Vienna Mapping Function Research Articles

Comparing Discrete and Empirical Troposphere Delay Models: A Global IGS‐Based Evaluation

AbstractZenith tropospheric delay (ZTD) is an important atmospheric parameter in radio‐space‐geodetic techniques such as Global Navigation Satellite System (GNSS), which is pivotal for GNSS positioning, navigation and meteorology. The Vienna Mapping Function (VMF) data server is a widely utilized source for implementing ZTD, offering two types of models, that is, the empirical one and the discrete one with Grid‐wise and Site‐wise models. Therefore, to evaluate the accuracy of these models becomes the focus of this article. Specifically, this study investigates their performances in terms of calculation of ZTD, using the hourly values derived from the International GNSS Service data as references. The results show that the root mean square err (RMSE) of the Site‐wise, Grid‐wise and global pressure and temperature 3 model are 11.71/13.03/38.56 mm, respectively, indicating the discrete model performs generally better than the empirical model, and the Site‐wise model is the better of the two discrete models. From the perspective of spatial resolution, the performance of these three models in ZTD calculation shows obvious influences of latitude changes and elevation differences. From the temporal analysis, the accuracy of the discrete model shows differences over different UTC epochs, while the empirical model can only express the seasonal ZTD characteristics with the average RMSE at different epochs being similar, the specifically values are 39.67, 39.26, 39.38 and 39.18 mm at UTC 0:00, 6:00, 12:00 and 18:00, respectively. The histogram and boxplot well indicate the accuracy differences of the three models in different seasons. Additionally, the time series of three models at different latitudes were also explored in this research. These explorations are conducive to the selection of appropriate models for calculating ZTD based on specific requirements.

Troposphere delay modeling in SLR based on PMF, VMF3o, and meteorological data

Satellite laser ranging (SLR) requires accurate troposphere delay models to properly correct the observed distances to satellites and derive fundamental geodetic and geodynamic parameters. The currently used models for the tropospheric delay employ in situ meteorological data collected simultaneously with laser measurements. However, the standard models assume full symmetry of the atmosphere above the SLR stations because all meteorological data come from one sensor. In this study, we evaluate various methods of troposphere delay modeling based on numerical weather models, such as the Potsdam Mapping Function (PMF) and Vienna Mapping Function for optical frequencies (VMF3o), in situ measurements, the Wrocław gradient model (WGM) and the combination of different models. We found large discrepancies between pressure, temperature, and humidity records between in situ measurements and numerical models. The best results for the zenith delays are obtained when using in situ meteorological data with the estimation of tropospheric biases. For stations with some deficiencies in proper humidity measurements, e.g., Zimmerwald in Switzerland, the best results are obtained when using hydrostatic zenith delays based on in situ data and wet delays based on numerical weather models. Finally, we found that using horizontal gradients of the tropospheric delay is indispensable to avoid biases in the SLR-based Earth rotation parameters of approximately 20 μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upmu$$\\end{document}as for polar motion. The horizontal gradients successfully account for the asymmetry of the troposphere above SLR stations and can be derived from PMF, VMF3o, or a parameterized WGM model with similar accuracies.

An initial investigation of the non-isotropic feature of GNSS tropospheric delay

Tropospheric delay is a significant error source in Global Navigation Satellite Systems (GNSS) positioning. Slant Path Delay (SPD) is commonly derived by multiplying Zenith Tropospheric Delay (ZTD) with a mapping function. However, mapping functions, assuming atmospheric isotropy, restrict the accuracy of derived SPDs. To improve the accuracy, a horizontal gradient correction is introduced to account for azimuth-dependent SPD variations, treating the atmosphere as anisotropic. This study uncovers that, amidst atmospheric dynamics and spatiotemporal changes in moisture content, the SPD deviates from that based on traditional isotropy or anisotropy assumption. It innovatively introduces the concept that SPD exhibits non-isotropy with respect to azimuth angles. Hypothesis validation involves assessing SPD accuracy using three mapping functions at five International GNSS Service (IGS) stations, referencing the SPD with the ray-tracing method. It subsequently evaluates the SPD accuracy with horizontal gradient correction based on Vienna Mapping Function 3 (VMF3) estimation. Lastly, the non-isotropic of SPD is analyzed through the ray-tracing method. The results indicate the smallest residual (1.1–82.7 mm) between the SPDs with VMF3 and those with the ray-tracing. However, introducing horizontal gradient correction yields no significant improvement of SPD accuracy. Considering potential decimeter-level differences in SPD due to non-isotropic tropospheric delay across azimuth angles, a precise grasp and summary of these variations is pivotal for accurate tropospheric delay modeling. This finding provides vital support for future high-precision tropospheric delay modeling.

Impact analysis of processing strategies for long-term GPS zenith tropospheric delay (ZTD)

Abstract. Homogenized atmospheric water vapour data are an important prerequisite for climate analysis. Compared with other techniques, GPS has an inherent homogeneity advantage, but it still requires reprocessing and homogenization to eliminate impacts of applied strategy and observation environmental changes where a selection of proper processing strategies is critical. This paper comprehensively investigates the influence of the mapping function, the elevation cut-off angle and homogenization on long-term reprocessing results, in particular for zenith tropospheric delay (ZTD) products, using GPS observations at 44 IGS (International GNSS Service) stations during 1995 to 2014. In the analysis, for the first time, we included the mapping function (the latest Vienna mapping function, VMF3) and exploited homogenized radiosonde data as a reference for ZTD trend evaluations. Our analysis shows that both site position and ZTD solutions achieved the best accuracy when using VMF3 and the 3∘ elevation cut-off angle. Regarding the long-term ZTD trends, homogenization reduced the trend inconsistency among different elevation cut-off angles. ZTD trend results show that the impact of mapping functions is very small, with a maximum difference of 0.19 mm yr−1. On the other hand, the discrepancy can reach 0.60 mm yr−1 using different elevation cut-off angles. Low-elevation cut-off angles (3∘ or 7∘) are suggested for the best estimates of ZTD reprocessing time series when compared to homogenized radiosonde data or ERA5 reference time series.

Machine Learning-Based Calibrated Model for Forecast Vienna Mapping Function 3 Zenith Wet Delay

An accurate estimation of zenith wet delay (ZWD) is crucial for global navigation satellite system (GNSS) positioning and GNSS-based precipitable water vapor (PWV) inversion. The forecast Vienna Mapping Function 3 (VMF3-FC) is a forecast product provided by the Vienna Mapping Functions (VMF) data server based on the European Centre for Medium-Range Weather Forecasts (ECMWF)-based numerical weather prediction (NWP) model. The VMF3-FC can provide ZWD at any time and for any location worldwide; however, it has an uneven accuracy distribution and fails to match the application requirements in certain areas. To address this issue, in this study, a calibrated model for VMF3-FC ZWD, named the XZWD model, was developed by utilizing observation data from 492 radiosonde sites globally from 2019–2021 and the eXtreme Gradient Boosting (XGBoost) algorithm. The performance of the XZWD model was validated using 2022 observation data from the 492 radiosonde sites. The XZWD model yields a mean bias of −0.03 cm and a root-mean-square error (RMSE) of 1.64 cm. The XZWD model outperforms the global pressure and temperature 3 (GPT3) model, reducing the bias and RMSE by 94.64% and 58.90%, respectively. Meanwhile, the XZWD model outperforms VMF3-FC, with a reduction of 92.68% and 6.29% in bias and RMSE, respectively. Furthermore, the XZWD model reduces the impact of ZWD accuracy by latitude, height, and seasonal variations more effectively than the GPT3 model and VMF3-FC. Therefore, the XZWD model yields higher stability and accuracy in global ZWD forecasting.

Tropospheric Delay Model Based on VMF and ERA5 Reanalysis Data

The global tropospheric zenith delay grid products of VMF1 and VMF3 (Vienna mapping functions) with different resolutions are used to calculate the tropospheric zenith delay of eight IGS (International GNSS Service) stations in China, and the accuracy of the two products under different interpolation methods is analyzed. As a result, the accuracy of utilizing different interpolation methods shows no obvious differences. The interpolation accuracy of the VMF3 grid model is slightly higher than that of the VMF1, and the interpolation accuracy of tropospheric delay is related to the elevation difference of grid points. In addition, according to ERA5 (the fifth generation of the Global Climate Information Analysis data set), the atmospheric stratification tropospheric delay is obtained, and a ZTD (the zenith tropospheric delay) height change grid model is constructed using the least squares exponential fitting method. The accuracy of the model is verified using the tropospheric delay product provided by the IGS. Finally, the constructed ZTD height change grid model is used as ZTD height reduction to solve the problem of large tropospheric delay errors in the VMF interpolation when the height change is large. The model accuracy of URUM station improve from 96.47 mm.to 32.97 mm (65.82%).

Refinement of global gridded ray-traced Zenith tropospheric delay over Nigeria based on local GNSS network observations

• Application of deep structured supervised neural network modelling technique. • Data derived from the local GNSS observation within the study area. • A refinement model is developed to predict ZTD parameters at user-defined points. • Improvement in the accuracy of the local refine VMF1-ZTD estimates. Recent studies on the application of Zenith Tropospheric Delay (ZTD) in meteorology and geodesy have shown improved accuracy with the use of locally estimated ZTD compared to Global ZTD. This paper focuses on the refinement of global gridded ray-traced Vienna Mapping Functions 1 (VMF1) ZTD over Nigeria, using ZTD estimates derived from local GNSS Continuously Operating Reference Stations (CORS) observations. Three (3) years of GNSS observations collected at 13 CORS located within the study area were post-processed using GNSS Analysis and Processing Software (GAPS) to derive the local GNSS-ZTD estimates. The global VMF1-ZTD were interpolated at the locations of the local GNSS-CORS. An optimal deep structured supervised Neural Network (NN) was derived by training feedforward networks using the Levenberg-Marquardt backpropagation algorithm, with spatiotemporal variables and VMF1-ZTD as inputs and the GNSS-ZTD as the target. The outputs from the NN model were assessed using internal and external data. Results from internal data evaluations indicated a robust and accurate model output with optimum network performances indicating MSE of ∼7.22 × 10 −6 m and ∼4.56 × 10 −4 m for hydrostatic and wet ZTD networks respectively. The external data evaluations performed using the International GNSS Services (IGS) Zenith Path Delay (ZPD), indicated an RMSE range of ∼0.041 m to 0.095 m for global-VMF1 and ∼0.030 m to 0.037 m for the Local Refine (LR)-VMF1. The results suggest that the application of ZTD estimates derived from local GNSS observations in the refinement of global gridded VMF1-ZTD yields an improved accuracy over Nigeria. Consequently, the LR-VMF1 ZTD estimates are recommended for improved accuracy, required for geodetic and meteorological applications in and around Nigeria :

Performance of ray-traced VMF3 products in retrieving Zenith Tropospheric Delay over the African tropical region

The availability of Global Navigation Satellite Systems (GNSS) networks allowed for the precise determination of the Zenith Tropospheric Delay (ZTD) with high temporal and spatial resolution. For the proper assessment of the atmospheric water vapour, which has a significant impact on weather forecasting and prediction, the availability of high spatial resolution ZTD measurements is absolutely essential. However, the quality and uses of GNSS ZTD products are impacted by the dispersed GNSS receiver stations and the significant data gaps in the GNSS observations over the tropical region of Africa. To solve the issues of data gaps and spatial variability, a number of methods have been devised to estimate the ZTD. The Vienna Mapping Functions 3 (VMF3) which applies the ray-tracing technique to get ZTD is one of the preferred methods for post processing. To evaluate the precision and viability of VMF3 in extracting the ZTD in the tropical region of Africa, validation is necessary. Therefore, the goal of this work is to give a statistical evaluation and performance of the Nevada Geodetic Laboratory (NGL) products and the site dependent VMF3 ZTD (V3GR_FC, V3GR_EI and V3GR_OP) products. To assess the level of agreement between the two datasets, the study used six different statistical criteria. These include the mean bias (MnB), coefficient of determination (R2), root mean square error (RMSE), correlation coefficient (r), the index of agreement (IA) and the Taylor skill score (TSS). The results demonstrate that VMF3 datasets can generate ZTD results that are comparable to NGL ZTD. However, it was discovered that the V3GR_OP ZTD (overall average values, RMSE = 9.77 mm, r = 0.953, R2 = 0.908, IA = 0.971, TSS = 0.948) outperforms the V3GR_EI ZTD (overall average values, RMSE = 13.12 mm, r = 0.912, R2 = 0.832, IA = 0.950, TSS = 0.915) and the V3GR_FC ZTD (overall average values, RMSE = 13.27 mm, r = 0.907, R2 = 0.824, IA = 0.942, TSS = 0.903). The performance of V3GR_EI ZTD and V3GR_FC ZTD varies from station to station.

Effects of Tropospheric Refraction on Precise Point Positioning in Brazil

Troposphere is one of the most limiting sources of error in the accuracy of 15 Precise Point Positioning method. This work aims to analyze the effects of tropospheric 16 delay on this positioning technique by applying the Bernese scientific program to GNSS 17 data belonging to the Brazilian Network for Continuous Monitoring of GNSS Systems. 18 GNSS data are related to several climatic zones, including Amazon region, referring to four 19 seasons. The data were processed considering six strategies, each one using different 20 troposphere models and mapping functions for analysis. Results were evaluated according 21 to the Root Mean Square, estimated for 15 processing days for each season, involving 89 22 GNSS stations. Results show that the greatest effects of the tropospheric delay occurred in 23 the equatorial region, related to the altimetric component, in all seasons of the year. This 24 climatic zone is under a strong influence of the Amazon region, which presents high annual 25 humidity values. In addition, it can occur a great humidity variation in this region, which 26 can compromise the process of estimating wet component of tropospheric delay. Finally, 27 results showed that the best processing strategy was the use of the Vienna mapping function 28 in conjunction with corrections based on the Numerical Weather Forecast model.

Airborne Coherent GNSS Reflectometry and Zenith Total Delay Estimation over Coastal Waters

High-precision GNSS (global navigation satellite e system) measurements can be used for remote sensing and nowadays play a significant role in atmospheric sounding (station data, radio occultation observations) and sea surface altimetry based on reflectometry. A limiting factor of high-precision reflectometry is the loss of coherent phase information due to sea-state-induced surface roughness. This work studies airborne reflectometry observations recorded over coastal waters to examine the sea-state influence on Doppler distribution and the coherent residual phase retrieval. From coherent observations, the possibility of zenith total delay inversion is also investigated, considering the hydrostatic mapping factor from the Vienna mapping function and an exponential vertical decay factor depending on height receiver changes. The experiment consists of multiple flights performed along the coast between the cities of Calais and Boulogne-sur-Mer, France, in July 2019. Reflected signals acquired in a right-handed circular polarization are processed through a model-aided software receiver and passed through a retracking module to obtain the Doppler and phase-corrected signal. Results from grazing angle observations (elevation < 15°) show a high sensitivity of Doppler spread with respect to sea state with correlations of 0.75 and 0.88 with significant wave height and wind speed, respectively. An empirical Doppler spread threshold of 0.5 Hz is established for coherent reflections supported by the residual phase observations obtained. Phase coherence occurs in 15% of the observations; however, the estimated zenith total delay for the best event corresponds to 2.44 m, which differs from the typical zenith total delay (2.3 m) of 5%.

Comparative evaluation and analysis of different tropospheric delay models in Ghana

Tropospheric delay prediction models have become increasingly important in Global Navigation Satellite System (GNSS) as they play a critical role in GNSS positioning applications. Due to the different atmospheric conditions over the earth regions, tropospheric effect on GNSS signals also differs, influencing the performance of these prediction models. Thus, the choice of a particular prediction model can significantly degrade the positioning accuracy especially when the model does not suit the user’s environs. Therefore, a performance assessment of existing prediction models in various regions for a suitable one is very imperative. This paper evaluates and analyses seven commonly used tropospheric delay models in Ghana in terms of performances in Zenith Tropospheric Delay (ZTD) estimation and baseline positional accuracies using data from six selected Continuously Operating Reference Stations (CORS). The 1˚x1˚ gridded Vienna Mapping Functions 3 (VMF3) ZTD product and coordinates solutions from the CSRS-PPP positioning service were respectively used as references. The results show that the Black model performed better in estimating the ZTD, followed by Askne and Nordius model. The Saastamoinen, Marini and Murray, Niell, Goads and Goodman and Hopfield models respectively performed poorly. However, the result of the baseline solutions did not show much variation in the coordinate difference provided by the use of the prediction models, nonetheless, the Black and Askne and Nordius models continue to dominate the other models. Of all the models evaluated, either Black or Askne and Nordius model is recommended for use to mitigate the ZTD in the study area, however, the choice of the Black model will be more desirable.

Water vapor monitoring with IGS RTS and GPT3/VMF3 functions over Turkey

Precise Point Positioning (PPP) has been proven as a cost-effective and efficient technique for monitoring the precipitable water vapor (PWV), which plays a critical role in weather forecasting. The recent developments of the International Global Navigation Satellite Systems Service (IGS) Real-Time Pilot Project (RTPP) ease the difficulties of real-time determination of PWV. In this study, the quality of the real-time PPP-derived PWVs and their correlation with precipitation estimates from different global products are investigated. The accuracies of the orbits and clocks derived from the IGS02 streams are evaluated and their performance outperforms the predicted part of IGS Ultra-rapid (IGU) orbits and clocks. A year of GNSS observations from 9 stations located in Turkey are processed to estimate the PWV in the real-time. Since there are no stations to collect meteorological data near GNSS stations, the real-time solutions employ empirical Global Pressure and Temperature 3 (GPT3) and its fully consistent discrete Vienna Mapping Function 3 (VMF3) to model the troposphere. The performance of the PWV estimates is assessed by comparing with Radiosonde-derived PWV values and post-processed PPP-derived PWV estimates employing the Center for Orbit Determination in Europe (CODE) final orbits and clocks. At most sites, the root-mean-square (RMS) differences of the estimated PWVs in comparison with the radiosonde-derived PWVs are less than 2.4 mm with correlation coefficients greater than 0.90. The RMS differences between RT and post-processed PPP PWV estimates are found to be less than 1.2 mm. The estimated RT zenith total delays (ZTDs) at two IGS sites agree well with the ZTDs provided by the IGS final products. A method of precipitation monitoring is proposed based on the daily standard deviation (SD) of the PWV estimates. Experimental analyses of the proposal are undertaken by comparing the daily SDs of the PWV estimates and the daily accumulated precipitation estimates from the global precipitation grids produced from rain gauges, satellite images, and reanalyses product sets. The results show that the daily variation of the PWVs and the high precipitation estimates present a good agreement for some days but not always. The PWVs overestimate and underestimate a significant number of events. Spatial refinement of GPT3 grids and gridded precipitation products with higher sampling over Turkey may offer better results.

Development and evaluation of the refined zenith tropospheric delay (ZTD) models

The tropospheric delay is a significant error source in Global Navigation Satellite System (GNSS) positioning and navigation. It is usually projected into zenith direction by using a mapping function. It is particularly important to establish a model that can provide stable and accurate Zenith Tropospheric Delay (ZTD). Because of the regional accuracy difference and poor stability of the traditional ZTD models, this paper proposed two methods to refine the Hopfield and Saastamoinen ZTD models. One is by adding annual and semi-annual periodic terms and the other is based on Back-Propagation Artificial Neutral Network (BP-ANN). Using 5-year data from 2011 to 2015 collected at 67 GNSS reference stations in China and its surrounding regions, the four refined models were constructed. The tropospheric products at these GNSS stations were derived from the site-wise Vienna Mapping Function 1 (VMP1). The spatial analysis, temporal analysis, and residual distribution analysis for all the six models were conducted using the data from 2016 to 2017. The results show that the refined models can effectively improve the accuracy compared with the traditional models. For the Hopfield model, the improvement for the Root Mean Square Error (RMSE) and bias reached 24.5/49.7 and 34.0/52.8 mm, respectively. These values became 8.8/26.7 and 14.7/28.8 mm when the Saastamoinen model was refined using the two methods. This exploration is conducive to GNSS navigation and positioning and GNSS meteorology by providing more accurate tropospheric prior information.

Regression models for predicting daily IGS zenith tropospheric delays in West Africa: Implication for GNSS meteorology and positioning applications

AbstractThe ability to precisely and accurately model and predict tropospheric delay is essential for precise global navigation satellite system (GNSS) and meteorological applications. The International GNSS Service (IGS) provides highly accurate and highly reliable daily time series zenith tropospheric delay (ZTD) products for all its member sites using data from each IGS site. Nevertheless, if for reasons such as poor internet connectivity, equipment failure, and power outages the IGS station is inaccessible, gaps are created in the data archive, resulting in degrading the quality of the ZTD estimation, as well as inhibits the quality of precipitable water vapour (PWV) estimation, needed for precise positioning applications, meteorological studies, and weather forecasting. To address this challenge, five regression models are proposed in this study to model and predict daily ZTDs using daily datasets from four IGS stations in West Africa over a period of 5 years (2015–2019). The site‐specific Vienna Mapping Functions 3 (VMF3) products (ZTD, pressure, temperature, water vapour partial pressure) and stations' coordinates (latitudes and longitudes) are used as the predictors, while the IGS final ZTD product as the response variable in fitting the models. Several performance measures are calculated to compare the predictive performance of the models. The results show that the five regression models performed outstandingly and agree very well with the IGS‐ZTD data, and hence provide a useful alternative for ZTD predictions and also in the event the West African IGS stations' ZTD data are unavailable. Nonetheless, the support vector regression model outperformed the remaining four models.

Kinematic Zenith Tropospheric Delay Estimation with GNSS PPP in Mountainous Areas

The use of global navigation satellite systems (GNSS) precise point positioning (PPP) to estimate zenith tropospheric delay (ZTD) profiles in kinematic vehicular mode in mountainous areas is investigated. Car-mounted multi-constellation GNSS receivers are employed. The Natural Resources Canada Canadian Spatial Reference System PPP (CSRS-PPP) online service that currently processes dual-frequency global positioning system (GPS) and Global’naya Navigatsionnaya Sputnikovaya Sistema (GLONASS) measurements and is now capable of GPS integer ambiguity resolution is used. An offline version that can process the above and Galileo measurements simultaneously, including Galileo integer ambiguity resolution is also tested to evaluate the advantage of three constellations. A multi-day static data set observed under open sky is first tested to determine performance under ideal conditions. Two long road profile tests conducted in kinematic mode are then analyzed to assess the capability of the approach. The challenges of ZTD kinematic profiling are numerous, namely shorter data sets, signal shading due to topography and forests of conifers along roads, and frequent losses of phase lock requiring numerous but not always successful integer ambiguity re-initialization. ZTD profiles are therefore often only available with float ambiguities, reducing system observability. Occasional total interruption of measurement availability results in profile discontinuities. CSRS-PPP outputs separately the zenith hydrostatic or dry delay (ZHD) and water vapour content or zenith wet delay (ZWD). The two delays are analyzed separately, with emphasis on the more unpredictable and highly variable ZWD, especially in mountainous areas. The estimated delays are compared with the Vienna Mapping Function 1 (VMF1), which proves to be highly effective to model the large-scale profile variations in the Canadian Rockies, the main contribution of GNSS PPP being the estimation of higher frequency ZWD components. Of the many conclusions drawn from the field experiments, it is estimated that kinematic profiles are generally determined with accuracy of 10 to 20 mm, depending on the signal harshness of the environment.

Assessment of the Troposphere Products Derived From VMF Data Server With ERA5 and IGS Data Over China

AbstractThe accurate modeling of zenith troposphere delay (ZTD) is significant for data processing of radio‐space‐geodetic techniques such as GNSS. The delay can be estimated by classic tropospheric models, which need the surface meteorological parameters. Considering the difficulty of obtaining meteorological data caused by various environment factors and the modeling errors in the models themselves, the above methods are often difficult to provide accurate products for GNSS data processing. Based on continuously updated NWP data from the European Centre for Medium‐Range Weather Forecasts (ECMWF), the Vienna Mapping Functions (VMF) data server provides a widely used data‐source for achieving ZTD with two types of models, that is, the discrete (VFM3) and empirical model (GPT3). Little or no accuracy comparison of these two models with different horizontal resolutions in GNSS community has been conducted over China. This study investigates the performance of the models in terms of calculation of ZHD, ZWD and ZTD, using the hourly values derived from the ERA5 and IGS data as references. The results show that the discrete model outperforms the empirical model and the increasing of horizontal resolution can improve the accuracy of the models. The performance of the empirical model in ZHD/ZWD calculation shows obvious influences of latitude changes and land‐sea differences, while the discrete model improves this phenomenon. The empirical distribution function and boxplot well indicate the accuracy differences of the two models in different seasons. We also found that the diurnal variation can only be detected in the ZHD calculation by the discrete model. The ZTD comparison gives the details of the two types of models at GNSS stations.

An investigation into real-time GPS/GLONASS single-frequency precise point positioning and its atmospheric mitigation strategies

One of the challenges in the innovative application of global navigation satellite systems (GNSS) lies in real-time single-frequency precise point positioning (RT-SFPPP). The well-known problems associated with SFPPP are its slow convergence and lower positioning accuracy due to the effects of various errors inherited by the GNSS positioning, including the atmospheric error. In order to mitigate the two above-mentioned problems, several scenarios for the reduction of the atmospheric delays in RT-SFPPP, including the ionospheric constraint and tropospheric constraint, were investigated in this research. The performance of the RT-SFPPP utilizing the above-mentioned methods was evaluated using one-week observations from multi-GNSS experiment stations in a simulated kinematic mode. Results showed that the convergence time of the RT-SFPPP was significantly shortened when the slant ionospheric delays derived from the real-time ionospheric products provided by Centre National d’Études Spatiales were applied as the pseudo-observations. Results also showed that the convergence time of GPS + GLONASS RT-SFPPP can be reduced by modeling the frequency-dependent part of the GLONASS receiver uncalibrated code delay as a quadratic polynomial function of GLONASS frequency number. The a priori zenith wet delays (ZWDs) derived from forecast Vienna Mapping Functions 3 (VMF3_FC) were compared to the reference ZWDs from globally distributed radiosonde stations. The mean root mean square error of the a priori ZWDs resulting from VMF3_FC at 381 radiosonde stations in 2020 was 1.53 cm compared to the radiosonde-based ZWDs. When the ZWDs derived from VMF3_FC were used as pseudo-observations in the position estimation system, the convergence time for the vertical positioning reaching a 0.3 m accuracy was considerably shortened.

Sparse representation of tropospheric grid data using compressed sensing

With the widespread application of global navigation satellite system, increasing amounts of gridded tropospheric data have been generated, increasing the difficulty of real-time transmission. We present that the global tropospheric grid data (GTGD) provided by Vienna mapping functions open access data have approximate sparse characteristics, and the compressed sensing (CS) method is used for sparse reconstruction for the first time. To reduce the memory and number of calculations required for the K-SVD (K-means and SVD) algorithm, the mini-batch K-SVD algorithm is proposed to speed up the calculation process. This article discusses several key problems of CS processing and application in GTGD, such as signal sparsity, reconstruction accuracy, iteration times, etc. We use mini-batch K-SVD algorithm train 1436 GTGD historical files from 2018 to establish a sparse representation model. To evaluate the accuracy of the new model, sparse reconstruction is performed on 1460 GTGD files from 2019. The experimental results show that the average root-mean-square (RMS) error, the BIAS, the maximum absolute error (MAAE), and the mean absolute error (MAE) of the compressed sensing are 2.16, 2.00E-04, 17.99, and 1.44 mm, respectively. The average RMS, BIAS, MAAE, and MAE of the spherical harmonic expansion (72-degree) are 27.39, − 9.71E-14, 273.93, and 16.67 mm. The results show that the CS approach yields a more accurate solution than spherical harmonic expansion. In summary, the established sparse representation model saves real-time transmission cost and enables high-precision sparse reconstruction and can achieve data encryption and compression simultaneously.

Global Assessment of the GNSS Single Point Positioning Biases Produced by the Residual Tropospheric Delay

The tropospheric delay is one of the main error sources that degrades the accuracy of Global Navigation Satellite Systems (GNSS) Single Point Positioning (SPP). Although an empirical model is usually applied for correction and thereby to improve the positioning accuracy, the residual tropospheric delay is still drowned in measurement noise, and cannot be further compensated by parameter estimation. How much this type of residual error would sway the SPP positioning solutions on a global scale are still unclear. In this paper, the biases on SPP solutions introduced by the residual tropospheric delay when using nine conventionally Zenith Tropospheric Delay (ZTD) models are analyzed and discussed, including Saastamoinen+norm/Global Pressure and Temperature (GPT)/GPT2/GPT2w/GPT3, University of New Brunswick (UNB)3/UNB3m, European Geostationary Navigation Overlay System (EGNOS) and Vienna Mapping Functions (VMF)3 models. The accuracies of the nine ZTD models, as well as the SPP biases caused by the residual ZTD (dZTD) after model correction are evaluated using International GNSS Service (IGS)-ZTD products from around 400 globally distributed monitoring stations. The seasonal, latitudinal, and altitudinal discrepancies are analyzed respectively. The results show that the SPP solution biases caused by the dZTD mainly occur on the vertical direction, nearly to decimeter level, and significant discrepancies are observed among different models at different geographical locations. This study provides references for the refinement and applications of the nine ZTD models for SPP users.

Evaluation of Zenith Tropospheric Delay Derived from Ray-Traced VMF3 Product over the West African Region Using GNSS Observations

The growing demand for Global Navigation Satellite System (GNSS) technology has necessitated the establishment of a vast and ever-growing network of International GNSS Service (IGS) tracking stations worldwide. The IGS provides highly accurate and highly reliable daily time-series Zenith Tropospheric Delay (ZTD) products using data from the member sites towards the use of GNSS for precise geodetic, climatological, and meteorological applications. However, if for reasons like poor internet connectivity, equipment failure, and power outages, the IGS station is inaccessible or malfunctioning, and gaps are created in the data archive resulting in degrading the quality of the ZTD and precipitable water vapour (PWV) estimation. To address this challenge as a means of providing an alternative data source to improve the continuous availability of ZTD data and as a backup data in the event that the IGS site data are missing or unavailable in West Africa, this paper compares the sitewise operational Vienna Mapping Functions 3 (VMF3) ZTD product with the IGS final ZTD product over five IGS stations in West Africa. Eight different statistical evaluation metrics, such as the mean bias (MB), mean absolute error (MAE), root mean squared error (RMSE), Pearson correlation coefficient (r), coefficient of determination (r2), refined index of agreement (IAr), Nash–Sutcliffe coefficient of efficiency (NSE), and the fraction of prediction within a factor of two (FAC2), are employed to determine the degree of agreement between the VMF3 and IGS tropospheric products. The results show that the VMF3-ZTD product performed excellently and matches very well with the IGS final ZTD product with an average MB, MAE, RMSE, r, r2, NSE, IAr, and FAC2 of 0.38 cm, 0.87 cm, 1.11 cm, 0.988, 0.976, 0.967, 0.992, and 1.00 (100%), respectively. This result is an indication that the VMF3-ZTD product is accurate enough to be used as an alternative source of ZTD data to augment the IGS final ZTD product for positioning and meteorological applications in West Africa.