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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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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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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The provision of accurate ionospheric corrections in PPP-RTK enormously improves the performance of single-receiver user integer ambiguity resolution (IAR), thus enabling fast high precision positioning. While an external provider can disseminate such corrections to the user with a time delay, it is the task of the user to accurately time-predict the corrections so that they become applicable to the user positioning time. Accurate time prediction of the corrections requires a dynamic model in which the process noise of the corrections has to be correctly specified. In this contribution, we present an estimation method to determine the process noise variance of PPP-RTK corrections using single-receiver GNSS data. Our focus is on variance estimation of the first-order slant ionospheric delays, which allows one to analyze how the ionospheric process noise changes as a function of the solar activity, receiver local time, and receiver geographic latitude. By analyzing 11-year GNSS datasets, it is illustrated that estimates of the ionospheric process noise are strongly correlated with the solar flux index F10.7. These estimates also indicate a seasonal variation, with the highest level of variation observed during the spring and autumn equinoxes.

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To obtain single-receiver Global Navigation Satellite System (GNSS) parameter solutions, the PPP-RTK user-filter combines measurements with time-correlated corrections that are separately computed by the filter of an external provider. The consequence of exercising such double-filtering is that the Kalman filter’s standard assumption of having uncorrelated measurements in time becomes violated. This leads the user-filter to lose its ‘minimum variance’ property, thereby delivering imprecise parameter solutions. The solutions’ precision-loss becomes more pronounced when one experiences an increase in the correction latency, i.e., the delay in time after the corrections are estimated and the time they are applied to the user measurements. In this contribution, we propose a new multi-epoch formulation for the PPP-RTK user-filter upon which both the uncertainty and the temporal correlation of the corrections are incorporated. By a proper augmentation of the user-filter state-vector, the corrections are jointly measurement-updated with the user parameter solutions. Supported by numerical results, the proposed formulation is shown to outperform its commonly used counterpart in the minimum-variance sense.

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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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In this contribution, we generalize PPP–RTK theory by allowing the transmitters to transmit on different frequencies. The generalization is based on the integer-estimability theory of Teunissen (A new GLONASS FDMA model. GPS Solutions, 2019). As the theory and associated algorithms provided are generally applicable, they apply to satellite-based carrier-phase positioning as well as to terrestrial interferometric sensory networks. Based on an identification of the constraints imposed on the admissible ambiguity transformations by PPP–RTK, a fundamental network+user condition is found that determines whether PPP–RTK is possible or not. The discriminating contributions of both the network and user observation equations to this PPP–RTK condition are analysed, followed by a description of PPP–RTK enabling classes of measurement scenarios.

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The corrections needed to realize integer ambiguity resolution-enabled precise point positioning (PPP-RTK) at a single-receiver user are often treated as if they are deterministic quantities. The present contribution aims to study and analyze the effect the neglected uncertainty of these corrections, which are subject to time delay, has on the PPP-RTK user ambiguity resolution and positioning performance. Next to the analyses of the estimation results, we emphasize their quality information and show to what extent the assumed positioning precision that the user is provided with differs from the minimum-variance counterpart under an incorrectly specified user stochastic model. We develop and present two alternatives to the fully populated error variance matrix of the PPP-RTK corrections that the user can reconstruct with limited information from the provider so as to properly weigh his corrected data and achieve close-to-optimal performance for high latencies. Supported by numerical results, our study demonstrates that the alternative variance matrices are sufficient enough for the user to obtain improved instantaneous PPP-RTK performance and a realistic precision description in the positioning domain.

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An assessment of standalone GLONASS RTK performance is provided using its FDMA and CDMA signals. The new integer-estimable GLONASS FDMA model (Teunissen 2019), which guarantees the integer-estimability of its ambiguities, is employed. We introduce the combined integer-estimable GLONASS FDMA+CDMA model and compare its strength against the FDMA model for instantaneous integer ambiguity resolution and positioning. Various combinations of GLONASS signals are considered including FDMA L1, FDMA+CDMA L1+L3, FDMA L1+L2 and FDMA+CDMA L1+L2+L3. To provide insight into the current RTK performance of GLONASS, we used observations of a short baseline to analyze the integer ambiguity resolution success rate and positioning precision, formally and empirically. To provide insight into the future RTK performance of GLONASS, we present a formal analysis of the integer ambiguity resolution success rate and ADOP, assuming that all the GLONASS satellites transmit FDMA L1, L2 and CDMA L3 signals. A formal analysis of standalone GLONASS ambiguity resolution based on current and future GLONASS constellation is then presented for different locations around the world.

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In this contribution, we introduce a generalized Kalman filter with precision in recursive form when the stochastic model is misspecified. The filter allows for a relaxed dynamic model in which not all state vector elements are connected in time. The filter is equipped with a recursion of the actual error-variance matrices so as to provide an easy-to-use tool for the efficient and rigorous precision analysis of the filter in case the underlying stochastic model is misspecified. Different mechanizations of the filter are presented, including a generalization of the concept of predicted residuals as needed for the recursive quality control of the filter.@en

Although ionosphere-weighted GNSS parameter estimation is a popular technique for strengthening estimator performance in the presence of ionospheric delays, no provable rules yet exist that specify the needed weighting in dependence on ionospheric circumstances. The goal of the present contribution is therefore to develop and present the ionospheric conditions that need to be satisfied in order for the ionosphere-weighted solution to be mean squared error (MSE) superior to the ionosphere-float solution. When satisfied, the presented conditions guarantee from an MSE performance view, when (a) the ionosphere-fixed solution can be used, (b) the ionosphere-float solution must be used, or (c) an ionosphere-weighted solution can be used.@en

Estimability and S-systems are important concepts when dealing with rank-defect models. In this contribution, we generalize the concept of estimability to integer estimability and determine the necessary and sufficient conditions that need to be satisfied for parameter functions to be integer estimable. This is then worked out and applied to the integer estimability analysis of GNSS observation equations. We hereby consider both network ambiguity resolution and single-receiver PPP-RTK ambiguity resolution. In our analyses, use is made of graph theory and properties of the ambiguity incidence matrices of the bipartite and connected network graphs.@en

Various studies have been performed to investigate the accuracy of troposphere zenith wet delays (ZWDs) determined from GPS. Most of these studies use dual-frequency GPS data of large-scale networks with long baselines to determine the absolute ZWDs. For small-scale networks the estimability of the absolute ZWDs deteriorates due to high correlation between the solutions of the ZWDs and satellite-specific parameters as satellite clocks. However, as relative ZWDs (rZWDs) can always be estimated, irrespective of the size of the network, it is of interest to understand how the large-scale network rZWD-performance of dual-frequency GPS using an ionosphere-float model compares to the small-scale network rZWD-performance of single-frequency GPS using an ionosphere-weighted model. In this contribution such an analysis is performed using undifferenced and uncombined network parametrization modelling. In this context we demonstrate the ionosphere weighted constraints, which allows the determination of the rZWDs independent from signals on the second frequency. Based on an analysis of both simulated and real data, it is found that under quiet ionosphere conditions, the accuracy of the single-frequency determined rZWDs in the ionosphere-weighted network is comparable to that of the large-scale dual-frequency network without ionospheric constraints. Making use of the real data from two baselines of 15 days, it was found that the absolute differences of the rZWDs applying the two strategies are within 1 cm in over 90% and 95% of the time for ambiguity-float and -fixed cases, respectively.@en

In view of the recent proliferation of low-cost mass-market receivers, the number of network receivers and GNSS users will be growing rapidly, demanding an efficient way of data processing in terms of computational power and capacity. One way of improving the computational capacity is to decentralize the underlying data processing and distribute the task of the computer center across individual network receivers. In this invited contribution we review the problem of distributed estimation and present an algorithm for distributed least-squares estimation using the alternating direction method of multipliers. Applying the algorithm to a network of GNSS receivers, we show how the distributed data processing of individual receivers can deliver parameter solutions comparable to their centralized network-derived counterparts. With distributed estimation techniques, GNSS single-receiver users can therefore obtain high-precision solutions without the need of having a centralized computing center.@en

A new integer-estimable GLONASS FDMA model is studied and analysed. The model is generally applicable, and it shows a close resemblance with the well-known CDMA models. The analyses provide insights into the performance characteristics of the model and concern a variety of different ambiguity-resolution critical applications. This will be done for geometry-free, geometry-fixed and several geometry-based formulations. Next to the analyses of the model’s instantaneous ambiguity-resolved positioning and attitude determination capabilities, we show the ease with which the model can be combined with existing CDMA models. We thereby present the instantaneous ambiguity-resolution performances of integrated L1 GPS + GLONASS, both for high-grade geodetic and mass-market receivers. We also consider the potential of the single-frequency combined model for mixed-receiver processing, particularly for the case the between-receiver GLONASS pseudorange data are biased. In all cases, the speed of successful ambiguity resolution is studied as well as the precision with which positioning is determined. Software routines for constructing the model are also provided.@en

Successful implementation of integer ambiguity resolution enabled precise point positioning (aka PPP-RTK) algorithms is inextricably linked to the ability of a user to perform near real-time positioning by quickly and reliably resolving the integer carrier-phase ambiguities. In the PPP-RTK technique, a major barrier to successful ambiguity resolution is the unmodelled impact of the ionosphere. We present a 4D ionospheric tomographic model that computes in real time the ionospheric electron density as a linear combination of basis functions, namely B-splines. The results show that when the ionospheric estimates are provided as atmospheric corrections for a PPP-RTK end-user, the time to fix its horizontal position below 10 cm is around 20 epochs (the sample rate is 30 s) at the 90 % of the cumulative distribution function (CDF), as opposed to the time it takes when no external corrections are provided, which is around 80 epochs at 90 % of the CDF.

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For long-baseline over several hundreds of kilometers, the ionospheric delays that cannot be fully removed by differencing observations between receivers hampers rapid ambiguity resolution. Compared with forming ionospheric-free linear combination using dual- or triple-frequency observations, estimating ionospheric delays using uncombined observations keeps all the information of the observations and allows extension of the strategy to any number of frequencies. As the number of frequencies has increased for the various GNSSs, it is possible to study long-baseline ambiguity resolution performance using up to five frequencies with uncombined observations. We make use of real Galileo observations on five frequencies with a sampling interval of 1 s. Two long baselines continuously receiving signals from six Galileo satellites during corresponding test time intervals were processed to study the formal and empirical ambiguity success rates in case of full ambiguity resolution (FAR). The multipath effects are mitigated using the measurements of another day when the constellation repeats. Compared to the results using multipath-uncorrected Galileo observations, it is found that the multipath mitigation plays an important role in improving the empirical ambiguity success rates. A high number of frequencies are also found to be helpful to achieve high ambiguity success rate within a short time. Using multipath-uncorrected observations on two, three, four and five frequencies, the mean empirical success rates are found to be about 73, 88, 91, and 95% at 10 s, respectively, while the values are increased to higher than 86, 95, 98, and 99% after mitigating the multipath effects.

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The real-time kinematic precise point positioning (PPP-RTK) technique enables integer ambiguity resolution by providing singlereceiver users with information on the satellite phase biases next to the standard PPP corrections. Using undifferenced and uncombined observations, rank deficiencies existing in the design matrix need to be eliminated to formestimable parameters. In this contribution, the estimability of the parameters was studied in single-frequency ionosphere-weighted scenario, given a dynamic satellite-clock model in the network Kalman filter. In case of latency of the network corrections, the estimable satellite clocks, satellite phase biases, and ionospheric delays need to be predicted over short time spans. With and without satellite-clock models incorporated in the network Kalman filter, different approaches were used to predict the network corrections. This contribution shows how the predicted network corrections responded to the presence and absence of satellite-clock models. These differences in the predicted network corrections were also reflected in the user positioning results. Using three different 1-Hz global positioning system (GPS) single-frequency data sets, two user stations in one small-scale network were used to compute the positioning results, applying predicted network corrections. The latency of the network products ranges from 3 to 10 s. It was observed that applying strong satellite-clock constraints in the network Kalman filter (i.e., with the process noise of 1 or 0.5mm per square root of second) reduced the root-mean squares (RMS) of the user positioning results to centimeters in the horizontal directions and decimeters in the vertical direction for latencies larger than 6 s, compared to the cases without a satellite-clock model.@en

Precise point positioning (PPP) and its integer ambiguity resolution-enabled variant, PPP-RTK (real-time kinematic), can benefit enormously from the integration of multiple global navigation satellite systems (GNSS). In such a multi-GNSS landscape, the positioning convergence time is expected to be reduced considerably as compared to the one obtained by a single-GNSS setup. It is therefore the goal of the present contribution to provide numerical insights into the role taken by the multi-GNSS integration in delivering fast and high-precision positioning solutions (sub-decimeter and centimeter levels) using PPP-RTK. To that end, we employ the Curtin PPP-RTK platform and process data-sets of GPS, BeiDou Navigation Satellite System (BDS) and Galileo in stand-alone and combined forms. The data-sets are collected by various receiver types, ranging from high-end multi-frequency geodetic receivers to low-cost single-frequency mass-market receivers. The corresponding stations form a large-scale (Australia-wide) network as well as a small-scale network with inter-station distances less than 30 km. In case of the Australia-wide GPS-only ambiguity-float setup, 90% of the horizontal positioning errors (kinematic mode) are shown to become less than five centimeters after 103 min. The stated required time is reduced to 66 min for the corresponding GPS + BDS + Galieo setup. The time is further reduced to 15 min by applying single-receiver ambiguity resolution. The outcomes are supported by the positioning results of the small-scale network.

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In PPP-RTK network processing, the wet component of the zenith tropospheric delay (ZTD) cannot be precisely modelled and thus remains unknown in the observation equations. For small networks, the tropospheric mapping functions of different stations to a given satellite are almost equal to each other, thereby causing a near rank-deficiency between the ZTDs and satellite clocks. The stated near rank-deficiency can be solved by estimating the wet ZTD components relatively to that of the reference receiver, while the wet ZTD component of the reference receiver is constrained to zero. However, by increasing network scale and humidity around the reference receiver, enlarged mismodelled effects could bias the network and the user solutions. To consider both the influences of the noise and the biases, the mean-squared errors (MSEs) of different network and user parameters are studied analytically employing both the ZTD estimation strategies. We conclude that for a certain set of parameters, the difference in their MSE structures using both strategies is only driven by the square of the reference wet ZTD component and the formal variance of its solution. Depending on the network scale and the humidity condition around the reference receiver, the ZTD estimation strategy that delivers more accurate solutions might be different. Simulations are performed to illustrate the conclusions made by analytical studies. We find that estimating the ZTDs relatively in large networks and humid regions (for the reference receiver) could significantly degrade the network ambiguity success rates. Using ambiguity-fixed network-derived PPP-RTK corrections, for networks with an inter-station distance within 100 km, the choices of the ZTD estimation strategy is not crucial for single-epoch ambiguity-fixed user positioning. Using ambiguity-float network corrections, for networks with inter-station distances of 100, 300 and 500 km in humid regions (for the reference receiver), the root-mean-squared errors (RMSEs) of the estimated user coordinates using relative ZTD estimation could be higher than those under the absolute case with differences up to millimetres, centimetres and decimetres, respectively.@en