Estimation and Mitigation of the Main Errors for Centimetre-level Compass RTK Solutions over Medium-Long Baselines

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.

The Global Navigation Satellite System (GNSS) precise point positioning (PPP) technology is currently used to process GNSS water vapor observations in real time or near real time. Further developments are required to improve the accuracy and real-time performance of processing tropospheric delays from which the water vapor observations are extracted. In real-time BDS/GPS precise clock correction estimation with square-root information filtering and PPP solution with Kalman filtering, a fixed variance is often assigned for the process noise of the troposphere dynamics model. However, this fixed value may deviate from reality as the weather conditions change, especially for extreme weather. In this paper, a new adaptive Kalman filter is proposed for tropospheric delay processing, in which the variance of the process noise for the zenith tropospheric delay (ZTD) dynamics model is tuned in real time using the least-squares variance component estimation technique. MGEX/IGS data of 15 consecutive days were processed in a stepwise manner. Namely: Real-time BDS/GPS precise clock corrections were estimated firstly, followed by comparison among ZTD solutions of four schemes based on these real-time clocks of first step and GFZ multi-GNSS precise clock (GBM) final clocks: (1) real-time solution of ZTD (G(GPS), GC(GPS and BDS)) without VCE; (2) real-time solution of ZTD (G, GC) with VCE; (3) final solution of ZTD (G, GC, GR (GPS and GLONASS), GE (GPS and GALILEO), GRC (GPS, GLONASS and BDS), GREC (GPS, GLONASS, GALILEO and BDS) without VCE; and (4) final solution of ZTD (G, GC, GR, GE, GRC, GREC) with VCE. The performance of the ZTD and positioning solution was analyzed. Results showed that the accuracy of estimated real-time satellite clock correction was 0.27, 1.31, 0.29, and 0.21 ns for GPS, BDS/GEO, BDS/IGSO, and BDS/MEO, respectively. For the ZTD solutions, the results of schemes 1 and 2 were 12.8 mm (GC) and 10.1 mm (GC) in terms of mean root-mean-square (RMS) values and 2.1 mm (GC) and 1.6 mm (GC) in terms of minimum RMS values, respectively, thereby showing an improvement for scheme 2 of 1.8–81.4% over scheme 1 with average increasing rates of 20.7% (GC) and 20.2% (G). The results for schemes 3 and 4 were 7.6 mm (G) and 6.3 mm (GRC) in terms of mean RMS values and 2.1 mm (G) and 1.9 mm (GRCE) in terms of minimum RMS values, respectively, thereby showing an improvement in scheme 4 over scheme 3 by 1.3–85.6% with average increasing rates of 22.1% (GRCE), 21.9% (GRC), 18.4% (GR), 15.9% (GC), 15.2% (GE), and 12.1% (G). Similar results can be observed for the positioning solution, especially in the height component. These findings clearly show the advantage of the proposed method, which is consistent with the theoretical analysis. Notably, the advantage of the adaptive VCE becomes significant with the inclusion of additional satellite systems.

The electromagnetic signals of the Global Navigation Satellite Systems (GNSS) satellites suffer delays while propagating through the troposphere. The tropospheric delay is a significant systematic error of GNSS positioning. For safety-of-life applications of positioning many systematic error effects are either mitigated or eliminated in the positioning solution. Space based augmentation systems provide corrections for the orbital and satellite clock error, the ionospheric effects, etc. Moreover advanced GNSS provide dual frequency code observations for civilian users to eliminate the ionospheric delays caused by the electron content of the upper atmosphere. Nevertheless tropospheric delays are still taken into account using empirical models.For safety-of-life applications besides the accuracy of the positioning, the integrity of the positioning service is an important factor, too. The integrity information includes the maximal positioning error at an extremely rare probability level, called protection level to ensure highly reliable position solution in the aviation. The Radio Technical Commission for Aeronautics Minimum Operational Performance Standard (RTCA MOPS) recommends 0.12 m as the maximum zenith tropospheric error in terms of standard deviation. Previous studies show that this recommendation seems to be too conservative leading to a lower service availability. Therefore a more realistic integrity model has to be derived for the estimation of maximal residual tropospheric delay error.In the recent years many advanced empirical tropospheric delay models have been formulated compared to the one recommended by the RTCA. Recently new integrity models have been derived for estimating the maximum residual tropospheric delay error using numerical weather models under real extreme weather.The aim of this paper is to study the reliability of these models conditions. In order to achieve this, high-resolution numerical weather models were ray-traced using an improved ray-tracing algorithm to evaluate the slant and zenith tropospheric delays with the geographical resolution of 0.1° × 0.1°.

Existing apriori tropospheric models are not sufficiently accurate to remove tropospheric delay from GPS observations. Remaining effects of residual tropospheric delay need to be estimated to ensure high accuracy and reliability of GPS positioning. Other researchers have shown that implementations of network- based positioning techniques can adequately model the residual tropospheric delay as well as ionospheric delay and orbit biases. However, the effectiveness in removing residual tropospheric delay is highly dependent on the degree to which the wet component from the troposphere can be estimated or mitigated, an effect which shows strong variation with time and space. The aim of this paper is to illustrate the performance of an existing apriori tropospheric model and to discuss some issues concerning the estimation of the (total) tropospheric delay in the equatorial area. Finally, the network approach is applied to mitigate the effect of residual tropospheric delay. Some preliminary results from test experiments using GPS network data from an equatorial region, a location with the highest effect of tropospheric delay, are presented.

Tropospheric delay acts as a systematic error source in the Global Navigation Satellite Systems (GNSS) positioning. Empirical models UNB3, UNB3m, UNB4 and EGNOS have been developed for use in Satellite-Based Augmentation Systems (SBAS). Model performance, however, is limited due to the low spatial resolution of the look-up tables for meteorological parameters. A new design has been established in this study for improving performance of the tropospheric delay model by more effectively eliminating the error produced by tropospheric delay. The spatiotemporal characteristics of the Zenith Tropospheric Delay (ZTD) were analyzed with findings that ZTD exhibits different annual variations at different locations and decreases exponentially with height increasing. Spherical harmonics are utilized based on the findings to fit the annual mean and amplitude of the ZTD on a global scale and the exponential function is utilized for height corrections, yielding the ZTrop model. On a global scale, the ZTrop features an average deviation of -1.0 cm and Root Mean Square (RMS) of 4.7 cm compared with the International GNSS Service (IGS) ZTD products, an average deviation of 0.0 cm and RMS of 4.5 cm compared with the Global Geodetic Observing System (GGOS) ZTD data, and an average deviation of -1.3 cm and RMS of 5.2 cm compared with the ZTD data from the Constellation Observing System of Meteorology, Ionosphere, and Climate (COSMIC). The RMS of the ZTrop model is 14.5% smaller than that of UNB3, 6.0% smaller than that of UNB3m, 16% smaller than that of UNB4, 14.5% smaller than that of EGNOS and equivalent to the sophisticated GPT2+Saas model in comparison with the IGS ZTD products. The ZTrop, UNB3m and GPT2+Saas models are finally evaluated in GPS-based Precise Point Positioning (PPP), as the models act to aid in obtaining PPP position error less than 1.5 cm in north and east components and relative large error (>5 cm) in up component with respect to the random walk approach.

Tropospheric delay accounts for one source of error in global navigation satellite systems (GNSS). To better characterize the tropospheric delays in the temporal and spatial domain and facilitate the safety-critical use of GNSS across China, a method is proposed to generate a grid-based tropospheric product (GTP) using the GNSS network with an empirical tropospheric model, known as IGGtrop. The prototype system generates the GTPs in post-processing and real-time modes and is based on the undifferenced and uncombined precise point positioning (UU-PPP) technique. GTPs are constructed for a grid form ( $$2.0{^{\circ }}\times 2.5{^{\circ }}$$ latitude–longitude) over China with a time resolution of 5 min. The real-time GTP messages are encoded in a self-defined RTCM3 format and broadcast to users using NTRIP (networked transport of RTCM via internet protocol), which enables efficient and safe transmission to real-time users. Our approach for GTP generation consists of three sequential steps. In the first step, GNSS-derived zenith tropospheric delays (ZTDs) for a network of GNSS stations are estimated using UU-PPP. In the second step, vertical adjustments for the GNSS-derived ZTDs are applied to address the height differences between the GNSS stations and grid points. The ZTD height corrections are provided by the IGGtrop model. Finally, an inverse distance weighting method is used to interpolate the GNSS-derived ZTDs from the surrounding GNSS stations to the location of the grid point. A total of 210 global positioning system (GPS) stations from the crustal movement observation network of China are used to generate the GTPs in both post-processing and real-time modes. The accuracies of the GTPs are assessed against with ERA-Interim-derived ZTDs and the GPS-derived ZTDs at 12 test GPS stations, respectively. The results show that the post-processing and real-time GTPs can provide the ZTDs with accuracies of 1.4 and 1.8 cm, respectively. We also apply the GTPs in real-time kinematic GPS PPP, and the results show that the convergence time of the PPP solutions is shortened. These results confirm that the GTPs can act as an efficient information source to augment GNSS positioning over China.

Based on the precise point positioning, the impact of IGU predicted orbit and clock error on the accuracy of real-time obtaining zenith tropospheric delay (ZTD) of a single receiver was compared and analyzed. Theoretic study and experiment analysis shown that the IGU predicted orbit can satisfy the real-time ZTD accuracy. But, as to the IGU predicted clock errors, its real-time obtained ZTD accuracy (~ 2cm) can't meet the need for the accuracy of GPS remote sensing of water vapor. Therefore, we need a new method to real-time estimate the precise predicted satellite clock errors.

Regional modeling of tropospheric delay considering vertically and horizontally separation of station for regional augmented PPP

GGOS tropospheric delay forecast product performance evaluation and its application in real-time PPP

Tropospheric delay is one of the main sources of measurement error in global navigation satellite systems. It is usually compensated by using an empirical correction model. In this paper, temporal and spatial variations of the global zenith tropospheric delay (ZTD) are further analyzed by ZTD time series from global International GNSS Service stations and annual ZTDs derived from global National Centers for Environmental Prediction reanalysis data, respectively. A new ZTD correction model, named IGGtrop, is developed based on the characteristics of ZTD. Experimental results show that this new 3D-grid-based model that accommodates longitudinal as well as latitudinal variations of ZTD performs better than latitude-only based models (such as UNB3, EGNOS, and UNB3m). The global average bias and RMS for IGGtrop are about −0.8 cm and 4.0 cm, respectively. Bias values for UNB3, EGNOS, and UNB3m are 2.0, 2.0, and 0.7 cm, respectively, and respective RMS values 5.4, 5.4, and 5.0 cm. IGGtrop shows much more consistent prediction errors for different areas than EGNOS and UNB3m. In China, the performance of IGGtrop (bias values from −2.0 to 0.4 cm and RMS from 2.1 to 6.4 cm) is clearly superior to those of EGNOS and UNB3m. It is also demonstrated that IGGtrop biases vary little with height, and its RMS values tend to decrease with increasing height. In addition, IGGtrop generally estimates ZTD with greater accuracy than EGNOS and UNB3m in the Southern Hemisphere.

In this study, three years of the zenith tropospheric delay (ZTD) data observed from 46 International GNSS system (IGS) sites distributed in Asian area used to assess the effectiveness and accuracy of ZTD calculated from EGNOS model, and the application of the EGNOS model are also analyzed in Asian area. Relative to IGS observed ZTD, the bias and root mean square (RMS) for ZTD calculated from EGNOS model presents an obvious variation in temporal and spatial. These results provide a reference for the study of the tropospheric delay correction model, the real-time GNSS navigation and positioning.

Tropospheric delay is a primary error source in earth observations and a variety of radio navigation technologies. However, the main problem still present is that not all strategic points around the world will have a GPS receiver. To overcome the shortcoming, a Fusion model was proposed to compensate for errors in prediction of zenith tropospheric delay (ZTD), by incorporating back propagation neural network (BPNN). The input parameters include the surface meteorological data (altitude, atmospheric pressure, absolute temperature and water vapour partial pressure) and Hopfield model predicted ZTD, the output is the difference between the Hopfield predicted and high altitude balloon derived ZTD (also called true ZTD). The datasets covered ∼250 meteorological observation stations within 8∶00 and 20∶00 for May 1st, 10th, 30th, and July 1st, August 1st and September 1st, 2010. Data from 16 uniformly distributed stations were used for training at 8∶00 on May 1st, and the remaining for validation. Modelling results from the Hopfield and current model were compared by reference to the true ZTD. It shows that the current model generates an average RMSE value of 0.0040 m, compared with 0.3445 m for the Hopfield model. In overall, the current model can improve ZTD prediction accuracy by more than 90%. In addition, ZTD predictions from the current model were compared to those obtained directly from GPS data, indicating that our model provides a good alternative for ZTD prediction when the GPS receiver at a specific location is absent.

Satellite orbit error and clock bias are the keys to precise point positioning (PPP). The traditional PPP algorithm requires precise satellite products based on worldwide permanent reference stations. Such an algorithm requires considerable work and hardly achieves real-time performance. However, real-time positioning service will be the dominant mode in the future. IGS is providing such an operational service (RTS) and there are also commercial systems like Trimble RTX in operation. On the basis of the regional Continuous Operational Reference System (CORS), a real-time PPP algorithm is proposed to apply the coupling estimation of clock bias and orbit error. The projection of orbit error onto the satellite-receiver range has the same effects on positioning accuracy with clock bias. Therefore, in satellite clock estimation, part of the orbit error can be absorbed by the clock bias and the effects of residual orbit error on positioning accuracy can be weakened by the evenly distributed satellite geometry. In consideration of the simple structure of pseudorange equations and the high precision of carrier-phase equations, the clock bias estimation method coupled with orbit error is also improved. Rovers obtain PPP results by receiving broadcast ephemeris and real-time satellite clock bias coupled with orbit error. By applying the proposed algorithm, the precise orbit products provided by GNSS analysis centers are rendered no longer necessary. On the basis of previous theoretical analysis, a real-time PPP system was developed. Some experiments were then designed to verify this algorithm. Experimental results show that the newly proposed approach performs better than the traditional PPP based on International GNSS Service (IGS) real-time products. The positioning accuracies of the rovers inside and outside the network are improved by 38.8% and 36.1%, respectively. The PPP convergence speeds are improved by up to 61.4% and 65.9%. The new approach can change the traditional PPP mode because of its advantages of independence, high positioning precision, and real-time performance. It could be an alternative solution for regional positioning service before global PPP service comes into operation.

In order to solve the problem of low precision of tropospheric delay interpolation under the long-distance sparse reference station, a method of tropospheric delay interpolation estimation based on support vector machine (SVM) theory was proposed. Firstly, the troposphere was assumed to be an infinitely thin single-layer membrane, and with the known information of reference stations, the puncture point coordinates and zenith tropospheric delay of each reference station on each visible satellite in the single-layer membrane were obtained. Then, the puncture point coordinates and zenith tropospheric delay were taken as training samples to optimize the appropriate core parameters of SVM and establish the SVM model of tropospheric delay. Finally, two sets of network RTK data with different lengths were selected to compare and analyze the effect of the interpolated tropospheric delay. The results showed that under the long-distance reference station, with the SVM tropospheric delay model established in this paper, the tropospheric delay accuracy of the interpolation estimation is better than 2cm, and the estimation error is generally stable, and the interpolation effect of satellite in different systems is basically the same.

Real-time retrieval of high-precision ZTD maps using GNSS observation