Orbit Errors Research Articles

Stochastic modeling of physical drag coefficient – Its impact on orbit prediction and space traffic management

Ambitious satellite constellation projects by commercial entities and the ease of access to space in recent times have led to a dramatic proliferation of low-Earth space traffic. It jeopardizes space safety and long-term sustainability, necessitating better space domain awareness (SDA). Correct modeling of uncertainties in force models and orbital states, among other things, is an essential part of SDA. For objects in the low-Earth orbit (LEO) region, the uncertainty in the orbital dynamics mainly emanate from limited knowledge of the atmospheric drag-related parameters and variables. In this paper, which extends the work by Paul et al. (2021), we develop a feed-forward deep neural network model for the prediction of the satellite drag coefficient for the full range of satellite attitude (i.e., satellite pitch ∈ (-90°,+90°) and satellite yaw ∈ (0°,+360°)). The model simultaneously predicts the mean and the standard deviation and is well-calibrated. We use numerically simulated physical drag coefficient data for training our neural network. The numerical simulations are carried out using the test particle Monte Carlo method using the diffuse reflection with incomplete accommodation gas-surface interaction model. Modeling is carried out for the well-known CHAllenging Minisatellite Payload (CHAMP) satellite. Finally, we use the Monte Carlo approach to propagate CHAMP over a three-day period under various modeling scenarios to investigate the distribution of radial, along-track, and cross-track orbital errors caused by drag coefficient uncertainty. The key takeaways of this paper are - (a) a constant drag coefficient cannot be used for reliable SDA purposes, and (b) stochastic machine learning models allow for the computation of drag coefficients in a timely manner while providing reliable uncertainty estimates.

Doppler Positioning of Dynamic Targets with Unknown LEO Satellite Signals

A new concept of global navigation based on Doppler measurements from a large low Earth orbit (LEO) constellation is investigated that has potential to serve as a complement or backup to global navigation satellite systems (GNSSs) to provide navigation and positioning services in GNSS denial environments. In this work, we investigate the potential of LEO communication satellite opportunity signals in dynamic navigation, establish LEO satellite Doppler positioning equations, derive the main error sources affecting the Doppler positioning results of a dynamic target, and analyze the impact of each error source on positioning accuracy through a simulation. The results show that the orbit error and clock drift had a large impact on positioning accuracy. This navigation scheme would be more competitive if it could provide high-precision satellite orbits and accurate Doppler measurements. The obtained results show that the LEO satellite signals used as navigation opportunity signals are an attractive alternative in GNSS rejection environments for high dynamic targets.

A Design of Differential-Low Earth Orbit Opportunistically Enhanced GNSS (D-LoeGNSS) Navigation Framework

Considering the problem of GNSS service interruption caused by the insufficient number of available satellites in complex environments, Low Earth Orbit (LEO) satellites can supplement GNSS effectively. To eliminate the unknown satellite clock error and the atmospheric delay error with spatial correlation in LEO observations, a Differential-Low Earth Orbit opportunistically enhancing GNSS (D-LoeGNSS) navigation framework is proposed. Firstly, because of the uncertainty of the LEO orbit, we derive the effect of the LEO orbit error on the differential measurement model. Secondly, aiming at the noise amplification and correlation in double-difference (DD), we propose a Householder-Based D-LoeGNSS (HB-DLG) algorithm, which suppresses noise by introducing an orthogonal matrix. Thirdly, in D-LoeGNSS, the typical measurement of LEO is Doppler, which is heterogeneous with the GNSS pseudorange, rendering the Dilution of Precision (DOP) evaluation method unsuitable. Given the unbiasedness of differential measurements, the Cramer Rao Lower Bound (CRLB) is derived as a metric to characterize the positioning accuracy and satellite spatial distribution. Finally, a field experiment using Orbcomm (ORB) and GPS is conducted. The experimental results show that the performance of the HB-DLG algorithm is superior to DD. Especially when the number of satellites is insufficient or the measurement redundancy is poor; the D-LoeGNSS framework has advantages of rapid convergence and high accuracy compared with a single constellation.

A Multi-Scale Spatial Difference Approach to Estimating Topography Correlated Atmospheric Delay in Radar Interferograms

The Interferometric Synthetic Aperture Radar (InSAR) has been widely used as a powerful technique for monitoring land surface deformations over the last three decades. InSAR observations can be plagued by atmospheric phase delays; some have a roughly linear relationship with the ground elevation, which can be approximated using a linear model. However, the estimation results of this linear relationship are sometimes affected by phase ramps such as orbital errors, tidal loading, etc. In this study, we present a new approach to estimate the transfer function of vertical stratification phase delays and the transfer function of phase ramps. Our method uses the idea of multi-scale spatial differences to decompose the atmospheric phase delay into the vertical stratification component, phase ramp component, and other features. This decomposition makes the correlation between the vertical stratification phase delays and topography more significant and stable. This can establish the correlation between the different scales and phase ramps. We demonstrate our approach using a synthetic test and two real interferograms. In the synthetic test, the transfer functions estimated by our method were closer to the design values than those estimated by the full interferogram–topography correlation approach and the band-pass filtering approach. In the first real interferogram, out of the 9 sub-regions corrected by the proposed method, 7 sub-regions were outperformed the full interferogram–topography correlation approach, and 8 sub-regions were superior to the band-pass filtering method. Our technique offers a greater correction effect and robustness for coseismic deformation signals in the second real interferogram.

Statistical Assessments of InSAR Tropospheric Corrections: Applicability and Limitations of Weather Model Products and Spatiotemporal Filtering

Tropospheric correction is a crucial step for interferometric synthetic aperture radar (InSAR) monitoring of small deformation magnitude. However, most of the corrections are implemented without a rigorous evaluation of their influences on InSAR measurements. In this paper, we present three statistical metrics to evaluate the correction performance. Firstly, we propose a time series decomposition method to estimate the tropospheric noise and mitigate the bias caused by ground displacement. On this basis, we calculate the root-mean-square values of tropospheric noise to assess the general performance of tropospheric corrections. Then, we propose the use of semi-variograms with model-fitted range and sill to investigate the reduction of distance-dependent signals, and Spearman’s rank correlation between phase and elevation to evaluate the mitigation of topography-correlated signals in hilly areas. The applicability and limitations were assessed on the weather model-derived corrections, a representative spatiotemporal filtering method, and the integration of the two mainstream methods. Furthermore, we notice that the persistent scatter InSAR processing resulted in two components, the primary and secondary images’ contribution to the tropospheric and orbit errors. To the best of our knowledge, this paper for the first time analyzes the respective roles of the two components in the InSAR tropospheric corrections.

PPP/INS Tight Integration with BDS−3 PPP−B2b Service in the Urban Environment

To provide continuous and reliable real-time precise positioning services in challenging environments and poor internet conditions, the real-time precise corrections of the BeiDou global navigation satellite system (BDS-3) PPP-B2b signal are utilized to correct the satellite orbit errors and clock offsets. In addition to this, using the complementary characteristics of the inertial navigation system (INS) and the global navigation satellite system (GNSS), a PPP-B2b/INS tight integration model is established. With observation data collected in an urban environment, the results show that PPP-B2b/INS tight integration can ensure a decimeter-level positioning accuracy; the positioning accuracies of the E, N, and U components are 0.292, 0.115, and 0.155 m, respectively, which can provide a continuous and secure position during short interruptions in the GNSS. However, there is still a gap of about 1 dm compared with the three-dimensional (3D) positioning accuracy obtained from Deutsche GeoForschungsZentrum (GFZ) real-time products, and a gap of about 2 dm compared with the GFZ post-precise products. Using a tactical inertial measurement unit (IMU), the velocimetry accuracies of the tightly integrated PPP-B2b/INS in the E, N, and U components are all about 0.3 cm/s, and the attitude accuracy of yaw is about 0.1 deg, while the pitch and roll show a superior performance of less than 0.01 deg. The accuracies of the velocity and attitude mainly depend on the performance of the IMU in the tight integration mode, and there is no significant difference between using real-time products and post products. The performance of the microelectromechanical system (MEMS) IMU and tactical IMU is also compared, and the positioning, velocimetry, and attitude determinations with the MEMS IMU are significantly worsened.

The impacts of reducing atmospheric and oceanic de-aliasing model error on temporal gravity field model determination

SUMMARYThe Gravity Recovery and Climate Experiment (GRACE) and its successor GRACE Follow-On mission have been estimating temporal gravity field models for various geoscience applications since 2002. While the errors in atmospheric and oceanic de-aliasing (AOD) model are still the key limitation for GRACE-type missions especially taking advantages of improved laser ranging interferometer (LRI), this paper explores the realistic assumption of continuously reducing AOD model error in global scale and regional scale (e.g. ocean, Greenland, Qinghai-Tibet plateau and South America) in the anticipated future. For this, using a realistic orbit scenario and error assumptions both for instrument and background model errors, the 5-yr full-scale simulations are implemented in the context of 26 scenarios with different AOD model errors. Our overall simulation results demonstrate that: (1) when the AOD model errors are reduced in the global scale, the corresponding gain in temporal gravity field determination is significant, with the cumulative geoid height error decrease from 11.28 to 6.25 mm in spectral domain, as well as 26.3–65.2 per cent noise reduction in terms of mean RMS residuals over ocean in spatial domain. (2) When the AOD model errors are reduced in the regional scale, the improvements of temporal gravity field estimations are diverse in different simulation scenarios. The most notable noise reduction, with 22.9–43.9 per cent noise reduction in spatial domain, is observed when the AOD model error decreases in ocean. The limited noise reductions (0.1 per cent and 1.3 per cent respectively in spectral and spatial domain) are observed in the temporal gravity field estimations if the AOD model is refined in other typical regions such as the Qinghai-Tibet plateau, the Greenland and the South America. (3) However, when the satellites passing over the AOD model refined regions, the along-orbit range rate analysis indicates that, there are visible differences by about 50.0 nm s−1 in terms of range rate residuals as well as 11.0–48.5 nm s−1 in terms of the mean RMS of range rate residuals. These results reflect the benefits of reducing AOD model error in both global and regional scale for improving GRACE-type temporal gravity estimation, especially considering the development of LRI technology.

ASSESSING EFFICIENCY OF GPS EPHEMERIDES IN DIFFERENT REGION USING PRECISE POINT POSITIONING

Abstract. Precise Point Positioning (PPP) is a technique that process Global Positioning System (GPS) observation that can achieve sub-decimetre accuracy. Precise ephemerides that contain satellite clock and position plays a vital role in minimising orbital and clock error in PPP. This study hypothesize that the efficiency of ephemerides varies across region, influenced by the density of stations involved in generating the ephemerides. GPS observations from stations in five different regions were processed using PPP technique by using RTKLIB. Four types of ephemerides were used namely broadcast, ultra-rapid, rapid, and final ephemerides during the processing of each station. Root-mean-squared error (RMSE) of 3D error of the PPP solution for the stations in each region is quantified. Among the five regions selected, Europe has the lowest RMSE of 155.67 cm, 27.93 cm, 27.38 cm and 27.06 cm for broadcast, ultra-rapid, rapid, and final ephemeris respectively. This high accuracy can be attributed to the density of International GNSS Service (IGS) stations in Europe that are available for the generation of ephemerides. Position Dilution of Precision (PDOP) was also found to contribute to the accuracy of PPP solution in a region, whereby a low PDOP often promises higher positioning accuracy. More regional stations are suggested to be involved in the generation of ephemerides to improve the efficiency of ephemerides in that region.

Signal-in-Space Range Error of the Global BeiDou Navigation Satellite System and Comparison with GPS, GLONASS, Galileo, and QZSS

Based on the standard of precise multisystem orbit and clock offset, the evaluation methods of satellite orbit, satellite clock offset and the signal-in-space range error (SISRE) from broadcast ephemeris were given in this study. The SISRE was used as a key performance index to compare the global BeiDou navigation satellite system (BDS-3) with the regional navigation satellite system (BDS-2), Global Positioning System (GPS), Galileo, Globalnaya Navigazionnaya Sputnikovaya Sistema (GLONASS), and Quasi-Zenith Satellite System (QZSS). Through the combination of domestic tracking stations and inter satellite links, the orbit determination accuracy of BDS-3 has been significantly improved. The radial (R), tangential (T) and normal (N) direction orbit accuracies of BDS-3 medium Earth orbit (MEO) satellites were 0.08, 0.37, and 0.34 m, respectively, which were considerably better than those of other satellite navigation systems. The R-, T-, and N-direction orbit accuracies of BDS-3 inclined geosynchronous orbit (IGSO) satellites were 0.17, 0.59, and 0.56 m, respectively, which were better than those of QZSS IGSO satellites. The R-, T-, and N-direction orbit accuracies of BDS-3 geostationary earth orbit (GEO) satellites were 0.27, 3.42, and 1.29 m, respectively, revealing considerable accuracy advantages in the R and N directions compared with QZSS GEO, although performance in the T direction was inferior. For the clock offset, the BDS-3 MEO satellites had an accuracy of 1.83 ns, which was noticeably superior, similar, slightly inferior, and superior to those of BDS-2, GPS, Galileo, and GLONASS. The clock offset accuracies of BDS-3 IGSO and GEO satellites were 3.13 and 3.17 ns, respectively, which were slightly better than those of QZSS IGSO and GEO satellites but did not have an obvious advantage over BDS-2. The primary reason could be that the launch time of the two types of BDS-3 satellites was close to the analysis period, and therefore, the clocks may be still in the aging process. Considering only orbit error, the average SISRE (orb) of BDS-3 MEO, Galileo, GPS, and GLONASS was 0.10, 0.28, 0.57, and 1.03 m, respectively. The SISRE (orb) of BDS-3 IGSO was 0.19 m, which was marginally greater than that of BDS-3 MEO but smaller than those of the other systems and considerably smaller than that (0.90 m) of QZSS IGSO. The SISRE (orb) of BDS-3 GEO was 0.60 m, which was comparable to that of GPS (0.59 m) and considerably smaller than that of QZSS GEO (0.91 m) satellites. Considering both orbit and clock offset error, the SISRE of BDS-3 MEO satellites was 0.52 m, which was slightly greater than that of Galileo (0.40 m), marginally smaller than that of GPS (0.59 m), and considerably smaller than that of GLONASS (2.33 m). The BDS-3 IGSO satellites had a SISRE of 0.90 m, which was greater than those of GPS and Galileo but comparable to that of QZSS IGSO (0.92 m). The average SISRE of BDS-3 GEO satellites was 1.15 m, which was greater than those of GPS and Galileo and slightly greater than that of QZSS0.95 m GEO satellites, but smaller than that of GLONASS.

Random error propagation on electron beam dynamics for a 50 MeV S-band linac

The stability and the quality of particle beams are of utmost importance for many emerging linac installations. The impact on beam properties damage of beam electromagnetic element misalignments and jitter/fluctuations in various accelerator sub-systems should be properly known, as usually such shot-to-shot fluctuations cannot be avoided. On top of that, knowing which parameters the machine is most sensitive to is of utmost to take precautionary measures to reduce the beam degradation and thus improve beam stability and quality. This simulation work focuses on a 50 MeV S-band linear accelerator based on RF photoinjector electron source. The sensitivity of the beam parameters towards several errors has been studied collectively as well as individually for each accelerator element. While the emittance at the end of the linac is dominated by the laminar behavior in the accelerating section, the main emittance degradation comes mainly from orbit errors located at the linac entrance.

Effect of the low-order coefficients of the Earth gravity model in calculating the satellite orbit

The Earth's gravity model is a crucial factor in determining satellite orbits. Scientific organizations such as GFZ Potsdam in Germany, GRGS Toulouse in France, and AIUB in Switzerland have established Earth gravity models with increasing precision of spherical harmonic coefficients. Low-order coefficients, including C ̅_21,( S) ̅_21, C ̅_10,C ̅_11,( S) ̅_11and C ̅_20, play a vital role in describing changes in the Earth's poles, geometric center, and flattening. To evaluate the impact of these coefficients and understand altimeter satellite orbital error, the Propagerror program was developed. This program calculates satellite orbital error from the differential components of Earth gravity model spherical harmonic coefficients (dC/Slm), which can be obtained from the difference between two gravity models or from seasonal and annual components of spherical harmonics. By separating appropriate low-order components in the Earth gravity models, the Propagerror program enables the estimation of satellite orbital error. In this study, we isolate C ̅_10,C ̅_11,( S) ̅_11, and C ̅_21,( S) ̅_21 coefficients in the EIGEN-GRGS.RL02bis.MF and EIGEN-6S gravity models to assess the geophysical impact on satellite orbits. The influence of the geometry center elements results in a 2 cm error in the Jason-2 satellite, while the rotational axis elements have no effect. The C ̅_31,( S) ̅_31 coefficient has a 6-7 mm impact on the accuracy of the Jason-2 satellite, as demonstrated by the satellite error map in two situations with and without the C ̅_31,( S) ̅_31 harmonic coefficient. This study highlights the significance of regulating function coefficients in satellite orbit determination, particularly the low-level harmonic parameters. The Propagerror program provides insights into the impact of each spherical harmonic parameter on satellite orbits, contributing to the improvement of orbit accuracy and the understanding of the Earth's gravity model.

Relative Orbital Motion Control of Spacecraft Based on Multi-Objective Optimization

In this paper, we consider the relative motion of a follower spacecraft orbiting the Earth in a near circular orbit with respect to a leader spacecraft. The relative orbital motion-control problem of the follower spacecraft is studied here. In order to allow the follower spacecraft to succeed in its target orbital motion maneuver, we proposed a multi-objective optimization method to solve the relative orbital motion-control problem. Firstly, a Hill–Clohessy–Wiltshire equation was used to describe the continuous relative orbital motion-control system. Then, the system was discretized into a discrete system using numerical methods. Next, a multi-objective optimization model of the relative orbital motion-control problem was formulated. In the model, two objectives, i.e., the orbital motion error and the energy consumption, were minimized simultaneously. Furthermore, the ε-constraint method was used to solve the multi-objective optimization problem and the Pareto front, which demonstrates that the trade-off between the two objectives can be achieved. Finally, numerical experiments were carried out to validate the effectiveness of the proposed multi-objective optimization approach.

Comprehensive Assessment of BDS-2 and BDS-3 Precise Orbits Based on B1I/B3I and B1C/B2a Frequencies from iGMAS

The BeiDou Global Navigation Satellite System (BDS), including the second generation (BDS-2) and the third generation (BDS-3), has been widely used in areas of positioning, navigation, and timing (PNT). One of the essential prerequisites for accurate PNT service is the precise satellite orbits of multi-frequency and multi-constellation BDS-2 and BDS-3 satellites. As usual, the precise orbit products can be obtained from analysis centers (ACs) of the international GNSS Service (IGS). The precise orbits can also be downloaded from the international GNSS Monitoring and Assessment System (iGMAS). Compared with the IGS ACs, the iGMAS can provide featured services such as satellite orbits based on the new B1C/B2a BDS signals. Considering the indispensability of the new signals, the performance of all BDS precise orbits from iGMAS needs to be known. However, there is no comprehensive assessment of BDS-2 and BDS-3 precise orbits based on B1I/B3II and B1C/B2a frequencies from iGMAS, especially for the period after the BDS entered the stable operation stage. In this paper, BDS-2/BDS-3 final (ISC), rapid (ISR), and ultra-rapid (ISU) products based on B1I/B3I and B1C/B2a frequencies from iGMAS are all assessed comprehensively. Specifically, at first, the precise orbits from iGMAS are compared with the ones from the IGS ACs. Based on this, the satellite laser ranging inspects the precise orbits from iGMAS. Finally, the orbit errors are discussed systematically by considering the beta and elongation angles. Using one year of data, the orbit accuracy of geostationary orbit, inclined geosynchronous orbit, and medium earth orbit (MEO) satellites can almost reach meter to decimeter level, decimeter to sub-decimeter level, and centimeter level, respectively, where the ISC products are the best. The ISC, ISR, and ISU products based on B1I/B3I frequencies are generally better than the ones based on B1C/B2a frequencies. Additionally, according to the SLR data, the results show that the accuracy of precise orbits of BDS-3 is better than that of BDS-2. The mean values of orbit biases of BDS-3 MEO satellites are approximately 2.88 cm. In addition, the orbit errors are related to the beta angle and elongation angle to some extent, and the manufacturers may also have an influence on the orbit errors.

Advanced ensemble modeling method for space object state prediction accounting for uncertainty in atmospheric density

For objects in the low Earth orbit region, uncertainty in atmospheric density estimation is an important source of orbit prediction error, which is critical for space traffic management activities such as the satellite conjunction analysis. This paper investigates the evolution of orbit error distribution in the presence of atmospheric density uncertainties, which are modeled using probabilistic machine learning techniques. The recently proposed “HASDM-ML,” “CHAMP-ML,” and “MSIS-UQ” machine learning models for density estimation (Licata and Mehta, 2022b; Licata et al., 2022b) are used in this work. The investigation is convoluted because of the spatial and temporal correlation of the atmospheric density values. We develop several Monte Carlo methods, each capturing a different spatiotemporal density correlation, to study the effects of density uncertainty on orbit uncertainty propagation. However, Monte Carlo analysis is computationally expensive, so a faster method based on the Kalman filtering technique for orbit uncertainty propagation is also explored. It is difficult to translate the uncertainty in atmospheric density to the uncertainty in orbital states under a standard extended Kalman filter or unscented Kalman filter framework. This work uses the so-called “consider covariance sigma point (CCSP)” filter that can account for the density uncertainties during orbit propagation. As a test-bed for validation purposes, a comparison between CCSP and Monte Carlo methods of orbit uncertainty propagation is carried out. Finally, using the HASDM-ML, CHAMP-ML, and MSIS-UQ density models, we propose an ensemble approach for orbit uncertainty quantification for four different space weather conditions.

Analysis of Baseline Impact on Differential Doppler Positioning and Performance Improvement Method for LEO Opportunistic Navigation

The positioning system based on the signals of opportunity (SOPs) of low-Earth-orbit (LEO) satellites can meet the positioning requirements of the receiver in the global navigation satellite system (GNSS) denial environment. However, the received signal does not contain or cannot extract the orbit information and signal emission time of the radiation source, which has become the main error source in this positioning system. Taking the Iridium satellites as an example, a Doppler differential (DD)-LEO (DD-LEO) positioning framework is proposed in this article, and the baseline impact analysis of the differential Doppler positioning system (DDPS) and a performance improvement method are carried out. First, the workflow between a base and a rover is introduced, and the error elimination of the DDPS is analyzed. Second, the equivalent Doppler error models formed by the orbit error at the base and the rover are established, respectively, and the relationship between the baseline and the mean or variance of equivalent Doppler residual in the DDPS is derived. Then, aiming at the problem of poor positioning accuracy due to the inconsistency of space–time reference in the long-baseline DDPS, a signal emission time estimation (SETE) algorithm based on maximum likelihood estimation (MLE) is proposed to further improve the positioning accuracy. Finally, a simulation experiment of the influence of baseline length and direction on the DDPS is implemented, and several groups of field differential experiments with different baseline lengths are completed. The experimental results of long-baseline positioning show that the DDPS can reduce the 3-D positioning error by 48.0% compared with the single station without differential positioning. At the same time, the positioning accuracy of the rover station is further enhanced by 86.3% when the base station adopts the SETE algorithm, which verifies the effectiveness of the proposed algorithm.

Image-Based Baseline Correction Method for Spaceborne InSAR With External DEM

An accurate baseline of synthetic aperture radar (SAR) interferometry (InSAR) is an important parameter for the geodetic application of the InSAR data. Although some advanced SAR satellites have precise orbit determination, there are still many SAR satellites suffering from baseline inaccuracies, such as GF-3. In this article, an image-based estimator for baseline correction is proposed, which requires only the external digital elevation model (DEM) data. The idea of the method is to project the orbit error phase onto the phase components carrying the baseline error information, which is called orbit error phase bases in this article, and to correct the baseline according to the projection coefficients. Since the pure orbit error phase is unavailable, the residual phase of the interferogram is used to approximate the orbit error phase, and a series of processes are introduced to weaken the effect of this approximation. Both the simulated and real data from GF-3 SAR are used to validate the proposed method, and a comparison with the conventional nonlinear least-square and the latest proposed flat-Earth phase-based baseline refinement methods are made. The results indicated the superior accuracy and robustness of our method, especially in areas with higher relief and wider coverage.

Doppler Positioning of LEO Satellites Based on Orbit Error Compensation and Weighting

Doppler Positioning of LEO Satellites Based on Orbit Error Compensation and Weighting

Performance Assessment of Multi-GNSS Real-Time Products from Various Analysis Centers

The performance of real-time precise point positioning (PPP) relies primarily on the availability and quality of orbit and clock corrections. In this research, we collected data streams from 12 real-time mount points of IGS Real-Time Service (RTS) or analysis centers for a one-month period and conducted a performance assessment, including product latency and data availability, accuracy of orbit, clock and positioning performance. The epoch availability of GPS, GLONASS, Galileo and BDS was more than 98.5%, 95.79%, 94.20% and 85.9%, respectively. In addition, the orbit and clock errors of different real-time corrections was investigated. Then, PPP in static and kinematic for 16 IGS stations was conducted. The results show the real-time PPP for different products has a longer convergence time and a slightly worse accuracy than those of the post-processing PPP. For static PPP over 24 h, the real-time products of WHU had the best performance, with a mean RMSE of 1.0 cm in the horizontal and vertical directions and a median convergence time of 12.0 min. The products of CAS had the faster convergence speed due to the shortest product latency. Regarding real-time kinematic PPP for GPS only in an hourly batch, the real-time products of WHU and ESA performed best with a mean RMSE of 10.8 cm and 9.5 cm in the horizontal and vertical directions, respectively. Additionally, the PPP for different real-time products with the multi-GNSS combination obtained higher accuracy than those with GPS only in post-processing or real-time mode, and the PPP with the GPS/GLONASS/Galileo/BDS combination had the fastest convergence speed and best positioning performance. The hourly based kinematic PPP results of CAS, DLR, GFZ and WHU with the GREC combination had positioning errors smaller than 5.2 cm.

Real-Time LEO Satellite Orbits Based on Batch Least-Squares Orbit Determination with Short-Term Orbit Prediction

The augmentation of the Global Navigation Satellite System (GNSS) by Low Earth Orbit (LEO) satellites is proposed as an effective method to improve the precision and shorten the convergence time of Precise Point Positioning (PPP). Serving as navigation satellites in the future, LEO satellites need to be provided with their high-accuracy orbits in real-time. This would potentially enable the high-accuracy real-time LEO satellite clock determination, and eventually facilitate the high-accuracy ground-based positioning. Studies have been performed to achieve such real-time orbits using a Kalman filter in both the kinematic and reduced-dynamic modes. Batch Least-Squares (BLS) adjustment delivers more stable orbits in near-real-time, as it performs better phase screening. However, it suffers from longer delays compared to the Kalman filter. With the LEO satellite orbit prediction strategies improved over time, this latency can be bridged by short-term orbit prediction. In this study, using real-time GNSS satellite products, the real-time LEO satellite orbits are obtained based on the batch least-squares adjustment and short-term prediction. LEO ephemeris parameters are generated within specific prediction time windows. Using real data from the 500 km GRACE C satellite and 810 km Sentinel-3B satellite, the near-real-time BLS Precise Orbit Determination (POD) results exhibit good accuracy with an Orbital User Range Error (OURE) of 2–4 cm using different real-time GNSS products. A range of delays of the BLS POD processes are assumed, based on tests performed on different processing machines, leading to various prediction windows, from 3–8 min to 12–17 min that correspond to the real-time usage. The orbital prediction errors are shown to be highly correlated with the orbital height and the prediction time. The computational efficiency thus becomes essential to reduce the prediction errors for a certain LEO satellite. For advanced processing units leading to a prediction window shorter or equal to 6–11 min, one can expect a total real-time orbital error budget of 3–5 cm, provided that an appropriate prediction strategy is applied and high-quality GNSS products are used. For a given fitting interval, the ephemeris fitting errors are generally related to the number of ephemeris parameters and the orbital height. Compared with the prediction errors, the ephemeris fitting errors do not play a significant role in the total error budget when using 22 ephemeris parameters.

Simulation of the Use of Variance Component Estimation in Relative Weighting of Inter-Satellite Links and GNSS Measurements

Inter-satellite links (ISLs) can improve the performance of the Global Navigation Satellite System (GNSS) in terms of precise orbit determination, communication, and data-exchange capabilities. This research aimed to evaluate a simulation-based processing strategy involving the exploitation of ISLs in orbit determination of Galileo satellites, which are not equipped with operational ISLs. The performance of the estimation process is first tested based on relative weighting coefficients obtained with methods of variance component estimation (VCE) varying in the complexity of the calculations. Inclusion of biases in the ISL measurements allows evaluation of the processing strategy and assessment of the impact of three different sets of ground stations: 44 and 16 stations distributed globally and 16 located in Europe. The results indicate that using different VCE approaches might lower orbit errors by up to 20% with a negligible impact on clock estimation. Depending on the applied ISL connectivity scheme, ISL range bias can be estimated with RMS between 10% to 30% of initial bias values. The accuracy of bias estimation may be associated with weighting approach and the number of ground stations. The results of this study show how introducing VCE with various simulation parameters into the processing chain might increase the accuracy of the orbit estimation.