Search Results (16)

The G-modified Helmholtz equation is a partial differential equation that allows gravity intensity g to be expressed as a series of spherical harmonics, with the radial distance r raised to irrational powers. In this study, we consider a non-rotating triaxial ellipsoid parameterized by the geodetic latitude φ and geodetic longitude λ, and eccentricities e_e, e_x, e_y. On its surface, the value of gravity potential has a constant value, defining a level triaxial ellipsoid. In addition, the gravity intensity is known on the surface, which allows us to formulate a Dirichlet boundary value problem for determining the gravity intensity as a series of spherical harmonics. This expression for gravity intensity is presented here for the first time, filling a gap in the study of triaxial ellipsoids and spheroids. Given that the triaxial ellipsoid has very small eccentricities, a first order approximation can be made by retaining only the terms containing e_e² and e_x². The resulting expression in spherical harmonics contains even degree and even order harmonic coefficients, along with the associated Legendre functions. The maximum degree and order that occurs is four. Finally, as a special case, we present the geometrical degeneration of an oblate spheroid. Full article

The equatorial plasma bubbles (EPBs) are depleted plasma density regions in the ionosphere occurring during the post-sunset hours, associated with the signal fading and scintillation signatures in the trans-ionospheric radio signals. Severe scintillations may critically affect the performance of dynamic systems relying on global navigation satellite system (GNSS)-based services. Furthermore, the occurrence of scintillations in the equatorial and low latitudes can be triggered or inhibited during space weather events. In the present study, the possible presence of the EPBs during the geomagnetic storm periods under the 25th solar cycle is investigated using the GNSS-derived total electron content (TEC) depletion characteristics at a low-latitude equatorial ionization anomaly location, i.e., KL University, Guntur (Geographic 16°26′N, 80°37′E and dip 22°32′) in India. The detrended TEC with a specific window size is used to capture the characteristic depletion signatures, indicating the possible presence of the EPBs. Moreover, the TEC depletions, amplitude (S4) and phase scintillation (σ_φ) indices from multi-constellation GNSS signals are probed to verify the vulnerability of the signals towards the scintillation effects over the region. Observations confirm that all GNSS constellations witness TEC depletions between 15:00 UT and 18:00 UT, which is in good agreement with the recorded scintillation indices. We report characteristic depletion depths (22 to 45 TECU) and depletion times (28 to 48 min) across different constellations confirming the triggering of EPBs during the geomagnetic storm event on 23 April 2023. Unlikely, but the other storm events evidently inhibited TEC depletion, confirming suppressed EPBs. The results suggest that TEC depletions from the traditional geodetic GNSS stations could be used to substantiate the EPB characteristics for developing regional as well as global scintillation mitigation strategies. Full article

To obtain a higher accuracy for the real-time Zenith Tropospheric Delay (ZTD), a refined tropospheric delay correction model was constructed by combining the tropospheric delay correction model based on meteorological parameters and the GPT3 model. The meteorological parameters provided by the Global Geodetic Observing System (GGOS) Atmosphere and the zenith tropospheric delay data provided by Centre for Orbit Determination in Europe (CODE) were used as references, and the accuracy and spatial–temporal characteristics of the proposed model were compared and studied. The results show the following: (1) Compared with the UNB3m, GPT and GPT2w models, the accuracy and stability of the GPT3 model were significantly improved, especially the estimation accuracy of temperature, the deviation (Bias) of the estimated temperature was reduced by 90.60%, 32.44% and 0.30%, and the root mean square error (RMS) was reduced by 42.40%, 11.02% and 0.11%, respectively. (2) At different latitudes, the GPT3 + Saastamoinen, GPT3 + Hopfield and UNB3m models had great differences in accuracy and applicability. In the middle and high latitudes, the Biases of the GPT3 + Saastamoinen model and the GPT3 + Hopfield model were within 0.60 cm, and the RMS values were within 4 cm; the Bias of the UNB3m model was within 2 cm, and the RMS was within 5 cm; in low latitudes, the accuracy and stability of the GPT3 + Saastamoinen model were better than those of the GPT3 + Hopfield and UNB3m models; compared with the GPT3 + Hopfield model, the Bias was reduced by 22.56%, and the RMS was reduced by 5.67%. At different heights, the RMS values of the GPT3 + Saastamoinen model and GPT3 + Hopfield model were better than that of the UNB3m model. When the height was less than 500 m, the Biases of the GPT3 + Saastamoinen, GPT3 + Hopfield and UNB3m models were 3.46 cm, 3.59 cm and 4.54 cm, respectively. At more than 500 m, the Biases of the three models were within 4 cm. In different seasons, the Bias of the ZTD estimated by the UNB3m model had obvious global seasonal variation. The GPT3 + Saastamoinen model and the GPT3 + Hopfield model were more stable, and the values were within 5 cm. The research results can provide a useful reference for the ZTD correction accuracy and applicability of GNSS navigation and positioning at different latitudes, at different heights and in different seasons. Full article

Uniform grid data are widely used in climate science and related interdisciplinary fields. Such data usually describe the hydrometeorological states averaged over uniform latitude–longitude grids. While these data have larger grid areas in the tropics than other high-latitude regions, less attention has been paid to the areal weights of these grid data. Here, we revisited two methods available for processing these uniform grid data, including weighted sample statistics and grid interpolation. The former directly considers the grid area differences using geodetic weights; the latter converts the uniform grids to equal-area grids for conventional data analysis. When applied to global temperature and precipitation data, we found larger differences between weighted and unweighted samples and smaller differences between weighted and interpolated samples, highlighting the importance of areal weights in grid data analysis. Given the different results from various methods, we call for explicit clarification of the grid data processing methods to improve reproducibility in climate research. Full article

As a frequently-occurred phenomenon in the high-latitude region, ionospheric scintillations affect the stable service of the positioning navigation and timing service of the Global Navigation Satellite System (GNSS), calling for an urgent need of monitoring the scintillations accurately. The monitoring of scintillations usually adopts a special type of receiver, called an ionospheric scintillation monitoring receiver (ISMR), which cannot cover the whole high-latitude region due to its loss distribution. Geodetic receivers are densely distributed, but set at a 30s-sampling-interval usually. It is a controversial issue, namely, the accuracy of the scintillation index extracted from 30s-sampling-interval observations. This paper evaluates the accuracy of two 30s-sampling-interval indices in monitoring scintillations from both the time and space aspects using observations collected in the whole year of 2020. The accuracy in the time aspect is assessed with the phase scintillation index from ISMR as the reference through the following three-pronged approaches, i.e., the accuracy of the daily scintillation occurrence rates in the year 2020, the correlation with space weather parameters, and the variation pattern of the scintillation occurrence rate with the local time and day of the year 2020. The accuracy in space is studied based on the scintillation grid model considering the following two aspects, i.e., the scintillation monitoring performance in a Swarm satellite observation arc, and the statistical scintillation occurrence rate in the whole research region throughout the year 2020. The results of this paper reveal the efficiency of the 30s-sampling-interval scintillation indices in monitoring scintillations and detecting the occurrence patterns in the high-latitude region. The outcome of this paper can provide a basic idea for introducing the widely distributed geodetic receivers to monitor and model the scintillations in the high-latitude region. Full article

The bright stars in the clear night sky with weak background lights should be observed in the traditional deflection of the vertical (DOV) measurement so that the DOV cannot be observed under all-weather conditions, which limits its wide applications. An all-weather DOV measurement method combining three-dimensional (3D) laser tracking free-fall and multi-GNSS baselines is proposed in this paper. In a vacuum environment, the 3D laser tracking technique is used to continuously track and observe the motion of free-fall with high frequency and precision for obtaining 3D coordinate series. The plumb line vector equation is established to solve the gravity direction vector in the coordinate system of the laser tracker at the measuring point using least squares fitting coordinate series. Multi-GNSS observations are solved for obtaining the precise geodetic cartesian coordinates of the measuring point and GNSS baseline information. A direction transformation method based on the baseline information proposed in this paper is used to convert the gravitational direction vector in the laser tracker coordinate system into the geodetic cartesian coordinate system. The geodetic cartesian coordinates of the measuring point are used to calculate the ellipsoid normal vector, and the angle between this and the gravity direction vector in the geodetic cartesian coordinate system is estimated to obtain the astrogeodetic DOV. The DOV is projected to the meridian and prime vertical planes to obtain the meridian and prime vertical components of the DOV, respectively. The astronomical latitude and longitude of the measuring point are calculated from these two components. The simulation experiments were carried out using the proposed method, and it was found that the theoretical precision of the DOV measured by the method could reach 0.2″, which could realise all-weather observation. Full article

Using the symbolic calculation program Mathematica and based on the power series expansions of the common latitude with geodetic latitude as a variable, power series expansions of the common latitude with geocentric latitude as the variable are derived. The coefficients of the two groups of formulas are based on the ellipsoid eccentricity e and the ellipsoid third flattening n, which make the expansions more uniform. Taking the CGCS2000 as an example, numerical analysis is applied to verify the accuracy and reliability of the derived power series expansions. By analyzing and calculating the truncation error of the common latitude based on ellipsoidal eccentricity e and the third flattening n expansion to different orders, we obtain simplified, practical formulas for the common latitude that satisfy the requirement of geodesic accuracy. Moreover, we show that the practical formula derived has higher calculation efficiency and easier dissemination, enriches the theory of map projection, and provides a basis for better display of remote sensing images. Full article

Precipitable water vapor (PWV) is an important component in the climate system and plays a pivotal role in the global water and energy cycles. Over the years, many approaches have been devised to accurately estimate the PWV. Among them, global navigation satellite systems (GNSS) have become one of the most promising and fastest-growing PWV acquisition methods because of its high accuracy, high temporal and spatial resolution, and ability to acquire PWV in all weather and in near real time. We compared GNSS-derived PWV with a 5 min resolution globally distributed over 14,000 stations from the Nevada Geodetic Laboratory (NGL) from 1994 to 2020 with global radiosonde (RS) data, temperature anomalies, and sea height variations. Then, we examined the temporal and spatial variability of the global PWV and analyzed its climate implications. On a global scale, the average bias and root mean square error (RMSE) between GNSS PWV and RS PWV were ~0.72 ± 1.29 mm and ~2.56 ± 1.13 mm, respectively. PWV decreased with increasing latitude, and the rate of this decrease slowed down at latitudes greater than 35°, with standard deviation (STD) values reaching a maximum at latitudes less than 35°. The global average linear trend was ~0.64 ± 0.81 mm/decade and strongly correlated with temperature and sea height variations. For each 1 °C and 1 mm change, PWV increased by ~2.075 ± 0.765 mm and ~0.015 ± 0.005 mm, respectively. For the time scale, the PWV content peaked ~40 days after the maximum solar radiation of the year (the summer solstice), and the delay was ~40 days relative to the summer solstice. Full article

Global Navigation Satellite System (GNSS) signals generate slant tropospheric delays when they pass through the atmosphere, which is recognized as the main source of error in many spatial geodetic applications. The zenith tropospheric delay (ZTD) derived from radio occultation data is of great significance to atmospheric research and meteorology and needs to be assessed in the use of precision positioning. Based on the atmPrf, sonPrf, and echPrf data from the Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) Data Analysis and Archiving Center (CDAAC) from 1 January to 31 December 2008 and 2012, we obtained the ZTDs of the radio occultation data (occZTD) and the corresponding radiosonde (sonZTD) and ECWMF data (echZTD). The ZTDs derived from ground-based global positioning system (GPS) observations from the International GNSS Service (IGS) were corrected to the lowest tangent point height of the matched radio occultation profile by the barometric height formula (gnsZTD). The statistical results show that the absolute values of the bias between occZTD and echZTD, sonZTD, or gnsZTD are less than 5 mm, and the standard deviations are approximately 20 mm or less, indicating that occZTD had significant accuracy in the GNSS positioning model even when the local spherical symmetry assumption error was introduced when the Abel inversion algorithm was used to obtain the refractive index profile of atmPrf. The effects of the horizontal/vertical matching resolution and the variation in the station height/latitude on the biases of occZTD and gnsZTD were analyzed. The results can be used to quantify the performance of radio occultation data for tropospheric delay error correction in dynamic high-precision positioning. Full article

Zenith tropospheric delay (ZTD) plays an important role in high-precision global navigation satellite system (GNSS) positioning and meteorology. At present, commonly used ZTD forecasting models comprise empirical, meteorological parameter, and neural network models. The empirical model can only fit approximate periodic variations, and its accuracy is relatively low. The accuracy of the meteorological parameter model depends heavily on the accuracy of the meteorological parameters. The recurrent neural network (RNN) is suitable for short-term series data prediction, but for long-term series, the ZTD prediction accuracy is clearly reduced. Long short-term memory (LSTM) has superior forecasting accuracy for long-term ZTD series; however, the LSTM model is complex, cannot be parallelized, and is time-consuming. In this study, we propose a novel ZTD time-series forecasting utilizing transformer-based machine-learning methods that are popular in natural language processing (NLP) and forecasting global ZTD, the training parameters provided by the global geodetic observing system (GGOS). The proposed transformer model leverages self-attention mechanisms by encoder and decoder modules to learn complex patterns and dynamics from long ZTD time series. The numeric results showed that the root mean square error (RMSE) of the forecasting ZTD results were 1.8 cm and mean bias, STD, MAE, and R 0.0, 1.7, 1.3, and 0.95, respectively, which is superior to that of the LSTM, RNN, convolutional neural network (CNN), and GPT3 series models. We investigated the global distribution of these accuracy indicators, and the results demonstrated that the accuracy in continents was superior to maritime space transformer ZTD forecasting model accuracy at high latitudes superior to that at low latitude. In addition to the overall accuracy improvement, the proposed transformer ZTD forecast model also mitigates the accuracy variations in space and time, thereby guaranteeing high accuracy globally. This study provides a novel method to estimate the ZTD, which could potentially contribute to precise GNSS positioning and meteorology. Full article

The Google Smartphone Decimeter Challenge (GSDC) was a competition held in 2021, where data from a variety of instruments useful for determining a phone’s position (signals from GPS satellites, accelerometer readings, gyroscope readings, etc.) using Android smartphones were provided to be processed/assessed in regard to the most accurate determination of the longitude and latitude of user positions. One of the tools that can be utilized to process the GNSS measurements is RTKLIB. RTKLIB is an open-source GNSS processing software tool that can be used with the GNSS measurements, including code, carrier, and doppler measurements, to provide real-time kinematic (RTK), precise point positioning (PPP), and post-processed kinematic (PPK) solutions. In the GSDC, we focused on the PPK capabilities of RTKLIB, as the challenge only required post-processing of past data. Although PPK positioning is expected to provide sub-meter level accuracies, the lower quality of the Android measurements compared to geodetic receivers makes this performance difficult to achieve consistently. Another latent issue is that the original RTKLIB created by Tomoji Takasu is aimed at commercial GNSS receivers rather than smartphones. Therefore, the performance of the original RTKLIB for the GSDC is limited. Consequently, adjustments to both the code-base and the default settings are suggested. When implemented, these changes allowed RTKLIB processing to score 5th place, based on the performance submissions of the prior GSDC competition. Detailed information on what was changed, and the steps to replicate the final results, are presented in the paper. Moreover, the updated code-base, with all the implemented changes, is provided in the public repository. This paper outlines a procedure to optimize the use of RTKLIB for Android smartphone measurements, highlighting the changes needed given the low-quality measurements from the mobile phone platform (relative to the survey grade GNSS receiver), which can be used as a basis point for further optimization for future GSDC competitions. Full article

The eastern Tien Shan hosts substantial mid-latitude glaciers, but in situ glacier mass balance records are extremely sparse. Haxilegen Glacier No. 51 (eastern Tien Shan, China) is one of the very few well-measured glaciers, and comprehensive glaciological measurements were implemented from 1999 to 2011 and re-established in 2017. Mass balance of Haxilegen Glacier No. 51 (1999–2015) has recently been reported, but the mass balance record has not extended to the period before 1999. Here, we used a 1:50,000-scale topographic map and long-range terrestrial laser scanning (TLS) data to calculate the area, volume, and mass changes for Haxilegen Glacier No. 51 from 1964 to 2018. Haxilegen Glacier No. 51 lost 0.34 km² (at a rate of 0.006 km² a⁻¹ or 0.42% a⁻¹) of its area during the period 1964–2018. The glacier experienced clearly negative surface elevation changes and geodetic mass balance. Thinning occurred almost across the entire glacier surface, with a mean value of −0.43 ± 0.12 m a⁻¹. The calculated average geodetic mass balance was −0.36 ± 0.12 m w.e. a⁻¹. Without considering the error bounds of mass balance estimates, glacier mass loss over the past 50 years was in line with the observed and modeled mass balance (−0.37 ± 0.22 m w.e. a⁻¹) that was published for short time intervals since 1999 but was slightly less negative than glacier mass loss in the entire eastern Tien Shan. Our results indicate that Riegl VZ^®-6000 TLS can be widely used for mass balance measurements of unmonitored individual glaciers. Full article

The atmospheric humidity in the Polar Regions is an important factor for the global budget of water vapour, which is a significant indicator of Earth’s climate state and evolution. The Global Navigation Satellite System (GNSS) can make a valuable contribution in the calculation of the amount of Precipitable Water Vapour (PW). The PW values retrieved from Global Positioning System (GPS), hereafter PW_GPS, refer to 20-year observations acquired by more than 40 GNSS geodetic stations located in the polar regions. For GNSS stations co-located with radio-sounding stations (RS), which operate Vaisala radiosondes, we estimated the PW from RS observations (PW_RS). The PW values from the ERA-Interim global atmospheric reanalysis were used for validation and comparison of the results for all the selected GPS and RS stations. The correlation coefficients between times series are very high: 0.96 for RS and GPS, 0.98 for RS and ERA in the Arctic; 0.89 for RS and GPS, 0.97 for RS and ERA in Antarctica. The Root-Mean-Square of the Error (RMSE) is 0.9 mm on average for both RS vs. GPS and RS vs. ERA in the Arctic, and 0.6 mm for RS vs. GPS and 0.4 mm for RS vs. ERA in Antarctica. After validation, long-term trends, both for Arctic and Antarctic regions, were estimated using Hector scientific software. Positive PW_GPS trends dominate at Arctic sites near the borders of the Atlantic Ocean. Sites located at higher latitudes show no significant values (at 1σ level). Negative PW_GPS trends were observed in the Arctic region of Greenland and North America. A similar behaviour was found in the Arctic for PW_RS trends. The stations in the West Antarctic sector show a general positive PW_GPS trend, while the sites on the coastal area of East Antarctica exhibit some significant negative PW_GPS trends, but in most cases, no significant PW_RS trends were found. The present work confirms that GPS is able to provide reliable estimates of water vapour content in Arctic and Antarctic regions too, where data are sparse and not easy to collect. These preliminary results can give a valid contribution to climate change studies. Full article

Tropospheric delay is one of the major error sources in GNSS (Global Navigation Satellite Systems) positioning. Over the years, many approaches have been devised which aim at accurately modeling tropospheric delays, so-called troposphere models. Using the troposphere data of over 16,000 global stations in the last 10 years, as calculated by the Nevada Geodetic Laboratory (NGL), this paper evaluates the performance of the empirical troposphere model GPT3, which is the latest version of the GPT (Global Pressure and Temperature) series model. Owing to the large station number, long time-span and diverse station distribution, the spatiotemporal properties of the empirical model were analyzed using the average deviation (BIAS) and root mean square (RMS) error as indicators. The experimental results demonstrate that: (1) the troposphere products of NGL have the same accuracy as the IGS (International GNSS Service) products and can be used as a reference for evaluating general troposphere models. (2) The global average BIAS of the ZTD (zenith total delay) estimated by GPT3 is −0.99 cm and the global average RMS is 4.41 cm. The accuracy of the model is strongly correlated with latitude and ellipsoidal height, showing obviously seasonal variations. (3) The global average RMS of the north gradient and east gradient estimated by GPT3 is 0.77 mm and 0.73 mm, respectively, which are strongly correlated with each other, with values increasing from the equator to lower latitudes and decreasing from lower to higher latitudes. Full article

Low-cost GPS receivers provide geodetic positioning information using the NMEA protocol, usually with eight digits for latitude and nine digits for longitude. When these geodetic coordinates are converted into Cartesian coordinates, the positions fit in a quantization grid of some decimeters in size, the dimensions of which vary depending on the point of the terrestrial surface. The aim of this study is to reduce the quantization errors of some low-cost GPS receivers by using a Kalman filter. Kinematic tractor model equations were employed to particularize the filter, which was tuned by applying Monte Carlo techniques to eighteen straight trajectories, to select the covariance matrices that produced the lowest Root Mean Square Error in these trajectories. Filter performance was tested by using straight tractor paths, which were either simulated or real trajectories acquired by a GPS receiver. The results show that the filter can reduce the quantization error in distance by around 43%. Moreover, it reduces the standard deviation of the heading by 75%. Data suggest that the proposed filter can satisfactorily preprocess the low-cost GPS receiver data when used in an assistance guidance GPS system for tractors. It could also be useful to smooth tractor GPS trajectories that are sharpened when the tractor moves over rough terrain. Full article