Slant Ionospheric Delays Research Articles

Regional Real-Time between-Satellite Single-Differenced Ionospheric Model Establishing by Multi-GNSS Single-Frequency Observations: Performance Evaluation and PPP Augmentation

The multi-global navigation satellite system (GNSS) undifferenced and uncombined precise point positioning (UU-PPP), as a high-precision ionospheric observables extraction technology superior to the traditional carrier-to-code leveling (CCL) method, has received increasing attention. In previous research, only dual-frequency (DF) or multi-frequency (MF) observations are used to extract slant ionospheric delay with the UU-PPP. To reduce the cost of ionospheric modeling, the feasibility of extracting ionospheric observables from the multi-GNSS single-frequency (SF) UU-PPP was investigated in this study. Meanwhile, the between-satellite single-differenced (SD) method was applied to remove the effects of the receiver differential code bias (DCB) with short-term time-varying characteristics in regional ionospheric modeling. In the assessment of the regional real-time (RT) between-satellite SD ionospheric model, the internal accord accuracy of the SD ionospheric delay can be better than 0.5 TECU, and its external accord accuracy within 1.0 TECU is significantly superior to three global RT ionospheric models. With the introduction of the proposed SD ionospheric model into the multi-GNSS kinematic RT SF-PPP, the initialization speed of vertical positioning errors can be improved by 21.3% in comparison with the GRAPHIC (GRoup And PHase Ionospheric Correction) SF-PPP model. After reinitialization, both horizontal and vertical positioning errors of the SD ionospheric constrained (IC) SF-PPP can be maintained within 0.2 m. This proves that the proposed SDIC SF-PPP model can enhance the continuity and stability of kinematic positioning in the case of some GNSS signals missing or blocked. Compared with the GRAPHIC SF-PPP, the horizontal positioning accuracy of the SDIC SF-PPP in kinematic mode can be improved by 37.9%, but its vertical positioning accuracy may be decreased. Overall, the 3D positioning accuracy of the SD ionospheric-constrained RT SF-PPP can be better than 0.3 m.

Uncertainties of Interpolating Satellite-Specific Slant Ionospheric Delays and Impacts on PPP-RTK

Uncertainties of Interpolating Satellite-Specific Slant Ionospheric Delays and Impacts on PPP-RTK

Improving GNSS PPP-RTK through global forecast system zenith wet delay augmentation

The precise point positioning real-time kinematic (PPP-RTK) is a high-precision global navigation satellite system (GNSS) positioning technique that combines the advantages of wide-area coverage in precise point positioning (PPP) and of rapid convergence in real-time kinematic (RTK). However, the PPP-RTK convergence is still limited by the precision of slant ionospheric delays and tropospheric zenith wet delay (ZWD), which affects the PPP-RTK network parameters estimation and user positioning performance. The present study aims to construct a PPP-RTK model augmented with a priori ZWD values derived from the global forecast system (GFS) product (a global numerical weather prediction (NWP) model) to improve the PPP-RTK performance. This study gives a priori ZWD values and conversion based on the GFS products, and the full-rank GFS-augmented undifferenced and uncombined (UDUC) PPP-RTK network model is derived. To verify the performance of GFS-augmented UDUC PPP-RTK, a comprehensive evaluation using 10-day GNSS observation data from three different GNSS station networks in the United States (US), Australia, and Europe is conducted. The results show that with the GFS ZWD a priori information, PPP-RTK performance significantly improves at the initial filtering stage, but this advantage gradually decays over time. Based on 10-day positioning results for all user stations, the GFS ZWD-augmented PPP-RTK approach reduces the average convergence time by 46% from 10.0 to 5.4 min, the three-dimensional root-mean-square (3D-RMS) error by 5.7% from 3.5 to 3.3 cm, and the time to first fix (TTFF) value by 35.8% from 6.7 to 4.3 min, all when compared to the traditional PPP-RTK without GFS ZWD constraints.

Quasi-4-dimension ionospheric modeling and its application in PPP

Ionospheric delay modeling is not only important for Global Navigation Satellite System (GNSS) based space weather study and monitoring, but also an efficient tool to speed up the convergence time of Precise Point Positioning (PPP). In this study, a novel model, denoted as Quasi-4-Dimension Ionospheric Modeling (Q4DIM) is proposed for wide-area high precision ionospheric delay correction. In Q4DIM, the Line Of Sight (LOS) ionospheric delays from a GNSS station network are divided into different clusters according to not only the location of latitude and longitude, but also satellite elevation and azimuth. Both Global Ionosphere Map (GIM) and Slant Ionospheric Delay (SID) models that are traditionally used for wide-area and regional ionospheric delay modeling, respectively, can be regarded as the special cases of Q4DIM by defining proper grids in latitude, longitude, elevation, and azimuth. Thus, Q4DIM presents a resilient model that is capable for both wide-area coverage and high precision. Four different sets of clusters are defined to illustrate the properties of Q4DIM based on 200 EUREF Permanent Network (EPN) stations. The results indicate that Q4DIM is compatible with the GIM products. Moreover, it is proved that by inducting the elevation and azimuth angle dependent residuals, the precision of the 2-dimensional GIM-like model, i.e., Q4DIM 2-Dimensional (Q4DIM-2D), is improved from around 1.5 Total Electron Content Units (TECU) to better than 0.5 TECU. In addition, treating Q4DIM as a 4-dimensional matrix in latitude, longitude, elevation, and azimuth, whose sparsity is less than 5%, can result in its feasibility in a bandwidth-sensitive applications, e.g., satellite-based Precising Point Positioning Real-Time Kinematic (PPP-RTK) service. Finally, the advantages of Q4DIM in PPP over the 2-dimensional models are demonstrated with the one month's data from 30 EPN stations in both high solar activity year 2014 and low solar activity year 2020.

PPP-RTK considering the ionosphere uncertainty with cross-validation

With the high-precision products of satellite orbit and clock, uncalibrated phase delay, and the atmosphere delay corrections, Precise Point Positioning (PPP) based on a Real-Time Kinematic (RTK) network is possible to rapidly achieve centimeter-level positioning accuracy. In the ionosphere-weighted PPP–RTK model, not only the a priori value of ionosphere but also its precision affect the convergence and accuracy of positioning. This study proposes a method to determine the precision of the interpolated slant ionospheric delay by cross-validation. The new method takes the high temporal and spatial variation into consideration. A distance-dependent function is built to represent the stochastic model of the slant ionospheric delay derived from each reference station, and an error model is built for each reference station on a five-minute piecewise basis. The user can interpolate ionospheric delay correction and the corresponding precision with an error function related to the distance and time of each reference station. With the European Reference Frame (EUREF) Permanent GNSS (Global Navigation Satellite Systems) network (EPN), and SONEL (Système d'Observation du Niveau des Eaux Littorales) GNSS stations covering most of Europe, the effectiveness of our wide-area ionosphere constraint method for PPP-RTK is validated, compared with the method with a fixed ionosphere precision threshold. It is shown that although the Root Mean Square (RMS) of the interpolated ionosphere error is within 5 cm in most of the areas, it exceeds 10 cm for some areas with sparse reference stations during some periods of time. The convergence time of the 90th percentile is 4.0 and 20.5 min for horizontal and vertical directions using Global Positioning System (GPS) kinematic solution, respectively, with the proposed method. This convergence is faster than those with the fixed ionosphere precision values of 1, 8, and 30 cm. The improvement with respect to the latter three solutions ranges from 10 to 60%. After integrating the Galileo navigation satellite system (Galileo), the convergence time of the 90th percentile for combined kinematic solutions is 2.0 and 9.0 min, with an improvement of 50.0% and 56.1% for horizontal and vertical directions, respectively, compared with the GPS-only solution. The average convergence time of GPS PPP-RTK for horizontal and vertical directions are 2.0 and 5.0 min, and those of GPS + Galileo PPP-RTK are 1.4 and 3.0 min, respectively.

Modeling and assessment of atmospheric delay for GPS/Galileo/BDS PPP-RTK in regional-scale

Combining advantages of precise point positioning (PPP) and real-time kinematic (RTK), PPP-RTK has become a promising technology in the mass-market positioning. When performing PPP-RTK, atmospheric corrections enable the user fast ambiguity resolution as well as centimeter accuracy. However, the legacy method of obtaining atmospheric corrections, i.e., interpolating the atmospheric delay based on nearby reference stations, is not regular in space and needs the reference station coordinate information, thereby limiting real-time professional applications. In this contribution, a low-degree polynomial with regular grid residuals is investigated. Forming between-satellite single-differenced slant ionospheric delay, we theoretically analyze the reference network size. Afterwards, the zenith tropospheric delay and slant ionospheric delay accuracy using zero-, first- and second-degree polynomials in calm atmospheric periods are about (5.6, 4.7 and 4.7) mm and (1.9, 1.3 and 1.3) cm. It is therefore recommended to adopt the first-degree polynomial model in terms of the accuracy and transmission load.

Wide-Area Grid-Based Slant Ionospheric Delay Corrections for Precise Point Positioning

Introducing ionospheric information into a precise point positioning (PPP) solution enables faster ambiguity resolution and significantly improves positioning accuracy. To compute such corrections over wide areas, sparse networks with potentially irregular station distributions are often used. This aspect brings a new level of complexity as ionospheric corrections should be weighted appropriately in the PPP filter. This paper presents a possible implementation of grid-based wide-area slant ionospheric delay corrections, with a focus on the reported uncertainties. A balance is obtained between obtaining corrections with formal errors small enough to enable fast convergence, while large enough to overbound most errors. Based on least-squares collocation, the method uses satellite-specific variograms based on the 99th percentile values in each distance bin. Tested in southern Canada over a 53-week period in 2020, ionospheric grids allowed dual-frequency receivers to obtain around 5 cm accuracy in each horizontal component within 5 min of static data collection. For single-frequency solutions using data from geodetic receivers, positioning errors were reduced by over 60% for both static and kinematic processing.

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.

Real-time GNSS precise point positioning for low-cost smart devices

Global Navigation Satellite System raw measurements from Android smart devices make accurate positioning possible with advanced techniques, e.g., precise point positioning (PPP). To achieve the sub-meter-level positioning accuracy with low-cost smart devices, the PPP algorithm developed for geodetic receivers is adapted and an approach named Smart-PPP is proposed in this contribution. In Smart-PPP, the uncombined PPP model is applied for the unified processing of single- and dual-frequency measurements from tracked satellites. The receiver clock terms are parameterized independently for the code and carrier phase measurements of each tracking signal for handling the inconsistency between the code and carrier phases measured by smart devices. The ionospheric pseudo-observations are adopted to provide absolute constraints on the estimation of slant ionospheric delays and to strengthen the uncombined PPP model. A modified stochastic model is employed to weight code and carrier phase measurements by considering the high correlation between the measurement errors and the signal strengths for smart devices. Additionally, an application software based on the Android platform is developed for realizing Smart-PPP in smart devices. The positioning performance of Smart-PPP is validated in both static and kinematic cases. Results show that the positioning errors of Smart-PPP solutions can converge to below 1.0 m within a few minutes in static mode and the converged solutions can achieve an accuracy of about 0.2 m of root mean square (RMS) both for the east, north and up components. For the kinematic test, the RMS values of Smart-PPP positioning errors are 0.65, 0.54 and 1.09 m in the east, north and up components, respectively. Static and kinematic tests both show that the Smart-PPP solutions outperform the internal results provided by the experimental smart devices.

Ionospheric VTEC and satellite DCB estimated from single-frequency BDS observations with multi-layer mapping function

The ionospheric delays and satellite differential code biases (DCBs) act as the significant error sources in the global navigation satellite system (GNSS) positioning, navigation and timing (PNT) services, and are still challenging to estimate correctly. In this study, the ionospheric vertical total electron content (VTEC) and satellite DCBs are estimated by a refining single-frequency precise point positioning (SFPPP) method based on the multi-layer ionosphere mapping function (MF), as well as the dual-frequency methods, including the carrier-to-code leveling (CCL) and dual-frequency PPP (DFPPP). The solutions isolate the ionospheric VTEC values from the slant ionospheric delay with the generalized trigonometric series function (GTSF) and precisely estimate the satellite DCB with a zero-mean condition. The SFPPP-derived VTEC estimates are validated and evaluated by comparing with the International GNSS Service (IGS) products and using ionosphere-corrected (IC) SFPPP in both static and kinematic scenarios. Using the 74 experimental stations collected from the multi-GNSS experiment (MGEX) network from January to March 2020, the results show that the VTEC estimation precision by applying the multi-layer MF is improved for the SFPPP approach. The positioning performances of the static and kinematic BDS IC SFPPP with the ionospheric correction derived from the multi-layer MF SFPPP are better when compared to the single-layer MF. The estimated BDS DCB with the SFPPP is stable and of high accuracy. The SFPPP approach with multi-layer MF is demonstrated as a promising and reliable method to retrieve the VTEC and satellite DCB with the low-cost property for the GNSS users.

Time delay bias between the second and third generation of BeiDou Navigation Satellite System and its effect on precise point positioning

To realize the deep integration of BeiDou-2 regional navigation satellite system (BDS-2) and BeiDou-3 global navigation satellite system (BDS-3), the time delay bias between BDS-2 and BDS-3 at the receiver side is analyzed for the joint BDS-2/BDS-3 precise point positioning (PPP). Twenty Multi-GNSS (Global Navigation Satellite System) Experiment tracking stations with different receiver types were selected to investigate the time-varying characteristics of time delay bias. We also make a comprehensive analysis about the influence of time delay bias on the ionosphere-free PPP and uncombined PPP. The results show that the daily mean values of time delay bias for consecutive days vary within 1 ns without apparent day-boundary discontinuities. In addition, the time delay bias for the receivers of the same brand can differ by several nanoseconds, especially for the TRIMBLE receivers. Without considering the time delay bias between BDS-2 and BDS-3, the pseudorange residuals, slant ionospheric delays, float ambiguities are subjected to change with the magnitude of the time delay bias, while the phase residuals and positioning accuracy are slightly affected. Moreover, the deviation of float ambiguity in ionosphere-free PPP caused by neglecting the time delay bias is larger than that of uncombined PPP.

Mitigation of receiver biases in ionospheric observables from PPP with ambiguity resolution

PPP (Precise Point Positioning) is a GNSS (Global Navigation Satellite Systems) positioning method that requires SSR (State Space Representation) corrections in order to provide solutions with an accuracy of centimetric level. The so-called RT-PPP (Real-time PPP) is possible thanks to real-time precise SSR products, for orbits and clocks, provided by IGS (International GNSS Service) and its associate analysis centers such as CNES (Centre National d'Etudes Spatiales). CNES SSR products also enable RT-PPP with integer ambiguity resolution. In GNSS related literature, PPP with ambiguity resolution (PPP-AR) in real-time is often referred as PPP-RTK (PPP – Real Time Kinematic). PPP-WIZARD (PPP - With Integer and Zero-difference Ambiguity Resolution Demonstrator) is a software that is made available by CNES. This software is capable of performing PPP-RTK. It estimates slant ionospheric delays and other GNSS positioning parameters. Since ionospheric effects are spatially correlated by GNSS data from active networks, it is possible to model and provide ionospheric delays for any position in the network coverage area. The prior knowledge ionospheric delays can reduce positioning convergence for PPP-RTK users. Real-time ionospheric models could benefit from highly precise ionospheric delays estimated in PPP-AR. In this study, we demonstrate that ionospheric delays obtained throughout PPP-AR estimation are actu ally ionospheric observables. Ionospheric observables are biased by an order of few meters caused by the receiver hardware biases. These biases prohibit the use of PPP-WIZARD ionospheric delays to produce ionospheric models. Receiver biases correction is essential to provide ionospheric delays while using PPP-AR based ionospheric observables. In this contribution, a method was implemented to estimate and mitigate receiver hardware biases influence on slant ionospheric observables from PPP-AR. In order to assess the proposed approach, PPP-AR data from 12 GNSS stations were processed over a two-month period (March and April 2018). A comparison between IGS ionospheric products and PPP-AR based ionospheric observables corrected for receiver biases, resulted in a mean of differences of −39 cm and 51 cm standard deviation. The results are consistent with the accuracy of the IGS ionospheric products, 2–8 TECU, considering that 1 TECU is ~16 cm in L1. In another analysis, a comparison of ionospheric delays from 5 pairs of short baselines GNSS stations found an agreement of 0.001 m in mean differences with 22 cm standard deviation after receiver biases were corrected. Therefore, the proposed solution is promising and could produce high quality (1–2 TECU) slant ionospheric delays. This product can be used in a large variety of modeling approaches, since ionospheric delays after correction are unbiased. These results indicate that the proposed strategy is promising, and could benefit applications that require accuracy of 1–2 TECU (~16–32 cm in L1).

On the application of the raw-observation-based PPP to global ionosphere VTEC modeling: an advantage demonstration in the multi-frequency and multi-GNSS context

The ionospheric delay accounts for one of the major errors that the Global Navigation Satellite Systems (GNSS) suffer from. Hence, the global ionosphere Vertical Total Electron Content (VTEC) map has been an important atmospheric product within the International GNSS Service (IGS) since its early establishment. In this contribution, an enhanced method has been proposed for the modeling of the global VTECs, in which the enhancements include two aspects. Firstly, to cope with the rapid development of the newly established Galileo and BeiDou constellations in recent years, we extend the current dual-system (GPS/GLONASS) solution to a quad-system (GPS/GLONASS/Galileo/BeiDou) solution. More importantly, instead of using dual-frequency observations based on the Carrier-to-Code Leveling method, all available triple-frequency signals are utilized with a general raw-observation-based multi-frequency Precise Point Positioning model, which can process dual-, triple- or even arbitrary-frequency observations compatibly and flexibly. Benefiting from this, quad-system slant ionospheric delays can be retrieved based on multi-frequency observations in a more flexible, accurate and reliable way, which are finally used to establish global VTEC models with the spherical harmonic function. In this process, multi-GNSS Differential Code Biases (DCBs) are also estimated as by-products. More than 400 globally distributed stations from the IGS and the Multi-GNSS Experiment (MGEX) networks have been processed in both 2014 (with high solar activity) and 2018 (with low solar activity). Global VTECs have been compared with the IGS final products, and the over-ocean VTECs are validated with the results from the JASON altimeter. The mean RMS values of the VTEC differences are 1.84 (2014) and 1.23 (2018) TECUs with respect to the IGS final products. The standard deviations of the VTEC differences with respect to the JASON results are 4.71 (2014) and 2.82 (2018) TECUs, outperforming all the other products generated with the spherical harmonic function. Additionally, multi-GNSS satellite DCBs have also been validated with the existing products from the Center for Orbit Determination in Europe and MGEX. All the results prove that the proposed method can be used as an effective and accurate approach for global VTEC modeling and DCB estimation, especially in the future multi-frequency and multi-GNSS context.

Modeling of multi-sensor tightly aided BDS triple-frequency precise point positioning and initial assessments

The BeiDou global navigation satellite system (BDS), which is the first satellite navigation system that provides triple-frequency signals on B1, B2, and B3 for civil applications, has been applied widely around the Asian-Pacifica region. The current BDS precise point positioning (PPP) approaches are mainly based on the B1&B2 dual-frequency observations. To make full use of BDS’ triple-frequency observations, the motion sensors measurements, and the platform motion information, this paper proposes an inertial sensor, odometer, and heading measurement tightly aided BDS triple-frequency PPP model. In this model, inertial sensor biases, odometer scale factor, residuals of slant ionospheric delays, and inter-frequency code biases are estimated simultaneously in a unique extended Kalman filter. The Rauch-Tung-Striebel (RTS) smoother is further adopted to reduce the solution's noises and enhance the stability and relative measuring accuracy. To evaluate the capability of this method, a set of triple-frequency BDS raw observations, inertial measurements, odometer data, and heading measurements collected by a customized hardware system on Lanzhou-Urumqi high speed railway track in China, are processed and analyzed. Results illustrated that both positioning accuracy and cycle slip detection capability are upgraded significantly by applying B3 frequency observations in BDS PPP. About 13–55% position accuracy enhancements from B3 observations, inertial sensors, and RTS smoother, and over 70% heading improvements from the aids of heading measurements can be obtained. Moreover, such multi-sensor tight integration system can directly provide millimeter-level positioning accuracy in term of repeatability and provide sub-millimeter-level accuracy indirectly by transforming attitude solutions into distance solutions. Such accuracy is much higher than the state-of-art GNSS and such method presents potential capability in 3D geometry measuring.

Computationally efficient dual-frequency uncombined precise orbit determination based on IGS clock datum

Using uncombined (UC) or undifferenced original observations has become a research topic in the global navigation satellite system (GNSS) community. This far, precise orbit determination (POD) with ionospheric-free (IF) observation combination has been favored. Using the uncombined strategy, i.e., using the raw observations to perform POD, the slant ionospheric delay is regarded as an unknown epoch parameter. We assume that the slant ionospheric delay depends on the frequency having the same magnitude for pseudoranges and carrier phases but opposite sign. Since there are, at each station and each epoch, as many ionospheric delay parameters as the observed satellites, a large number of ionospheric parameters need to be estimated for uncombined POD. The usual approch is that, when the observations at an epoch are accumulated, the epoch parameters consisting of clock and ionospheric parameters are eliminated simultaneously, which results in low computational efficiency. We develop a high-efficiency method by eliminating ionospheric parameters one by one, where each of them is eliminated when the observations of a station and a satellite are accumulated and no matrix inversion is required. In this uncombined POD algorithm, the IF-combined clock datum adopted by the international GNSS service (IGS) clock products is used to keep compatibility with existing IGS products by using the re-parameterized observation equations. We deduce the ambiguity resolution strategy for uncombined POD. In the experimental part, the modified parameter elimination method is first validated. With GPS observations from 58 global IGS stations, the first iteration processing time of parameter estimation for IF POD, UC POD and modified UC POD are 3.6, 62.5 and 9.2 min, respectively. The computational efficiency of modified UC POD strategy is greatly improved compared to UC POD. Then, the IF POD and the improved UC POD strategies with ambiguity float and fixed solutions are analyzed by comparing orbits, clocks, station coordinates and tropospheric delays with respect to IGS products. Results show that the accuracy of UC and IF solutions is comparable. Afterward, the repeatability of station coordinates is analyzed. With the solutions of 13 POD arcs, the repeatability over 13 days is consistent for UC and IF strategy. In addition, the ambiguity fixing rate, data usage rate, as well as residuals of code and phase observations, is analyzed. It is concluded that the slight difference between UC and IF strategy may be related to the data usage rate.

Corrections of BDS Code Multipath Error in Geostationary Orbit Satellite and Their Application in Precise Data Processing.

Multipath error is a main error source in Global Navigation Satellite System (GNSS) data processing, which cannot be removed by a differential technique because of the strong relationship with the environment around the station. The multipath effect of the code observables is more complex than that of the carrier-phase observables, especially for BeiDou Navigation Satellite System (BDS) geostationary orbit (GEO) satellites. In this contribution, we deeply analyzed the characteristic and effect on the precise data processing of GEO satellite multipath errors based on a large number of permanent GNSS stations. A linear combination of code and carrier-phase observables was used to analyze the characteristics of repeatability for BDS GEO’s multipath. Then, a correction method was proposed to eliminate the multipath error of the GEO code observables, based on wavelet transform. The experiment data were collected at 83 globally distributed stations, from multi-GNSS experiments and national BDS augmentation systems, from days 32 to 66 in 2017. The results show that the systematic multipath variation component of the GEO code observables can be obtained with wavelet transform, which can significantly contribute to correcting the multipath error of GEO satellites. The average root mean square error (RMSE) of the multipath series is decreased by approximately 19.5%, 20.2%, and 7.5% for B1, B2, and B3, respectively. In addition, some experiments, including ionospheric delay extraction and satellite clock estimation, were conducted in simulated real-time mode in order to validate the effect of the correction methods. For the ionospheric delay estimation, the average RMSE of the slant ionospheric delay is reduced by approximately 15.5%. Moreover, the multipath correction can contribute greatly to shortening the convergence time of the satellite clock estimation of the BDS GEO satellites.

A New Method for converting Slant Ionospheric Delays to Vertical for SBAS Ionospheric corrections

SBAS gives ionospheric correction data to GNSS users for more accurate positioning. SBAS uses obliquity factor of thin shell model to convert slant ionospheric delays to vertical delays which is not precise especially at lower elevation angle. This paper suggests a new method for estimating the vertical delays which are needed for generating the ionospheric corrections. The new method uses Chapman profile assumption by which the vertical electron density distribution can be modeled. It is also assumed that Chapman profile’s parameters are given and the vertical ionospheric delay can be linearly modeled. We divided ionosphere into multi-layer and mapped ionospheric vertical delays with a linear function. Converting vertical delays of each layer to slant delays, we can set a linear equation for the vertical ionospheric delay at IPP. We used IRI model to get information about ionosphere for simulation to verify our new method. Estimation error of vertical ionospheric delay at IPP was decreased about 40% in average when using our new method.

Multi-GNSS triple-frequency differential code bias (DCB) determination with precise point positioning (PPP)

Differential code biases (DCBs) account for the most significant systematical biases when sensing the earth’s ionosphere with GNSS observations and are also important correction parameters in GNSS applications of positioning, navigation and timing. With the continuous modernization of the American GPS and Russian GLONASS systems, and also the rapid developments of the European Galileo and Chinese BeiDou systems, there is a strong demand of precise satellite DCB products for multiple constellations and frequencies. This study proposes a new method for the precise determination of multi-GNSS triple-frequency DCBs, which can be divided into three steps. The first step is to precisely retrieve slant ionospheric delays and “additional code biases” based on a newly established full-rank triple-frequency precise point positioning (PPP) model with raw observations. Both the slant ionospheric delays and “additional code biases” containing the DCBs need to be estimated. Then, an enhanced IGGDCB (IGG stands for Institute of Geodesy and Geophysics) method is used to estimate the DCBs between the first and second frequency bands with the PPP-derived slant ionospheric delays. At last, the previously estimated DCBs between the first and second frequency bands are substituted into the “additional code biases” and DCBs between the first and third frequency bands are estimated. Multi-GNSS slant ionospheric delays from the triple-frequency PPP method are compared with those from the traditional dual-frequency carrier-to-code level (CCL) method, in terms of formal precision and zero-baseline experiment. Quad-system average formal precisions are 0.08 and 0.41 TECU, for PPP and CCL methods, respectively, indicating the obvious improvements of PPP over CCL. One month of data from 60 globally distributed multi-GNSS experiment stations are selected, and totally eight types of DCBs are estimated for GPS, GLONASS, Galileo and BeiDou. Multi-GNSS satellite DCBs generated with the proposed method are compared with the products from different agencies, including Center for Orbit Determination in Europe (CODE), Deutsches zentrum fur Luft-und Raumfahrt (DLR) and Chinese Academy of Sciences (CAS). For GPS C1WC2 W DCBs, RMS values with respect to CODE products are 0.24, 0.07 and 0.09 ns for DLR, CAS and IGG (this study), respectively. RMS values are 0.31/0.25 and 0.19/0.15 ns, for GPS C1WC5X and C1WC5Q DCBs and with respect to DLR/CAS, respectively. For GLONASS C1PC2P DCBs, RMS values with respect to CODE are 0.68, 0.49 and 0.33 ns for DLR, CAS and IGG, respectively. For Galileo, RMS values are 0.16/0.20 and 0.13/0.14 ns, for C1XC5X and C1XC7X DCBs and with respect to DLR/CAS, respectively. For BeiDou, RMS values are 0.32/0.25 and 0.34/0.41, for C2IC7I and C2IC6I DCBs and with respect to DLR/CAS. These results show that the proposed method can provide multi-GNSS and multi-frequency satellite DCB estimation with high precision, processing efficiency and flexibility.

GLONASS real-time wide-lane ambiguity resolution with an enhanced geometry-based model for medium-range baselines

Double-difference wide-lane integer ambiguities are determined first to assist narrow-lane ambiguity resolution in real-time medium-range baseline resolution. The Hatch-Melbourne-Wübbena (HMW) is the conventional strategy for wide-lane ambiguity resolution, which works well for GPS/Galileo/BDS. However, the un-canceled inter frequency bias (IFB) on double-difference measurement causes that the HMW combination is invalid on GLONASS wide-lane ambiguity resolution. The residual IFB in double-difference measurement may be several meters, especially between inhomogeneous stations. There is no effective method to model or tabulate ranging-codes IFB. In this paper, we propose a device-independent geometry-based model to achieve GLONASS real-time wide-lane ambiguity resolution for medium-range baselines. Since zenith tropospheric delay and slant ionospheric delay of satellites are estimated as unknown parameters, we utilize atmosphere-weighted algorithm to enhance the model strength and decrease the dependence on pseudorange measurement. For medium-range baseline, GLONASS wide-lane ambiguity float solutions can converge quickly by the geometry-based model. The cumulative frequency of WL AR can exceed 99% during 10 epochs. Benefiting from rapid and reliable wide-lane ambiguity resolution, the positioning accuracy of GPS/GLONASS RTK fixed solution are greater than 3 cm in level direction and 5 cm in upward direction for 40–100 km baselines. An approximately 35% improvement rate is observed in each direction compared with that of single GPS RTK. Since less available satellites for single-GLONASS mode, the convergence of wide-lane ambiguity becomes slower. For base stations with known coordinates, the enhanced geometry-based model can be developed to the enhanced geometry-fixed model and are applied to longer baselines. Assisting with the geometry-fixed model, more than 90% of GLONASS narrow-lane ambiguities can be fixed quickly for 80–160 km baselines. It should be noted that the accuracy of external atmospheric delay is critical to the performance of the geometry-based method. Its benefit may weaken with the increasing of baseline length, when external atmospheric delays are set as 0 directly.

Real-Time Precise Point Positioning (RTPPP) with raw observations and its application in real-time regional ionospheric VTEC modeling

Precise Point Positioning (PPP) is an absolute positioning technology mainly used in post data processing. With the continuously increasing demand for real-time high-precision applications in positioning, timing, retrieval of atmospheric parameters, etc., Real-Time PPP (RTPPP) and its applications have drawn more and more research attention in recent years. This study focuses on the models, algorithms and ionospheric applications of RTPPP on the basis of raw observations, in which high-precision slant ionospheric delays are estimated among others in real time. For this purpose, a robust processing strategy for multi-station RTPPP with raw observations has been proposed and realized, in which real-time data streams and State-Space-Representative (SSR) satellite orbit and clock corrections are used. With the RTPPP-derived slant ionospheric delays from a regional network, a real-time regional ionospheric Vertical Total Electron Content (VTEC) modeling method is proposed based on Adjusted Spherical Harmonic Functions and a Moving-Window Filter. SSR satellite orbit and clock corrections from different IGS analysis centers are evaluated. Ten globally distributed real-time stations are used to evaluate the positioning performances of the proposed RTPPP algorithms in both static and kinematic modes. RMS values of positioning errors in static/kinematic mode are 5.2/15.5, 4.7/17.4 and 12.8/46.6 mm, for north, east and up components, respectively. Real-time slant ionospheric delays from RTPPP are compared with those from the traditional Carrier-to-Code Leveling (CCL) method, in terms of function model, formal precision and between-receiver differences of short baseline. Results show that slant ionospheric delays from RTPPP are more precise and have a much better convergence performance than those from the CCL method in real-time processing. 30 real-time stations from the Asia-Pacific Reference Frame network are used to model the ionospheric VTECs over Australia in real time, with slant ionospheric delays from both RTPPP and CCL methods for comparison. RMS of the VTEC differences between RTPPP/CCL method and CODE final products is 0.91/1.09 TECU, and RMS of the VTEC differences between RTPPP and CCL methods is 0.67 TECU. Slant Total Electron Contents retrieved from different VTEC models are also validated with epoch-differenced Geometry-Free combinations of dual-frequency phase observations, and mean RMS values are 2.14, 2.33 and 2.07 TECU for RTPPP method, CCL method and CODE final products, respectively. This shows the superiority of RTPPP-derived slant ionospheric delays in real-time ionospheric VTEC modeling.