Carrier Phase Observations Research Articles

A robust GNSS velocity estimation method combining Doppler and carrier phase observations in complex urban environments

Accurate and reliable velocity information is crucial for kinematic positioning, as it not only characterizes the carrier state but also serves as constraint information for positioning. However, due to the influence of low-cost Global Navigation Satellite System (GNSS) chips and complex environments, GNSS signals are prone to attenuation and interruptions. Consequently, frequent gross errors and cycle slips occur in observations, significantly degrading the velocity precision and reliability. To address this challenge, we have proposed a method combining Doppler and Time-Differenced Carrier Phase Velocity Estimation (D-TDCPVE). First, we assess the accuracy of both Doppler and carrier phase observations to determine the appropriate weight ratio for their combination. Second, we analyze the correlation between the two observation types and propose an adaptive parameter estimation strategy for estimating one or two clock bias variations to address their potential differences. Finally, we incorporate a combination of pre- and post-detection quality control measures along with additional cycle slip parameters to mitigate the impact of gross errors and cycle slips in complex environments. Experimental results conducted with three low-cost devices in urban environments demonstrate the superior performance of the D-TDCPVE method over both Doppler velocity estimation (DVE) and time-differenced carrier phase velocity estimation (TDCPVE). It enhances the success rate of velocity estimation by 3.3% to 20.1% compared to TDCPVE and improves velocity estimation accuracy by 16.1% to 60.9% compared to DVE. Moreover, the method does not necessitate the combination of dual-frequency observations, making it particularly valuable for cycle slip detection in scenarios involving mixed single- and dual-frequency observations, which is particularly useful in urban environments.

Comprehensive analysis of different GNSS receivers performance based on PPP-AR and positioning accuracy during 22 geomagnetic storms in 2023

Geomagnetic storms induced ionospheric disturbances can degrade the positioning accuracy and Ambiguity Resolution (PPP-AR) of GPS Precise Point Positioning (PPP), and this negative effect varies among different GNSS receivers. Under current conditions with frequent geomagnetic storms occurrences during the peak of solar cycle 25, selecting GNSS receivers with strong resistance to ionospheric disturbances is meaningful for precise GNSS positioning and ionospheric research. Therefore, it is necessary to conduct a comprehensive comparison and evaluation of the performance for different GNSS receivers under geomagnetic storms. Based on 22 geomagnetic storms in 2023, we investigated the three-dimensional root mean square error (3D RMS) and ambiguity resolved percentage (ARP), which is defined as the ratio of the number of carrier-phase observations with fixed ambiguities to the total number of phase measurements of GPS kinematic PPP, across ten groups of collocated stations around the world. For the groups using different receiver brands, the performance ranking for positioning accuracy and PPP-AR during geomagnetic storms are as follows: TPS > JVAVD > SEPTENTRIO > TRIMBLE. The station with a larger average ARP generally has a smaller average 3D RMS error than the another GNSS station. The average 3D RMS error between collocated stations using different receiver brands is typically greater than 0.010 m and even larger. The largest 3D RMS error difference is observed between collocated stations with SEPTENTRIO and JAVAD receivers, with 3D RMS errors of 0.089 m and 0.019 m, respectively. These identical-brand receivers with different types are as follows: TPS NET-G5 > TPS NET-G3A, TRIMBLE ALLOY > TRIMBLE NETRS, SEPT POLARX5TR > SEPT POLARX5S, and LEICA GR10 > LEICA GR30, respectively. The average differences in 3D RMS and ARP can reach up to 0.021 m and 6.0%, respectively. We found that the choice of antennas does not significantly affect PPP positioning performance during storms, with differences in average 3D RMS below 0.005 m and ARP differences below 0.2%.. Higher latitudes have more satellites affected by ionospheric disturbances, while this effect is typically only observed during strong storms (Dstmin≤-100nT) in mid-latitudes. Adequate available GPS satellites are crucial for achieving high PPP-AR and positioning accuracy during storms. The variations of disturbed satellites and the average Rate of Total Electron Content Index (ROTI) for collocated stations are generally consistent. However, there are differences in the average of ROTI values for different receivers under geomagnetic storms. The differences in PPP-AR performance and ROTI values among different receivers under geomagnetic storms can be attributed to the variations in satellite tracking algorithms, tracking loop designs, firmware, and multipath mitigation methods used by different receivers. These findings can provide a reference value and future research direction for selection, application and enhancement for different receivers to achieve GNSS high precision positioning under complex space weather conditions.

An Improved Velocity-Aided Method for Smartphone Single-Frequency Code Positioning in Real-World Driving Scenarios

The availability of Global Navigation Satellite System (GNSS) raw observations in smartphones has driven research into low-cost GNSS solutions, especially in challenging urban environments, which have garnered significant attention from scholars in recent years. This study proposes an improved smartphone-based velocity-aided positioning method and conducts vehicle-mounted experiments in urban roads representing typical scenarios. The results show that when transitioning from low- to high-multipath environments, the number of visible satellites and carrier phase observations are highly sensitive to environmental factors, with frequent multipath effects. The introduction of robust pre-fit and post-fit residual algorithms has proven to be an effective quality control method. Additionally, using more refined observation models and appropriate parameter estimation algorithms led to a slight 6% improvement in velocity performance. The improved Kalman filter position estimation model (KFSPP-P) strategy, by incorporating velocity uncertainty into the state estimation process, overcomes the limitations of conventional velocity-aided smartphone positioning methods (KFSPP-V) in complex urban environments. In low-multipath environments, the accuracy of the KFSPP-P strategy is comparable to that of KFSPP-V, with an approximate 8% improvement in horizontal accuracy. However, in more challenging environments, such as tree-lined roads and urban environments, the KFSPP-P strategy shows significant improvements, particularly enhancing horizontal positioning accuracy by approximately 50%. These advancements demonstrate the potential of using smartphones to provide reliable positioning services in complex urban environments.

Continuous High-Precision Positioning in Smartphones by FGO-Based Fusion of GNSS-PPK and PDR.

The availability of raw Global Navigation Satellites System (GNSS) measurements in Android smartphones fosters advancements in high-precision positioning for mass-market devices. However, challenges like inconsistent pseudo-range and carrier phase observations, limited dual-frequency data integrity, and unidentified hardware biases on the receiver side prevent the ambiguity resolution of smartphone GNSS. Consequently, relying solely on GNSS for high-precision positioning may result in frequent cycle slips in complex conditions such as deep urban canyons, underpasses, forests, and indoor areas due to non-line-of-sight (NLOS) and multipath conditions. Inertial/GNSS fusion is the traditional common solution to tackle these challenges because of their complementary capabilities. For pedestrians and smartphones with low-cost inertial sensors, the usual architecture is Pedestrian Dead Reckoning (PDR)+ GNSS. In addition to this, different GNSS processing techniques like Precise Point Positioning (PPP) and Real-Time Kinematic (RTK) have also been integrated with INS. However, integration with PDR has been limited and only with Kalman Filter (KF) and its variants being the main fusion techniques. Recently, Factor Graph Optimization (FGO) has started to be used as a fusion technique due to its superior accuracy. To the best of our knowledge, on the one hand, no work has tested the fusion of GNSS Post-Processed Kinematics (PPK) and PDR on smartphones. And, on the other hand, the works that have evaluated the fusion of GNSS and PDR employing FGO have always performed it using the GNSS Single-Point Positioning (SPP) technique. Therefore, this work aims to combine the use of the GNSS PPK technique and the FGO fusion technique to evaluate the improvement in accuracy that can be obtained on a smartphone compared with the usual GNSS SPP and KF fusion strategies. We improved the Google Pixel 4 smartphone GNSS using Post-Processed Kinematics (PPK) with the open-source RTKLIB 2.4.3 software, then fused it with PDR via KF and FGO for comparison in offline mode. Our findings indicate that FGO-based PDR+GNSS-PPK improves accuracy by 22.5% compared with FGO-based PDR+GNSS-SPP, which shows smartphones obtain high-precision positioning with the implementation of GNSS-PPK via FGO.

An approach to improve smartphone-based PPP performance in constrained environments using Doppler observations

ABSTRACT The smartphone-based precise point positioning (PPP) technique is usually used in limited satellite visibility environments, resulting in discontinuities in carrier phase observations. This study proposes an approach to improve the smartphone-based PPP performance by predicting the missed carrier phase observations with Doppler observations. Kinematic smartphone-based PPP experiments in the tree canopy and high-building environments are conducted, and results show that the proposed PPP approach improves the horizontal positioning accuracy by over 31% and 14% and vertical positioning accuracy by over 23% and 10% when compared to the PPP approaches without predictions and with linear predictions, respectively.

Improving the stochastic model for code pseudorange observations from Android smartphones

In recent years, there has been increasing attention to positioning, navigation, and timing applications with smartphones. Because of frequently disrupted carrier phase observations, code observations remain critical for smartphone-based positioning. Considering a realistic stochastic model is mandatory to obtain the utmost positioning performance, this study proposes a sound stochastic approach for code observations from Android smartphones. The proposed approach includes a modified version of the SIGMA-ɛ variance model with different coefficients for each GNSS constellation and a robust Kalman filter method. First the noise characteristics of observations from the Xiaomi Mi 8 smartphone are analyzed utilizing code-minus-phase combinations to estimate the coefficients for each GNSS constellation. This includes the determination of a variance model as well as a check of the probability distribution. Finally, the proposed approach is validated in the positioning domain using single-frequency code observation-based real-time standalone positioning. The results show that more than 95% of observations follow the normal distribution when the proposed approach is applied. Compared with the conventional stochastic approach, including a C/N0-dependent model and standard Kalman filter, it improves the positioning accuracy by 45.8% in a static experiment, while its improvement is equal to 26.6% in a kinematic experiment. For the static and kinematic experiments, in 50% of the epochs, the 3D positioning errors are smaller than 3.0 m and 3.4 m for the proposed stochastic approach. The results exhibit that the stochastic properties of code observations from smartphones can be successfully represented by the proposed approach.

GLONASS-K2 signal analysis

K2 is a new generation of GLONASS satellites that provides code division multiple access (CDMA) signals in the L1, L2 and L3 frequency bands in addition to legacy L1 and L2 signals based on frequency division multiple access (FDMA) modulation. The first GLONASS-K2 satellite was launched in August 2023 and started signal transmission in early September 2023. Based on measurements with a 30-m high-gain antenna, spectral characteristics of the various signal components are described and relative power levels are identified. A 3 dB (L1) to 4 dB (L2) higher total power is determined for the CDMA signal compared to the legacy FDMA signal and an equal power of the open service and secured CDMA signal components is found. The ranging code of the L2 channel for service information, which has not been publicly disclosed so far, is identified as a Gold code sequence consistent with the data channel of the L1 open service CDMA signal. The high-gain antenna measurements are complemented by tracking data from terrestrial receivers that enable a first assessment of user performance. An up to 50% improvement in terms of noise and multipath performance is demonstrated for the new L1 and L2 CDMA signals in comparison with their legacy counterpart, but no obvious differences between the different binary phase-shift keying and binary offset carrier modulations of the data and pilot components of these signals could be identified for the test stations. Triple-frequency carrier phase observations from L1, L2, and L3 CDMA signals exhibit good consistency at the noise and multipath level, except for small variations that can be attributed to slightly different antenna phase patterns on the individual frequencies. Overall, the new CDMA signals are expected to notably improve and facilitate precise point positioning applications once fully deployed across the GLONASS constellation.

Correlation between rate of TEC index and positioning error during solar flares and geomagnetic storms using navigation with Indian constellation receiver measurements

Abstract The real-time position accuracy of the Global Navigation Satellite System (GNSS)/Navigation with Indian Constellation (NavIC) receiver is limited by the dynamic behavior of the ionosphere, particularly in adverse conditions like solar flares and geomagnetic storms. The NavIC satellites broadcast dual coherent radio beacon signals on L5 (1,164.5 MHz) and S (2,472.5 MHz) bands for providing position, velocity, and timing services in all weather conditions. The Total Electron Content (TEC) and Rate of TEC Index (ROTI) are the potential indicators for characterizing the ionosphere and its irregularities. In this research work, the TEC and ROTI are computed from the code and carrier phase observations of the NavIC receiver located at Kurnool low latitude station (15.79° N, 78.07° E) with geomagnetic coordinates (7.30° N, 151.65° E). This paper presents a statistical study of TEC, ROTI, and the correlation between ROTI and NavIC positioning error during highly intense solar flares (X9.3 and X2.2) and geomagnetic storm conditions. Compared to quiet days mean TEC, the enhancement is 3 TECU due to X9.3 flares, and the maximum peak of TEC on storm day (September 8, 2017) is 80.92 TECU. Moreover, the correlation coefficient between ROTI and position error is 0.76 on a quiet day (September 4, 2017), 0.54 on an intense solar flares day (September 6, 2017), and 0.24 on a storm day (September 8, 2017), this indicates positional accuracy degradation on a geomagnetic storm day. The outcome of this research work would be helpful for investigating characteristics of the northern low latitude ionospheric irregularities and, in turn, useful for developing suitable ionospheric nowcasting/prediction models for GNSS applications.

An EOF-Based Global Plasmaspheric Electron Content Model and Its Potential Role in Vertical-Slant TEC Conversion

Topside total electron content (TEC) data measured by COSMIC/FORMAT-3 during 2008 and 2016 were used to analyze and model the global plasmaspheric electron content (PEC) above 800 km with the help of the empirical orthogonal function (EOF) analysis method, and the potential role of the proposed PEC model in helping Global Navigation Satellite System (GNSS) users derive accurate slant TEC (STEC) from existing high-precision vertical TEC (VTEC) products was validated. A uniform gridded PEC dataset was first obtained using the spherical harmonic regression method, and then, it was decomposed into EOF basis modes. The first four major EOF modes contributed more than 99% of the total variance. They captured the pronounced latitudinal gradient, longitudinal differences, hemispherical differences, diurnal and seasonal variations, and the solar activity dependency of global PEC. A second-layer EOF decomposition was conducted for the spatial pattern and amplitude coefficients of the first-layer EOF modes, and an empirical PEC model was constructed by fitting the second-layer basis functions related to latitude, longitude, local time, season, and solar flux. The PEC model was designed to be driven by whether solar proxy or parameters derived from the Klobuchar model meet the real-time requirements. The validation of the results demonstrated that the proposed PEC model could accurately simulate the major spatiotemporal patterns of global PEC, with a root-mean-square (RMS) error of 1.53 and 2.24 TECU, improvements of 40.70% and 51.74% compared with NeQuick2 model in 2009 and 2014, respectively. Finally, the proposed PEC model was applied to conduct a vertical-slant TEC conversion experiment with high-precision Global Ionospheric Maps (GIMs) and dual-frequency carrier phase observables of more than 400 globally distributed GNSS sites. The results of the differential STEC (dSTEC) analysis demonstrated the effectiveness of the proposed PEC model in aiding precise vertical-slant TEC conversion. It improved by 18.52% in dSTEC RMS on a global scale and performed better in 90.20% of the testing days compared with the commonly used single-layer mapping function.

GNSS carrier phase common-view precision time transfer using the ionospheric-weighted single-differenced model

ABSTRACT High-precision global navigation satellite system (GNSS) time transfer depends on the carrier phase observations, and ionospheric delay affects rapid integer ambiguity resolution. Therefore, this work applies the ionospheric-weighted single-differenced model. Experimental results indicate that for a time-link of 67.9 km, the time transfer accuracy of different single-system is better than 0.03 ns. Compared with single-system, multi-system demonstrates 8% to 14% improvement in accuracy and improvement of no more than 31% in frequency stability. Additionally, compared with the case without ionospheric constraint, imposing ionospheric constraints can improve performance of time transfer, but the gain decreases as the time-link length increases.

A tightly coupled GNSS RTK/IMU integration with GA-BP neural network for challenging urban navigation

Abstract Intelligent transportation system is increasing the importance of real-time acquisition of positioning, navigation, and timing information from high-accuracy global navigation satellite systems (GNSS) based on carrier phase observations. The complexity of urban environments, however, means that GNSS signals are prone to reflection, diffraction and blockage by tall buildings, causing a degraded positioning accuracy. To address this issue, we have proposed a tightly coupled single-frequency multi-system single-epoch real-time kinematic (RTK) GNSS/inertial measurement unit (IMU) integration algorithm with the assistance of genetic algorithm back propagation based on low-cost IMU equipment for challenging urban navigation. Unlike the existing methods, which only use IMU corrections predicted by machine learning as a direct replacement of filtering corrections during GNSS outages, this algorithm introduces a more accurate and efficient IMU corrections prediction model, and it is underpinned by a dual-check GNSS assessment where the weights of GNSS measurements and neural network predictions are adaptively adjusted based on duration of the integrated system GNSS failure, assisting RTK/IMU integration in GNSS outages or malfunction conditions. Field tests demonstrate that the proposed prediction model results in a 68.69% and 69.03% improvement in the root mean square error in the 2D and 3D component when the training and testing data are collected under 150 s GNSS signal-blocked conditions. This corresponds to 52.43% and 51.27% for GNSS signals discontinuously blocked with 500 s.

Influence of Inter-System Biases on Combined Single-Frequency BDS-2 and BDS-3 Pseudorange Positioning of Different Types of Receivers

The BeiDou Navigation Satellite System (BDS) has developed rapidly, and the combination of BDS Phase II (BDS-2) and BDS Phase III (BDS-3) has attracted wide attention. It is found that there are code ISBs between BDS-2 and BDS-3, which may have a certain impact on the BDS-2 and BDS-3 combined positioning. This paper focuses on the performance of BDS-2/BDS-3 combined B1I single-frequency pseudorange positioning and investigates the positioning performance with and without code ISBs correction for different types of receivers, include geodetic GNSS receivers and low-cost receivers. The results show the following: (1) For geodetic GNSS receivers, the code ISBs of each receiver is about −0.3 m to −0.8 m, and the position deviation is reduced by 7% after correcting code ISBs. The code ISBs in the baseline with homogeneous receivers has a little influence on the positioning result, which can be ignored. The code ISBs in the baseline with heterogeneous receivers is about −0.5 m, and the position deviation is reduced by 4% after correcting code ISBs. (2) The code ISBs in the low-cost receivers are significantly larger than those in the geodetic GNSS receivers, and the impact on the positioning performance of the low-cost receivers is significantly greater than that on the geodetic GNSS receivers. After correcting the code ISBs, the position deviation of low-cost receivers can be reduced by around 12% for both undifferenced and differenced modes. (3) For low-cost receivers, correcting the code ISBs can increase the number of epochs successfully solved, which effectively improves the low-cost navigation and positioning performance. (4) The carrier-phase-smoothing method can effectively reduce the distribution dispersion of code ISBs and make the estimation of ISBs more accurate. The STD values of estimated code ISBs in geodetic GNSS receivers are reduced by about 40% after carrier-phase smoothing, while the corresponding values are reduced by about 7% in low-cost receivers due to their poor carrier-phase observation quality.

Intercomparison of multi-GNSS signals characteristics acquired by a low-cost receiver connected to various low-cost antennas

With the increasing number of low-cost GNSS antennas available on the market, there is a lack of comprehensive analysis and intercomparison of their performance. Moreover, multi-GNSS observation noises are not well recognized for low-cost receivers. This study characterizes the quality of GNSS signals acquired by low-cost GNSS receivers equipped with eight types of antennas in terms of signal acquisition, multipath error and receiver noise. The differences between various types of low-cost antennas are non-negligible, with helical antennas underperforming in every respect. Compared with a geodetic-grade station, GPS and Galileo signals acquired by low-cost receivers are typically weaker by 3–9 dB-Hz. While the L1, E1 and E5b signals are well-tracked, only 72% and 86% of L2 signals are acquired for GPS and GLONASS, respectively. The signal noise for pseudoranges varies from 0.12 m for Galileo E5b to over 0.30 m for GLONASS L1 and L2, whereas for carrier-phase observations it oscillates around 1 mm for both GPS and Galileo frequencies, but exceeds 3 mm for both GLONASS frequencies. Antenna phase center offsets (PCOs) vary significantly between frequencies and constellations, and do not agree between two antennas of the same type by up to 25 mm in the vertical component. After a field calibration a of low-cost antenna and consistent application of PCOs, the horizontal and vertical accuracy is improved to a few millimeter and a few centimeter level for the multi-GNSS processing with double-differenced and undifferenced approach, respectively. Last but not least, we demonstrate that PPP-AR is possible also with low-cost GNSS receivers and antennas, and improves the precision and convergence time. The results prove that selection of low-cost antenna for a low-cost GNSS receiver is of great importance in precise positioning applications.

Path planning of substation inspection robot based on high-precision positioning and navigation technology

Abstract Outdoor substation is an important part of power system. Substation inspection robot based on intelligent autonomous inspection system has become the research focus of substation unmanned inspection. In order to improve the positioning accuracy and speed of the system, a high-precision positioning algorithm of transformer detection robot is proposed in this paper. Tikhonov regularization is used to correct the pathological problem of the localization algorithm model. The observation amount of the receiver is increased by using four signals of a single base station with double frequency and double antenna, and the position is solved by using single difference carrier phase observation and the integer ambiguity is fixed. The input–output mapping of the neural network is designed according to the information acquisition and two-wheel angular velocity control of the detection robot. Using the hyperbolic tangent function as the activation function of MLP neural network, the MLP neural network with 32 neurons in each of the three hidden layers is determined. By optimizing reinforcement learning reward function, adding scoring rules, and reward parameters, this paper carries out the following simulation exploration work. The high-precision positioning algorithm of transformer inspection robot is compared with the existing algorithm, and the superiority of the algorithm is verified. The basic motion ability of the robot installed with the system was tested.

Manufacturer calibrations of GPS transmit antenna phase patterns: a critical review

Over the past decade, the Global Positioning System has released pre-flight calibrations for the transmit antennas of the Block IIR/IIR-M, Block IIF, and GPS III satellites that make up the current GPS constellation. Frequency-specific phase variations (PHVs) provided as part of these data sets are of key interest for an accurate and consistent modeling of GNSS carrier phase observations in precise point positioning applications as well as orbit and clock offset determination of the GPS satellites themselves. For proper utilization of the manufacturer calibrations, complementary information on the phase center offset (PCO) from the spacecraft center-of-mass is required. We describe necessary processing steps for converting the raw phase calibrations of Lockheed Martin and Boeing into a representation compatible with antenna models of the International GNSS Service (IGS), and provide a detailed discussion of inherent assumptions for combining PHVs and PCOs from different sources. Comparison with estimated antenna data from globally distributed monitoring stations shows good consistency of PHVs and suggests the use of manufacturer-calibrated, azimuth-dependent patterns in future releases of the IGS antenna model. In terms of PCOs, the new Block IIF calibrations exhibit a systematic bias of about 12 cm from PCOs estimates based on the IGS20 reference frame. This value closely matches the bias observed for manufacturer calibrations of GPS III and Galileo satellites, and suggests a careful review of the contribution that GNSS can make to the scale definition of the International Terrestrial Reference Frame (ITRF).

FAST CONVERGING LIDAR-AIDED PRECISE POINT POSITIONING: A CASE STUDY WITH LOW-COST GNSS

Abstract. With the growing interest in autonomous driving, accurate vehicle positioning remains an open problem, especially in urban environments. According to regulatory organisations, the vehicle positioning accuracy is required to be centimetre-level. As the most used positioning technique which provides globally referenced positioning solutions, GNSS is the fundamental component for realising real-time vehicle positioning, usually through the RTK approach. However, RTK requires a nearby reference station to enable integer ambiguity resolution for the ultra-precise carrier phase observations. In comparison, PPP makes use of State-Space Representation (SSR) corrections produced by global networks for satellite orbits and clocks to facilitate phase-based positioning. Moreover, IGS now offers Real-Time Service (RTS) to transmit such corrections. Notably, the major drawback of PPP is that it takes a long time to converge to precise solutions due to the carrier phase ambiguities being real-valued, which can be severely elongated when real-time corrections and low-cost GNSS receivers are used. In this paper, a tightly coupled positioning method is proposed, which shortens RT-PPP convergence to seconds by using lidar measurements referenced from an HD map through deep learning. The lidar measurements are generated by point cloud registration and weighted by their intensity values and geometric distributions, and are then combined with RT-PPP in an Extended Kalman-Filter (EKF), thus achieving fast convergence. Experimental results show that the proposed method achieves and maintains centimetre-level accuracy within 2 seconds using a low-cost UBLOX F9P receiver, which is a significant improvement as compared to the decimetre-level accuracy obtained from standalone RT-PPP.

Reconstructed estimation method of the multi-frequency GNSS inter-frequency clock bias

In consideration of contributions from both carrier phase and pseudorange observations during computation, this study introduces a reconstructed method for estimating the multi-frequency global navigation satellite system (GNSS) inter-frequency clock biases (IFCB). Diverging from conventional approaches that separately calculate the time-varying and constant parts of IFCBs using carrier phase and pseudorange observations, the reconstructed method directly utilizes their combination to estimate satellite IFCBs. To validate the efficacy of the presented approach, 7 d observations from 87 International GNSS Service (IGS) stations are analyzed, with a specific focus on triple-frequency GPS, multi-frequency BDS-3, and Galileo IFCBs, aiming to scrutinize their distinctive characteristics. Furthermore, the performances of satellite IFCB estimation are investigated in both precise point positioning (PPP) and single-point positioning (SPP) using 35 IGS stations over 3 d. The results demonstrate that IFCBs of GPS BLOCK IIIA satellites exhibit centimeter-level variations, distinguishing them from BLOCK IIF counterparts. The Galileo IFCBs vary from millimeter to centimeter level, while those of BDS-3 reach a centimeter level. These variations significantly impact GPS PPP convergence performances but have minimal effects on Galileo and BDS-3 PPP. SPP performances are slightly enhanced when the time-varying IFCB part are taken into account. Additionally, we note disparities in the constant parts of satellite IFCBs computed with differential code bias (DCB) products, reconstructed, and pseudorange observation-based methods, particularly for BDS-3 C2I/C6I-C1P/C5P and C2I/C6I-C1P/C6I combinations. The differences of IFCB estimated with different strategies, SPP and PPP performances show that the reconstructed method is better than others, and the IFCB accuracy decreases when computed with satellite DCB products.

An Improved Carrier-Smoothing Code Algorithm for BDS Satellites with SICB

Carrier Smoothing Code (CSC), as a low-pass filter, has been widely used in GNSS positioning processing to reduce pseudorange noise via carrier phases. However, current CSC methods do not consider the systematic bias between the code and carrier phase observation, also known as Satellite-induced Code Bias (SICB). SICB has been identified in the BDS-2 and the bias will reduce the accuracy or reliability of the CSC. To confront bias, an improved CSC algorithm is proposed by considering SICB for GEO, IGSO, and MEO satellites in BDS constellations. The correction model of SICB for IGSO/MEO satellites is established by using a 0.1-degree interval piecewise weighted least squares Third-order Curve Fitting Method (TOCFM). The Variational Mode Decomposition combined with Wavelet Transform (VMD-WT) is proposed to establish the correction model of SICB for the GEO satellite. To verify the proposed method, the SICB model was established by collecting 30 Multi-GNSS Experiment (MGEX) BDS stations in different seasons of a year, in which the BDS data of ALIC, KRGG, KOUR, GCGO, GAMG, and SGOC stations were selected for 11 consecutive days to verify the effectiveness of the algorithm. The results show that there is obvious SICB in the BDS-2 Multipath (MP) combination, but the SICB in the BDS-3 MP is smaller and can be ignored. Compared with the modeling in the references, TOCFM is more suitable for IGSO/MEO SICB modeling, especially for the SICB correction at low elevation angles. After the VMD-WT correction, the Root Mean Square Error (RMSE) of SICB of B1I, B2I, and B3I in GEO satellites is reduced by 53.35%, 63.50%, and 64.71% respectively. Moreover, we carried out ionosphere-free Single Point Positioning (IF SPP), Ionosphere-free CSC SPP (IF CSC SPP), CSC single point positioning with the IGSO/MEO SICB Correction based on the TOCFA Method (IGSO/MEO SICB CSC), and CSC single point positioning with the IGSO/MEO/GEO SICB correction based on VMD-WT and TOCFA (IGSO/MEO/GEO SICB CSC), respectively. Compared to IF SPP, the average improvement of the IGSO/MEO/GEO SICB CSC algorithm in the north, east, and up directions was 24.42%, 27.94%, and 24.98%, respectively, and the average reduction in 3D RMSE is 24.54%. Compared with IF CSC SPP, the average improvement of IGSO/MEO/GEO SICB CSC is 7.03%, 6.50%, and 10.48% in the north, east, and up directions, respectively, while the average reduction in 3D RMSE was 9.86%. IGSO/MEO SICB mainly improves the U direction positioning accuracy, and GEO SICB mainly improves the E and U direction positioning accuracy. After the IGSO/MEO/GEO SICB correction, the overall improvement was about 10% and positioning improved to a certain extent.

100 Picosecond/Sub-10−17 Level GPS Differential Precise Time and Frequency Transfer

A Global Navigation Satellite System (GNSS) is a high-precision method for comparing clocks and time transfer. The GNSS carrier phase can provide more precise observable information than pseudorange. However, the carrier phase is ambiguous, and only pseudorange can provide the absolute time difference between two clocks. In our study, by taking full advantage of GNSS pseudorange and carrier-phase observables, a differential precise time transfer (DPT) method with a clustering constraint was employed to estimate the time difference between two clocks, aiming to achieve accurate and precise time and frequency transfer. Using this method, several time transfer results were analyzed for different baselines. For the common clock experiment, the time transfer results showed good consistency across different days, with an intra-day accuracy of within 20 ps. Furthermore, we evaluated the self-consistency of DPT using closure among three stations. DPT closure of the three stations had a peak-to-peak value of closure of about 25 ps. The closure did not change over time, indicating the self-consistency of the DPT processing in time transfer. Moreover, our results were compared to station clock solutions provided by the International GNSS Service (IGS), and the standard deviations (STDs) of the four baselines were all less than 100 ps within one month, confirming the time and frequency stability of the DPT method. In addition, we found that the time stability of DPT was less than 20 ps within one week. As for frequency stability, DPT achieved a 10−16 level of modified Allan deviation (MDEV) at an averaging time of about 1 day and a sub-10−17 level at an averaging time of one week.

Enhancing BDS-3 PPP-AR with observable-specific signal biases

In global navigation satellite system (GNSS) data processing, precise point positioning (PPP) with ambiguity resolution (PPP-AR) is a versatile technique that aims to achieve centimetre-level accuracy by resolving integer ambiguities in carrier phase observations. However, the inherent errors and biases in the satellite signals can degrade the performance of PPP-AR solutions. To mitigate such errors, this research proposed to argument PPP-AR using third-generation BeiDou Navigation Satellite System (BDS-3) multi-frequency observations and the observable-specific signal biases (OSBs) generated at the Centre National D’Etudes Spatiales (CNES). To test the proposed technique, both BDS-3 and Galileo observations from the multi-GNSS experiment network were used, in consideration that the latter also transmits multi-frequency signals. Before demonstrating the impact of CNES bias products on PPP-AR, the quality of BDS-3 and Galileo signals was assessed. The results indicated that the modernised frequencies had the best signal strength. The mean standard deviations for the estimated OSB for different receivers were close to each other in both constellations. Besides, the positioning results in different processing schemes unveiled a comparable positioning accuracy, and slightly better in the quad-PPP strategy using the Galileo constellation in both static and kinematic modes. Galileo also attained better ambiguity fixing rates and convergence time than BDS-3. Finally, there were slight differences in the magnitude of the estimated phase residuals for distinct frequency signals between BDS-3 and Galileo, including the interoperable and compatible signals.