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).

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.

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.

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.

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.

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.

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.

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.

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.

In this contribution, we introduce the ambiguity-resolved (AR) detector and study its distributional characteristics. The AR-detector is a new detector that lies in between the commonly used ambiguity-float (AF) and ambiguity-known (AK) detectors. As the ambiguity vector can seldomly be known completely, usage of the AK-detector is questionable as reliance on its distributional properties will then generally be incorrect. The AR-detector resolves the shortcomings of the AK-detector by treating the ambiguities as unknown integers. We show how the detector improves upon the AF-detector, and we demonstrate that the, for ambiguity-resolved parameter estimation, commonly required extreme success rates can be relaxed for detection, thus showing that improved model validation is also possible with smaller success rates. As such, the AR-detector is designed to work for mixed-integer GNSS models.

Statistical testing procedures employed in geodetic quality control often consist of two steps: detection and identification. In the detection step, the null hypothesis (working model) ℋ₀ undergoes a validity check. If the outcome of the detection step is the rejection of ℋ₀, identification of potential source of model error is exercised through a search among the specified alternative hypotheses. The testing performance is thus not only led by its ability to detect biases but to correctly identify them as well. The detection capability of a testing regime is usually assessed by its Minimal Detectable Bias (MDB) given a certain correct detection probability. The information provided by the MDB only concerns correct detection and not correct identification. The testing identification performance should be evaluated by its Minimal Identifiable Bias (MIB) given a certain correct identification probability. In this contribution, we demonstrate the difference between MDB and MIB. It is hereby highlighted that a small MDB (or a high probability of correct detection) does not necessarily imply a small MIB (or a high probability of correct identification). The factors driving the difference between detection and identification performance are illustrated using a simple example. Our analysis is then continued in the framework of deformation monitoring.

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.

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.

Ambiguity resolution plays a critical role in fast and high-precision applications of the Global Navigation Satellite System (GNSS). The parameter estimation of high-precision GNSS can benefit from ambiguity resolution when its success rate is very close to 1 (e.g., larger than 0.995); otherwise it is better, in order to avoid a substantial probability of incorrect resolution, to ignore the integer property of the ambiguity, and use the float solution. Nonetheless, model validation and fault detection can still benefit from the integer property of the ambiguity with relatively low ambiguity resolution success rates (e.g. between 0.8 and 0.995) by applying the ambiguity-resolved (AR) detector test statistic based on the ambiguity-resolved residual. Due to the integer property of the resolved (so-called fixed) ambiguity estimator, the distribution of the ambiguity-resolved residual cannot be evaluated analytically. Consequently, the critical value of the AR detector has to be obtained numerically via Monte Carlo simulation of the quantile. Due to the inherent uncertainty in the Monte Carlo simulation process, the implementation of the AR detector also needs to evaluate the uncertainty associated with the simulated critical value. If the simulation uncertainty is large, the actual significance level of the detector may deviate significantly from the target value. In this study, we first describe the process of simulating the samples of the AR test statistic and obtaining the AR critical value for a given significance level through Monte Carlo simulation of the quantile. A histogram of the AR test statistic samples will be shown as an example to illustrate the irregular shape of the distribution of this test statistic. Furthermore, we introduce three methods that can be used to evaluate the uncertainty of the simulated critical value: 1) variance based on the asymptotic normality of the Monte Carlo quantile estimator, 2) confidence interval based on a distribution-free approach, and 3) variance obtained numerically by repeating the simulation. We conduct experiments to compare the above three methods in terms of the consistency between the simulation uncertainties reported by these methods. It will also be shown how the uncertainty of the critical value simulation is affected by the specified significance level. Moreover, we provide the uncertainties of the critical value simulations for nine observation models with various numbers of simulation samples and significance levels, offering insights into the number of samples that should be used for simulating the ARD critical value with the desired uncertainty when applying the AR detector.

This book is a follow-up on Adjustment theory. It extends the theory to the case of time-varying parameters with an emphasis on their recursive determination. Least-squares estimation will be the leading principle used. A least-squares solution is said to be recursive when the method of computation enables sequential, rather than batch, processing of the measurement data. The recursive equations enable the updating of parameter estimates for new observations without the need to store all past observations. Methods of recursive least-squares estimation are therefore particularly useful for applications in which the time-varying parameters need to be instantly determined. Important examples of such applications can be found in the fields of real-time kinematic positioning, navigation and guidance, or multivariate time series analysis. The goal of this book is therefore to convey the necessary knowledge to be able to process sequentially collected measurements for the purpose of estimating time-varying parameters.

When determining time-varying parameters from sequentially collected measurement data, one can discriminate between three types of estimation problems: filtering, prediction and smoothing. Filtering aims at the determination of current parameter values, while smoothing and prediction aim at the determination of respectively past and future parameter values. The emphasis in this book will be on recursive least-squares filtering. The theory is worked out for the important case of linear(ized) models. The measurement-update and time-update equations of recursive least-squares are discussed in detail. Models with sequentially collected data, but time-invariant parameters are treated first.

In this case only the measurement-update equations apply. State-space models for dynamic systems are discussed so as to include time-varying parameters. This includes their linearization and the construction of the state transition matrix. Elements from the theory of random functions are used to describe the propagation laws for linear dynamic systems. The theory is illustrated by means of many worked out examples. They are drawn from applications such as kinematic positioning, satellite orbit determination and inertial navigation.

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.

In this contribution we introduce the dual mixed-integer least-squares problem and study it in relation to its primal counterpart. The dual differs from the primal formulation in the order in which the integer ambiguity vector a∈Zⁿ and baseline vector b∈R^p are estimated. As not the ambiguities, but rather the entries of b are usually the parameters of interest, the attractiveness of the dual formulation stems from its direct computation of b. It is shown that this potential advantage relies on the ease with which an implicit integer least-squares problem of the dual can be solved. For the convoluted cases, we introduce two methods of simplifying approximations. To be able to describe their quality, we provide a complete distributional analysis of their estimators, thus allowing users to judge whether or not the approximations are acceptable for their application. It is shown that this approach implicitly introduces a new class of admissible integer estimators of which we also determine the pull-in regions. As the dual function is shown to lack convexity, special care is required to be able to compute its global minimizer bˇ. Our proposed method, which has finite termination with a guaranteed ϵ-tolerance, is constructed from combining the branch-and-bound principle, with a special convex-relaxation of the dual, to which the projected-gradient-descent method is applied to obtain the required bounds. Each of the method’s three constituents are described, whereby special emphasis is given to the construction of the required continuously differentiable, convex lower bounding function of the dual.

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.

GNSS-based positioning plays an important role in safety-critical applications (e.g., automotive, aviation, shipping, rail) where positioning safety is paramount. Safety analyses typically include a probability-based formulation, such as calculating the probability that the position estimator falls outside a safety region (probability of positioning failure). Once this probability is being computed, it can be compared to an application specific requirement to decide whether or not this requirement is met. In this context, we carry out a component-wise analysis of the probability of positioning failure for the Detection, Identification, and Adaptation (DIA) estimator. The probability of positioning failure is formulated based on the DIA estimator’s probability density function (PDF) which accounts for the dependence between parameter estimation and statistical hypothesis testing. The probability of positioning failure can be further expressed in terms of its conditional components via the law of total probability which enables the assessment of the Most Impactful Components (MICs). Knowledge about the MICs can be useful to determine the main contributors to the probability of positioning failure. Using a dual GNSS (GPS and Galileo) positioning model for an automated vehicle, we compute the MICs based on the conditional PDFs components of the DIA estimator. Furthermore, we determine the worst-case scenario by computing the maximum total probability of positioning failure over a range of magnitudes for the outliers in the observables and over different orientations of the vehicle. These types of analyses can be useful during the design stage of DIA estimators to verify whether the safety requirements for positioning are met. Lastly, we summarize our contributions and provide an outlook on future work.