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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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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Until now, RTK (real-time kinematic) and NRTK (Network-based RTK) have been the most popular cm-level accurate positioning approaches based on Global Navigation Satellite System (GNSS) signals in real-time. The tropospheric delay is a major source of RTK errors, especially for medium and long baselines. This source of error is difficult to quantify due to its reliance on highly variable atmospheric humidity. In this paper, we use the NRTK approach to estimate double-differenced zenith tropospheric delays alongside ambiguities and positions based on a complete set of multi-GNSS data in a sample 6-station network in Europe. The ZTD files published by IGS were used to validate the estimated ZTDs. The results confirmed a good agreement, with an average Root Mean Squares Error (RMSE) of about 12 mm. Although multiplying the unknowns makes the mathematical model less reliable in correctly fixing integer ambiguities, adding a priori interpolated ZTD as quasi-observations can improve positioning accuracy and Integer Ambiguity Resolution (IAR) performance. In this work, weighted least-squares (WLS) were performed using the interpolation of ZTD values of near reference stations of the IGS network. When using a well-known Kriging interpolation, the weights depend on the semivariogram, and a higher network density is required to obtain the correct covariance function. Hence, we used a simple interpolation strategy, which minimized the impact of altitude variability within the network. Compared to standard RTK where ZTD is assumed to be unknown, this technique improves the positioning accuracy by about 50%. It also increased the success rate for IAR by nearly 1.@en

With the advancement of multi-constellation and multi-frequency global navigation satellite systems (GNSSs), more observations are available for high precision positioning applications. Although there is a lot of progress in the GNSS world, achieving realistic precision of the solution (neither too optimistic nor too pessimistic) is still an open problem. Weighting among different GNSS systems requires a realistic stochastic model for all observations to achieve the best linear unbiased estimation (BLUE) of unknown parameters in multi-GNSS data processing mode. In addition, the correct integer ambiguity resolution (IAR) becomes crucial in shortening the Time-To-Fix (TTF) in RTK, especially in challenging environmental conditions. In general, it is required to estimate various variances for observation types, consider the correlation between different observables, and compensate for the satellite elevation dependence of the observable precision. Quality control of GNSS signals, such as GPS, GLONASS, Galileo, and BeiDou can be performed by processing a zero or short baseline double difference pseudorange and carrier phase observations using the least-squares variance component estimation (LS-VCE). The efficacy of this method is investigated using real multi-GNSS data sets collected by the Trimble NETR9, SEPT POLARX5, and LEICA GR30 receivers. The results show that the standard deviation of observations depends on the system and the observable type in which a particular receiver could have the best performance. We also note that the estimated variances and correlations among different observations are also dependent on the receiver type. It is because the approaches utilized for the recovery techniques differ from one type of receiver to another kind. The reliability of IAR will improve if a realistic stochastic model is applied in single or multi-GNSS data processing. According to the results, for the data sets considered, a realistic stochastic model can increase the computed empirical success rate to 100% in multi-GNSS as well as a single system. As mentioned previously, the realistic precision of the solution can be achieved with a realistic stochastic model. However, using the estimated stochastic model, in fact, leads to better precision and accuracy for the estimated baseline components, up to 39% in multi-GNSS.

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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 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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Guaranteeing safety of navigation within the Netherlands Continental Shelf (NCS), while efficiently using its ocean mapping resources, is a key task of Netherlands Hydrographic Service (NLHS) and Rijkswaterstaat (RWS). Resurvey frequencies depend on seafloor dynamics and the aim of this research is to model the seafloor dynamics to predict changes in seafloor depth that would require resurveying. Characterisation of the seafloor dynamics is based on available time series of bathymetry data obtained from the acoustic remote sensing method of both single-beam echosounding (SBES) and multibeam echosounding (MBES). This time series is used to define a library of mathematical models describing the seafloor dynamics in relation to spatial and temporal changes in depth. An adaptive, functional model selection procedure is developed using a nodal analysis (0D) approach, based on statistical hypothesis testing using a combination of the Overall Model Test (OMT) statistic and Generalised Likelihood Ratio Test (GLRT). This approach ensures that each model has an equal chance of being selected, when more than one hypothesis is plausible for areas that exhibit varying seafloor dynamics. This ensures a more flexible and rigorous decision on the choice of the nominal model assumption. The addition of piecewise linear models to the library offers another characterisation of the trends in the nodal time series. This has led to an optimised model selection procedure and parameterisation of each nodal time series, which is used for the spatial and temporal predictions of the changes in the depths and associated uncertainties. The model selection results show that the models can detect the changes in the seafloor depths with spatial consistency and similarity, particularly in the shoaling areas where tidal sandwaves are present. The predicted changes in depths and uncertainties are translated into a probability risk-alert map by evaluating the probabilities of an indicator variable exceeding a certain decision threshold. This research can further support the decision-making process when optimising resurvey frequencies.

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In this contribution, we extend the principle of integer bootstrapping (IB) to a vectorial form (VIB). The mathematical definition of the class of VIB-estimators is introduced together with their pull-in regions and other properties such as probability bounds and success rate approximations. The vectorial formulation allows sequential block-by-block processing of the ambiguities based on a user-chosen partitioning. In this way, flexibility is created, where for specific choices of partitioning, tailored VIB-estimators can be designed. This wide range of possibilities is discussed, supported by numerical simulations and analytical examples. Further guidelines are provided, as well as the possible extension to other classes of estimators.

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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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Tropospheric wet delay, the main source of which is water vapor, is one of the major factors affecting the accuracy of positioning techniques using microwave. Tropospheric tomography is a powerful method to reconstruct the water vapor content in four-dimensional (4D) space. This paper studies the effect of using function-based and voxel-based tropospheric tomography methods on the positioning accuracy. This examination is performed on the static and kinematic positioning modes using Global Navigation Satellite Systems (GNSS) stations under different weather conditions. After validating the results of tomography methods using radiosonde observations, the tomography-based positioning solutions, including function-based and voxel-based approaches, are compared with the positions obtained using tropospheric models. The results of two GPS stations show that the accuracy increases when applying tomography approaches. The function-based tomography is able to increase the accuracy of the up component of the static and kinematic modes by about 0.42 and 0.79 cm, respectively, compared to the voxel-based method. In addition, the use of the function-based tropospheric tomography can decrease the convergence time of the kinematic Precise Point Positioning (PPP) solution.

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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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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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Precise point positioning (PPP) is one of the well-known applications of Global Navigation Satellite System (GNSS) and provides precise positioning solutions using accurate satellite orbit and clock products. The tropospheric delay due to the neutral atmosphere for microwave signals is one of the main sources of measurement error in PPP. As one component of this delay, the hydrostatic delay is usually compensated by using an empirical correction model. However, how to eliminate the effects of the wet delay during a weather event is a challenge because current troposphere models are not capable of considering the complex atmosphere around the receiver during situations such as typhoons, storms, heavy rainfall, et cetera. Thus, how positioning results can be improved if the residual wet delays are taken into account needs to be investigated . In this contribution, a real-time procedure of recursive detection, identification and adaptation (DIA) is applied to detect the model errors which have the same effects on both phase and code observables; e.g., the model error caused by the tropospheric delay. Once the model errors are identified, additional parameters are added to the functional model to account for the measurement residuals. This approach is evaluated with Global Positioning System (GPS) data during two rainfall events in Darwin, Australia, proving the usefulness of compensated residual slant wet delay for positioning results. Comparisons with the standard approach show that the precision of the up component is improved significantly during the periods of the weather events; for the two case studies, 72.46% and 64.41% improvements of root mean squared error (RMS) resulted, and the precision of the horizontal component obtained by the proposed approach is also improved more than 30% compared to the standard approach. The results also show that the identified model errors are concentrated at the beginning of both heavy rainfall processes when the front causes significant spatial and temporal gradients of the integrated water vapor above the receiver.

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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

Tropospheric tomography is one of the most important techniques to reconstruct three-dimensional (3D) images of the tropospheric water vapor fields using a local GNSS network. In the conventional tropospheric tomography method, called voxel-based tropospheric tomography, the 3D space is divided into many voxels and the amount of water vapor is estimated for each voxel. This method suffers from three disadvantages. First, it needs empirical constraints in order to fix the rank deficiency of the coefficient matrix. Second, the amount of water vapor is assumed to be constant in the 3D space of a voxel despite the large spatial variations of this parameter. Third, the number of unknown parameters is high compared to the number of observations. Therefore, an approach based on mathematical functions, called function-based tropospheric tomography, is presented to overcome these problems. The tropospheric tomography using the voxel-based and function-based approaches is performed using 17 GPS stations. Radiosonde observations and precise point positioning results are used to validate the obtained results. A comparison of the results with the radiosonde data indicates that using the function-based method reduces the mean RMSE by about 0.3 gr/m³. Validation using positioning under different wet conditions shows that in wet weather conditions the difference between the RMSE of the two tropospheric tomography approaches is significant. All the validations show the ability and applicability of the function-based tropospheric tomography approach.

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The water vapor content in the atmosphere can be reconstructed using the all-weather condition troposphere tomography technique. In common troposphere tomography, the water vapor of each voxel is represented by an unknown parameter. This means that when the desired spatial resolution is high or study area is large, there will be a huge number of unknown parameters in the problem that need to be solved. This defect can reduce the accuracy of troposphere tomography results. In order to overcome this problem, an optimal voxel-based troposphere tomography using the Weather Research and Forecasting (WRF) model is proposed. The new approach reduces the number of unknown parameters, the number of empty voxels and the role of constraints required to enhance the spatial resolution of tomography results in required areas. Furthermore, the effect of considering the topography of the study area in the tomography model is examined. The obtained water vapor is validated by radiosonde observations and Global Positioning System (GPS) positioning results. Comparison of the results with the radiosonde observations shows that using the WRF model outputs and topography of the area can reduce the Root Mean Square Error (RMSE) by 0.803 gr/m3. Validation using positioning shows that in wet weather conditions, the WRF model outputs and topography reduce the RMSE of the east, north and up components by about 17.42, 10.46 and 20.03 mm, which are equivalent to 46.01%, 35.78% and 53.93%, respectively.@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