GPS Solutions

Real Time Precise Point Timing (RT-PPT) receivers can achieve sub-nanosecond accuracy through precise clock offset estimation and receiver clock discipline. However, when satellite signals are lost, the receiver clock drifts, causing timing errors due to clock noise and environmental factors. Strict autonomous holdover is therefore essential for high-precision timing receivers. This paper proposes a time holdover method based on online clock modelling. At the timing stage, the frequency bias compared to the time reference is estimated and eliminated by the closed control loop. Meanwhile, the recursive least square algorithm is used to estimate the parameters of the clock temperature and ageing effects online. In the event of signal interruption, the initial frequency bias is reduced by one-step prediction, while the clock offset caused by temperature and ageing is compensated for using the estimated parameters. To verify the validity of the method, real experiments were carried out using an RT-PPT receiver with different types of oscillators. The results show that with the method proposed in this paper, we can achieve holdover accuracy better than 1ns in 1h using a rubidium clock after correcting for the initial frequency bias, with an improvement of more than 67% over the free-running clock. And the holdover accuracy of the receiver using an OCXO after correcting for temperature and ageing is better than 1us in 24h, with a performance improvement of more than 70% over the free-running clock.

Employing high precise positioning ability, the global navigation satellite systems (GNSS) could accurately capture subtle deformations in bridges, supplying critical data for structural health monitoring. However, GNSS antennas placed on bridges often encounter multipath interference generated by bridge deck and water surface reflections, compromising the quality of observation data and positioning accuracy. Through the analysis of GNSS data collected from an operational bridge and an oceanic construction platform, we discovered a strong correlation between pseudorange multipath and carrier multipath components. Subsequently, through a series of steps including phase adjustment, decorrelation, and resolution, we successfully mitigated the multipath within residuals, and then reconstructed the corrected residuals into displacements; in addition, the method does not require advance deployment of GNSS equipment to collect calibration information. In terms of positioning results, the proposed method improves the accuracy by 27.1–32.1% in the WLMS (Weighted Least-Mean-Square) resolution and by18.8–32.9% in KF (Kalman Filter) resolution, effectively weakening the multipath interference in carrier phase observations and significantly improving positioning accuracy.

With the rapid development of multi-GNSS and multi-frequency observations, the code observable‑specific bias (OSB) emerges due to its flexibility and convenience compared to the traditional differential code bias (DCB). To estimate the GNSS multi-frequency code OSBs, the geometry-free model is commonly used, where ionospheric error handling is a key issue. For ionospheric handling, previous studies mostly focus on the ionospheric-float approach based on ionospheric modeling and the ionospheric-fixed approach using an external empirical model. We proposed an improved estimation approach aided by geometry-free and ionospheric-free (GFIF) observations to estimate multi-frequency code OSBs. The proposed approach expands the DCB to multiple code bias (MCB) using the GFIF model, eliminating the first-order ionospheric error. Then, the satellite-plus-receiver MCB observations with high precision and reliability are extracted. Finally, a full-rank estimable functional model after the identification and elimination of rank deficiencies is proposed. The effectiveness of the proposed approach is verified, where the multi-frequency code OSBs of GPS, BDS-3, and Galileo are estimated. Taking the code OSB products provided by the Chinese Academy of Sciences (CAS) as the references, the estimated code OSBs exhibit good consistency, showing an average RMS of 0.115, 0.130, and 0.161 ns for GPS, BDS-3, and Galileo, respectively. The average STD of all code OSBs for GPS, BDS-3, and Galileo is 0.187, 0.296, and 0.331 ns, respectively, which is smaller than that of CAS products for GPS and Galileo. In conclusion, the proposed approach can estimate the reliable GNSS multi-frequency code OSBs with high consistency and stability.

To characterize the impact of a RFI signal on the effective carrier-to-noise power density ratio, C/N0, of a GNSS receiver is very important for civil aviation standardization procedures in order to define Radio Frequency Interference (RFI)/jamming robustness test for two reasons. First, the prediction of the effective C/N0 is fundamental to predict the minimum performance attainable by an airborne GNSS receiver and to adjust the tests accordingly. Second, the receiver’s estimation of the effective C/N0 can allow the detection of the jamming presence and thus can help the receiver to adapt to the RFI situation. This problematic is analyzed with the introduction of three C/N0 definitions, statistical C/N0 which is the statistical prediction of the effective C/N0, real C/N0 which is the true effective C/N0 faced by the receiver and estimated C/N0 which is the value provided by a C/N0 estimator. Three conditions of validity are given to establish a relationship of equality between the three terms: ergodicity, centered RFI contribution at the correlator output and C/N0 estimator characteristics. To evaluate this last condition, Narrow-Wideband Power Ratio (NWPR), Signal-to-Noise Variance (SNV), Beaulieu and Moments Methods (MM) estimators are inspected and their expected mean value in presence of a generic RFI is theoretically derived. Finally, a full analysis is conducted for a RFI chirp signal showing that not all chirp signals are able to allow a statistical prediction of its effective C/N0 (condition of validity 1) and that NWPR and SNV are the more robust estimators (condition of validity 3) against this type of RFI.

Real-time time transfer is the basis for the time synchronization and establishment and maintenance of time scales. In this contribution, we present a regional multi-station real-time time transfer approach, in which observations from multi-global navigation satellite system (GNSS) received at regional multi-station are integrated to conduct an undifferenced network solution. Consequently, the simultaneous estimation of receiver clock offsets for multiple stations becomes possible, enabling the execution of real-time time transfer. One week of observation data from five multi-GNSS Experiment (MGEX) stations located in Europe are selected to generate four links (distances from 919.7 to 2153.4 km), and four schemes are designed, i.e., GPS-only, BDS-3-only, Galileo-only, and GPS/BDS-3/Galileo (GCE) solutions, respectively. Experiment results show that the mean number of observations, time dilution of precision (TDOP) values, and standard deviation (STD) of clock offset difference time series between the Center for Orbit Determination in Europe (CODE) and the estimated solutions for four time links are 90, 76, 72, 239 and 0.34, 0.38, 0.39, 0.21 and 0.039, 0.050, 0.043, 0.036 ns for GPS-only, BDS-3-only, Galileo-only, and GCE solutions, respectively. When the averaging time is shorter than 7680 s, the frequency stability of GCE solutions shows the best performance. The mean frequency stability of four time links at 7680 s is 6.59 × 10–15,7.25 × 10–15, 6.88 × 10–15 and 5.96 × 10–15 for GPS-only, BDS-3-only, Galileo-only, and GCE solutions, respectively. The GCE solution shows the best performance among the four schemes in terms of the number of observations, TDOP, STD, and frequency stability. The experiment results demonstrate that this approach is suitable for regional multi-station real-time time transfer.

The fifth GLONASS-K1 satellite with space vehicle number R807 was launched in October 2022. It represents the first spacecraft of the K1+ generation, which offers various technical innovations. Compared to previous K1 satellites, R807 also transmits a code-division multiple access (CDMA) signal in the L2 frequency band in addition to L1 and L2 frequency-division multiple access (FDMA) and L3 CDMA signals. Thus, R807 is the first spacecraft of the K1+ generation. A geometry- and ionosphere-free triple carrier combination is used to analyze the GLONASS R807 clock consistency at different frequencies. Significant inconsistencies were found showing up as variations with a peak-to-peak amplitude of up to 40 cm and periods between 15 min and a few days. Whereas the ultimate explanation for these variations is not known, it is likely that they originate from cross-talk of two oscillators with similar frequency. A short-term clock analysis for integration times up to 100 s based on the one-way carrier phase (OWCP) method shows a superior stability of the R807 clock compared to all other GLONASS satellites including the new K2 generation. The Allan deviation computed from 5 s clock estimates confirms this finding for integration times up to 600 s but shows a significant bump at longer integration times due to the periodic variations mentioned above. Single-frequency OWCP processing confirms consistency of the L1 and L2 FDMA signals whereas the L3 CDMA signal shows a slight phase shift. Although the spurious variations mask the true performance of the K1+ atomic frequency standard, its behavior at short integration times points at a new type of GLONASS satellite clock.

In urban canyons, the reflections and obstructions of Global Navigation Satellite System (GNSS) signals frequently lead to significant errors in measurements, the number of which can be larger than that of the correct measurements. This leads to a severe degradation of GNSS performance in urban canyons. Various fault detection and exclusion (FDE) algorithms have been developed to cope with these outliers caused by multipath effects. Most of these FDE algorithms check the consistency among measurements. However, in urban canyons, their effectiveness is significantly compromised by the lack of fault-free measurements. There is an urgent need to develop new constraints for enhancing GNSS FDE performance. In recent years, the advent of Chip-Scale Atomic Clock (CSAC), known for their affordability and high frequency stability, offers a promising solution for accurately predicting receiver clock errors. Additionally, using city maps to establish height constraints is another way to increase redundancy. The purpose of this study is to improve the GNSS positioning accuracy in urban canyons with the aid of CSAC and city map data. A novel FDE algorithm is developed to search for positions through the constraints of height and receiver clock. Extensive tests were conducted in urban canyons to evaluate the performance of the system. Results showed that the positioning accuracy can be improved from tens of meters to less than 6 m.

The radio frequency interference (RFI) in global navigation satellite system (GNSS) signals has recently received much attention in the GNSS community because of frequent jamming issues. The carrier-to-noise density ratio (C/N0) is one of the common parameters to indicate the signal quality. In this work, we propose a real-time RFI analysis based on windowing and normalization of C/N0 observations. Specifically, the percentage of RFI values are analyzed based on the modified RFI detection. The steps to analyze the RFI levels (low, medium, high) are highlighted. In addition, we analyzed the occurrences of local RFI effects in areas surrounding the Suvarnabhumi International Airport as well as remote areas. We validate the modified RFI detection by using the GNSS reference stations at the urban, suburban, and outside the capital city in Thailand. The user positioning errors with the high (severe) RFI levels are investigated based on the single point positioning (SPP) and real-time kinematics (RTK). From the experimental simulations, the high RFI levels at the urban are higher than those at the suburban. As expected, the statistical analysis covering COVID-19 (2019 to 2023) shows that the high RFI levels in June 2023 (post COVID-19) are more than those in June 2020 and 2021 (lockdown COVID-19) by about twofold. Additionally, the SPP positioning errors with the medium/high RFI levels are clearly seen. There are more floating solutions in the RTK system in the year with more RFI presence.

With the increasing scale and complexity of network RTK, the reliability of ambiguity resolution becomes particularly crucial. Undetected incorrect fixings may trigger a chain reaction in subsequent atmospheric delay extraction and fitting stages, thereby affecting the reliability of user positioning services. Current methods for checking abnormal ambiguities suffer from issues such as inflexible threshold selection, excessive exclusion, and overlooking observational anomalies. Addressing these concerns, we propose a new method for ambiguity detection of reference stations, referred to as the “Box-Cox transformation & Secondary screening combining Chi-square Test” (BS-CT). Firstly, the tropospheric residuals after ambiguities fixing are extracted and they are unitized through their corresponding co-variance matrix. Secondly, the Box-Cox method is employed to transform the distribution of the unitized residuals, making them conform to a standard normal distribution. This enables the use of a chi-square test to eliminate satellites with observation anomalies. Finally, a secondary-screening method is applied to ensure the reliability of the fixed ambiguity quantity. In the experimental section, the BS-CT method was contrasted with the ordinary chi-square test, Partial Ambiguity Resolution method (PAR), and a method utilizing a decision function g for enhanced fixed fraction and variance strategy. The results indicate that, compared to the other three methods, lower fall-out ratio in anomaly ambiguity testing is observed with the BS-CT method. It is comparable to PAR in terms of omission ratio and performs lower than the other two methods.

To address the background noise interference in GNSS-RTK during prolonged structural monitoring, the integration of the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and the standard deviation (STD)-based principal component analysis (PCA) method (CEEMDAN-PCA-S) is proposed for de-noising. The CEEMDAN-PCA-S aims to de-noise the internal random noise by leveraging its white noise characteristics and subsequently extract the multipath error by leveraging its periodic repetition characteristics. The efficiency of CEEMDAN-PCA-S is verified with a simulation and a field experiment. The simulation results demonstrate that the STD-based PCA (PCA-S) could achieve superior amplitude consistency with the original signal compared to traditional PCA. Additionally, the correlation coefficients of the multipath error obtained with different algorithms are PCA-S (0.9243) < CEEMDAN (0.9582) < CEEMDAN-PCA-S (0.9945). Moreover, the residual error of CEEMDAN-PCA-S is minimal in terms of amplitude and root mean square error (RMSE), confirming its superior de-noising performance compared to CEEMDAN and PCA-S. The outcomes of a field experiment utilizing the GNSS-RTK indicate that the correlation coefficients between the multipath error extracted with CEEMDAN-PCA-S and the CEEMDAN residual errors are all above 0.94. The CEEMDAN-PCA-S decreases the RMSE of residual error to 0.1599 cm, marking a 79% reduction from the original signal. Moreover, both the amplitude and mean value are reduced by 83% and 16%. In conclusion, the proposed CEEMDAN-PCA-S could effectively remove white noise and subsequently extract the multipath error, enhancing the accuracy of GNSS-RTK for long-term structural monitoring and safety warnings.

Precise orbit, clock and observable-specific phase bias products play a pivotal role in facilitating precise point positioning (PPP) with ambiguity resolution (AR). The model and strategy used to generate Wuhan University Multi-GNSS (WUM) hourly updated ultra-rapid products is summarized in this study. A refined iteration of the block-to-block operation method is introduced, which demonstrates a substantial increase in efficiency, achieving enhancements of 13-fold and 29-fold for Central Processing Unit and Graphics Processing Unit platforms, respectively. Subsequently, a pre-integration approach is introduced for numerical orbit integration, resulting in a noticeable improvement of efficiency ranging from 6 to 28 times. To enhance the solution strength and precision, a single-differenced-based ambiguity fixing method is explored in our routine processing, which improves for all the satellites, except for GLONASS, and multi-GNSS products are released with a 1-h latency. The orbit assessment is conducted through a comparative analysis with International GNSS Service (IGS) and the Wuhan University Multi-GNSS Combined orbits. The three-dimensional precision of the near real-time orbits reaches 2.4 cm, 4.6 cm, 3.8 cm, and 4.9 cm for GPS, GLONASS, Galileo, and BDS3 MEO satellites, respectively. Regarding IGSO satellites of BDS and QZSS, the corresponding values range between 11.4 cm and 15.7 cm. In contrast, the GEOs of BDS and QZSS exhibit comparatively inferior performance, demonstrating consistencies of 234.7 cm and 70.9 cm, respectively. The validation of the near real-time products is performed utilizing PPP-AR. The statistical analysis across a global network of 340 stations with 20 days reveals a wide-lane fixing rate exceeding 91.1% for multi-GNSS, accompanied by a narrow-lane fixing rate of approximately 97.0%. The PPP-AR solution demonstrates the accuracy of 1.6 mm, 1.7 mm, and 5.0 mm in the east, north, and up directions for IGS weekly solutions, respectively. The results are promising and further confirm that the products are effective.

We examine the decadal evolution of GPS, GLONASS, and Galileo satellite orbital elements, including the semi-major axis, inclination, eccentricity, right ascension of the ascending node, and the argument of perigee. We focus on the long-term changes in Keplerian elements by averaging them over several complete revolutions forming mean orbital elements giving an explanation of the main perturbing forces for each Keplerian parameter. The combined International GNSS Service (IGS) orbits are employed which were derived in the framework of IGS Repro3 for ITRF2020 preparation spanning eight years from 2013 to 2021. The semi-major axis for GPS satellites is affected by a strong resonance with Earth’s gravity field resulting in a long-period perturbation similar to a secular drift. The semi-major axes of Galileo and GLONASS do not show any large-scale rates, however, Galileo satellites are affected by the Y-bias resulting in semi-major axis drifts. A significant perturbations due to solar radiation pressure affect the semi-major axis, eccentricity, and the argument of perigee. Notably, for Galileo satellites in eccentric orbits, the signal with a one-draconitic year is evident in the semi-major axis. The evolution of the mean right ascension of the ascending node and argument of perigee is primarily characterized by nearly linear regression mainly due to even zonal harmonics of the Earth’s gravity field. The long-term evolution of eccentricity and inclination does not follow a linear trend but exhibits clear oscillations dependent on the secular drift of the right ascension of the ascending node (for inclination) or the argument of perigee (for eccentricity). Additionally, the long-term perturbation of inclination reaches its maximum when the absolute value of the Sun’s elevation angle above the orbital plane ( \(\beta\) angle) is at its minimum, while the eccentricity reaches its minimum simultaneously with the minimum of the \(\beta\) angle.

Real-time service (RTS) products are an important guarantee for real-time precise point positioning (RT-PPP), and the RTS outages caused by loss of network connection are a concern. In this paper, a multivariate CNN-LSTM model is proposed for short-term BDS satellite clock offset prediction during the discontinuity in receiving RTS clock offsets, which utilizes the superior feature of convolution neural network (CNN) and long short-term memory (LSTM) for simultaneous prediction of multiple satellite clock offsets by considering the inter-satellite correlation. First, the correlation between satellite clock offsets was analyzed to identify satellites suitable for parallel prediction. Then, to preserve the sequential structure of the features extracted from multiple parallel satellite clock offsets, remove the pooling layer of traditional CNN, and use the convolution layer to learn the relationships and dependencies between clock offsets of different satellites and the LSTM layer to model the temporal dependencies in satellite clock offsets. The experiment results show that the computational efficiency of the proposed model is significantly better than that of autoregressive integrated moving average (ARIMA), wavelet neural network (WNN), and LSTM models. Compared with the linear polynomial (LP), quadratic polynomial model (QP), ARIMA, WNN, and the LSTM models, the prediction accuracy of the multivariate CNN-LSTM model for 5 min, 15 min, 30 min, and 1 h is improved by approximately (84.0, 76.6, 1.5, 8.3, 8.3)%, (72.0, 62.6, 6.0, 15.3, 18.7)%, (57.1, 48.5, 11.3, 18.4, 23.3)%, and (34.9, 35.1, 27.3, 21.8, 26.3)%, respectively.

Global navigation satellite system reflectometry (GNSS-R) has emerged as a pivotal remote sensing (RS) technology, widely utilized for retrieving crucial oceanic parameters such as wind speed, sea surface height, and sea ice detection. However, the retrieval of ocean swell height remains an underexplored area within this domain. The complexity of constructing multivariate regression models for swell height retrieval poses a significant challenge, particularly in contrast to existing empirical models. For this purpose, this article proposes a novel deep learning (DL) hybrid model, namely Multi-scale Conv-BiLSTM, which combines multi-scale convolution and bidirectional long short-term memory (BiLSTM) networks for the first time to retrieve ocean swell height using spaceborne GNSS-R data. This innovative hybrid model comprises three fundamental modules: a multi-scale feature extraction module, a feature relationship inference module based on BiLSTM network, and a manual extraction of multiple feature parameters module encompassing GNSS-R variable and auxiliary variable. Specifically, the multi-scale feature extraction module leverages deep convolutional neural network (DCNN) to extract spatial features surrounding the specular reflection point (SP) from the two-dimensional matrix of the bistatic radar scattering cross-section (BRCS) image and effective scattering area. Subsequently, the feature relationship inference module employs the BiLSTM network to engage in the inference process between feature relationships. This module excels in considering critical information associated with temporal characteristics, effectively capturing preceding and subsequent information. Validation was conducted using ERA5 and WaveWatch III (WW3) data by comparingthe proposed Multi-scale Conv-BiLSTM model against seven traditional machine learning (ML) models, including support vector machine (SVM), decision tree (DTR), light gradient boosting machine (Lightgbm), eXtreme gradient boosting (XGBoost), adaptive boosting (AdaBoost), gradient boosting decision tree (GBDT) and DCNN. The results show that when ERA5 is used as reference data, the proposed Multi-scale Conv-BiLSTM model achieves a reduction in root mean square error (RMSE) by 23.67%, 28.63%, 9.77%, 8.91%, 28.50%, 16.46%, and 15.05% compared to the SVM, DTR, Lightgbm, XGBoost, AdaBoost, GBDT, and DCNN models, respectively. When WW3 is used as reference data, the proposed Multi-scale Conv-BiLSTM model exhibits an improvement in RMSE by 35.99%, 36.62%, 25.49%, 24.74%, 41.37%, 30.82%, and 29.61% compared to the SVM, DTR, Lightgbm, XGBoost, AdaBoost, GBDT, and DCNN models, respectively.

Real-time precise orbits of low earth orbit (LEO) satellites are becoming indispensable with the rapid development of real-time Earth observation application and LEO enhanced precise point positioning. Currently, GNSS-based precise orbit determination is a widely used method for LEO onboard navigation. However, the real-time LEO satellite orbits are usually obtained by ambiguity-float solution even when the GNSS augmentation corrections are considered. In this study, we perform LEO ambiguity-fixed multi-GNSS real-time precise orbit determination (RTPOD) based on square root information filter. One month of onboard GPS + Galileo observations from Sentinel-6A and real-time products of the Centre National d’Etudes Spatiales (CNES) are used to investigate the contribution of integer ambiguity resolution (IAR). The benefit of dual-system combination on LEO RTPOD is firstly evaluated. The combination of GPS and Galileo dual-system contributes to more visible GNSS satellites and better observation geometry for LEO RTPOD, which results in an evident accuracy improvement of 19% over GPS and a convergence time reduction of 43% and 41% compared to the GPS and Galileo solutions respectively. Considering the short arc of onboard GNSS observations and imperfections in GNSS real-time products, we propose a strict IAR quality control method to avoid fixing the ambiguity to the wrong values. The results indicate that the IAR quality control method we used can effectively reduce the wrong fixing risk and increase the robustness of the IAR solution. Using onboard GPS and Galileo observations, the 3D orbit accuracy of the ambiguity-fixed solution is significantly improved from 5.17 to 3.61cm, by 30%, compared to the ambiguity-float solution. Furthermore, the application of IAR also achieves a faster convergence to the centimeter-level orbit.

This study proposes a data-driven troposphere zenith total delay (ZTD) modeling method that takes into account the distance of GNSS continuously operating reference station (CORS) to the coast. Using ZTD data from 106 CORS stations, the strong correlation between the CORS-to-coast distance and ZTD is first identified. Then, this CORS-to-coast distance is incorporated as a new input data, along with the traditional time epoch and location input data, to develop a deep learning-based neural network model for ZTD prediction. This model is trained using 96 CORS stations spaced an average of 92 km apart. Ten testing CORS stations are divided into five inner stations and five outer stations from the CORS network to evaluate the ZTD prediction accuracy. Results from the study show that the proposed method improves the accuracy of ZTD prediction over traditional methods for a four-month period in 2018. At inner testing stations, the average ZTD prediction Root-mean-square errors (RMSEs) of the proposed method is 29.5 mm, which is smaller than the 34.2 mm of the traditional method. For outer testing stations, the average ZTD prediction RMSEs are 31.0 mm and 41.3 mm for the proposed method and traditional method respectively, resulting in a 5/17% ZTD prediction accuracy improvement. To sum up, the proposed method, which considers the CORS-to-coast distance for ZTD modeling, is demonstrated to enhance ZTD prediction accuracy over the traditional method.

Ground-based Global Navigation Satellite Systems (GNSS) augmentation services such as real-time service (RTS) facilitate smartphones in achieving real-time lane-level precise point positioning (PPP). However, in mountainous areas and similar regions, ground-communication network blind spots are common, leading to GNSS-augmented positioning failures. Additionally, although the integration of GNSS and Visual-Inertial Navigation System (VINS) can maintain high precision and continuous positioning, VINS consumes excessive battery power. Therefore, this paper proposes a lightweight seamless lane-level positioning method for smartphones with the integration of PPP and VINS based on BeiDou navigation satellite system (BDS-3) PPP-B2b services. The You Only Look Once V5 (YOLOv5) algorithm is employed to recognize road signs. When structures like bridges or highway toll booths are recognized, the factor graph algorithm combines PPP-B2b/VINS to achieve positioning. The experimental findings reveal that real-time PPP-B2b achieves a horizontal positioning accuracy of 1.48 m in dynamic scenarios. Upon implementing the PPP-B2b/VINS model, this accuracy improves to 0.99 m. Moreover, in instances where the PPP-B2b positioning solution is unavailable, smartphone VINS demonstrates the capability to maintain a horizontal positioning accuracy of 1 m within 20-30 s. This symbiotic relationship between PPP-B2b and VINS establishes a lightweight and seamless solution ideal for applications requiring continuous and precise positioning of smartphone, such as highway navigation, and various lane-level services.

Positioning technology based on signals of opportunity (SOPs) of low-earth-orbit (LEO) satellites has become an effective global positioning backup in GNSS denial environments. Currently, Doppler measurements are widely used because they are easy to obtain. However, the Doppler positioning has poor accuracy and stability, which is mainly due to the low Doppler measurement accuracy and the complex system errors caused by the satellite position and velocity errors of the two-line element (TLE) and the simplified general perturbation 4 (SGP4) model. Therefore, we first design a carrier tracking loop based on squaring and code phase assistance (S-CPA) for Orbcomm SOPs to obtain accurate carrier phase measurements under lower carrier-to-noise ratio (CNR) conditions. Second, we analyze the observability of main system errors in the carrier phase observation model. Subsequently, a carrier phase positioning algorithm with additional system parameters is proposed to reduce the influence of system errors. Additionally, three typical epoch selection schemes are considered for the multi-epoch positioning system. Simulation results show that the CNR threshold of the S-CPA loop is approximately 10 dB Hz lower than that of the conventional Costas loop. Experimental results show a three-dimensional (3D) positioning root mean squared error (RMSE) of 77.5 m; the positioning accuracy and stability of the proposed algorithm are 48% and 45% higher than that of the Doppler positioning, respectively.

Low-cost single-frequency receivers are increasingly applied to high-precision positioning in support of high-performance GNSS market applications. However, frequent and multiple small cycle slips (CS) detection and repair is a limiting factor for single-frequency receiver-based high-precision positioning techniques, particularly in challenging environments. In this contribution, a single-frequency CS detection and repair method for a standalone GNSS receiver is proposed by using a common-antenna-based dual-board design. This design can produce two independent phase measurements received from the two boards at the same frequency for each satellite. The proposed method can construct between-board, between-epoch, and between-satellite triple differences by using these measurements to eliminate interference errors as statistics for detecting and repairing CS. If the reference satellite has a CS, the CS is transferred among the statistics caused by the between-satellite difference operation, which inevitably leads to incorrect CS detection. To solve this problem, the proposed method further considers adding a common clock design between the dual boards sharing one antenna. In this way, the between-board difference operation can directly eliminate receiver clock errors without performing the between-satellite difference operation, thereby avoiding the problem of the reference satellite. A real-world static and kinematic experiment was conducted to test the proposed method. Compared with traditional methods, the proposed method can achieve better CS detection and repair performance in terms of different multipath effects, dynamics, and sampling intervals, making it possible to obtain better positioning performance in carrier phased-based positioning techniques.

Zenith Tropospheric Delay (ZTD) is one of the main atmospheric errors in the Global Navigation Satellite System (GNSS). In this study, we propose a novel ZTD forecasting model based on the deep-learning method named Informer-based ZTD (IBZTD) forecasting model using the European Centre for Medium-Range Weather Forecasts (ECMWF) fifth generation reanalysis data (ERA5) from 2011 to 2021. With 72-hour historical GNSS-derived ZTDs as prior information, the subsequent 24-hour ZTDs can be forecasted. The IBZTD forecasting model achieves the best regression fit with post GNSS-derived ZTDs compared with GPT3 (Global Pressure and Temperature 3) and HGPT2 (Hourly Global Pressure and Temperature 2) models, especially in winter with a Root Mean Square Error (RMSE) of 1.51 cm and a Mean Absolute Error (MAE) of 1.15 cm. With the post GNSS-derived ZTDs as reference, in terms of the overall 24-hour forecasting accuracy for 9 GNSS stations in 2022, IBZTD forecasting model achieves a MAE of 1.66 cm and a RMSE of 2.21 cm, significantly outperforming the GPT3 model (MAE: 2.60 cm, RMSE: 3.37 cm), HGPT2 model (MAE: 3.23 cm, RMSE: 4.03 cm) and Long Short-Term Memory (LSTM) model (MAE: 2.65 cm, RMSE: 3.65 cm). An average time improvement of 17.8% and comparable forecasting precisions are achieved in the IBZTD forecasting model compared with the Transformer-based ZTD (TBZTD) forecasting model. Using predicted ZTD as prior constraints in Precise Point Positioning (PPP), the vertical convergence speed exhibits a significant improvement of 14.20%, 20.24%, 18.48%, and 19.39% in four seasons.