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.

Multipath propagation represents a dominant error source limiting the accuracy of radio-based ranging and positioning in urban environments. Conventional ranging estimators typically estimate only the range of the first arriving path while neglecting the multipath components. This introduces a bias and causes the estimator's variance to diverge from the Cramér-Rao Bound (CRB). While extensive research has established performance bounds for estimators that jointly estimate all paths, the specific impact on the variance of ignoring these components remains largely unexplored. This paper derives bounds on the bias, variance, and mean square error (MSE) for ranging in multipath channels for the purpose of positioning. We consider Orthogonal Frequency Division Multiplexing (OFDM) signals in multipath channels with deterministic path delays and gains. The paper focuses on time-delay estimation (TDE) in the mid-to-high signal-to-noise ratio (SNR) regime under two scenarios: (i) when the receiver jointly estimates all propagation paths, and (ii) when the receiver underestimates the number of paths. Specifically, for the latter case, we derive a semi closed-form expression for the variance of the time-delay estimator that considers a single-path when in reality there are L paths. This derivation is based on the misspecified Cramér-Rao bound (MCRB). Although unmodelled multipath has been traditionally viewed as detrimental to time-delay estimation, we reveal that in some cases the estimation variance improves. For a two-path channel we show that the variance depends on the relative gain, carrier phase, and separation of the paths. Additionally, the estimation bias is upper-bounded by constructive and destructive path interference. Finally, we empirically process a six-path simulated channel frequency response with fixed path delays and gains, to demonstrate that the derived MSE bound is tight for mid-to-high SNRs. This work characterizes the impact of multipath propagation on the variance of time-delay estimation, which is essential for designing accurate ranging signals and estimators in urban scenarios.

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.

Navigation with radio-signals using GNSS or a terrestrial positioning system in urban environments is susceptible to multipath propagation, which can severely degrade positioning accuracy. In a Line-of-Sight (LOS) multipath channel, the received signal is composed of a direct path component and a sum of time-shifted and attenuated replicas of the transmitted signal. When these multipath components are not accounted for in the time-delay estimation (TDE) model, they may introduce substantial estimation bias. For positioning, only the first arriving path is of interest. Therefore, it is crucial to focus on estimating the reflections that most significantly affect the TDE of this primary path, while ignoring others with negligible impact. To reduce the impact of close-in multipath in TDE, we propose Maximum Likelihood estimators that account for the strongest reflections, with models considering either one or two multi-path components. The Maximum Likelihood Estimation (MLE) problem is optimized using the Space Alternating Generalized Expectation-Maximization (SAGE) method. To reduce computational load, the delay search space for each path is constrained based on the maximum bias observed in the multipath error envelope (MPEE). To assess the ranging accuracy for the various MLE estimators that account for multiple paths, we utilize a synthetically generated channel based on the Saleh-Valenzuela model. Additionally, we benchmark the positioning performance of these estimators using channel impulse responses recorded with a terrestrial positioning prototype system tested at The Green Village on the TU Delft campus.

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.

Inspired by the attractive features of least-squares theory in many practical applications, this contribution introduces least-squares-based deep learning (LSBDL). Least-squares theory connects explanatory variables to predicted variables, called observations, through a linear(ized) model in which the unknown parameters of this relation are estimated using the principle of least-squares. Conversely, deep learning (DL) methods establish nonlinear relationships for applications where predicted variables are unknown (nonlinear) functions of explanatory variables. This contribution presents the DL formulation based on least-squares theory in linear models. As a data-driven method, a network is trained to construct an appropriate design matrix of which its entries are estimated using two descent optimization methods: steepest descent and Gauss–Newton. In conjunction with interpretable and explainable artificial intelligence, LSBDL leverages the well-established least-squares theory for DL applications through the following three-fold objectives: (i) Quality control measures such as covariance matrix of predicted outcome can directly be determined. (ii) Available least-squares reliability theory and hypothesis testing can be established to identify mis-specification and outlying observations. (iii) Observations’ covariance matrix can be exploited to train a network with inconsistent, heterogeneous and statistically correlated data. Three examples are presented to demonstrate the theory. The first example uses LSBDL to train coordinate basis functions for a surface fitting problem. The second example applies LSBDL to time series forecasting. The third example showcases a real-world application of LSBDL to downscale groundwater storage anomaly data. LSBDL offers opportunities in many fields of geoscience, aviation, time series analysis, data assimilation and data fusion of multiple sensors.

This paper presents a terrestrial networked positioning system that obtains a reliable time reference from a national time scale realization and distributes it in a prototype to six roadside base stations through a fiber-optic Gigabit Ethernet network. Wireless wideband signals are transmitted by the base stations, thereby enabling positioning by a mobile receiver with an accuracy of one decimeter in a multipath urban environment. The scalability and compatibility of this system with existing telecommunication-network technologies paves the way for wide-area global navigation satellite system-independent back-up systems for timing and positioning with improved coverage and performance. The results presented in this paper are based on research carried out within the scope of a project funded by the Dutch Research Council (NWO, project 13970).

Global navigation satellite systems (GNSS) are widely used for navigation and time distribution^1–3, features that are indispensable for critical infrastructure such as mobile communication networks, as well as emerging technologies such as automated driving and sustainable energy grids^3,4. Although GNSS can provide centimetre-level precision, GNSS receivers are prone to many-metre errors owing to multipath propagation and an obstructed view of the sky, which occur particularly in urban areas where accurate positioning is most needed^1,5,6. Moreover, the vulnerabilities of GNSS, combined with the lack of a back-up system, pose a severe risk to GNSS-dependent technologies⁷. Here we demonstrate a terrestrial positioning system that is independent of GNSS and offers superior performance through a constellation of radio transmitters, connected and time-synchronized at the subnanosecond level through a fibre-optic Ethernet network⁸. Using optical and wireless transmission schemes similar to those encountered in mobile communication networks, and exploiting spectrally efficient virtual wideband signals, the detrimental effects of multipath propagation are mitigated⁹, thus enabling robust decimetre-level positioning and subnanosecond timing in a multipath-prone outdoor environment. This work provides a glimpse of a future in which telecommunication networks provide not only connectivity but also GNSS-independent timing and positioning services with unprecedented accuracy and reliability.