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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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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Precise Point Positioning (PPP) is a Global Navigation Satellite Systems (GNSS) modelling and processing method that provides single-receiver users with high positioning accuracy anywhere on the globe, without the explicit dependence on reference receivers. The realization of PPP is based on undifferenced code and phase measurements, a priori correction models, as well as on precise satellite orbits and clocks. Although PPP delivers highly accurate positioning results, a relatively long timespan is needed to achieve such accurate results. This long convergence time is mainly due to the presence of the carrier-phase ambiguities and ionospheric delays, and can be significantly reduced if one can do away with these unknown parameters using integer-estimation and external corrections, respectively. The integer ambiguity resolution-enabled variant of PPP, namely PPP-RTK, is the GNSS positioning mode that is capable of delivering ambiguity-resolved parameter solutions on the basis of single-receiver user data and state-space corrections, which include, next to satellite orbits and clocks, information about the satellite phase and code biases. These corrections, when properly provided from either a multi- or a single-station setup, enable recovery of the integer property of the user ambiguities, thus enabling single-receiver integer ambiguity resolution and, therefore, reduced convergence times compared to those experienced with ambiguity-float PPP. A considerable observational time span of 30-60 min is, however, still needed to integer-resolve the ambiguities with sufficiently large success rate in the presence of ionospheric delays, which cannot compete with that achieved with relative positioning techniques over short baselines. The lack of any ionospheric information necessitates that the user utilizes the ionosphere-float model – a model that treats the slant ionospheric delays as unknown parameters – that is known to be relatively weak in the sense of its ambiguity resolution capabilities. Faster ambiguity resolution and, therefore, improved convergence time are expected when such information can be provided to the user’s model. The augmentation with ionospheric information, though, requires dense network infrastructure that is often not available either because of spatial restrictions or due to the high-cost and complex operation requirements involved. In such cases, a user’s model strengthening can be alternatively substantiated through the integration of multi-constellation multi-frequency measurements. The increased number of satellites and frequencies paves the way for accelerating successful ambiguity resolution and, therefore, convergence times. Next to the rapid centimeter-level convergence that is of top priority to the users, positioning reliability is critical as well for the user performance. The commonly used practice in PPP-RTK to neglect the correctional uncertainty may have considerable effects not only on the ambiguity resolution performance but, most importantly, on the precision description the user is provided with to judge his real-time performance. To obtain the optimal positioning performance, the users need to incorporate the quality description of the corrections into their estimation process. Obviously, the PPP-RTK user positioning convergence time and reliability are still open problems. In order to overcome the aforementioned limitations, three approaches are investigated in this PhD thesis. The first method utilizes ionospheric information from regional multi-scale networks to aid the user model in increasing its redundancy, thus allowing for faster PPP-RTK ambiguity resolution. An extensive formal analysis revealed that such an acceleration would be possible only if the precision of the provided ionospheric corrections is equal to or better than 5 cm. It was observed, though, that this quality level may not be achieved with a function-based two-dimensional ionosphere model that considers a single-layer model and a slant-to-vertical mapping function. To overcome this, a methodology was introduced that uses the slant delays directly as estimated from the PPP-RTK network processing and predicts, by means of the best linear unbiased prediction framework, the slant ionospheric corrections per satellite and per epoch at the user’s location. It was shown how the user’s model needs to be extended to its ionosphere-weighted variant in order to incorporate these corrections, and how their quality can be reliably evaluated. The empirical analysis of a sufficiently large number of positioning solution samples showed that near-instantaneous centimeter-level positioning is feasible in case the corrections are provided by a small-scale network. Further analysis of networks with varying density revealed, for the first time in terms of PPP-RTK, the impact the network density has on the achieved convergence times and their linear relationship with the mean inter-station distance. Then, the approach of integrating multi-GNSS multi-frequency data, as an alternative to the ionospheric corrections augmentation, was analyzed for improving PPP-RTK convergence. The advantage of this approach compared to the previous is that it dispenses with the stringent requirement of operating a dense network infrastructure and also the necessity for the user to be located within the network’s operating range to utilize the provided ionospheric signals. A formal performance analysis of globally distributed user stations showed the impact of the increased number of satellites and frequencies on the expected ambiguity resolution and positioning performance. Although both factors bring considerable improvements, it was revealed that the satellite redundancy plays a more crucial role in speeding up the convergence time due to the improved geometry strength. Analysis of various simulated datasets revealed that the sensitivity of the user’s performance, in response to changes in the measurement precision, becomes less pronounced for multi-GNSS multi-frequency models. In addition, the impact of the number and spacing of frequencies on the multi-frequency PPP-RTK user performance was investigated, for the first time in terms of PPP-RTK. It was both formally and empirically evidenced that frequency spacing contributes to a larger extent, compared to the number of frequencies, to the user ambiguity resolution and, therefore, to the convergence times. The role of the estimable satellite code biases in multi-frequency data processing was highlighted and their impact on the achieved performance was evaluated. The positioning results using multi-frequency Galileo-plus-GPS data showed that centimeter-level positioning can be achieved almost instantaneously, even in the absence of ionospheric information. Finally, the PPP-RTK user positioning reliability was analyzed in terms of the precision description the user is provided with when the user stochastic model is misspecified. A generalized Kalman-filter was introduced that is capable of, first, rigorously processing dynamic systems when only a subset of the state-vector elements are linked in time and, second, recursively providing the actual precision in case of a misspecified stochastic model as is the case when neglecting the uncertainty of PPP-RTK corrections. Analysis of the behavior of the filter-precision indicated that the actual error-variance, in response to changes in the assumed stochastic model, is difficult to predict a priori. The effects of such a misspecification on the data quality control mechanisms was discussed and analyzed with illustrative examples. The impact of the neglected PPP-RTK correctional uncertainty on the user ambiguity resolution and positioning performance was empirically evaluated for nonzero correction latencies. It was evidenced that, apart from the reduced ambiguity success rates, the inconsideration of the corrections’ quality may lead to significant deviation between the formal and empirical positioning errors, thereby misleading the users with incorrect standard deviations. Mitigation methods were developed and their performance was numerically demonstrated for varying latency and for both single- and multi-constellation models.@en

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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The tropospheric delay is one of many error sources that affect the Global Navigation Satellite System (GNSS) positioning solutions. The widely used troposphere models assume a homogeneous atmosphere so that only the zenith delay needs to be determined and is mapped through an elevation-dependent mapping function. This procedure is to reduce the computational burden and keep the positioning model full-rank. However, this assumption fails for a realistic description of the troposphere, which is always asymmetrical at a certain elevation angle, especially during a weather event when the weather conditions are very complex. These imperfectly modelled tropospheric delays may influence the positioning accuracy and integer ambiguity resolution performance. In this case, this contribution aims to investigate the effects of the model errors due to the asymmetrical troposphere on GNSS estimations. The Numerical Weather Prediction (NWP) model is applied to generate the actual ray-tracing tropospheric delay in Western Europe, and the tropospheric model errors are calculated in a normal weather condition and a weather event condition by comparing the slant delay calculated from the NWP model and the mapping function. Case studies on the same GNSS station are conducted in two weather conditions: a normal troposphere condition and a weather event with heavy rainfall. The results based on the case studies show that the troposphere in the normal weather condition is nearly homogeneous that the azimuthal-dependent discrepancies of the tropospheric delay are less than 1cm at a very low elevation angle; meanwhile, the discrepancies between different azimuthal angles can reach to more than 25cm in the weather event. A single-frequency Single Point Positioning (SPP) model and a Precise Point Positioning (PPP) model that preserves the integer property of ambiguity are chosen for studying the estimation biases caused by the troposphere model errors. It turns out that almost all horizontal positioning biases of SPP and PPP are less than 1cm in the normal weather condition; however, the scales of the horizontal and 3D biases are concentrated in 1 to 10cm in the weather event for these two models. This contribution also contains the study of the actual integer ambiguity resolution success rate in the presence of the tropospheric model errors by applying the Monte Carlo simulation, and the success rates of PPP in the normal weather condition are consistent with the theoretical values calculated with the ideal troposphere which is totally symmetrical. However, the actual success rates in the weather event are extremely low at some epochs due to the tropospheric model errors, which means that wrong fixing may occur since the theoretical values cannot take into account these model errors. Note that the horizontal tropospheric gradients are not involved in the processing, which means that an optimistic performance might be expected if the gradients are considered.

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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

The single-receiver integer ambiguity resolution-enabled variant of precise point positioning (PPP), namely PPP-RTK, has proven to be crucial in reducing the long convergence time of PPP solutions through the recovery of the integerness of the user-ambiguities. The proliferation of global navigation satellite systems (GNSS) supports various improvements in this regard through the availability of more satellites and frequencies. The increased availability of the Galileo E6 signal from GNSS receivers paves the way for speeding up integer ambiguity resolution, as more frequencies provide for a stronger model. In this contribution, the Galileo-based PPP-RTK ambiguity resolution and positioning convergence capabilities are studied and numerically demonstrated as a function of the number and spacing of frequencies, aiming to shed light on which frequencies should be used to obtain optimal performance. Through a formal analysis, we provide insight into the pivotal role of frequency separation in ambiguity resolution. Using real Galileo data on up to five frequencies and our estimated PPP-RTK corrections, representative kinematic user convergence results with partial ambiguity resolution are presented and discussed. Compared to the achieved performance of dual-frequency fixed solutions, it is found that the contribution of multi-frequency observations is significant and largely driven by frequency separation. When using all five available frequencies, it is shown that the kinematic user can achieve a sub-decimeter level convergence in 15.0 min (90% percentile). In our analysis, we also show to what extent the provision of the estimable satellite code biases as standard PPP-RTK corrections accelerates convergence. Finally, we numerically demonstrate that, when integrated with GPS, the kinematic user solution achieves convergence in 3.0 and 5.0 min on average and at 90%, respectively, in the presence of ionospheric delays, thereby indicating the single-receiver user’s fast-convergence capabilities.

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The technology of integer ambiguity resolution-enabled precise-point-positioning (also referred to as PPP-AR) has been proven capable of providing comparable accuracy, efficiency, and productivity to long-baseline real-time kinematic positioning (RTK) during the last decade. Commercial PPP-AR services have been provided by different institutions and companies and have been widely used in geodetic missions. However, the usage and research of the PPP-AR mostly concentrated on nonaviation applications, e.g., vehicle navigation, surveying, and mapping, and monitoring crustal motions. Few of them focused on fixing the ambiguities during an aircraft flight. In this contribution, we implemented the PPP-AR technique for the first time in an airplane flight test to investigate how much the fixed ambiguities could contribute to airplane positioning solutions in challenging circumstances, including high velocity and severe maneuvers. We first looked into the influences of the tropospheric delay on the positioning and ambiguity solutions because the height of the airplane may dramatically change within a narrow time span, and thus, a proper constraint of this parameter was crucial for the computation of the tropospheric effects. Then, how to fix the ambiguities successfully and reliably in challenging circumstances was discussed. Finally, the airplane data was processed in 15 and 1s intervals with ambiguity float and fixed solutions under different configurations to illustrate in which condition and to what extent the fixed ambiguities can improve the airplane positioning accuracy.

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A single-receiver integer ambiguity resolution-enabled precise point positioning (PPP-RTK) user experiences a long convergence time when the rather weak single-constellation dual-frequency ionosphere-float model is used. Nowadays, the rapid development of Global Navigation Satellite Systems (GNSS) provides a multitude of available satellites and frequencies that can serve in improving the user's model strength and, therefore, its ambiguity resolution and positioning capabilities. In this study, we provide insight into and analyze the global impact of a multi-GNSS (GPS, Galileo, BeiDou-3) multi-frequency integration on the expected ambiguity resolution and positioning performance of the ionosphere-float uncombined PPP-RTK user model, and demonstrate whether it is the increased number of satellites or frequencies, or a combination thereof, that speeds up ambiguity-resolved positioning. Moreover, we explore the capabilities of both full (FAR) and partial (PAR) ambiguity resolution, considering the full ambiguity information content with the LAMBDA method, and investigate whether PAR is an efficient solution to the multi-dimensional ambiguity case. The performance of our solutions is assessed in terms of the ambiguity success rate (ASR), the number of epochs (TTFA) to achieve both an ASR criterion and a horizontal positioning precision better than 10 cm, as well as the gain in precision improvement. Based on multi-system multi-frequency simulated data from nine globally distributed stations and a large number of kinematic solutions over a day, we found that the increase in number of frequencies enhances the ambiguity resolution performance, with PAR achieving a TTFA reduction of 70% when five instead of two Galileo frequencies are used, while the ambiguity-float solution is only slightly improved. Further, our numerical results demonstrated that the increase in number of satellites leads to an improvement in both the positioning and ambiguity resolution performance, due to the improved geometry strength. It is shown that the GPS+Galileo+BeiDou solutions can achieve a TTFA of 6.5 and 4.5 min (at 90%) on a global scale when two and three frequencies are used, respectively, without any a priori information on the ionospheric delays. Finally, we analyzed the sensitivity of the PPP-RTK user's performance to changes in the precision of the measurements.

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This contribution implements the Kriging interpolation in predicting the tropospheric wet delays using global navigation satellite system networks. The predicted tropospheric delays can be used in strengthening the precise point positioning models and numerical weather prediction models. In order to evaluate the performances of the Kriging interpolation, a sparse network with 8 stations and a dense network with 19 stations from continuously operating reference stations (CORS) of the Netherlands are selected as the reference. In addition, other 15 CORS stations are selected as users, which are divided into three blocks: 5 stations located approximately in the center of the networks, 5 stations on the edge of the networks and 5 stations outside the networks. The zenith tropospheric wet delays are estimated at the network and user stations through the ionosphere-free positioning model; meanwhile, the predicted wet delays at the user stations are generated by the Kriging interpolation in the use of the tropospheric estimations at the network. The root mean square errors (RMSE) are calculated by comparing the predicted wet delays and estimated wet delays at the same user station. The results show that RMSEs of the stations inside the network are at a sub-centimeter level with an average value of 0.74 cm in the sparse network and 0.69 cm in the dense network. The stations on edge and outside the network can also achieve 1-cm level accuracy, which overcomes the limitation that accurate interpolations can only be attained inside the network. This contribution also presents an insignificant improvement of the prediction accuracy from the sparse network to the dense network over 1-year’s data processing and a seasonal effect on the tropospheric wet delay predictions.

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The benefits of an increased number of global navigation satellite systems (GNSS) in space have been confirmed for the robustness and convergence time of standard precise point positioning (PPP) solutions, as well as improved accuracy when (most of) the ambiguities are fixed. Yet, it is still worthwhile to investigate fast and high-precision GNSS parameter estimation to meet user needs. This contribution focuses on integer ambiguity resolution-enabled Precise Point Positioning (PPP-RTK) in the use of the observations from four global navigation systems, i.e., GPS (Global Positioning System), Galileo (European Global Navigation Satellite System), BDS (Chinese BeiDou Navigation Satellite System), and GLONASS (Global’naya Navigatsionnaya Sputnikova Sistema). An undifferenced and uncombined PPP-RTK model is implemented for which the satellite clock and phase bias corrections are computed from the data processing of a group of stations in a network and then provided to users to help them achieve integer ambiguity resolution on a single receiver by calibrating the satellite phase biases. The dataset is recorded in a local area of the GNSS network of the Netherlands, in which 12 stations are regarded as the reference to generate the corresponding corrections and 21 as the users to assess the performance of the multi-GNSS PPP-RTK in both kinematic and static positioning mode. The results show that the root-mean-square (RMS) errors of the ambiguity float solutions can achieve the same accuracy level of the ambiguity fixed solutions after convergence. The combined GNSS cases, on the contrary, reduce the horizontal RMS of GPS alone with 2 cm level to GPS + Galileo/GPS + Galileo + BDS/GPS + Galileo + BDS + GLONASS with 1 cm level. The convergence time benefits from both multi-GNSS and fixing ambiguities, and the performances of the ambiguity fixed solution are comparable to those of the multi-GNSS ambiguity float solutions. For instance, the convergence time of GPS alone ambiguity fixed solutions to achieve 10 cm three-dimensional (3D) positioning accuracy is 39.5 min, while it is 37 min for GPS + Galileo ambiguity float solutions; moreover, with the same criterion, the convergence time of GE ambiguity fixed solutions is 19 min, which is better than GPS + Galileo + BDS + GLONASS ambiguity float solutions with 28.5 min. The experiments indicate that GPS alone occasionally suffers from a wrong fixing problem; however, this problem does not exist in the combined systems. Finally, integer ambiguity resolution is still necessary for multi-GNSS in the case of fast achieving very-high-accuracy positioning, e.g., sub-centimeter level.

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The long convergence time required to achieve high-precision position solutions with integer ambiguity resolution-enabled precise point positioning (PPP-RTK) is driven by the presence of ionospheric delays. When precise real-time ionospheric information is available and properly applied, it can strengthen the underlying model and substantially reduce the time required to achieve centimeter-level accuracy. In this study, we present and analyze the real-time PPP-RTK user performance using ionospheric corrections from multi-scale regional networks during a day with medium ionospheric disturbance. It is the goal of this contribution to measure the impact the network dimension has on the ambiguity-resolved user position through the predicted ionospheric corrections. The user-specific undifferenced ionospheric corrections are computed at the network side, along with the satellite phase biases needed for single-receiver ambiguity resolution, using the best linear unbiased predictor. Such corrections necessitate the parameterization of an estimable user receiver code bias, on which emphasis is given in this study. To this end, we process GPS dual-frequency data from four four-station evenly distributed CORS networks in the United States with varying station spacings in order to evaluate if and to what extent the ionospheric corrections from multi-scale networks can improve the user convergence times. Based on a large number of samples, our experimental results showed that sub-10 cm horizontal accuracy can be achieved almost instantaneously in the ionosphere-weighted partially-ambiguity-fixed kinematic PPP-RTK solutions based on corrections from a network with 68 km spacing. Most of the solutions (90%) were shown to require less than 6.0 min, compared to the ionosphere-float PPP solutions that needed 68.5 min. In case of sparser networks with 115, 174 and 237 km spacing, 50% of the horizontal positioning errors are shown to become less than one decimeter after 1.5, 4.0 and 7.0 min, respectively, while 90% of them require 10.5, 16.5 and 20.0 min. We also numerically demonstrated that the user’s convergence times bear a linear relationship with the network density and get shorter as the density increases, for both full and partial ambiguity resolution.@en

This paper presents a first analysis on the positioning performance using raw GNSS dual-frequency measurements acquired by the Xiaomi Mi 8 smartphone. In our study, we used the code-only-based Single Point Positioning technique for epoch-by-epoch positioning, and the Precise Point Positioning technique, which is based on both code and carrier-phase measurements, on precise satellite orbit and clock corrections, as well as on a dynamic model to exploit the time-constant property of the phase ambiguities. The functional model used in both our own SPP and PPP algorithms is based on the uncombined GNSS observation equations. As the Xiaomi Mi 8 smartphone is able to track both GPS and Galileo systems in two frequencies, we will present results based on single and combined GNSS solutions in single- and dual-frequency mode. The performance of these solutions will be assessed in terms of their repeatability and accuracy with respect to the ground-truth, while the improvement in Android-based positioning with multiple frequencies and GNSSs will be assessed compared to the GPS L1-only SPP case. An optimal combination of the elevation cut-off angle and the carrier-to-noise density ratio mask will be explored in order to have a reasonable balance between availability and quality of observations. Both real-time and post-processing positioning results will be presented.@en

This paper presents an analysis of the ionospheric corrections required to get a significant improvement in PPP-RTK performance. The main aim was to determine the improvement in the position precision and Time-To-First-Fix in the PPP-RTK user side using ionospheric corrections computed from a network. The study consists of two main steps. The first one includes an empirical investigation of the ionosphere model precision necessary to greatly improve the PPP-RTK performance in a simulated environment in terms of precision and convergence time. In the second one, an optimal ionosphere representation was developed to provide precise ionospheric corrections by parameterizing the ionospheric slant delays after the PPP-RTK network processing in terms of ionosphere model coefficients and differential code biases using real GNSS measurements. Experimental results demonstrate that the proposed methodology can be used for reliable regional ionosphere modeling and satellite code bias estimation, due to the consistency of the satellite code bias estimates with those provided from the International GNSS Service Analysis Centres, the high stability of the estimated receiver and satellite code biases and the low least-squares residuals of the network-based ionosphere modeling solution. Finally, it has been shown that the precision of ionospheric corrections at zenith needs to be better than 5 cm to enable faster PPP-RTK solutions.@en

Officially released on August 2016, the Android 7.0 has been a breakthrough for the Global Navigation Satellite System (GNSS) community. Starting with the introduction of its Application Programming Interface 24 (API 24), Android users have been able to directly access raw GNSS measurements within their smartphone, independently on the specific phone design or manufacturer. In this research we consider as a case-study device the Xiaomi Mi 8, equipped with the BCM47755 location hub (first dual-frequency GNSS chipset), after being upgrade to Android 9.0 in order to access new functionalities implemented in the API 28. The analysis of raw measurements and their quality is fundamental in smartphone-based positioning, since biased or extremely noisy estimates of the pseudorange measurements will hamper the precise positioning performances. Here, we report on several tests performed in static mode over a geodetic pillar, under similar conditions and covering time-spans up to several hours. Main quantities retrieved from the API were first described and then characterized based on the statistics retrieved from all these long dataset. After some considerations, a dependency w.r.t. the carrier-to-noise density ratio (C/N0) has been investigated for some of these variables, as well as for the impact of the API multipath indicator on the available observations. Finally, a preliminary C/N0-based trade-off between data ‘quantity’ and ‘quality’ is here proposed in order to support smartphone-based precise positioning.@en