Unmanned Aerial Vehicles (UAVs) support, or are planned to support, a wide range of operations, including emergency response, environmental research, urban air mobility, and (commercial) air transportation, where positioning safety is paramount. This contribution presents a framework for assessing positioning safety of UAVs by computing the probability of positioning failure, rather than conservative upper bounds, while accounting for time-varying positioning models. In contrast to existing studies, we (i) explicitly adopt UAV safety regions and target probability of positioning failure requirements as specified by the European Union Agency for the Space Programme (EUSPA) for Specific Assurance and Integrity Levels (SAIL) 3 (10^-4/hour) and 4 (10^-5/hour), and (ii) use representative positioning models for the UAV GPS receiver which are consistent with Technical Standard Order (TSO) specifications. For the computation of the probability of positioning failure, we use a method based on rare event simulation techniques while accounting for the dependence between parameter estimation and statistical hypothesis testing. We apply the framework to simulation-based positioning safety analysis across authorized European airspace regions in eight countries using real GPS satellite orbit data. The probability of positioning failure is computed over a 24-hour period, then connected to per-hour requirements using one-hour moving averages, and compared against the EUSPA SAIL 3 and 4 requirements. The time-dependent analysis further reports best-case and worst-case probabilities of positioning failure and quantifies sensitivity to key hypothesis-testing design parameters, such as the level of significance. This analysis can help UAV operators and regulators verify compliance with EUSPA safety standards, supporting management of safe UAV operations.

The Detection, Identification, and Adaptation (DIA)-estimator integrates parameter estimation and hypothesis testing for model misspecifications. This contribution presents a positioning safety analysis approach grounded in the DIA-estimator framework, with a particular emphasis on multidimensional model misspecifications, such as simultaneous outliers in the observations. While recent work has focused on the performance of the detection and identification of multidimensional model misspecifications, we turn our attention to how they affect the probability density function (PDF) of the DIA-estimator and, consequently, the probability of positioning failure–an indicator relevant for safety-of-life applications (e.g., automotive, aviation, rail, maritime). This work formulates and quantifies the probability of positioning failure and its conditional components. A representative simulation-based study is presented for a UAV equipped with a GPS receiver configured to achieve performance comparable to Technical Standard Order (TSO)-certified receivers. The analysis is carried out for two scenarios: a fixed GPS satellite geometry at a single time snapshot, and for a varying GPS satellite geometry over a 24-hour period over an authorized UAV airspace region in the Netherlands using real satellite ephemeris data. Together, these scenarios provide insights into the structure of the DIA-estimator’s PDF, such as multimodality and orientation with respect to the chosen positioning safety region, and support comprehensive evaluation of positioning safety. Although the current focus is on GPS-based positioning, the presented approach is general and can be extended to include multisensor configurations, additional GNSS constellations, and applied to other safety-critical applications, which are subjects of future work.

In this contribution, we introduce the concept of Fourier ambiguity resolution. We show how it is rooted in the principle of integer equivariant (IE) estimation and in its periodic representation. As a result, we present a general Fourier representation of IE-estimators. As the IE-class is the largest class of estimators used in GNSS ambiguity resolution, the periodic representation opens up a broad spectrum of new applications, both in the field of parameter estimation and in that of statistical testing. The representation also applies to the integer class, with its popular estimators of integer-rounding, integer-bootstrapping, and integer least-squares, as well as to their integer-aperture variants. In this contribution, we consider the periodic representation of the best integer equivariant (BIE) estimator. It is shown how this minimum mean squared error IE-estimator can be represented in both the spatial and frequency domains and how preference for one of the two representations should be based on the GNSS carrier-phase ambiguity precision. We also present a hybrid form of the BIE-estimator and show how the spatial and frequency representations can be mixed so as to do justice to the practical situation when carrier-phase ambiguity vectors consist of ambiguities having a wide range of varying precision.

Mixed-integer (MI) estimation theory is fundamental to carrier-phase GNSS, where real-valued parameters are coupled with integer ambiguities. The class of integer-equivariant (IE) estimators is the largest class that properly respects this structure. Its unbiased minimum mean-squared-error member is the best integer equivariant (BIE) estimator, whose precision is never worse than that of the BLUE and whose performance is often close to that of integer least-squares. Its practical drawback, however, is the need to evaluate an infinite sum. We introduce the projected-BIE (PBIE) estimator, a new unbiased minimum-variance IE-estimator with a finite representation. It is obtained by restricting the admissible periodic corrections in the canonical IE-representation and solving the resulting optimization problem. PBIE is valid for general observation distributions, has an exactly computable variance matrix, and is obtained from normal equations. We further show that PBIE has a dual optimality property: it is both minimum-variance in its class and the closest estimator in that class to the BIE. For normally distributed data, PBIE simplifies considerably and can be computed efficiently from the BLUE of the integer ambiguities.

The theory of the ambiguity-resolved parameter significance (ARs) and ambiguity-resolved parameter normed (ARn) tests has been developed for global navigation satellite system mixed-integer model validation. This contribution describes how to apply these tests and evaluates their performance by analyzing their power as a function of bias magnitude and comparing them with the ambiguity-float (AF) and ambiguity-known significance tests. When testing for blunders in pseudoranges and carrier-phase cycle slips, the ARn and ARs tests perform similarly and generally outperform the AF test. When testing for ionosphere and troposphere delays, the ARs and ARn tests perform differently, and the ARs power function is spiky. We explain this behavior and show the performance of two methods in mitigating reductions in the ARs power function. This contribution demonstrates that the ARs and ARn tests can perform better the AF test, even when the ambiguity resolution success rate is not close to one.

In the Original version of this article the title was incorrectly given as ‘DIA-Estimator and Multidimensional Model Misspecifications: GNSS-based Positioning Safety Analysis for UAV’ but should have been ‘DIA-Estimator and Multidimensional Model Misspecifications: GNSS-based Positioning Safety Analysis for UAVs’ and also a formatting Issue has been occurred through the entire article. The original article PDF has now been corrected.

While integer ambiguity resolution (IAR) enables GNSS to achieve real-time sub-centimeter-level positioning in open-sky environments, it can be easily hindered if the involved receivers are situated in areas with limited satellite visibility, such as in dense city environments. In such GNSS-challenged cases, commercial Low Earth Orbit (LEO) communication satellites can potentially augment GNSS by providing additional measurements. However, LEO satellites often lack code measurements, mainly transmitting satellite-specific frequency-varying carrier phase signals. This contribution aims to study the ambiguity-resolved baseline positioning performance of such phase-only signals, addressing the extent to which LEO constellations can realize near real-time positioning in standalone and GNSS-combined modes. Through a simulation platform, we analyze the distinct response of each LEO constellation (Iridium, Globalstar, Starlink, OneWeb, and Orbcomm) to IAR under various circumstances. Although achieving single-receiver high-precision positioning can be challenged by inaccuracies in the LEO satellite orbit products, the relative distance between two receivers can help overcome this limitation. As a result, centimeter-level relative positioning over short baselines can be made possible, even with a satellite elevation cut-off angle of 50 degrees, making it suitable for GNSS-challenged environments. This can be achieved with high-grade receiver clocks over very short baselines (∼5 km) and access to decimeter-level orbit products.

To fully utilize carrier phase measurements in high-precision interferometric positioning systems, such as global navigation satellite systems (GNSS), the corresponding integer ambiguities must be successfully resolved. Since the phase ambiguities are biased by non-integer phase delays, only specific combinations are allowed to serve as valid inputs for Integer Ambiguity Resolution (IAR) methods. Consequently, the resultant ambiguity-resolved phase data may not improve position precision as significantly as when all the ambiguities are resolved. The goal of this contribution is to study the role of phase biases in IAR and quantify the effect of bounding such biases in the ambiguity-resolved positioning performance. By identifying the interrelationship of the model's solutions, we show how constraining the phase biases has the potential to improve the precision of both the position and the ambiguities. With the aid of simulated results, it is illustrated that one can leverage the boundedness property of phase biases to obtain positioning results that are considerably more accurate than those obtained when the bias constraint is discarded.

In this contribution, we introduce some new theory for the classical GNSS ambiguity function (AF) method. We provide the probability model by means of which the AF-estimator becomes a maximum likelihood estimator, and we provide a globally convergent algorithm for computing the AF-estimate. The algorithm is constructed from combining the branch-and-bound principle, with a special convex relaxation of the multimodal ambiguity function, to which the projected-gradient-descent method is applied to obtain the required bounds. We also provide a systematic comparison between the AF-principle and that of integer least-squares (ILS). From this comparison, the conclusion is reached that the two principles are fundamentally different, although there are identified circumstances under which one can expect AF- and ILS-solutions to behave similarly.

GNSS model validation constitutes an essential part of any GNSS data processing scheme. With the inclusion of the very precise, but integer ambiguous carrier-phase data, the GNSS models become of the mixed-integer type. Although inference theory of mixed-integer models is well developed for parameter estimation, this is not yet the case for the validity testing of such models. It is the goal of this contribution to help close this gap by introducing the ambiguity-resolved parameter significance test. It differs from existing significance tests in that it takes the unknown integerness of the ambiguities rigorously into account. Our analysis shows that the proposed test can significantly outperform currently used tests.

The analysis of positioning safety often employs a probability-based formulation. This approach quantifies the probability of positioning failure, which is the probability of the position estimator being outside a safety-region, and compares it against an application specific requirement. The design of positioning algorithms for safety-critical applications, such as automated/autonomous vehicles, should consider the dependence between parameter or state estimation and statistical hypothesis testing for model misspecifications in the evaluation of positioning safety. If this dependence is not considered, as this article shows, the conclusions drawn from the positioning safety analysis might be overly-optimistic. Therefore, this article focuses on the aforementioned dependence through a vehicle positioning scenario based on an Extended Kalman Filter (EKF) and the Detection, Identification, and Adaptation (DIA) method for misspecifications in the motion and measurement models. Grounded in the distributional theory for the DIA method, our positioning safety analysis utilizes the conditional probability density functions (PDFs) of the combined EKF and DIA position error, which are generally nonnormal. We compute the probability of vehicle positioning failure in two cases 1) when the dependence is considered and 2) when it is not, to quantify the over-optimism introduced by ignoring this dependence. Finally, we present our conclusions and recommendations.

Teunissen (J Geod 98(83):1–16, 2024) proposed the ambiguity-resolved (AR) detection theory for GNSS mixed-integer model validation. In this contribution, we study the performance of the AR detector through analysis and simulation experiments and compare it with the ambiguity-float (AF) and ambiguity-known (AK) detectors. We describe how the detectors can be implemented and how to evaluate their performance by computing the power as functions of the model misspecifications’ size. We present two simulation experiments with single- and dual-frequency GPS models and demonstrate that the AR detector can provide a larger detection power than the AF detector, even if the success rate is not close to one. Then, we obtain power functions over 25 user locations with five observation models and 72 satellite geometries per location per model. We find that the AR detector increases the detection probability of ionosphere and troposphere delays by 47% and 60% on average when the success rate is larger than 97.5% and the level of significance is 0.01. We also find the AR detection power to be larger than that of the AF detector in case of multi-dimensional misspecifications.

This contribution investigates four members of the class of Detection, Identification, and Adaptation (DIA) estimators, which integrate parameter estimation with hypothesis testing. Using the framework of minimum mean penalty testing, we analyze and compare the misclosure-space partitionings of the traditional DIA procedure, which combines the overall model test with likelihood-ratio-based tests, and those maximizing the probabilities of correct hypothesis identification and parameter estimation. A constrained version of the latter, with the null hypothesis acceptance region fixed to the traditional procedure, is also examined. Our study focuses on cases where the biases under alternative hypotheses are fully known. Next to the conceptual comparison, we also assess, through a number of examples, misclosure-space partitionings and the probabilities of DIA estimators falling within a defined elliptical safety region. The results highlight the relationships and distinctions among the DIA estimators, revealing the influence of penalty functions, bias magnitude, safety region size, and false alarm probability.