In current conventional precise point positioning (PPP) processing strategies, the tropospheric zenith wet delay (ZWD) is usually dynamically estimated as a stochastic parameter. During the convergence period, ZWD estimates can appear to be negative or unrealistically large due to the low estimation precision, which adversely affects the estimation of other state parameters, especially the Up component of coordinates. To address this issue, we propose a method that incorporates physical constraints on ZWD in PPP processing. This method employs the inequality constrained least squares (ICLS), utilizing Karush–Kuhn–Tucker (KKT) conditions to add boundary conditions on ZWD. The boundary conditions of ZWD are calculated based on the relation between ZWD and relative humidity (RH). The use of physical constraints does not rely on external products or space state representation (SSR) corrections for ZWD during PPP processing and can improve the short-term accuracy of ZWD and coordinate Up component. The efficiency of this approach has been validated using GNSS data and products from GFZ operational networks. For real-time PPP solutions, there is a 30% improvement in short-term accuracy of Up component; for post-processing solutions, the short-term RMSE improvement is about 20% . After convergence, the ZWD upper bound is no longer applied as an ICLS constraint, but is instead used as a diagnostic indicator to identify ZWD anomalies. This indicator demonstrates high sensitivity and reliability under extreme weather conditions, highlighting its potential for application in meteorological hazard early-warning systems.

Theoretically, it is possible to reconstruct the 3D distribution of water vapor by means of GNSS tomography using troposphere estimates from a network of GNSS stations, i.e., zenith delays mapped back into satellite directions. However, this technique is still limited by restricted satellite-to-ground observation geometry, a simplified parameterization of troposphere delays in the observation model and mandatory usage of constraints to stabilize the equation system. We propose an alternative approach, called STEPPP, in which a network of ground-based GNSS receivers is used to process GNSS observations, based on the Precise Point Positioning (PPP) technique and, instead of estimating the zenith wet delay and horizontal gradients, the estimation of the whole wet refractivity field in a grid space is performed simultaneously. Contrary to GNSS tomography, STEPPP operates on raw observation data instead of products. Values of wet refractivity at grid nodes are estimated from all stations simultaneously, i.e., they appear as common parameters in PPP. We present the functional and stochastic model of STEPPP, as well as first results of the model performance. We use GPS and Galileo dual-frequency observations generated by the Spirent simulator for 20 evenly distributed stations. The simulated observations are intentionally free of troposphere delays. However, we use a numerical weather model to retrieve reference profiles of wet refractivity and calculate slant wet delays, which are added to the simulated observations. We define a voxel space above the network of stations up to 12 km height and we recover the wet refractivity profiles by means of the STEPPP model. Several numerical experiments are performed using homogenous, inhomogeneous, constant and dynamic wet refractivity profiles. After a convergence time of a few hours, the STEPPP model accurately recovers all model states, including the 3D wet refractivity field and station coordinates.

Thermospheric density uncertainty is the dominant source of uncertainty for orbit prediction in Low Earth Orbit (LEO) and Very Low Earth Orbit (VLEO). While analytical uncertainty propagation methods based on mean orbital elements accurately capture long-term secular drift, they fail to represent short-term dynamics, leading to a significant overestimation of along-track uncertainty for prediction horizons below one orbital period. These short-term dynamics are characterized by a zero-crossing of the along-track error caused by the initial differential drag acceleration counteracting the secular drift known as the drag paradox. This limitation is addressed in this paper in the form of an uncertainty propagation model based on linearized relative orbit mechanics using the Hill-Clohessy-Wiltshire (HCW) equations. An analytical solution is derived to characterize the zero-crossing phenomenon and evaluate the effect on the along-track uncertainty. To account for time-varying dynamics and temporally correlated atmospheric density errors, the model is extended using Linear Covariance Propagation with a First-Order Gauss-Markov Process (GMP). Validation with Monte Carlo simulations demonstrates that the numerical HCW model accurately reproduces the empirical uncertainty and the zero-crossing effect that the mean element approaches miss. A sensitivity analysis confirms the model's robustness as well as the general zero-crossing behavior across varying altitudes, correlation times and solar activities while identifying valid propagation timeframes and the limiting effects of orbital eccentricity. (c) 2026 The Author(s). Published by Elsevier B.V. on behalf of COSPAR. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

INS toolkit for integrated navigation concepts and training (INSTINCT) is an open-source positioning, navigation and timing (PNT) framework for global navigation satellite system (GNSS) navigation and sensor fusion written in C++. It uses flow-based programming to encapsulate functionality, enforce clean interfaces and promote reusability. Not only multi-constellation, multi-frequency single point positioning (SPP) and real-time kinematic positioning (RTK) algorithms are available, but also inertial navigation system (INS)/GNSS sensor fusion. Moreover, innovative concepts like multi inertial measurement unit (IMU) arrays and factor graph optimization are featured. Furthermore, most file formats common in the PNT field can be read with the software and converted between them. Also, simulation of trajectories and IMU data with different error models is possible. A graphical user interface allows the user to directly set parameters and analyze results in plots, which enables rapid prototyping and testing. A developer can easily extend the functionality with own algorithms and sensor interfaces building upon the existing modules. In order to evaluate the performance of the algorithms two experiments were performed. Analysis of a static dataset shows that the position accuracy of the RTK algorithm of INSTINCT is comparable to RTKLIB. Additionally, a dynamic dataset was generated using a Spirent GNSS simulator and INSTINCT’s IMU simulation capabilities. In-depth assessment confirms the high accuracy of the results and demonstrates that the INS/GNSS loosely coupled Kalman filter can compensate for GNSS outages.

Robust Precise Orbit Determination (POD) is essential for Low Earth Orbit (LEO) satellite missions. Based on observational data from Global Navigation Satellite System (GNSS), the orbit parameters are estimated by means of an adjustment process. To achieve accurate results, highly reliable navigation data are required. In this work, we investigate the application of Receiver Autonomous Integrity Monitoring (RAIM)-based methods - originally developed for aviation - to detect and exclude faulty GNSS measurements in a post-processed POD framework. Our approach utilizes redundant GNSS code measurements and evaluates the residuals from a weighted least squares (WLSQ) orbit solution. Based on a predefined false alarm rate, we set a threshold and compare it to a residual-based test statistic. The method enables detection and exclusion of outliers in post-processed solutions. Simulation and real GRACE-FO data results demonstrate the effectiveness of the applied RAIM-based method. It enhances the integrity and reliability of GNSS-based POD solutions for LEO satellites. Copyright (c) 2025 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

The Collaborative Research Centre (CRC) 1667 “Advancing Technologies of Very Low Altitude Satellites—ATLAS” was established in April 2024 with the scientific goal of addressing the fundamental challenges of making satellite operations in Very Low Earth Orbits (VLEO) sustainable. These orbits are beneficial for satellite services that have become indispensable to our modern society. Moreover, access to VLEO offers the opportunity to operate satellites without exposure or contribution to the increasing contamination of traditional orbits with space debris. Seventeen highly interlinked research projects have been selected to investigate and advance accurate numerical and experimental methods for gas–surface interactions, novel concepts utilising the residual atmosphere and minimising the satellite sizes, and mission-related challenges of a selected scenario. In addition, support projects cover topics related to public outreach and academic exchange and assist in achieving the strategic goal of positioning the University of Stuttgart as a key contributor to this internationally very important research area. In summary, the CRC ATLAS aims to constitute a research-oriented profile-building measure at the University of Stuttgart with a strong international reputation.

Troposphere’s asymmetry can introduce errors ranging from centimeters to decimeters at low elevation angles, which cannot be ignored in high-precision positioning technology and meteorological research. The traditional two-axis gradient model, which strongly relies on an open-sky environment of the receiver, exhibits misfits at low elevation angles due to their simplistic nature. In response, we propose a directional mapping function based on cyclic B-splines named B-spline mapping function (BMF). This model replaces the conventional approach, which is based on estimating Zenith Wet Delay and gradient parameters, by estimating only four parameters which enable a continuous characterization of the troposphere delay across any directions. A simulation test, based on a numerical weather model, was conducted to validate the superiority of cyclic B-spline functions in representing tropospheric asymmetry. Based on an extensive analysis, the performance of BMF was assessed within precise point positioning using data from 45 International GNSS Service stations across Europe and Africa. It is revealed that BMF improves the coordinate repeatability by approximately 10% horizontally and about 5% vertically. Such improvements are particularly pronounced under heavy rainfall conditions, where the improvement of 3-dimensional root mean square error reaches up to 13% .

Incorrect offsets between a satellite's center of mass and its global navigadetermination (POD) of many current Earth observation missions. Based on hardware-in-the-loop simulations, this paper demonstrates the more adverse effects on agile satellites, which perform frequent attitude maneuvers around all spacecraft axes. However, findings obtained from an observability analysis and Monte Carlo simulations indicate that rapid attitude changes enable the direct estimation of otherwise unobservable offsets. Application to the POD of agile satellites leads to a consistent and significant performance improvement in the presence of incorrect phase center offsets. Directly estimated corrections for the phase center offset of Sentinel-6A, which performs slews on several occasions, are consistent with values obtained from other studies via independent methods. These results underscore the possibility of estimating the lever arm for both agile and non-agile satellites in dedicated calibration maneuvers.

Zenith wet delay (ZWD) estimation is a key component for the global navigation satellite system (GNSS) meteorology. At present, the zenith hydrostatic delay can be computed with sufficient accuracy by means of empirical models, while the ZWD, which is induced by water vapor with the nature of highly spatio-temporal variability, is typically estimated as an unknown parameter in precise point positioning (PPP). Due to GNSS receiver noise and the system biases of GNSS receivers, the accuracy as well as the precision of ZWD estimates is limited. In this study, we propose a novel fusion model based on undifferenced GNSS pseudorange and carrier-phase observations for sites, which have several receivers connected to a single antenna or which are separated horizontally by only a few meters. By fusing GNSS measurements collected by multiple receivers on the observation level, our model can provide common ZWD estimates with a high temporal resolution, which can then be used for more accurate and reliable meteorologic applications on a local scale. According to results with simulated and real data, it is revealed that such combined ZWD estimates are superior to single receiver estimates in terms of precision and accuracy. On the other hand, it is confirmed that the estimation of a common ZWD parameter leads to an improvement in positioning accuracy and precision, especially in the vertical component.

Asymmetric troposphere modeling is crucial in Precise Point Positioning (PPP). The functional model of the asymmetric troposphere has been thoroughly studied, while the stochastic model lacks discussion. Currently, there is no suitable stochastic model for asymmetric tropospheric conditions, potentially degrading the positioning accuracy and the reliability of Zenith Total/Wet Delay (ZTD/ZWD) estimates. This paper introduces an Azimuth-Dependent Weighting (ADW) scheme that utilizes information from asymmetric mapping functions to adaptively weight Global Navigation Satellite System (GNSS) observations affected by azimuth-dependent errors. The concept of ADW has been validated using Numerical Weather Prediction data and International GNSS Service data. The results indicate that ADW effectively improves the coordinate repeatability of the PPP solution by approximately 10% in the horizontal and 20% in the vertical direction. Additionally, ADW appears to be capable to improve the ZWD estimates during the PPP convergence period and yields smoother ZWD estimates. Consequently, it is recommended to adopt this new weighting scheme in PPP applications when an asymmetric mapping functions is employed.

Satellite-based formation flying Earth observations are crucial for environmental monitoring and sustainable use of our Earth's resources. One example is the German mission TanDEM-X, for which the twin satellites TerraSAR-X and TanDEM-X are working in unison to derive highly precise digital elevation models. To interpret the measurement data of such satellites correctly, the post-processing algorithms depend on precise millimeter-level knowledge of their relative position, known as their baseline. One technique for deriving estimates of the required precision is Precise Baseline Determination (PBD) based on measurements from Global Navigation Satellite Systems (GNSS). While the latest advancements in PBD are suitable for meeting the current precision requirements of non-agile missions, future Synthetic Aperture Radar (SAR) projects will require the same precision for agile mission profiles. Additionally, the PBD for agile missions depends on precise knowledge of the satellite's attitude. Usually, the PBD and the attitude estimation are performed independently from each other. Whereas PBD utilizes GNSS observations with one receiver per spacecraft, the attitude is estimated via star trackers and inertial measurement units. This paper presents a novel approach of simultaneous GNSS-based baseline and attitude estimation, named Multi Receiver Precise Baseline Determination (MR-PBD). Two benefits are expected. Firstly, the achieved precision of the inter-spacecraft baseline estimate will improve, since multiple receivers are used per spacecraft. Secondly, the estimation delivers an additional attitude product, which can be used to improve the existing star-tracker-based attitude estimation. To validate the approach, the observations of a low-cost and low-power GNSS receiver (u-blox ZED-F9P) are utilized and plugged into a Spirent signal generator. By applying MR-PBD a baseline estimate was achieved that is 15% to 40% more precise than the single receiver PBD by utilizing 3 to 12 receivers per spacecraft.

The centimeter-level positioning accuracy of real-time kinematic (RTK) depends on correctly resolving integer carrier-phase ambiguities. To improve the success rate of ambiguity resolution and obtain reliable positioning results, an enhanced Kalman filtering procedure has been developed. Based on a posteriori residuals of measurements and state predictions, the measurement noise variance–covariance matrix for double-differenced measurements is adaptively estimated, rather than approximated by an empirical function which uses satellite elevation angle as input. Since, in real-world situations, unexpected outliers and carrier-phase outages can degrade the filter performance, a stochastic model based on robust Kalman filtering is proposed, for which the double-differenced measurement noise variance–covariance matrix is computed empirically with a modified version of the IGG (Institute of Geodesy and Geophysics) III method in order to detect and identify outliers. The performance of the proposed method is assessed by two tests, one with simulated data and one with real data. In addition, the performance of F-ratio and W-ratio tests as proxies for the success of ambiguity fixing is investigated. Experimental results reveal that the proposed method can improve the reliability and robustness of relative kinematic positioning for simulation scenarios as well as in a real urban test.

<p>In GNSS analysis, tropospheric modelling is done in the form of Zenith Hydrostatic Delay (ZHD), which can be empirically computed from surface pressure and temperature, and Zenith Wet Delay (ZWD), which is estimated together with the other unknown parameters.  Analysis methods based on undifferenced GNSS code- and carrier-phase observations like Precise Point Positioning (PPP), which can achieve millimeter-accurate positioning results, provide therefore also time-series of ZWD, which can be used for meteorologic applications. However, due to receiver noise and system characteristics like cycle-slips the accuracy as well as the precision of such ZWD estimates is limited. Thus, we propose a novel approach for sites, which have several receivers connected to a single antenna or which are separated horizontally by only a few meters. For such sites, one can simultaneously process multi-frequency GNSS data by fusing observations from several receivers, while estimating a common ZWD parameter.</p> <p>For this purpose, we have implemented a PPP algorithm based on an Extended Kalman Filter (EKF) approach, which has the advantage that ZWD estimates are available in real-time for meteorologic applications. We demonstrate that those combined ZWD estimates are superior to single receiver estimates in term of precision and accuracy. For the latter measure, we make use of a GNSS hardware simulator and show that the RMS between the simulated and estimated ZWD significantly decreases when having two or more receivers at that site. Based on real-data we show that this concept provides less noisy ZWD estimates which agree better with physical properties of the local wet refractivity field.</p> <p>Moreover, we demonstrate that fusing data from several receivers by estimating a common ZWD parameter improves also positioning accuracy and precision, in particular in the up-component. In order to properly combine observations from geodetic-grade and low-cost GNSS receivers, we present our adaptive Kalman filter approach, which adjusts the observation noise covariance matrix automatically during processing. The presentation concludes with an outlook on the usage of this approach for larger networks and answers the question how arrays of low-cost GNSS receivers can compete against geodetic-grade GNSS hardware in term of providing ZWD estimates for meteorology.</p>

<p>Multipath is a large systematic GNSS error source which can bias the Zenith Total Delay (ZTD) estimation of Precise Point Positioning (PPP). In this paper, we explore the magnitudes and systematics of the errors caused by multipath in ZTD estimates. We apply several process noise models based on an Extended Kalman Filter (EKF) and study whether those models are capable of partly mitigating the multipath error. Simulated data, generated with a commercial GNSS simulator, is thereby used to study the impact of multipath signals. All results are based on PPP solutions for which ambiguities are resolved (PPP-AR) and since the simulations provide us reference data, the degradation of ZTD accuracy can be studied in different scenarios where the multipath errors come from different reflection sources. The results reveal that the magnitude of ZTD errors due to multipath reaches millimeter to centimeter order, depending on the chosen scenario. In order to mitigate the effect of multipath errors on ZTD estimates, we study the use of the Continuous Wavelet Transform (CWT). We compute Code-minus-Carrier (CMC) observations and apply a CWT with the purpose to detect the periods during which multipath errors are affecting the observables. Once those periods are identified, it might be possible to mitigate the multipath error by using different process noise models or different function representations for the unknown parameters. In particular, we focus on the random walk model, the first-order Gaussian-Markov model, a noise-overbounding approach and a B-spline representation. We discuss how effective those models perform and reveal whether there is the possibility to improve troposphere estimates which would otherwise be biased by multipath effects.  Although all our findings relate to PPP post-processing, the suggested approaches can be mapped to real-time applications since multipath errors mitigation is done epoch wise with the help of an EKF.</p>

Flow-based programming (FBP) splits software functionality into modules which are triggered by data elements flowing from one module to the next. Thus, modules, which are not directly dependent on data from each other, can run independently and applications can be parallelized on a basic level what is a major improvement for the performance of multi-sensor data fusion algorithms. This paper discusses the flow-based software INSTINCT, which is a solid framework for implementing PNT algorithms and helps to reduce development times by making algorithms reusable. It provides an intuitive graphical user interface (GUI) that makes it usable in research and teaching. Inside the software, a variety of file formats and sensors for IMU und GNSS data processing are implemented. The realization of an INS/GNSS loosely-coupled Kalman filter flow is discussed and the results are compared to a flow representing a single point positioning (SPP) solution. Finally, a performance study of the algorithm on a 4th generation Intel CPU is presented, demonstrating that real-time capabilities can be ensured even on older systems.

Precise Orbit Determination (POD) and Precise Baseline Determination (PBD) are indispensable for many of today’s remote sensing satellite missions. With the advent of agile satellite missions, new solutions are needed to cope with frequently occurring maneuvers to fulfill the strict POD requirements imposed by scientific mission goals. The C++ software "Precise Orbit Determination for Complex and Agile Satellite Technology" (PODCAST) aims to create a flexible framework to investigate novel approaches related to POD and PBD of agile and non-agile satellites. PODCAST facilitates this by abiding by a modular concept for all core components needed for POD and thus ensures the complete interchangeability of algorithms and models in the development process. This study gives insights into the modular architecture of PODCAST and the underlying fundamental principles. The capabilities are demonstrated for a GNSS-based POD using simulated observations created for a Sentinel-3A reference trajectory. The obtained results indicate the accomplishable accuracy and precision for given measurement errors using the approaches presented in this work. We further showcase that PODCAST can serve as the foundation for future POD software development and suggest improvements that allow addressing of future mission-critical POD features.

Near real-time estimation of zenith total delays (ZTD) of Global Navigation Satellite Systems (GNSS) signals is operationally performed in Europe. We demonstrate that high accuracy ZTD can be provided even in real-time. Using a state-of-the-art processing strategy we estimate ZTD and horizontal gradients for 162 permanent stations in Europe over 100 days in 2019, covering the event of hurricane Lorenzo. The accuracy of real-time ZTD with respect to the final ZTD product varies from 3.3 to 10.4 mm. The accuracy of real-time ZTD with respect to the ICON numerical weather prediction model varies from 4 mm to 18 mm and we notice station-specific biases, reaching up to ±10 mm. We demonstrate that horizontal gradients show signatures under the severe weather during hurricane Lorenzo in 2019.

In the original version of this article, equation (9) was displayed incorrectly.

Remote sensing of water vapor using Global Navigation Satellite Systems (GNSS) is a well-established tool for weather and climate monitoring. The current challenges of GNSS meteorology are real-time performance and the inclusion of emerging GNSS, such as Galileo. We demonstrate that real-time GPS-only, Galileo-only, and GPS+Galileo solutions are consistent among each other. However, our results show that the Galileo-only solutions tend to underestimate Zenith Total Delay (ZTD) with respect to GPS. The Galileo-only real-time ZTD is less accurate as the one from GPS. The combination of both GNSS leads to a superior product. The daily solution availability increases by up to 50%, and the overall gain is 0.7% over the entire year. The accuracy improves by 3.7% to 8.5% and uncertainty is reduced by a factor of 1.5–2. A combined GPS and Galileo solution suppresses artifacts in a real-time ZTD product which otherwise would be attributed to high-frequency orbital effects.