In precise point positioning (PPP) solutions, stochastic modeling of the receiver clock parameter affects solution stability, especially the station up component. This study analyzes GPS + Galileo static and kinematic PPP solutions for 14 GNSS stations equipped with ultra-stable hydrogen masers, utilizing real-time orbit and clock products, such as HAS, IGS, and CNES, as well as final products from CODE. The solution with modeling of the receiver clock parameter enhances stability and precision compared to the reference solution, in which the receiver clocks are estimated independently for each epoch. In static PPP solutions, receiver clock modeling reduces spikes in the clock parameter and improves short-term stability. The most notable improvement of the station up component reaches up to 59, 21, 25, and 38

Heights provided by GNSS are affected by the quality of the geoid or quasigeoid model used for the transformation of the ellipsoidal heights to the orthometric or normal heights, as well as by the data processing techniques, including Real-Time Kinematic (RTK), Real-Time Network (RTN), static relative baseline, and absolute Precise Point Positioning (PPP) solutions. We employ two geoid models for the Tatra Mountains with constant and variable density of the lithosphere. We compare heights for 113 mountain peaks and passes directly measured using GNSS, applying two geoid models and two quasigeoid models – one dedicated to the Tatra Mountains and the second that is used as a national standard for GNSS applications in Poland. We also compare the results of height determination based on static GNSS measurements and post-processing to those based on RTK, RTN, and PPP. We found that the maximum differences from using different geoid and quasigeoid models reach up to 7.5 and 11.6 cm, respectively, whereas the standard deviations from height differences based on different GNSS processing techniques are just 0.8 cm with a maximum difference of 2.4 cm. Wrong tropospheric delay handling may result in an error of 17 cm. Hence, the geoid and quasigeoid models are crucial in GNSS height determination of the mountain peaks, whereas the GNSS data processing technique plays a minor role. Therefore, quick real-time RTN solutions are fully applicable for the GNSS measurements of mountain peaks and passes, even if the height difference between the reference station and the rover exceeds 1800 m, provided that the tropospheric delay is properly corrected by extrapolation or estimation.

In GNSS solutions, the receiver clock parameter (Clc), the station height component (Up), and the Zenith Tropospheric Delay (ZTD) are correlated. However, some of these parameters are considered known values to avoid correlations and reduce time convergence. In Precise Point Positioning (PPP), the receiver clock parameter is estimated at each measurement epoch together with the coordinates and potentially ZTD. However, coordinates can be taken from daily solutions, whereas ZTD can be taken from an external model and fixed. We study twelve PPP strategies using International GNSS Service (IGS) stations equipped with external hydrogen masers, employing two different ZTD models, and various configurations for estimating or fixing Up component and ZTD. Each day is analyzed independently utilizing multi-GNSS PPP, post-processing, and Forward–Backward–Forward estimation techniques. The most accurate results are obtained in four strategies, two of which estimate all three parameters: Clc, Up, and ZTD; whereas in the other two strategies, the station coordinates are derived from the last epoch of the daily pre-processed PPP solution. Fixing station coordinates to an a priori position from ITRF2020 leads to inferior results. Fixing troposphere parameters to GMF or VMF3 models also leads to worse results than estimating corrections to wet delay. Solutions using VMF3 are only slightly better than solutions based on GMF. Estimating all parameters simultaneously, however, provides the most comprehensive solution despite existing correlations. The best results for parameter stability and Time Transfer are achieved in strategy estimating Up, Clc, and ZTD using the VMF3 model updated every 6 h for ZTD.

Sports watches come equipped with Global Navigation Satellite System (GNSS) receivers capable of tracking GPS, GLONASS, and Galileo, and thus, can be considered low-cost GNSS receivers. In our research, we conducted three distinct experiments to evaluate the accuracy of 1) positioning, 2) distance determination, and 3) height determination based on sports watches using different GNSS combinations: GPS, GPS + GLONASS, and GPS + Galileo. For positioning, a professional GNSS receiver served as the reference. To determine height, we used a laser rangefinder. The assessment of distance measurement utilized a full-size athletics track. A noteworthy outcome of this study is that the increased cost of sports watches does not consistently correlate with higher GNSS positioning accuracy. The most accurate 2D results are achieved by the Polar M430 for the GPS case with the mean positioning error of 1.74 m, and the Garmin Fenix 6 PRO for GPS + GLONASS case with the mean error of 1.43 m.

In the multi-GNSS solutions integrating GPS, GLONASS, Galileo, and BeiDou, the receiver clock may be treated twofold; the clock parameter may be estimated for each GNSS separately or the common clock can be estimated, e.g., for GPS, with inter-system biases (ISBs) for other systems. The latter strategy reduces the number of estimated independent clock parameters per epoch almost by a factor of four because the clock parameters are estimated epoch-wise, whereas ISBs are estimated as constant values for the entire day or month. Due to the discontinuities in reference satellite clocks, the estimated ISBs and receiver clock parameters have also to be reinitialized at day boundaries. This raises questions about whether only the common clock has to be reset or all ISB values and what is the impact of the reinitialization of clock parameters with covariance values when estimating system-specific clock parameters. We analyze the effects of different types of stochastic modeling applied to the parameters of clocks and ISBs. In this study, we test five different strategies to clock handling in multi-GNSS kinematic Precise Point Positioning derived continuously for one month. We found that two solutions can be considered equivalent: (1) estimating system-specific clocks and (2) estimating the common clock with ISB and resetting at day boundaries the common clock parameter and ISBs. Oppositely, resetting only the common clock parameter or assuming that the ISB keep their stabilities over long periods is insufficient to obtain superior results of station coordinates and reliable time transfer results.

The quality of the International GNSS Service (IGS) station coordinates depends, among other things, on the type of oscillator located at the station in undifferenced GNSS solutions. A distinction of the clock standards can be made between internal, rubidium, cesium, and hydrogen maser clocks. The stability of time standards can vary by several orders of magnitude depending on the type of clock. High - accuracy clocks allow for the introduction of stochastic modeling in the absolute positioning algorithms which contributes to faster solution convergence and superior position accuracy. In the Precise Point Positioning (PPP) measurement technique based on multi - GNSS, the clock parameter may be determined in two ways. Either it is estimated separately for each system, or a single clock parameter is determined for all systems, taking into account the inter - system biases. In this paper, we analyze the impact of stochastic modeling imposed on clock parameters on positioning in a multi - GNSS PPP solution. We use selected IGS stations observing GPS, GLONASS, Galileo, and BeiDou satellites. All analyses for the clock parameter are conducted for different types of oscillators along with the performance characteristics for each system. An important advantage of modeling the clock parameter is to stabilize the station vertical coordinate component in multi - GNSS PPP solutions.

Global navigation satellite system (GNSS) receivers belonging to the International GNSS Service (IGS) are equipped with different types of clocks, such as internal crystal quartz clocks, rubidium and cesium atomic clocks, as well as hydrogen masers. These clocks are characterized by different phase and frequency accuracies and stabilities, resulting in different systematic clock time series patterns. We analyze the clock offsets between different GNSS systems, provide noise characteristics of the undifferenced and differenced clock parameters, and detect systematic patterns of the clocks. The time series of the receiver clocks are dominated by the diurnal, semidiurnal, and sometimes terdiurnal signals with amplitudes up to several meters. Hydrogen masers provide the highest clock stability, and the lowest is by internal clocks. However, there are also groups of very stable internal clocks that perform similarly to low-performing hydrogen masers and rubidium clocks. The interquartile ranges for epoch-differenced clock parameters fall between 3 and 250 mm for the best hydrogen masers and the worst internal clocks, respectively.

In Precise Point Positioning (PPP), the receiver clock parameter is typically estimated independently in each observation epoch, which increases the noise of the estimated station coordinates and troposphere parameters due to correlations. Applying stochastic modeling to the receiver clock parameter stabilizes PPP solutions and reduces clock noise for the time transfer. However, the receiver clock modeling is possible only for the GNSS receivers connected to the utmost stable atomic clocks. We propose receiver clock modeling that involves the Markov stochastic process in the form of a random walk. We test different levels of random walk constraints for GNSS stations equipped with different types of clocks for Galileo-only and multi-GNSS solutions in kinematic and static PPP. In multi-GNSS solutions, the common clock parameter is derived with inter-system biases (ISBs). This raises the question of the constraints that should be imposed on the common clock only or also on the ISBs. We found that similar results can be achieved by imposing constraints on the common clock parameter and estimating ISB as a constant parameter and when constraining the common clock parameter and ISBs with a ratio of 1:100. Other ratios of clock-to-ISB constraints, such as 1:1 and 1:10, give inferior results. In the kinematic PPP, stochastic clock modeling has a marginal impact on the North and East coordinate components, whereas the Up component is substantially improved for GNSS receivers equipped with hydrogen masers. In the static PPP, the clock modeling improves the time transfer, due to the reduced noise of the clocks.

It is difficult to imagine today's world without Global Navigation Satellite Systems (GNSS). The dynamic development of GNSS has contributed to the fact that current users are able to use four global systems that use more than 120 satellites. This progress was related not only to the space segment but also to the user segment. Modern technology and miniaturization have resulted in the users' disposal of different types of GNSS receivers, including geodetic receivers, gaining popularity low-cost receivers, or other devices using the GNSS signal, such as smartphones, sports trackers, or sports watches. Modern sports watches are equipped with many sensors, among which GNSS chipsets play an important role. Those GNSS chipsets make it possible to determine the distance traveled and other related parameters that are important from the point of view of athletes. The most modern constructions can track several constellations at the same time. However, it is difficult to find reliable information to determine the actual quality of positioning by these low-cost GNSS receivers. Most of the works use comparative methods of watches and visual analysis of the route covered. Due to the above-mentioned gap in this area, the positioning quality of leading manufacturers of sports watches was assessed in this study. Ten sports watches from Garmin, Polar, and Suunto were assessed in the study regarding the geodetic grade GNSS Trimble receiver. The watches were evaluated in three experiments: field positioning experiment, distance accuracy experiment conducted on the athletics track, and the accuracy of the altitude determination conducted on the 37 m high tower. The tests were performed for all the GNSS system options available in watches. The best positioning quality was obtained for the Polar M430 watch that uses only GPS for which almost all recorded epochs obtain positioning accuracy better than 5 m. When measuring distance, most watches had a result that was less than 1% from the theoretical value. Garmin Vivoactive 4s achieved the best results in height determination. For 11 different measured levels, located about 3 m apart, it obtained an average difference equal to 0.48 m. The results show also that the use of the additional GNSS system degrades the obtained results in some cases.

Although the full operational capability of the Galileo system has not been officially announced as yet, the European GNSS, Galileo, has already remarkably contributed to geodesy, positioning, navigation, timing, and fundamental physics. Galileo metadata with the details on the satellite construction and surface properties allow for the development of the high-accuracy satellite macro-models and precise orbit determination. Two integrated onboard observation techniques – satellite laser ranging (SLR) and microwave GNSS – allow for the integration of space geodetic techniques and co-location in space. Calibrated satellite and receiver antenna offsets allow for scale realization and scale transfer for the reference frames. GNSS orbits of superior quality constitute the basis for other geodetic products, such as Earth rotation parameters, station coordinates, geocenter motion, international terrestrial reference frames, tropospheric and ionospheric delays. Moreover, the high-quality orbits and clocks installed on a pair of Galileo satellites launched onto eccentric orbits allow for studying effects emerging from general relativity, both related to the time redshift, as well as to orbital Schwarzschild, Lense-Thirring, and de Sitter effects constituting the essential issues of fundamental physics. Finally, high-quality and frequently-updated broadcast orbits together with very stable clocks onboard Galileo assure the superior accuracy of the real-time positioning when compared to other GNSS. We discuss the advantages and limitations of the Galileo system in terms of its applicability to geodesy, concentrating on daily and sub-daily Earth rotation parameters – polar motion and length-of-day variability, station coordinates, and geocenter motion. We address the system-specific errors discovered in GPS, GLONASS, and Galileo time series due to different satellite revolution periods, aliasing effects, tidal constituents, and orbit modeling issues. Some orbit modeling issues related, e.g., to thermal effects, remain unresolved, however, their impact may be mitigated by estimating empirical parameters and the combination of laser and microwave observations. The co-location in space onboard Galileo paves new opportunities for the realization of the reference frames tied in space, onboard GNSS satellites. We provide results on the recent developments of precise orbit determination and co-location in space based on integrated SLR and GNSS observations. Eventually, we discuss the latest applications of high-accurate orbits of Galileo satellites in near-circular and eccentric orbits toward the verification of the effects emerging from general relativity.

Wydawnictwo SIGMA-NOT wydaje czasopisma fachowe informujące swoich czytelników o najnowszych osiągnięciach naukowych i nowoczesnych rozwiązaniach technicznych w Polsce i na świecie, popularyzuje problemy techniczne oraz poszerza wiedzę i kulturę techniczną.