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GLONASS ambiguity resolution

Thu, 07/18/2019 - 00:00
Abstract

A new integer-estimable GLONASS FDMA model is studied and analysed. The model is generally applicable, and it shows a close resemblance with the well-known CDMA models. The analyses provide insights into the performance characteristics of the model and concern a variety of different ambiguity-resolution critical applications. This will be done for geometry-free, geometry-fixed and several geometry-based formulations. Next to the analyses of the model’s instantaneous ambiguity-resolved positioning and attitude determination capabilities, we show the ease with which the model can be combined with existing CDMA models. We thereby present the instantaneous ambiguity-resolution performances of integrated L1 GPS + GLONASS, both for high-grade geodetic and mass-market receivers. We also consider the potential of the single-frequency combined model for mixed-receiver processing, particularly for the case the between-receiver GLONASS pseudorange data are biased. In all cases, the speed of successful ambiguity resolution is studied as well as the precision with which positioning is determined. Software routines for constructing the model are also provided.

A new GLONASS FDMA model

Thu, 07/18/2019 - 00:00
Abstract

We introduce a new formulation of the double-differenced GLONASS FDMA model. It closely resembles that of CDMA-based systems and it guarantees the estimability of the newly defined GLONASS ambiguities. The formulation is made possible because of our defining new concept of integer-estimability and the analytical construction of a special integer matrix canonical decomposition. As a result, an easy-to-compute new design matrix is created that automatically establishes the integer-estimability of the ambiguities. The presented model is generally applicable, and its close resemblance to its CDMA-counterparts implies that available CDMA-based GNSS software is easily modified and that existing methods of integer ambiguity resolution can be directly applied. Also, because of their similar structure, many of the available CDMA results with corresponding insights can now be directly translated to the GLONASS case. We make use of this property to provide insight into the ambiguity resolution capabilities of the model and to analyze the characteristics of the GLONASS ambiguity dilution of precision.

Kalman-filter-based undifferenced cycle slip estimation in real-time precise point positioning

Tue, 07/16/2019 - 00:00
Abstract

Global navigation satellite system (GNSS) precise point positioning (PPP) requires continuous carrier-phase observations to achieve a solution of high precision. Precisely correcting cycle slips caused by signal interruptions is crucial for recovering the data continuity. Most of the existing approaches usually employ only data of one epoch after the interruption for real-time cycle slip processing. In this study, we propose to introduce and estimate cycle slip parameters together with standard PPP parameters, such as position, ionospheric delay, and ambiguities in the case that possible cycle slips are detected, using a Kalman-filter-based procedure with the undifferenced and uncombined PPP model. The integer search strategy is used to fix cycle slips. To reduce the probability of wrong integer fixing, a strict integer validation threshold is suggested. As a result, it is not easy to fix all cycle slips with only one epoch of data. Our approach can be easily extended to use multi-epoch observations to enhance the cycle slip estimation. Once the cycle slips are correctly determined, continuous PPP can be achieved instantaneously. This new approach is tested and validated with three groups of experiments using GPS and GLONASS stations operated by the International GNSS Service from DOY 1–10, 2017, and a real vehicle kinematic data. Numerous experimental results showed that the proposed method can correctly fix the cycle slips for more than 99.5% of epochs suffering from re-convergence. On average, this method takes observation information from about 1.5–2.5 epochs to fix cycle slips and realize rapid re-convergence. Consequently, positioning performance is significantly improved.

Signal characterization and assessment of code GNSS positioning with low-power consumption smartphones

Mon, 07/15/2019 - 00:00
Abstract

It has been acknowledged that smartphone GNSS observations suffer not only from high measurement noise and multipath but also from anomalies such as duty cycling and gradual accumulation of phase errors. These phenomena importantly constrain the application of smartphone phase measurement to high-precision techniques such as RTK or PPP. Hence, we aim at a comprehensive characterization of smartphone signal quality, including carrier-to-noise density ratio, measurement noise and anomalies present in observables with the focus on the impact of duty-cycling mode. The analysis confirms the abnormal properties of smartphone measurements related to the divergence between code and phase data and poor quality of the latter. To address these limitations, the second objective is to assess the smartphone medium- to long-range code-based relative positioning. This task includes the validation of the weighting scheme suited for handling the low quality of smartphone observations. The results show that it is feasible to use a sparse countrywide GNSS network as reference stations for code-based relative positioning. Even with the baselines over 100 km, we can significantly enhance the positioning with respect to a stand-alone solution and reach the submeter level of horizontal coordinate accuracy. We have also noticed a discernible benefit from the C/N0-dependent weighting scheme, which is superior to the satellite elevation one.

Pivot single-difference ambiguity resolution for multi-GNSS positioning with non-overlapping frequencies

Mon, 07/15/2019 - 00:00
Abstract

Ambiguity resolution in real-time kinematic (RTK) positioning is increasingly dependent on the combination of multi-GNSS observations, especially in challenging signal environments such as urban canyon areas. By applying a priori calibration to the receiver-dependent differential inter-system bias (DISB), a DISB-fixed multi-GNSS combination method with overlapping frequencies is widely used to strengthen the positioning model. However, as for non-overlapping frequencies, the differential inter-frequency bias (DIFB), which correlates with ambiguities, needs to be further corrected in case of the DISB-fixed multi-GNSS combination method. Since the DIFB is related to a pivot single-difference (SD) ambiguity, the DIFB is generally reduced by means of a SD ambiguity resolution. Owing to the code multipath and the code–carrier inconsistency, the traditional carrier-minus-code combination method will result in a large SD ambiguity bias and affect the DIFB correction. Thus, a SD ambiguity resolution method based on the integer least squares ambiguity estimator is proposed to deal with the issue. A kinematic experiment is conducted in urban areas, which is based on single-frequency observations from GPS L1 and BDS B1. The result shows that the proposed method can improve the performance of ambiguity resolution and positioning continuity when compared with the traditional multi-GNSS RTK methods.

A new parameterized approach for ionospheric tomography

Mon, 07/15/2019 - 00:00
Abstract

GNSS ionospheric tomography technique is capable to reconstruct the high-quality 3D ionospheric electron density (IED) images with a relatively low cost. We present a new parameterized approach for refining the voxel-based ionospheric tomography modeling. This approach is different to most other voxel-based techniques as they assume a homogeneous IED distribution in each voxel that is unreasonable for the tomography modeling. In this method, IED of any point within a voxel is determined via vertically exponential interpolation and horizontally inverse distance weighted interpolation from the IED values at the eight corners of that voxel. The parameterized tomography is tested with real data collected over the period of June 1–30, 2015 from 45 GPS stations in south China. The superiority of the new parameterized method is verified by comparison with the traditional nonparametric method. The new parameterized method outperforms the traditional method by 12%, 10%, 5% and 2% for vertical resolutions of 25 km, 50 km, 75 km and 100 km, respectively, in the self-consistency validation by GPS data. Such improvements are 20%, 24%, 22% and 16%, respectively, when assessed by the Swarm in situ IEDs. In terms of the vertical layer discretization, configurations using the resolution of 25 km generally performs better than the other three vertical resolutions. Overall, the parameterized method using a vertical resolution of 25 km achieves the best performance from the comprehensive comparisons with ionospheric data derived by GPS, ionosonde and Swarm satellites.

Mean acquisition time analysis for GNSS parallel and hybrid search strategies

Wed, 07/10/2019 - 00:00
Abstract

Mean acquisition time (MAT) is one of the key performance indicators for the acquisition of global navigation satellite system (GNSS) signals. Usually, there are two stages in GNSS signal acquisition, called coarse acquisition and fine acquisition. A receiver starts the coarse acquisition with relatively large code-phase and frequency steps to obtain rough code-phase and Doppler estimates. Then, it follows a fine acquisition, in which the search steps are much smaller, to obtain the more accurate code-phase and Doppler estimates to meet the tracking loop requirements, and the search space is dramatically reduced. The existing MAT models only consider the coarse acquisition stage. We present the MAT expressions in consideration of both the coarse and fine acquisition stages. Also, our models are extended versions of the existing ones. The MAT expressions for three common GNSS signal acquisition schemes are provided, including parallel code-phase search (PCS), parallel frequency search, and hybrid search. The analytical expressions are verified by Monte Carlo simulations of GPS L1 C/A signals. We show how to optimize the acquisition parameters, including detection threshold, coherent integration time, and non-coherent integration time with the metric of MAT. The models facilitate not only MAT performance evaluation but also design parameter optimization of GNSS receivers.

Densification of the ITRF2014 position and velocity solution in the Nordic and Baltic countries

Wed, 07/10/2019 - 00:00
Abstract

The growing need for a consistent and densified GNSS position and velocity solution for the Nordic and Baltic countries resulted in development of the joint GNSS Analysis Centre of the Nordic Geodetic Commission (NKG) in 2012. We first developed the methods of the operational processing and combination of solutions and then reprocessed the full data history between 1997 and 2017. In this study, we present an ITRF2014 densification for the area including 252 stations having more than 3 years of data. We combined all 20 years of daily solutions with full covariance matrices instead of station-wise analysis and analyzed the noise characteristics of the residual time series. We concluded that the flicker plus white noise uncertainty estimates were more robust than the general power-law estimates. Additionally, we found significant horizontal velocity differences at the co-located stations, pointing out biases not included in the formal uncertainties. The solution is more accurate and denser than any previous estimate, and it will be of great benefit for maintaining the reference frames in the Nordic and Baltic countries, as well as for the geodynamic studies in the area.

FCB estimation with three different PPP models: equivalence analysis and experiment tests

Mon, 07/08/2019 - 00:00
Abstract

Precise and reliable fractional cycle bias (FCB) products are critical for ambiguity resolution of precise point positioning (PPP). Three PPP models are usually adopted to generate the FCB products, i.e., the traditional ionospheric-free combined PPP (IF-PPP) model, the uncombined and unconstrained PPP model (UU-PPP) as well as the ionospheric-constrained PPP (IC-PPP) model. Considering that different observation models and ionospheric delay constraints are used, the applicability and interoperability of the obtained FCB products are assessed. We presented the equivalent conversion formulas of different FCB products obtained with different PPP models, which are then converted and compared. The root mean square (RMS) of the wide-lane (WL) FCB differences for IF-UU, IF-IC and UU-IC is 0.021, 0.024 and 0.010 cycles, while the RMS of the narrow-lane FCB differences is 0.028, 0.018 and 0.021 cycles. The precision of the WL FCBs based on the uncombined PPP models is higher than that based on the IF-PPP model since the new WL ambiguities derived from the uncombined ambiguities are free of the pseudorange noise. The equivalence of the FCB products estimated from the three different PPP models is confirmed in theory and by experiment results. To further evaluate the performance of the three PPP models, the positioning accuracy, the convergence time and the ambiguity fixing success rate are calculated using IGS data. The maximum positioning difference is less than 0.9 mm among the three PPP models. Compared to the float solutions of the IF-PPP, UU-PPP, IC-PPP with GIM model or with re-injected ionospheric delay corrections, the convergence time is shortened by 38.5%, 46.2%, 50.0% and 87.8% and the positioning accuracy improved by 27.1%, 38.9%, 41.1% and 25.7%, respectively. The ambiguity fixing success rate of the uncombined PPP models is slightly higher than that of the combined model, and fast ambiguity-fixed solution can proceed with high-precise ionospheric delay corrections.

Signal analysis of the first GPS III satellite

Sat, 07/06/2019 - 00:00
Abstract

The signal-in-space of the first GPS Block III spacecraft is analyzed based on radio frequency measurements collected with a 30-m high-gain dish antenna as well as data from geodetic GPS receivers. The spectral properties and modulation characteristics are discussed with focus on the L1 band, which employs a novel interlaced majority voting technique for combination of the C/A, P(Y), and L1C data + pilot signal components. Compared to the preceding generation of Block IIF satellites, a modified shaping of the L1 transmit antenna gain pattern is found, which results in lower carrier-to-noise density ratios at mid to high elevations. Along with this, use of a separate transmission chain for the military M-code signal is evidenced through the analysis of in-phase/quadrature signal components and the derived transmit antenna gain variations. A high level of signal purity is demonstrated on all frequencies, which can be attributed to the use of a new, mostly digital, signal generation unit. Maps of code bias variations for selected signals are presented to quantify the achievable user tracking performance as a function of user receiver parameters. For the L5 signal, a notable reduction in digital distortions is obtained with respect to the Block IIF satellites, whereas analog distortions are found to be of similar magnitude. Thermally induced L5 phase variations found in the Block IIF satellites are no longer observed in GPS III. Using triple-frequency phase observations, a sub-centimeter consistency of the L1, L2, and L5 carriers is demonstrated.

PRIDE PPP-AR: an open-source software for GPS PPP ambiguity resolution

Fri, 07/05/2019 - 00:00
Abstract

The PRIDE Lab at GNSS Research Center of Wuhan University has developed an open-source software for GPS precise point positioning ambiguity resolution (PPP-AR) (i.e., PRIDE PPP-AR). Released under the terms of the GNU General Public License version 3 (GPLv3, http://www.gnu.org/licenses/gpl.html), PRIDE PPP-AR supports relevant research, application and development with GPS post-processing PPP-AR. PRIDE PPP-AR is mainly composed of two modules, undifferenced GPS processing and single-station ambiguity resolution. Undifferenced GPS processing provides float solutions with wide-lane and narrow-lane ambiguity estimates. Later, single-station ambiguity resolution makes use of the phase clock/bias products, which are released also by the PRIDE Lab at ftp://pridelab.whu.edu.cn/pub/whu/phasebias/, to recover the integer nature of single-station ambiguities and then carry out integer ambiguity resolution. PRIDE PPP-AR is based on a least-squares estimator to produce daily, sub-daily or kinematic solutions for various geophysical applications. To facilitate the usage of this software, a few user-friendly shell scripts for batch processing have also been provided along with PRIDE PPP-AR. In this article, we use 1 month of GPS data (days 001–031 in 2018) to demonstrate the performance of PRIDE PPP-AR software. The PRIDE Lab is committed to consistently improve the software package and keep users updated through our website.

Characteristics of raw multi-GNSS measurement error from Google Android smart devices

Thu, 07/04/2019 - 00:00
Abstract

Developers targeting Android platforms can obtain raw GNSS measurements, which can achieve submeter or even decimeter-level positioning accuracy. An accurate receiver measurement error model is an important prerequisite for precise positioning with smart devices. Therefore, we analyzed the measurement error characteristics of raw GNSS data from smart devices using both embedded and external antennas. We find that the GNSS signals produced by smart devices have non-uniform signal strengths, rapid C/N0 variations, and low C/N0 at high elevations. The pseudorange noise is about 10 times larger than that from geodetic receivers; the carrier phase noise of Nexus 9 is 3–5 times larger than that of geodetic receivers, and unexpectedly is half of that of μ-blox. We provide theoretical parameters for the noise versus C/N0 models of the GNSS chipset for different smart devices. Unfortunately, the carrier phase tracking of Samsung Galaxy S8 and Huawei Honor v8 are discontinuous due to the duty-cycle issue, which results in greater noise and carrier phase unavailability. Moreover, we found two unique error characteristics of the carrier phase available from Nexus 9 anomalous “jagged” distribution and random initial phase biases, which is evident in the controlled environment test. Finally, we obtained promising positioning results: the horizontal and vertical RMS of pseudorange single-point positioning are about 10–20 m; the static carrier phase relative positioning (CRP) solutions of Nexus 9 can achieve centimeter-level precision, whereas both horizontal and vertical STDs are about 1 cm or better but with decimeter-level biases. When using an external antenna, the resulting biases are as small as a few centimeters. Encouragingly, the actual vehicle test results showed that the STD of the Nexus 9 kinematic CRP 3D-distance error is 0.169 m, and the percentages of errors falling into ± 0.1 m and ± 0.5 m are 63.59% and 100%, respectively. Furthermore, multi-GNSS is able to provide more reliable position services in GNSS-adverse environments.

Temporal characteristics of triple-frequency GNSS scintillation during a visible aurora borealis event over the Faroe Islands amid a period of very low solar activity

Tue, 07/02/2019 - 00:00
Abstract

Ionospheric scintillation causes noise in GNSS measurements and hence affects the resulting coordinates or even the ability to track data. We investigate the characteristics of GNSS scintillation during a visible aurora borealis event over the Faroe Islands during October 7, 2018, which lasted for around 30 min. Data of 1 Hz rate from a typical geodetic GNSS receiver are analyzed during a 3-h window around the event, as well as during the same period the days before and after. Data from all four GNSS are analyzed, which consist of 31 satellites during the occurrence, resulting in a good overview of the temporal distribution of the scintillation. Two approaches are used, called the time difference code-minus-carrier and the time difference phase ionospheric residual. The results illustrate that there is little effect on the code measurements, but significant repercussion on the carrier phase observables, resulting in erroneous position solutions. The approaches we present can be used in GNSS processing software to detect scintillation noise in real time on individual satellites, allowing such noisy data to be rejected.

Evaluation of HY-2A satellite-borne water vapor radiometer with shipborne GPS and GLONASS observations over the Indian Ocean

Wed, 06/19/2019 - 00:00
Abstract

The Chinese Haiyang-2A (HY-2A) altimetry satellite is equipped with a calibration microwave radiometer (CMR) for correcting atmospheric water vapor path delay in radar altimeter observations. To evaluate the satellite-borne CMR, we retrieved 1 Hz high-frequency precipitable water vapor (PWV) using shipborne GPS and GLONASS from a two-month cruise in the Indian Ocean. This open-sea evaluation of the satellite-borne radiometer is free of the contamination of the satellite footprints induced by coastal lands, which occurs inevitably in ground-based or coastal stations. The estimates and errors of the retrieved PWV from shipborne GNSS kinematic precise point positioning were analyzed and then compared to the CMR-retrieved PWV. The results show that the shipborne GNSS kinematic precise point positioning can obtain marine PWV with an uncertainty of about 2.8 mm. When the HY-2A sub-satellite point and the ship cross, the HY-2A CMR-retrieved PWV is in good agreement with the GNSS PWV with the difference of about 0.8 mm, which demonstrates that the HY-2A satellite can contribute high-precision precipitable water vapor measurements to weather and atmospheric studies.

Triple-frequency carrier phase precise time and frequency transfer models for BDS-3

Wed, 06/19/2019 - 00:00
Abstract

The third-generation BeiDou navigation satellite system (BDS-3) began providing global positioning, navigation and timing service on December 27, 2018. We present three triple-frequency carrier phase (CP) precise time and frequency transfer models using the BDS-3 B1I/B3I/B2a signals, named IF-PPP1, IF-PPP2 and UC-PPP models, respectively. The BDS B1I/B3I dual-frequency ionospheric-free (IF) model is also introduced, known as IF-PPP0 model. The corresponding mathematical and stochastic models are developed. Two stations located at time laboratories and connected to a high-precision atomic clock are utilized to assess the performances of the proposed CP precise time and frequency transfer models. In addition, the number of visible satellites, position dilution of precision, time dilution of precision, estimated positioning errors, zenith tropospheric delay and inter-frequency bias for two stations are also analyzed. The results show that BDS CP precise time and frequency transfer can achieve better performances with increasing number of BDS-3 observations. The proposed models all can be applied for precise time and frequency transfer with the BDS-3 triple-frequency signals, with stability and accuracy identical to the BDS IF-PPP0 solution. The stability of 10,000 s for the proposed BDS CP precise time and frequency models is better than 1.5 × 10−14.

INS/magnetometer integrated positioning based on neural network for bridging long-time GPS outages

Wed, 06/19/2019 - 00:00
Abstract

In global position system (GPS) and inertial navigation system (INS) integrated navigation systems, the positioning method based on artificial intelligence (AI) learning algorithms has the disadvantage of position diverging as the GPS outages time continues. To solve this problem, INS/magnetometer integrated positioning based on neural network is proposed for bridging GPS outages over a long time. First, the possibility of using magnetometer for positioning is verified by analyzing the international geomagnetic reference field model and the magnetometer measurement model. Then, a magnetometer-assisted positioning solution is proposed. This solution includes four parts: the magnetic fields update module with adaptive extended Kalman filter (AEKF), predictor by AI, INS/GPS integration with KF, and INS/position integration with AEKF. The simulation and driving test results show that the proposed method can keep most of the position errors within a certain range and there is no tendency of divergence at all.

Impact of ambiguity resolution with sequential constraints on real-time precise GPS satellite orbit determination

Wed, 06/19/2019 - 00:00
Abstract

The real-time precise orbit is an essential prerequisite for a real-time precise point positioning service. Focusing on the impact of ambiguity resolution on real-time orbital precision, we propose an efficient approach for real-time ambiguity resolution, which consists of two modules. The first module resolves the double-differenced ambiguities according to their proximity to the nearest integer. The second module sequentially adds the resolved integer ambiguity constraints to the square-root information filter process, after confirming that the same constraints have not been imposed before the current epoch. To validate our method, GPS data collected from 100 globally distributed stations are used to simulate a real-time precise orbit determination. The convergence performance is analyzed, and the accuracy of the orbit is evaluated. The results show that: (1) Almost 90% of the double-differenced ambiguities are fixed correctly in real time for baselines shorter than 1000 km. (2) After application of the proposed approach, the root-mean-square errors of all the satellite orbits are reduced from (5.9, 3.4, 2.3) cm to (4.7, 2.6, 2.2) cm for the along-track, cross-track and radial directions, respectively, with improvements of about 22% for the along-track and cross-track directions, and 6% for the radial direction. (3) Simulated real-time orbits determined with this method can obtain almost the same accuracy as some ultra-rapid products and, particularly, better accuracy can be achieved for eclipsing satellites.

Retrieving geophysical signals from GPS in the La Plata River region

Sat, 06/08/2019 - 00:00
Abstract

Over the last few years, efforts to model short-term deformations of the Earth’s crust have multiplied. Sudden water level rise can cause sporadic, but significant, motions in the solid Earth’s surface. In this work, we address the problem of retrieving reliable estimates of the vertical displacement of a Global Positioning System (GPS) station located very close to the eastern shore of the La Plata River, during a strong storm surge event. Capturing sub-daily GPS displacements demands an elaborate processing strategy because several highly correlated parameters must be estimated simultaneously. We present a successful strategy that reduces the number of unknowns that have to be estimated simultaneously, by using an empirical model that describes the elastic response of the Earth’s crust to the hydrological load variations. We incorporate this model into the observation equations so that, instead of estimating the station position, we estimate every epoch a single parameter of the empirical model, i.e., the empirical elastic parameter EEP, that is assumed to be a constant of the Earth’s crust in the region of the GPS station. We verify that the estimated parameter agrees well with the value calculated from the CRUST 1.0-A model of the Earth’s crust. The GPS receiver was tied to an external cesium clock, which allowed us to process the data according to two different strategies: (a) estimating the receiver clock error (Δt) as an epoch-wise free parameter, which is equivalent to ignoring the presence of the external clock, and (b) conditioning the variability of that estimate with a small a priori variance compatible with the external clock’s variability. We find that, without having an external atomic clock, the estimation of all the parameters, i.e., the zenith tropospheric delay, Δt, and the EEP, worsens when the GPS station is affected by sub-daily vertical displacements.

Rectification of GNSS-based collaborative positioning using 3D building models in urban areas

Thu, 06/06/2019 - 00:00
Abstract

GNSS collaborative positioning receives great attention because of the rapid development of vehicle-to-vehicle communication. Its current bottleneck is in urban areas. During the relative positioning using GNSS double-difference pseudorange measurements, the multipath effects and non-line-of-sight (NLOS) reception cannot be eliminated, or even worse, both might be aggregated. It has been widely demonstrated that 3D map aided GNSS can mitigate or even correct the multipath and NLOS effects. We, therefore, investigate the potential of aiding GNSS collaborative positioning using 3D city models. These models are used in two phases. First, the building models are used to exclude NLOS measurements at a single receiver using GNSS shadow matching positioning. Second, the models are used together with broadcast ephemeris data to generate a predicted GNSS positioning error map. Based on this error map, each receiver will be identified as experiencing healthy or degraded conditions. The receiver experiencing degraded condition will be improved by the receiver experiencing the healthy condition, hence the aspect of collaborative positioning. Five low-cost GNSS receivers are used to conduct experiments. According to the result, the positioning accuracy of the receiver in a deep urban area improves from 46.2 to 14.4 m.

Multi-GNSS inter-system biases: estimability analysis and impact on RTK positioning

Wed, 06/05/2019 - 00:00
Abstract

Inter-system biases (ISB) are of great relevance for the combined processing of the code and phase data of multiple global navigation satellite systems (GNSSs). Calibrating the ISB makes it possible to enhance the interoperability among different GNSS constellations and thus benefits multi-GNSS-based positioning, navigation and timing applications. Initial investigations of the characteristics of ISB have been carried out, usually making use of overlapping frequencies and adopting the double-differenced (DD) model. However, this approach seems inapplicable when dealing with ISB for non-overlapping frequencies. We identify the estimability of the ISB by using the ionospheric-float, ionospheric-fixed and ionospheric-weighted models formulated on the basis of between-receiver single-differenced (SD) multi-GNSS observation equations, resulting in the so-called SD method, which is capable of estimating the ISB in case of both overlapping and non-overlapping frequencies. Using dual-frequency data for short and medium baselines, we analyze 30-s epoch-by-epoch estimates of the GPS–Galileo and GPS–BDS ISB. The quantitative results indicate that the same conclusion is reached using either the SD method or the customary method based on DD observations (called the DD method); that is, the code and phase ISB time series are both approximately constant on a time scale of a few days from a statistical perspective. However, the SD method has the advantage that it can be used to flexibly estimate ISB for both overlapping and non-overlapping frequencies and thus can be better applied for real-time kinematic positioning than the DD method. Furthermore, the multi-GNSS positioning accuracy using inter-system differencing can be improved by 20–35%, as compared to the SD classical differencing in which S-basis is selected per constellation, thanks to the reasonable calibration of the ISB.

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