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.

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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.

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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. @en

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.

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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. @en

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).

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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.

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GPS is undoubtedly the most popular system providing positioning, navigation and timing (PNT) services to a host of applications, industries and infrastructures. GPS is mass-adopted, has worldwide coverage, has an impressive up-time and can be used with a wide range of receiver devices, featuring low to high cost and low to high precision.@en

This book provides an introduction, at academic level, into the field of surveying and mapping. The book has been compiled based on hand-outs and readers written for the third-year course Surveying and Mapping, in the bachelor program Civil Engineering at Delft University of Technology. This book covers a wide range of measurement techniques, from land surveying, GPS/GNSS and remote sensing to the associated data processing, the underlying coordinate reference systems, as well as the analysis and visualization of the acquired geospatial information.@en

For validation and demonstration of high accuracy ranging and positioning algorithms and systems, a wideband radio signal generation and acquisition testbed, tightly synchronized in time and frequency, is needed. The development of such a testbed requires solutions to several challenges. Tight time and frequency synchronization, derived from a centrally distributed time-frequency reference signal, needs to be maintained in the hardware of the transmitter and receiver nodes, and wideband signal acquisition requires sustainable data throughput between the receiver and host PC as well as data storage at GB level. This article presents a testbed for wideband radio signal acquisition, for validation and demonstration of high accuracy ranging and positioning. It consists of multiple Ettus X310 universal software radio peripherals (USRPs) and supports high accuracy (<100 ps) time-deterministic, sustainable signal transmission and acquisition, with a bandwidth up to 320 MHz (in dual channel mode) and frequencies up to 6 GHz. Generation and processing of wideband arbitrary signal waveforms is done offline. To realize these features, radio frequency on chip (RFNoC) compatible HDL units were developed for integration in the X310 SDR platform. Wideband transmission and signal acquisition at a lower duty cycle is applied to reduce the data offloading throughput to the host's personal computer (PC). Benchmarking of the platform was performed to demonstrate sustainable long duration dual channel acquisition. Indoor range measurements with the synchronous operation of the testbed show a decimeter-level accuracy.

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This paper presents a methodology to design a sparse multiband ranging signal with a large virtual bandwidth, from which time delay and carrier phase are estimated by a low complexity multivariate maximum likelihood (ML) method. In the estimation model for a multipath channel, not all reflected paths are considered, and time delay and carrier phase are estimated in a step-wise manner to further reduce the computational load. By introducing a measure of dependence and a measure of bias for a multipath reflection, we analyse the bias, precision and accuracy of time delay and carrier phase estimation. Since these two indicators are determined by the signal spectrum pattern, they are used to formulate an optimization for signal design. By solving the optimization problem, only a few bands from the available signal spectrum are selected for ranging. Consequently, the designed signal only occupies a small amount of signal spectrum but has a large virtual bandwidth and can thereby still offer a high ranging precision with only a small bias, based on the low-complexity simplified ML method. Numerical and laboratory experiments are carried out to evaluate the ranging performance of the proposed estimation method based on sparsely selected signal bands. Relative positioning, in which we only measure a change in position, based on either the time delay estimates or the carrier phase estimates, is presented as a proof-of-concept for precise positioning. The results show that positioning based on only 7 out of 16 signal bands, sparsely placed in the available spectrum, achieves a decimeter level accuracy when using time delay estimates, and a millimeter level accuracy when using carrier phase estimates. Compared with the case of using all available bands, and without largely decreasing the positioning performance, the computational complexity when using the sparse multiband signal can be reduced by about 80%.

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Terrestrial positioning systems are being investigated as the complement to the global navigation satellite systems (GNSS), to provide precise and reliable positioning services in a GNSS-challenged environment. In this paper, we present the positioning performance of a ground-based positioning system, in which a multiband OFDM burst is used as a ranging signal to estimate carrier phase, and all transmitters are tightly synchronized by optically distributed time and frequency reference signals. The receiver, like in GNSS, runs on its own clock. An experiment has been carried out in an outdoor living lab environment to demonstrate the flexibility of precise positioning using carrier phase with the proposed ground-based system. During the experiment, the receiver was moved over a trajectory of 17 m forth and back, and acquired the ranging signal for 71 seconds. Without calibrating the different initial phase offsets among the transmitters, we keep the carrier phase cycle ambiguities as float numbers and compute the so called float position solutions. The root mean-squared error (RMSE) of the position solution in East and North direction are 4.22 cm and 4.63 cm, respectively, demonstrating the high-accuracy potential of the proposed burst oriented hybrid optical-wireless terrestrial positioning system.

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Smartphone accelerometers and low-cost Global Navigation Satellite System (GNSS) equipment have faced rapid and important advancement, opening a new door to deformation monitoring applications such as landslide, plate tectonics and structural health monitoring (SHM). The precision potential and operational feasibility of the equipment play an important role in the decision making of campaigning for affordable solutions. This paper focuses on the evaluation of the empirical precision, including (auto)time correlation, of a common smartphone accelerometer (Bosch BMI160) and a low-cost dual frequency GNSS reference-rover pair (u-blox ZED-F9P) set to operate at high rates (50 and 5 Hz, respectively). Additionally, a high-rate (5 Hz) GPS-only baseline-based multipath (MP) correction is proposed for effectively removing a large part of this error and allowing to correctly determine the instrumental noise of the GNSS sensor. Furthermore, the benefit of smartphone-based validation for the tracking of dynamic displacements is addressed. The estimated East-North-Up (ENU) precision values (σ^) of ±7.7, 8.1 and 9.6 mms2 are comparable with the declared precision potential (σ) of the smartphone accelerometer of ±8.8mms2. Furthermore, the acceleration noise shows only mild traces of (auto)correlation. The MP-corrected 3D (ENU) empirical precision values of ±2.6, 3.6 and 6.7 mm were found to be better by 30–40% than the straight-out-of box precision of the GNSS sensor, attesting the usefulness of the MP correction. The GNSS sensors output position information with time correlation of typically tens of seconds. The results indicate exceptional precision potential of these low-power-consuming, small-scale, affordable sensors set to operate at a high-rate over small regions. The smartphone-based dynamic displacement validation shows that GNSS data of a low-cost sensor at a 5 Hz sampling rate can be successfully used for tracking dynamic processes.@en

This paper presents an approach to analyse the quality, in terms of precision and reliability, of a system which integrates-at the observation-level-landmark positions and GNSS measurements, obtained with a single camera and a digital map, and a single frequency GNSS receiver respectively. We illustrate the analysis by means of design computations, and we present the actual performance by means of a small experiment in practice. It is shown that the integration model is able to produce a position solution even when both sensors individually fail to do so. With realistic assumptions on measurement noise, the proposed integrated, low-cost system can deliver a horizontal position with a precision of better than half a meter. The external reliability of the integrated system is at the few decimetre-level, showing that the impact of undetected faults in the measurements, for instance incorrectly identified landmarks in the image, on the horizontal position is limited and acceptable, thereby confirming the fault-robustness of the system.

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Global Navigation Satellite Systems (GNSS) are nowadays the most common solutions used to cope with Positioning-Navigation Timing (PNT) applications demands. GNSS are relied on in very diverse contexts and domains, yet the interest in systems such as GPS, GALILEO and Beidou is continuously increasing. However, and in particular for safety critical applications, GNSS are very vulnerable to unintentional interference and to intentional attacks such as spoofing or jamming. GNSS also provide degraded accuracy in dense multipath environments such as in urban canyons. Thus, solutions that could augment, back-up, complement, or surrogate GNSS, are actively sought after. In this paper, we introduce the concept of a hybrid optical wireless positioning system and present the initial experimental positioning results. The system uses optically distributed time and frequency reference signals for synchronization, and wideband radio signals for ranging. Initial results show that decimeter-level accuracy is obtained in urbanlike surroundings.@en

The goal of this contribution is to assess the impact statistical model selection has on confidence levels of parameter estimators in linear(ized) GNSS models. In the processing of observational data, parameter estimation and statistical testing are often combined. A testing procedure is exercised to select the most likely observational model among the hypothesized ones, which is then followed by the estimation of the identified model parameters. The resulting estimator will inherit the uncertainties involved in both estimation and testing which need to be properly taken into account when computing the corresponding confidence level. The approach that is usually followed in practice to determine the confidence level is to compute the probability of the estimator lying in a region around its true value conditioned on the identified hypothesis. Therefore, use is made of the estimator’s distribution under the identified hypothesis without regard to the conditioning process that led to the decision of accepting this hypothesis. In this contribution, it will be shown that for a proper computation of the confidence level in combined estimation-testing procedures, the associated probability should be conditioned not only on the identified hypothesis, but also on the testing outcome that led to the decision of accepting this hypothesis. Therefore, use need to be made of the conditional distribution of the estimator. We will provide numerical analysis of confidence levels with and without accounting for conditioning on testing decision using a number of examples in the context of GNSS single point positioning. It will be demonstrated that the customary practice which makes use of unconditional distributions to evaluate the confidence level, may give a too optimistic description of the estimator’s quality.@en

In this contribution, we propose a method for statistically evaluating the risk in a deformation monitoring system. When the structure under monitoring moves beyond tolerance, the monitor system should issue an alert. Only a very small probability is acceptable of the system telling us that no change beyond a critical threshold has taken place, while in reality it has. This probability is referred to as integrity risk. We provide a formulation of integrity risk where the interaction between estimation and testing is taken into account, implying the use of conditional probabilities. In doing so, we assumed different scenarios with the alerts being dependent on both the identified hypothesis and the threat that the estimated size of deformations entails. It is hereby highlighted that a correct risk evaluation requires estimation and testing being considered together, as they are typically intimately linked. In practice, one may, however, find it simpler computation-wise to neglect the estimation–testing link. For this case, we provide an approximation of the integrity risk. This approximation may provide a too optimistic or pessimistic description of the integrity risk depending on the testing procedure and tolerances of the structure at hand. Monitoring systems, besides issuing timely alerts, are also required to provide threat estimates together with their corresponding probabilistic properties. As the testing outcome determines how the threat gets estimated, the threat estimator will then inherit the statistical properties of both estimation and testing. We derive the threat estimator b¯ _j and its probability density function, taking the contributions from combined estimation and testing into account. It is highlighted that although the threat estimator under the identified hypothesis H_j, i.e., b^ _j, is normally distributed, the estimator b¯ _j is not. It is explained that working with b^ _j instead of b¯ _j, thus ignoring the estimation–testing link, may provide a too optimistic description of the threat estimator’s quality. The presented method is illustrated by means of two simple deformation examples.

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Time-based ranging accuracy is inversely proportional to the signal bandwidth. A larger the signal bandwidth leads to a higher accuracy of time delay estimation, but more complex hardware is needed. Alternatively, we explore the idea of using multiple narrow signal bands (e.g., 10 MHz of each) to create a large virtual signal bandwidth, which maintains the spectral efficiency but largely improves the ranging accuracy. Considering the impact of multipath, the propagation delay of the LoS path is computed from the estimated channel impulse response (CIR). In this paper, we propose an approach to sparsely select signal bands for ranging and positioning based on convex optimization. The Cramér-Rao lower bound (CRLB) for the propagation delay and gain estimators, as a performance criterion, is employed in the constraint of the optimization. The CRLB is derived in a two-path channel, so that the accuracy and the correlation between the LoS path and the reflection are taken into account. Experiments are conducted in a laboratory environment to illustrate the proposed signal design methodology dedicated for ranging with a sub-decimeter accuracy.

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In developing a high accuracy terrestrial radio navigation system, as a complement to a global navigation satellite system (GNSS), it is recognized that the performance of time delay estimation is proportional to, and thereby limited by, the signal bandwidth. Given a possibly narrow signal bandwidth, the central carrier phase can, alternatively, provide a better distance accuracy, though the central carrier phase cycle ambiguity should be resolved. In practice, the carrier phase may be perturbed by multipath. In this paper, considering an orthogonal frequency division multiplexing (OFDM) signal, we propose a two-step carrier phase estimation method to reduce the error introduced by multipath. First, the propagation delay of the LoS path is coarsely determined, then the carrier phase is estimated using the earlier determined coarse time delays. Furthermore, a positioning model only based on carrier phase estimates is presented in this paper. The proposed technique is evaluated by statistical analyses and a simulated OFDM-based terrestrial positioning system in different roadway multipath environments. The results show that the impact of multipath on carrier phase estimation can be largely mitigated, so that the carrier phase can be used for precise positioning. In addition, fixing the integer carrier phase cycle ambiguities can significantly reduce the time for the position solution to converge to high precision.

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