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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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.
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In this work we introduce the LAMBDA 4.0 toolbox, which provides an enhanced implementation for integer estimation, validation, and success rate evaluation. This free and open-source toolbox is a major update to LAMBDA 3.0 (2012), while it also integrates the functionalities from Ps-LAMBDA 1.0 (2013), thus respectively merging both estimation and evaluation capabilities. The new implementation provides redefined algorithms, such as an improved integer search strategy with a one-order reduction in the computational time, along with additional estimators: Vectorial Integer Bootstrapping (VIB), Integer Aperture Bootstrapping (IAB) and Best Integer Equivariant (BIE). This toolbox aims to become a valuable resource for researchers and/or practitioners dealing with mixed-integer models in high dimensions, e.g., terrestrial-based carrier-phase systems, multi-constellation Global Navigation Satellite Systems (GNSS), or other interferometric applications.

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The aim of computing a geodetic network is to determine the geometry of the configuration of a set of points from spatial observations (e.g. GPS baselines and/or terrestrial measurements). The configuration of points usually consists of newly established points, of which the coordinates still need to be determined, and already existing points, the so-called control points, of which the coordinates are known.

Network quality control deals with the qualitative aspects of network design, network adjustment, network validation and network connection. By means of a network adjustment the relative geometry of the new points is determined and integrated into the geometry of the existing control points. Prior to the network adjustment, the geometry of the network is designed on the basis of precision and reliability criteria.

The adjustment and validation of the overall geometry can be divided in two phases, the free network phase and the connected network phase. In the free network phase, the known coordinates of the control points do not take part in the adjustment and validation. The possible use of a free network phase is based on the idea that a good geodetic network should be sufficiently precise and reliable in itself, without the need of external control. Moreover, it allows one to validate the quality of the external control.

In the connected network phase, the geometry of the free network is integrated into the geometry of the control points. Adjustment and validation in this second phase differs from the free network phase. The adjustment in the second phase is a constrained connection adjustment, since it is often not practical to see the coordinates of the control points change every time a free network is connected to them. For the validation of the connected network however, the unconstrained connected adjustment is used as input. This allows one to take the intrinsic uncertainty of the coordinates of the control points in the connection phase into account.

The goal of this introductory text on network quality control is to convey the necessary knowledge for designing, adjusting and testing geodetic networks. For the purpose of network design, the precision and reliability theory is worked out in detail. This includes the minimal detectable biases and the bias-to-noise ratios. For the purpose of the network adjustment, the principles of unconstrained-, constrained-, and minimally constrained least-squares estimation, are treated. For the network testing, the principles of hypothesis testing are presented and worked out for the different network cases. For the free network phase this includes the overall model test, the w-test, and the data snooping procedure. For the connected network phase, it includes the T-test, with an emphasis on the detection and identification of errors in the control points.
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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.

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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.@en

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.

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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.

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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.@en

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

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.

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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

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.

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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.

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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.

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Carrier phase signals are considered among the key observations in global navigation satellite systems (GNSSs) and several other high-precision interferometric measurement systems. However, these ultra-precise measurements are not fully exploited when the integerness of their inherent ambiguities is discarded during the estimation process. Provided that the integer-estimable functions of their phase ambiguities are properly identified, integer ambiguity resolution (IAR) can be utilized to benefit their parameter solutions. For the GNSS code division multiple access systems with transmitters that broadcast carrier phase signals on identical frequencies, these integer-estimable functions have been characterized and are well-known as double differenced ambiguities. However, this is not the case with “frequency-varying” carrier phase signals that are broadcast by GLONASS satellites, Low-Earth-Orbiting communication satellites, or cellular long-term evolution (LTE) transmitters. This study aims to present full-rank models that can be used to identify integer-estimable ambiguity functions, thereby bringing the observation equations of frequency-varying carrier phase measurements into an IAR-applicable form. Our analytical results are supported by several numerical examples, including GNSS and terrestrial-based IAR as well as a new set of “inter-frequency” integer ambiguities that this study discovers in Galileo multi-frequency carrier phase signals.

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It is common practice to use the well-known concept of the minimal detectable bias (MDB) to assess the performance of statistical testing procedures. However, such procedures are usually applied to a null and a set of multiple alternative hypotheses with the aim of selecting the most likely one. Therefore, in the DIA method for the detection, identification and adaptation of model misspecifications, rejection of the null hypothesis is followed by identification of the potential source of the model misspecification. With identification included, the MDBs do not truly reflect the capability of the testing procedure and should therefore be replaced by the minimal identifiable bias (MIB). In this contribution, we analyse the MDB and the MIB, highlight their differences, and describe their impact on the nonlinear DIA-estimator of the model parameters. As the DIA-estimator inherits all the probabilistic properties of the testing procedure, the differences in the MDB and MIB propagation will also reveal the different consequences a detection-only approach has versus a detection+identification approach. Numerical algorithms are presented for computing the MDB and the MIB and also their effect on the DIA-estimator. These algorithms are then applied to a number of examples so as to analyse and illustrate the different concepts.

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In this contribution we present a review of the DIA-method to ensure navigational integrity. The DIA-method rigorously combines parameter estimation and statistical testing for the Detection, Identification and Adaptation of multivariate and multiple model misspecifications. We describe the statistical properties of the so-obtained DIA-estimator together with its probability density function. Numerical examples are given to highlight various aspects of the navigational DIA-estimator.

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In this contribution we will present and evaluate the joint probability density function (PDF) of the multivariate integer GNSS carrier phase ambiguity residuals, thereby assuming that the GNSS data belong to the very general class of elliptically contoured (EC) distributions. Examples of distributions belonging to this class are the multivariate normal distribution, the t-distribution and the contaminated normal distribution. Since the residuals and their properties depend on the integer estimation principle used, we will present the PDF of the ambiguity residuals for the whole class of admissible integer estimators. This includes the estimation principles of integer rounding, integer bootstrapping, and integer least squares. The probabilistic properties of these estimators vary with the distributions from the EC-class. In order to get a better understanding of the various features of the joint PDF of the ambiguity residuals we will use a step-by-step construction aided by graphical means.

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The present contribution aims to address why the stochastic model of the PPP-RTK user-filter is misspecified, and how one can limit the precision-loss associated with user parameter solutions. By developing tools for measuring the stated precision-loss under existing formulations of the user’s Kalman filter, we propose an alternative formulation that recursively delivers close-to-minimum-variance filtered solutions when certain conditions hold. Such conditions are discussed, and their impact on the user ambiguity-resolved positioning performance is illustrated by supporting numerical results.

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