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.

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.

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.

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.

The ambiguity-resolved detector (ARD) is developed for the validation of high-precision Global Navigation Satellite Systems (GNSS) mixed-integer observation models. The ARD releases the high success rate restriction of resolving the float ambiguities to integers, enabling ambiguity resolution to contribute to model validation even if the resolution success rate is not close to one. However, there are no closed-form expressions for the distribution of the ARD test statistic. The ARD critical value can only be obtained by Monte Carlo simulation, which requires heavy computations. To reduce the effort of applying the ARD, we provide a lookup table for the critical value of the ARD that utilizes the integer least-squares (ILS) estimator to resolve the ambiguities (ARDILS). We obtain 263 408 ARDILS critical values corresponding to GNSS observation models that vary in the precision of observables, satellite geometry, constellation, and the number of signal frequencies. We treat critical values as functions of the integer bootstrapping success rate and conduct curve fitting with second-order polynomials. The polynomial parameters are provided in the lookup table with which the ARDILS critical values can be simply computed rather than simulated. Numerical experiments demonstrate that the critical values computed with the lookup table provide significance levels close to the specified ones. The ARDILS can be implemented easily and efficiently with the lookup table. The lookup table with the polynomial coefficients to compute the ARD critical values can be downloaded from (Yin, 2024).

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.

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.

Although the theory of mixed-integer inference is well developed for GNSS parameter estimation, such is not yet the case for the validation and monitoring of mixed-integer GNSS carrier-phase models. It is the goal of this research to contribute to this field by introducing a class of mixed-integer model (MIM) tests for carrier-phase GNSS. Members from this class and their distributional properties are worked out for different model validation applications relevant to GNSS, such as detection, identification, significance testing, and integer testing. The power performance of the various tests is characterized, thereby showing how they are capable of significantly outperforming the customary ambiguity-float tests.

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.

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.

Positioning technologies are widely used in automotive, aviation, rail, and maritime safety-critical applications. Therefore, the computation of the probability of positioning failure for vehicles, which is the probability that the position estimator is outside a safety region, is of interest for positioning safety analyses. Since parameter estimation and statistical hypothesis testing for model misspecifications are commonly employed in positioning algorithms, the resulting position estimator is conditioned on the statistical hypothesis testing outcome. Hence, the probability density function (PDF) of the vehicle position estimator that accounts for the dependence between the two inference concepts should be used in the computations. In this contribution, we propose a method to compute the probability of positioning failure using the PDF of the vehicle position estimator, which accounts for the aforementioned dependence and is based on rare event simulation techniques, specifically Importance Sampling and the Cross-Entropy method. We apply the proposed method to a satellite-based positioning scenario, in decimeter precision, of an automated vehicle. The results show that the proposed method enables extensive positioning safety-analyses giving insights that can be used in the development of positioning algorithms and deciding whether safety targets and/or requirements are met. Finally, we discuss some limitations of the method and propose several further improvements.

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.

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.

Adjustment theory can be regarded as the part of mathematical geodesy that deals with the optimal combination of redundant measurements together with the estimation of unknown parameters. It is essential for a geodesist, its meaning comparable to what mechanics means to a civil engineer or a mechanical engineer. Historically, the first methods of combining redundant measurements originate from the study of three problems in geodesy and astronomy, namely to determine the size and shape of the Earth, to explain the long-term inequality in the motions of Jupiter and Saturn, and to find a mathematical representation of the motions of the Moon. Nowadays, the methods of adjustment are used for a much greater variety of geodetic applications, ranging from, for instance, surveying and navigation to remote sensing and global positioning.

The two main reasons for performing redundant measurements are the wish to increase the accuracy of the results computed and the requirement to be able to check for errors. Due to the intrinsic uncertainty in measurements, measurement redundancy generally leads to an inconsistent system of equations. Without additional criteria, such a system of equations is not uniquely solvable. In this introductory course on adjustment theory, methods are developed and presented for solving inconsistent systems of equations. The leading principle is that of least-squares adjustment together with its statistical properties.

The inconsistent systems of equations can come in many different guises. They could be given in parametric form, in implicit form, or as a combination of these two forms. In each case the same principle of least-squares applies. The algorithmic realizations of the solution will differ however. Depending on the application at hand, one could also wish to choose between obtaining the solution in one single step or in a step-wise manner. This leads to the need of formulating the system of equations in partitioned form. Different partitions exist, measurement partitioning, parameter partitioning, or a partitioning of both measurements and parameters. The choice of partitioning also affects the algorithmic realization of the solution. In this introductory text the methodology of adjustment is emphasized, although various samples are given to illustrate the theory. The methods discussed form the basis for solving different adjustment problems in geodesy.

In this contribution, we introduce, in analogy to penalized ambiguity resolution, the concept of penalized misclosure space partitioning, with the goal of directing the performance of the DIA-estimator towards its application-dependent tolerable risk objectives. We assign penalty functions to each of the decision regions in misclosure space and use the distribution of the misclosure vector to determine the optimal partitioning by minimizing the mean penalty. As each minimum mean penalty partitioning depends on the given penalty functions, different choices can be made, in dependence of the application. For the DIA-estimator, we introduce a special set of penalty functions that penalize its unwanted outcomes. It is shown how this set allows one to construct the optimal DIA-estimator, being the estimator that within its class has the largest probability of lying inside a user specified tolerance region. Further elaboration shows how these penalty functions are driven by the influential biases of the different hypotheses and how they can be used operationally. Hereby the option is included of extending the misclosure partitioning with an additional undecided region to accommodate situations when it will be hard to discriminate between some of the hypotheses or when identification is unconvincing. By extending the analogy with integer ambiguity resolution to that of integer-equivariant ambiguity resolution, we also introduce the maximum probability estimator within the similar larger class.

In this contribution we consider mixed-integer least-squares problems, where the integer ambiguities a∈Zⁿ and real-valued parameters b∈R^p are estimated. Both a primal and a dual formulation can be considered, with the latter concerning the ambiguity resolution process taking place into the parameters' domain. We study the p = 1 case, where an ad hoc 'P1' algorithm is introduced, and some geometrical insights are provided. It is demonstrated how the algorithm's complexity (i.e. number of candidate integer solutions to be evaluated) grows linearly with the ambiguity dimensionality n, differently from the primal formulation where an exponential growth is observed. By means of numerical simulations, here based on Global Navigation Satellite System (GNSS) models, we show the efficiency of this proposed 'P1' algorithm, meanwhile also demonstrating its quasi-optimal statistical performance.

To accommodate the presence of bounded biases in mixed-integer models, Khodabandeh (2022) extended integer estimation theory by introducing a new admissible integer estimator. The estimator follows the principle of integer least squares estimation and is computed via the integer search method of BEAT. In this contribution, we present the probability distributions of a class of estimators to which the proposed bias-constrained integer least squares estimation belongs. Some important interferometric measuring systems, whose estimation problems can be covered by BEAT, are identified. To show the proposed estimator at work, we apply BEAT to the problem of GLONASS single-differenced (SD) ambiguity resolution. Numerical results of several short-baseline datasets are presented to illustrate why one can achieve more accurate positioning solutions when considering between-receiver SD ambiguity resolution for the cases where carrier phase data are captured on frequency-varying signals with bounded SD receiver phase delays.

These lecture notes are a follow up on Adjustment theory. Adjustment theory deals with the optimal combination of redundant measurements together with the estimation of unknown parameters. There are two main reasons for performing redundant measurements. First, the wish to increase the accuracy of the results computed. Second, the requirement to be able to check for mistakes or errors. The present book addresses this second topic. Although one always will try one's best to avoid making mistakes, they can and will occasionally happen. It is therefore of importance to have ways of detecting and identifying such mistakes. Mistakes or errors can come in many different guises. They could be caused by mistakes made by the observer, or by the fact that defective instruments are used, or by wrong assumptions about the functional relations between the observables. When passed unnoticed, these errors will deteriorate the final results.

The goal of this introductory course on testing theory is therefore to convey the necessary knowledge for testing the validity of both the measurements and the mathematical model. Typical questions that will be addressed are: 'How to check the validity of the mathematical model? How to search for certain mistakes or errors? How well can errors be traced? And how do undetected errors affect the final results?' The theory is worked out in detail for the important case of linear(ized) models. Both the parametric form (observation equations) and the implicit form (condition equations) of linear models are treated. As an additional aid in understanding the basic principles involved, a geometric interpretation is given throughout. Attention is also paid to the performance of the testing procedures. The closely related concept of reliability is introduced and diagnostic measures are given to determine the size of the minimal detectable biases. In this introductory text the methodology of testing is emphasized, although various examples are given to illustrate the theory. The methods discussed form the basis for geodetic quality control and they provide the ingredients for the formulation of guidelines for the reliable design of measurement set-ups.

PPP-RTK corrections, aiding GNSS users to achieve single-receiver integer ambiguity-resolved parameter solutions, are often estimated in a recursive manner by a provider. Such recursive, multi-epoch, estimation of the corrections relies on a set of S-basis parameters that are chosen by the provider so as to make the underlying measurement setup solvable. As a consequence, the provider can only estimate estimable forms of the corrections rather than the original corrections themselves. It is the goal of the present contribution to address the consequence of the corrections’ dependency on the provider’s S-basis, showcasing the characteristics of their multi-epoch solutions, thereby identifying potential pitfalls which the PPP-RTK user should avoid when evaluating such solutions. To this end, we develop a simulation platform that allows one to have full control over the properties of PPP-RTK corrections and demonstrate various misleading temporal behaviors that exist when interpreting the multi-epoch solutions of their estimable forms. The roles of the correction latency and time correlation in the multi-epoch user positioning performance are quantified, while the deviation of the user-reported positioning precision description from its user-actual counterpart is measured under a misspecified user stochastic model.