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).

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.

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.

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.