Ionospheric Delay Research Articles

A Multi-Parameter Empirical Fusion Model for Ionospheric TEC in China’s Region

This article takes the measured Total Electron Content (TEC) from the GPS points of the China Regional Crust Observation Network as the starting point to establish a regional ionospheric empirical model. The model’s performance is enhanced by considering solar flux and geomagnetic activity data. The refinement function model of the ionospheric TEC diurnal variation component, seasonal variation component, and geomagnetic component is studied. Using the nonlinear least squares method to fit undetermined coefficients, MEFM-ITCR (Multi-parameter Empirical Fusion Model–Ionospheric TEC China Regional Model) is proposed to forecast the regional ionosphere TEC in China. The results show that the standard deviation of MEFM-ITCR residuals is 3.74TECU, and MEFM-ITCR fits the modeling dataset well. Analyses of geographic location variation, seasonal variation, and geomagnetic disturbance were carried out for MEFM-ITCR performance. The results indicate that in the Chinese region, MEFM-ITCR outperforms IRI2020 and NeQuick2 models in terms of forecast accuracy, linear correlation, and model precision for TEC measured using GPS points under different latitudes and longitudes, different seasons, and different geomagnetic disturbances. The empirical TEC model built for the Chinese region in this paper provides a new ionospheric delay correction method for GNSS single frequency users and is of great significance for establishing other new and improving existing ionospheric empirical models.

Research on Reliable Long-Baseline NRTK Positioning Method Considering Ionospheric Residual Interpolation Uncertainty

In the past few decades, network real-time kinematic (NRTK) positioning technology has developed rapidly. Generally, in the continuously operating reference stations (CORS) network, within a moderate baseline length, e.g., 80–100 km, atmospheric delay can be effectively processed through regional modeling and, thus, can support almost instantaneous centimeter-level NRTK positioning. However, in long-baseline CORS networks, especially during the active period of the ionosphere, ionospheric delays cannot be fully eliminated through modeling, leading to decreased NRTK positioning accuracy. To address this issue, this study proposes a long-baseline NRTK positioning method considering ionospheric residual interpolation uncertainty (IRIU). The method utilizes the ionospheric residual interpolation standard deviation (IRISTD) calculated during atmospheric delay modeling, then fits an IRISTD-related stochastic model through the fitting of the absolute values of the ionospheric delay modeling residuals and IRISTD. Finally, based on the ionosphere-weighted model, the IRISTD processed by the stochastic model is used to constrain the ionospheric pseudo-observations. This method achieves good comprehensive performance in handling ionospheric delay and model strength, and the advantage is validated through experiments using CORS data with baseline lengths ranging from 54 km to 106 km in western China and from 84 km to 180 km in AUSCORS data. Quantitative results demonstrate that, across the three sets of experiments, the proposed ionosphere-weighted model achieves an average increase in the fixed rate of 16.9% compared to the ionosphere-fixed model and 25.6% compared to the ionosphere-float model. In terms of positioning accuracy, the proposed model yields average improvements of 67.4%, 76.4%, and 66.0% in the N/E/U directions, respectively, compared to the ionosphere-fixed model, and average improvements of 21.0%, 32.0%, and 24.4%, respectively, compared to the ionosphere-float model. Overall, the proposed method can achieve better NRTK positioning performance in situations where ionospheric delay modeling is inaccurate, such as long baselines and ionospheric activity.

Analysis of Multifrequency GNSS Signals and an Improved Single-Epoch RTK Method for Medium-Long Baseline

In recent years, the quantity of visible satellites has increased significantly due to multiple satellite systems that leaped forward. The BeiDou Navigation Satellite System (BDS) and Galileo satellite navigation system (Galileo) broadcast triple-frequency signals and above to users, thus enhancing the reliability, continuity, and availability of the single-epoch real-time kinematic (RTK) positioning. In this study, an improved single-epoch multifrequency multisystem RTK method is successfully developed for the medium-long baseline. First, the Galileo and BDS extra-wide-lane (EWL) ambiguities are fixed at a high success rate, and the Galileo and BDS wide-lane (WL) ambiguity is achieved via the transformation process. Second, the fixed WL ambiguities of Galileo and BDS are exploited to elevate the fixed rate of GPS WL ambiguity. Third, the parametric strategies for ionospheric delay are carried out to upregulate the narrow-lane (NL) ambiguity-fixed rate of GPS. Further, the real-time data are adopted for verifying the feasibility of the method developed in this study. The experimental results demonstrate the optimal carrier-to-noise density ratio (C/N0) of full operational capability (FOC) E5a/E5b at all frequencies, followed by IIR-M L1, and IIR-A/B L2 exhibits the worst performance. Generally, the multipath combination (MPC) of Galileo signals shows root mean square (RMS) values within 0.4 m, ordered as follows: E 1 > E 5 b > E 5 a . For the BDS-2, the B3 signal exhibits optimal performance, while the B1 signal is the worst. The RMS of MPC errors of L1 signals is smaller than the L2 signals for the GPS. Furthermore, under the 50 km baseline, the GPS NL ambiguity-fixed rate using the ionosphere-free (IF) combination reaches only 47.74% at the ratio threshold of 2. Finally, compared to the ionosphere-free combination method, the GPS NL ambiguity-fixed rate is increased by 45.52% with the presented method. The proposed approach broadens the future application of deformation monitoring in medium-long baseline scenarios.

Improved Medium Baseline RTK Positioning Performance Based on BDS/Galileo/GPS Triple-Frequency-Only Observations

With the global service of the BeiDou Navigation Satellite System (BDS), the Galileo Navigation Satellite System (Galileo), and the modernization of the Global Positioning System (GPS), achieving high-precision positioning through triple-frequency-only observations in medium baseline real-time kinematics (RTK) is anticipated. This study investigates the impacts of double-difference (DD) troposphere delay and ionosphere delay on ambiguity resolution (AR) based on six medium baselines at a latitude of 30°. Additionally, it evaluates positioning accuracy, fixing rate, convergence time, and computational time using triple-frequency-only (B1I/B2a/B3I, E1/E5a/E5b, L1/L2/L5) data, comparing these results to those obtained from dual-frequency (B1I/B2a, E1/E5a, L1/L2) and combined dual-frequency and triple-frequency data. The experimental findings suggest that, for geometry-based wide-lane (WL) AR, the DD troposphere delay and ionosphere delay can be disregarded. However, they cannot be overlooked when aiming to resolve the raw ambiguity. Triple-frequency-only RTK exhibits comparable positioning accuracy to dual-frequency RTK, with its primary advantage lying in faster convergence. The probability of achieving convergence within 180 s is approximately 8.0% higher for triple-frequency-only RTK compared to dual-frequency RTK. In terms of computational time, the use of triple-frequency-only data reduces the required time by 8.26 s compared to the approach that simultaneously employs both dual-frequency and triple-frequency data, resulting in a computational time reduction of approximately 20%. Therefore, when conducting medium baseline RTK positioning, it is recommended to adopt the ambiguity resolution method proposed in this paper based on triple-frequency-only observations.

The Implementation of Precise Point Positioning (PPP): A Comprehensive Review.

High-precision positioning from Global Navigation Satellite Systems (GNSS) has garnered increased interest due to growing demand in various applications, like autonomous car navigation and precision agriculture. Precise Point Positioning (PPP) offers a distinct advantage over differential techniques by enabling precise position determination of a GNSS rover receiver through the use of external corrections sourced from either the Internet or dedicated correction satellites. However, PPP's implementation has been challenging due to the need to mitigate numerous GNSS error sources, many of which are eliminated in differential techniques such as Real-Time Kinematics (RTK) or overlooked in Standard Point Positioning (SPP). This paper extensively reviews PPP's error sources, such as ionospheric delays, tropospheric delays, satellite orbit and clock errors, phase and code biases, and site displacement effects. Additionally, this article examines various PPP models and correction sources that can be employed to address these errors. A detailed discussion is provided on implementing the standard dual-frequency (DF)-PPP to achieve centimeter- or millimeter-level positioning accuracy. This paper includes experimental examples of PPP implementation results using static data from the International GNSS Service (IGS) station network and a kinematic road test based on the actual trajectory to showcase DF-PPP development for practical applications. By providing a fusion of theoretical insights with practical demonstrations, this comprehensive review offers readers a pragmatic perspective on the evolving field of Precise Point Positioning.

Investigating the contribution of BDS-3 observations to multi-GNSS single-frequency precise point positioning with different ionospheric models

Due to the advantages of low-cost and high precision, single-frequency precise point positioning (SF-PPP) plays a vital role in the booming market of the SF global navigation satellite system (GNSS) receivers. With the full completion of the BeiDou global navigation satellite system (BDS-3), its contribution to single-GNSS SF-PPP is worthy of comprehensive exploration. In this study, three SF-PPP models including ionosphere-corrected (IC), ionosphere-free (IF), and ionosphere-weighted (IW) strategies are analyzed, and four ionospheric models including the GPS Klobuchar, BDGIM (BeiDou global ionospheric delay correction model), CNES real-time VTEC (vertical total electron content) and GIM (global ionospheric maps) are used for conducting SF-PPP. As to GPS/GLONASS/Galileo-only IC SF-PPP, its 3D positioning accuracy can be improved by 6.9–11.6 % with the adoption of BDS-3 observations. Similarly, for the IF SF-PPP solution, the corresponding positioning accuracy and convergence time can be improved by at least 17.3 and 38.3 %, respectively. In IW SF-PPP, the convergence performance of the CNES-VTEC-constrained SF-PPP is much better than that of both GPS Klobuchar- and BDGIM-constrained solutions. When using the BDGIM constraints, the horizontal/vertical convergence time of GPS/Galileo + BDS-3 solutions can be shortened by at least 16.5/27.9 % in comparison with GPS/Galileo-only solutions. While for GIM-constrained SF-PPP, the convergence improvement is only shown in the vertical component. In summary, the GIM-constrained SF-PPP with quad-system observations has the best performance, compared to IF solution, its convergence time can be shortened by 82.8 % to 5.5 min in horizontal and 29.7 % to 26.0 min in vertical when converging to 0.3 m. Note that the corresponding improvements in the BDGIM-constrained solution are relatively limited with only 5 %. At present, the quad-system IW SF-PPP has the ability to achieve cm-level fast positioning, with an RMS of 0.06 m in horizontal, showing promising applications in engineering and scientific fields.

Ionospheric TEC modeling using COSMIC-2 GNSS radio occultation and artificial neural networks over Egypt

Abstract The ionospheric delay significantly impacts GNSS positioning accuracy. To address this, an Artificial Neural Network (ANN) was developed using the high-quality COSMIC-2 ionospheric profile dataset to predict the Total Electron Content (TEC). ANNs are adept at addressing both linear and nonlinear challenges. For this research, eight distinct ANNs were cultivated. These ANNs were designed with the following inputs Year, Month, Day, Hour, Latitude, and Longitude. Along with solar and geomagnetic parameters such as the F10.7 solar radio flux index, the Sunspot Number (SSN), the Kp index, and the ap index. The goal was to discern the most influential parameters on ionosphere prediction. After pinpointing these key parameters, an enhanced model utilizing a pioneering technique of a secondary ANN was employed with the main ANN to predict TEC values for events in 2023. The study’s findings indicate that solar parameters markedly enhance the model’s accuracy. Notably, the augmented model featuring a prelude secondary network achieved a stellar correlation coefficient of 0.99. Distributionally, 41 % of predictions aligned within the (−1≤ ΔTEC ≤1) TECU spectrum, 28 % nestled within the (1< ΔTEC ≤2) and (−2≤ ΔTEC <−1) TECU ambit, while a substantial 30 % spanned the broader (2< ΔTEC ≤5) and (−5≤ ΔTEC <−2) TECU range. In essence, this research underscores the potential of incorporating solar parameters and advanced neural network techniques to refine ionospheric delay predictions, thus boosting GNSS positioning precision.

A revised BDS Klobuchar model with a non-symmetrical processing strategy and a high-latitude amplitude constraint

The BeiDou Navigation Satellite System (BDS) broadcasts the BDS Klobuchar model (BDSK) for ionospheric delay correction in both BDS-2 regional service phase and BDS-3 global service phase. Unlike the GPS Klobuchar model, the BDSK model adopts a simple “symmetrical strategy” due to the limited ionospheric input data within China, which results in inaccurate total electron content (TEC) estimation in the southern hemisphere. Besides, the “high latitude anomaly” occurs in the BDSK model due to the lack of latitude limitation strategy, which deteriorates the accuracy of the BDSK model in the southern hemisphere further. To solve this problem, we propose a revised BDS Klobuchar model (RBDK), in which the “non-symmetrical strategy” is adopted in the southern hemisphere and an amplitude constraint is introduced at high latitudes. For comparison, the fitted BDS Klobuchar model (BDFK) that is estimated using the same ionospheric input data and constraint algorithm for the RBDK model but with the “symmetrical strategy” is also included. First, we conduct a comparative analysis of the time series of the eight parameters for the BDSK, BDFK and RBDK models in 2021. The parameters of the BDFK and RBDK models are more stable, and there will be no abnormal jumping phenomenon that occurs in the BDSK model. Then, the performance of the broadcast BDSK model, the fitted BDFK model and the proposed RBDK model during the whole year of 2021 are evaluated by comparing them with the high-precision global ionospheric map (GIM) of the International GNSS Service (IGS). In the southern hemisphere, the root mean square (RMS) values of the BDSK and BDFK are 6.5 and 5.8 TEC unit (TECU), respectively, while that of the RBDK is 4.6 TECU, showing improvements of 29% and 21%, respectively. At high latitudes in the northern hemisphere, the RMS value of the BDSK is 7.6 TECU, while they are only 3.7 TECU for the RBDK and BDFK when the amplitude constraint is introduced. Final, we select 16 IGS stations to validate the performance of the BDSK, BDFK and RBDK models in the standard point positioning (SPP) domain in 2021. The average SPP accuracy of the BDSK is 8.88 m at 6 high-latitude stations, while they are 5.48 m for the RBDK and BDFK, showing significant improvement of 38%. Excluding the high-latitude stations, the SPP accuracy of the RBDK outperforms the BDFK and BDSK by 14% and 18%, respectively, at stations in the southern hemisphere. As expected, the proposed RBDK model shows significantly better performance in the southern hemisphere and at high latitudes in contrast to the broadcast BDSK model.

Spatio-Temporal Validation of GNSS-Derived Global Ionosphere Maps Using 16 Years of Jason Satellites Observations

Existing ionospheric models perform very well in mapping the calm state of the ionosphere. However, the problem is accurately determining the total electron content (TEC) for disturbed days. Knowledge of the exact electron density is essential for single−frequency receivers, which cannot eliminate the ionospheric delay. This study aims to investigate temporal and spatial variability in the distribution of TEC based on differences between maps of individual Ionospheric Associated Analysis Centers (IAACs) of the International GNSS Service (IGS) and aligned altimetry−TEC from 2005–2021. Based on the temporal distribution, we have observed a significant effect of solar activity on the mean and standard deviation behavior of the differences between global ionospheric maps (GIMs) and Jason−derived TEC. We determined the biases for the entire calculation period, through which it can be concluded that the upcg-Jason and igsg-Jason differences have the lowest standard deviation (±1.81 TECU). In addition, the temporal analysis made it possible to detect annual, semi−annual, and 117-day oscillations occurring in the Jason−TEC data, as well as 121-day oscillations in the GIMs. It also allowed us to analyze the potential sources of these cyclicities, solar and geomagnetic activity, in the case of the annual and semi−annual periodicities. When considering spatial variations, we have observed that the most significant average differences are in the intertropical areas. In contrast, the smallest differences were recorded in the southern hemisphere, below the Tropic of Capricorn (23.5°S). However, the slightest variations were noted for the northern hemisphere above the Tropic of Cancer (23.5°N). Our research presented in this paper allows a better understanding of how different methods of GNSS TEC approximation affect the model’s accuracy.

Characterizing Ionospheric Effects on GNSS Reflectometry at Grazing Angles from Space

Coherent observations in GNSS reflectometry are prominent in regions with smooth reflecting surfaces and at grazing elevation angles. However, within these lower elevation ranges, GNSS signals traverse a more extensive atmospheric path, and increased ionospheric effects (e.g., delay biases) are expected. These biases can be mitigated by employing dual-frequency receivers or models tailored for single-frequency receivers. In preparation for the single-frequency GNSS-R ESA “PRETTY” mission, this study aims to characterize ionospheric effects under variable parameter conditions: elevation angles in the grazing range (5° to 30°), latitude-dependent regions (north, tropic, south) and diurnal changes (day and nighttime). The investigation employs simulations using orbit data from Spire Global Inc.’s Lemur-2 CubeSat constellation at the solar minimum (F10.7 index at 75) on March, 2021. Changes towards higher solar activity are accounted for with an additional scenario (F10.7 index at 180) on March, 2023. The electron density associated with each reflection event is determined using the Neustrelitz Electron Density Model (NEDM2020) and the NeQuick 2 model. The results from periods of low solar activity reveal fluctuations of up to approximately 300 TECUs in slant total electron content, 19 m in relative ionospheric delay for the GPS L1 frequency, 2 Hz in Doppler shifts, and variations in the peak electron density height ranging from 215 to 330 km. Sea surface height uncertainty associated with ionospheric model-based corrections in group delay altimetric inversion can reach a standard deviation at the meter level.

Analysis of differential code biases for GPS receivers over the Indian region

Abstract The GPS Aided Geo Augmented Navigation (GAGAN) system provides the navigational services for single-frequency GNSS user via broadcasting the differential corrections with GEO stationary satellites. The significant differential correction contribution comes from ionospheric time delays and is necessary to be determined precisely. Dual-frequency GPS receivers measure the ionospheric time delays using GPS code and carrier phase measurements. The determination of absolute ionospheric Total Electron Content (TEC) requires the calibration of GPS satellites and receiver hardware biases due to different frequency-dependent signals (L1 and L2) due to environmental changes (Temperature and Humidity). In this paper, A receiver-based Differential Code Biases (DCB) algorithm is implemented to derive a joint estimation of TEC and RDCB parameters using the weighted Least Square (WLS) method. The daily averaged DCBs data for 26 GPS receivers are obtained for 3 years (2014–2016) from 26 GPS reeivers over Indian region. The receiver DCB algorithmis validated with the Fitted Receiver Biases (FRB) method. The correlation (R) between VTEC and RDCB is conducted to investigate the RDCB stability. The results would be useful for the accurate determination of ionospheric differential corrections to GAGAN users.

A Comparison of a GNSS‐GIM and the IRI‐2020 Model Over China Under Different Ionospheric Conditions

AbstractThe ionosphere is a crucial factor affecting Global Navigation Satellite System positioning. The Global Ionosphere Map (GIM) or the International Reference Ionosphere (IRI) model can be used for regional ionospheric correction. Since southern China is located near the electron density equatorial anomaly, this study evaluates the performance of the Wuhan University GIM (WHU‐GIM) and the IRI‐2020 from 2008 to 2020 over the China region. The comparison indicates that the Total Electron Content (TEC) from IRI‐2020 is lower than that from WHU‐GIM overall, the discrepancy is more obvious in high solar conditions and low‐latitude regions. The differential Slant TEC (dSTEC) during a phase‐arc with about 0.1 TECU accuracy derived from Global Positioning System (GPS) observations is used for model validation, the results show that the accuracies of WHU‐GIM and IRI‐2020 are 3.14 and 4.57 TECU, respectively. The dSTEC error is larger at low latitudes and decreases with increasing latitude. GPS‐derived TEC is taken for reference to evaluate the model reliability. Results show that both models can reproduce the diurnal TEC variations, but IRI‐2020 is more influenced by geomagnetic activities. The TEC correction percentage for IRI‐2020 is about 60%–80% under different ionospheric conditions, while for WHU‐GIM is 80%–90%. The Single‐Frequency Precise Point Positioning is performed with the ionosphere delay corrected by the two models, respectively. The positioning errors show that using IRI‐2020 has a lower accuracy, and the TEC discrepancy of the IRI‐2020 can cause a large bias in the up direction, especially at low‐latitude regions.

MLP-based user-applicable method for SBAS GIVD calibration: A comparative study based on CODE final GIM products

Satellite-Based Augmentation Systems (SBASs) are designed to enhance the performance of Global Navigation Satellite Systems (GNSSs) for civil aviation users. As SBAS Geostationary Earth Orbit (GEO) satellite signals are received in a wide area, far more non-aeronautical GNSS users utilize SBAS to improve navigation performance. Since ionospheric delay is the principal error source of GNSS and a leading correction aspect of SBAS, the GEO-broadcast ionospheric data quality of six SBASs over the past three years is assessed in this article. Results show that European Geostationary Navigation Overlay Service (EGNOS), System for Differential Correction and Monitoring (SDCM), and Wide-Area Augmentation System (WAAS) have the highest Grid Ionospheric Vertical Delay (GIVD) accuracy judged by the residual Standard Deviation (STD), then followed by BeiDou SBAS (BDSBAS) and Multi-functional Satellite Augmentation System (MSAS), while GPS-Aided GEO Augmented Navigation (GAGAN) is the worst. The 6 m Grid Ionospheric Vertical Error (GIVE) availability is over 90 percent of the time for most Ionospheric Grid Points (IGPs) of all tested SBASs. However, GAGAN and WAAS have a significant availability decline at 3 m GIVE. In addition, the integrity limits of MSAS and WAAS are fairly conservative. In contrast, the confidence bounds of BDSBAS and SDCM are excessively radical that fail to meet the integrity requirement. To improve the ionospheric data quality received from SBAS, a Multi-Layer Perceptron (MLP)-based user-applicable GIVD calibration method is proposed in the article. A well-trained MLP learned temporal and spatial correlations of the IGP delays and extracted transfer functions between broadcast and reference GIVDs to eliminate error. The SBAS processing errors in Differential Code Bias (DCB) compensation, GIVD estimation modeling, and GEO message quantization are rectified by the MLP trained by abundant historical data with Center for Orbit Determination in Europe (CODE) final Global Ionospheric Maps (GIMs) as reference. Test results on real-time data show that over 85 percent IGPs have STD decline after calibration. The STD declination rates of six SBASs except for SDCM are over 15 percent, and the Mean Absolute Error (MAE) and Root Mean Square Error (RMS) calibration effects are dramatic for all systems. Finally, the average GIVEs are reduced by more than 50 percent for EGNOS, GAGAN, MSAS, and WAAS with a 99.9 percent integrity level maintained. Despite no promotion on availability for BDSBAS and SDCM, the proposed method compensates for the integrity issues of the systems.

Research on single-epoch ambiguity resolution method based on BDS-3 ionosphere-reduced combinations

In conventional ambiguity resolution method for single-epoch, narrow-lane (NL) observations are often inaccurately fixed due to improper treatment of error sources such as ionospheric and tropospheric delays. Therefore, based on the multi-frequency observation data of the BeiDou Global Navigation Satellite System (BDS-3), this paper adopts a single-epoch resolution method based on the ionosphere-reduced combinations. Firstly, select the appropriate combination observations of extra wide-lane (EWL) and wide-lane (WL) and get its fixed ambiguity. Then the ambiguity of the WL combination with high observation accuracy is obtained by linear combination, which goes to constrain the fixation of NL ambiguity. Secondly, different from the conventional method, the ionosphere-reduced combinations observations were selected as NL, thus significantly weakening the influence of first-order ionospheric delay error on the fixation of narrow-lane ambiguity. The experimental results show that compared with the conventional methods, the method adopted in this paper can significantly improve the fixing success rate of the wide-lane ambiguity. In the NL ambiguity resolution part, using the ionosphere-reduced combinations as NL for solving improved the average fixing success rate from 68.6%/44.4%/35.7%/27.6% to 97.4%/95.4%/95.6%/87.3% compared to the conventional method of using the base frequency as NL for solving. Furthermore, the N/E/U positioning accuracy improved by an average of 63.05%/64.37%/52.27%. To sum up, this method effectively improves the success rate of NL ambiguity fixing and positioning accuracy.

Research on Soil Moisture Estimation of Multiple-Track-GNSS Dual-Frequency Combination Observations Considering the Detection and Correction of Phase Outliers.

Soil moisture (SM), as one of the crucial environmental factors, has traditionally been estimated using global navigation satellite system interferometric reflectometry (GNSS-IR) microwave remote sensing technology. This approach relies on the signal-to-noise ratio (SNR) reflection component, and its accuracy hinges on the successful separation of the reflection component from the direct component. In contrast, the presence of carrier phase and pseudorange multipath errors enables soil moisture retrieval without the requirement for separating the direct component of the signal. To acquire high-quality combined multipath errors and diversify GNSS-IR data sources, this study establishes the dual-frequency pseudorange combination (DFPC) and dual-frequency carrier phase combination (L4) that exclude geometrical factors, ionospheric delay, and tropospheric delay. Simultaneously, we propose two methods for estimating soil moisture: the DFPC method and the L4 method. Initially, the equal-weight least squares method is employed to calculate the initial delay phase. Subsequently, anomalous delay phases are detected and corrected through a combination of the minimum covariance determinant robust estimation (MCD) and the moving average filter (MAF). Finally, we utilize the multivariate linear regression (MLR) and extreme learning machine (ELM) to construct multi-satellite linear regression models (MSLRs) and multi-satellite nonlinear regression models (MSNRs) for soil moisture prediction, and compare the accuracy of each model. To validate the feasibility of these methods, data from site P031 of the Plate Boundary Observatory (PBO) H2O project are utilized. Experimental results demonstrate that combining MCD and MAF can effectively detect and correct outliers, yielding single-satellite delay phase sequences with a high quality. This improvement contributes to varying degrees of enhanced correlation between the single-satellite delay phase and soil moisture. When fusing the corrected delay phases from multiple satellite orbits using the DFPC method for soil moisture estimation, the correlations between the true soil moisture values and the predicted values obtained through MLR and ELM reach 0.81 and 0.88, respectively, while the correlations of the L4 method can reach 0.84 and 0.90, respectively. These findings indicate a substantial achievement in high-precision soil moisture estimation within a small satellite-elevation angle range.

A Modified Cycle Slip Detection Method with GNSS Doppler Assistance and Optimizing by Adaptive Threshold and Sliding Polynomial Fitting

Cycle slip determination plays an important role in high-precision data processing and application of global navigation satellite systems (GNSS). The TurboEdit method consists of the Melbourne-Wubbena (MW) and the geometry-free phase (GF) combination. It can correctly detect and repair cycle slip in most cases. Cycle slip detection (CSD) with GF is disturbed by severe ionospheric delay variations; moreover, CSD or cycle slip repair (CSR) with the MW faces the risk of the disturbance from large pseudorange errors. Hence, cycle slip determination would be difficult under some extreme conditions, e.g., cycle slips occur in low altitude satellite, low sampling rate of dual-frequency observations. To overcome the limitations, a new dual-frequency CSD and CSR method is proposed. The main contents are as follows: (1) compared with the MW method, the Doppler-assisted phase subtraction pseudorange (DAPSP) method that we proposed has no detection blind spot and can effectively reduce the influence of pseudorange noise at high sampling rates; thus, we replace MW by the DAPSP method to improve the detection accuracy. (2) An adaptive threshold model with root mean square (RMS) is established to effectively reduce the missing and false range detection of cycle slip. (3) The sliding polynomial fitting-assisted GF (SPFAGF) is carried out according to the satellite altitude angle. The trend of ionospheric delay and residual multipath effect error between adjacent epochs is extracted and suppressed by SPFAGF. The method combined with DAPSP and SPFAGF (DAPSP-SPFAGF) overcomes the situation that the TurboEdit method cannot effectively detect under extreme conditions. The experimental results of Beidou dual-frequency observation data show that the TurboEdit method and the DAPSP-SPFAGF method can perform CSD and CSR in most cases. At the sampling rate of 1 s, the detection speed of DAPSP-SPFAGF method is significantly faster than TurboEdit method. The number of false positives about CSD is reduced from 68 to 0. At the sampling rate of 30 s and under the condition of the observed satellite altitude angle below 30°, the false alarm rate of the DAPSP-SPFAGF method is 0, but the TurboEdit method’s false alarm rate is 71.2%. So DAPSP-SPFAGF method is prior to the TurboEdit method at the high sampling rates or under extreme conditions, especially it can accurately detect and repair cycle slip and reduce the false positives and false alarm rate.

Evaluation of Ionospheric Delay Extraction Model Using Dual-Frequency Multisystem Observations

Ionospheric delay presents complex spatiotemporal variation characteristics and is a critical error source restricting the improvement of global navigation satellite system (GNSS) positioning, navigation, and timing performance. It is a prerequisite to constructing an accurate ionospheric delay extraction model that supports the study of the ionospheric delay characteristics. By using modernized dual-frequency and multi-system signals and inter-system bias (ISB) calibration, the performance of the ionospheric delay extraction model can be enhanced further. This study focuses on quantitatively analyzing the influence of the geometric dilution of precision (GDOP) and the redundancy of observations on the accuracy of ionospheric delay extraction, by constructing two state-of-the-art high-precision dual-frequency multi-system ionospheric delay extraction models using the tightly coupled un-difference and un-combination precise point positioning (UPPP) and carrier-to-code leveling (CCL) methods. GNSS datasets from eight international GNSS service (IGS) stations were selected to evaluate their accuracy and analyze their influencing factors. The results show that the ionospheric delay extraction model using the UPPP method is more accurate than the CCL method for both single and multiple systems. The ionospheric delay extraction model using the CCL method is insensitive to time and coordinates. The ionospheric delay extraction model using the UPPP method shows inconsistent accuracy at different times and coordinates, which reflects a correlation with the GDOP effect.

An ionosphere-weighted calibration method with multi-station network corrections

Ionospheric delay is one of the most dominant error sources in GNSS fast relative positioning, especially for RTK positioning of medium and long baselines, which directly affects the ambiguity fixing successful rate and positioning accuracy. The strategy of combining ionospheric correction based on multi-reference station network with ionospheric weighted model is still one of the most effective methods for precise relative positioning. In this paper, a method of ionospheric weighted calibration model is presented. By using a large-scale reference station network at a certain period of time with ionospheric disturbance to obtain the prior ionospheric information required by the user, the GNSS precision relative positioning performance of the ionosphere-weighted calibration model in complex scenarios (the accuracy of prior ionospheric information will be reduced due to large-scale reference station network and ionospheric active scene) was systematically evaluated. The experimental results show that the ionosphere-weighted calibration (IWC) method can effectively reduce the influence of excessive ionospheric modeling residual on the positioning results, and significantly improve the fixed rate and positioning accuracy of RTK. Compared with the traditional ionosphere-fixed model based on virtual reference station (VRS) technology, the three-dimensional accuracy and fixed rate of the six rover stations are improved by 56% and 33.5% on average. Secondly, the prior ionospheric constraint intensity in IWC mode is controllable, which can effectively balance the relationship between initialization speed and positioning accuracy. On average, 94.3% of the time period of the six rover stations can complete the initialization and fixing within 10 epochs, which can basically meet the requirements of fast positioning. In addition, IWC model can ensure higher accuracy and better maintain fixed solution state after the ionospheric parameter convergence is completed. In the ionospheric disturbance stage, after a period of parameter convergence, high fixed rate and positioning accuracy can still be guaranteed.

A novel approach to enhancing the Klobuchar algorithm to mitigate the effect of ionospheric delay errors on static single-frequency receivers

Abstract The demand for real-time high-precision positioning for global navigation satellite system applications is difficult to satisfy. In this regard, a single-frequency receiver is found to play an important role in overcoming this challenge, especially in developing countries where economic factors are a major restriction. Hence, the development of built-in models, such as the Klobuchar model, is an important objective for single-frequency users to mitigate the effect of ionospheric delay errors in real-time applications. Accordingly, this study aims to devise a new approach to enhance the behavior of the Klobuchar model and increase its efficiency in resolving the aforementioned problem. The new approach seeks to enhance the behavior of the Klobuchar model without refining or increasing its coefficients. To eliminate the ionospheric delay disturbance, the proposed methodology applies normalization and filtration processes to the raw ionospheric delay probability distribution estimated by the unified least squares technique. A final assessment of the new method for enhancing the Klobuchar behavior in predicting the precise position of a single-frequency static receiver under different weather conditions around the globe is presented in this paper.

The Extraction and Characterization of Pseudorange Multipath Based on BDS-3 Multi-Frequency Observations.

Global Navigation Satellite System (GNSS) observations are subject to various errors during their propagation process. A reasonable correction of these errors can improve the positioning, navigation, and timing (PNT) service capability. The impact of multipaths on pseudorange observations can reach a decimeters or even meters level. However, their mechanism is complex and there is currently no universally accepted high-precision correction model. The correlation between the pseudorange multipaths (MP) of BDS-2 satellites and satellite elevation has been confirmed, while there have been fewer analyses of the MP characteristics for different frequencies of BDS-3 satellites. The broadcasting of multi-frequency observations in BDS-3 should theoretically make the extracted MP more accurate compared to traditional methods. Based on this, in this contribution, a multi-frequency MP extraction algorithm based on the least squares principle is proposed, which can simultaneously eliminate the influence of higher-order ionospheric delay. The analytical expression for only eliminating first-order ionospheric delay is successfully derived. Subsequently, the characteristics of the MPs extracted from different frequency combinations and the impact of combination noise on the extraction accuracy are discussed. The influence of second-order ionospheric delay on the MPs for each frequency under different combination noises, as well as the periodic behavior exhibited in long-term observations of the BDS-3 medium earth orbit (MEO) and inclined geosynchronous orbit (IGSO) satellites, are also analyzed. Finally, the correlations between the MPs of each frequency of BDS satellite and elevation are quantitatively analyzed based on observations from 35 stations. Overall, this work has positive implications for the study of the MP characteristics of BDS-3 and subsequent modeling efforts.