Research on Cm-level kinematic precise orbit determination for GRACE based on zero-difference method

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.

This paper describes a process by which Global Navigation Satellite System (GNSS) nominal statistics and fault rates can be assessed with high confidence. The use of a GNSS in Advanced Receiver Autonomous Integrity Monitoring (ARAIM) requires knowledge of the level of performance of the signal-in-space (SIS) user range error (URE) in both nominal and faulty conditions. The performance characterization is done through a careful analysis of historical data, from which a determination of whether or not it meets the performance commitments of the constellation service provider (CSP) can be made. Typically, this analysis is carried out though a comparison of the broadcast navigation messages with post-processed precise estimates of the satellite clock and orbit states. This paper highlights the issues with such an approach as well as mitigation strategies to those issues. The issues primarily impact the estimates of fault rates rather than nominal statistics, as the number of faults is small compared to the overall amount of nominal data. In particular, methods of ensuring integrity in the logged broadcast navigation messages and precise clock and orbit products used in the analysis are described. In order to verify precise clock and orbit products, a Kalman filter (KF) to produce independent GNSS clock bias estimates has been developed and tested. The KF leverages the International GNSS Service (IGS) receiver network to produce estimates of GNSS clock biases given precisely known positions of the receivers as well as precise satellite orbit products. The KF serves multiple purposes in detecting and verifying faults. First, when only low rate (5 or 15 minute) clock products are available, higher rate products can be produced to detect short faults. Second, when precise clock products are available, the KF serves as a secondary check to protect against erroneous external precise products, which have been observed. Initial testing of the filter has shown an RMS error of under 20 centimeters for a five day run of GPS clock bias estimates, which is sufficient performance for precise product verification and fault detection.

Due to an increasing requirement for high accuracy orbital information for low Earth orbit (LEO) satellites, precise orbit determination (POD) of LEO satellites is a topic of growing interest. To assure the safety and reliability of the applications requiring high accuracy LEO orbits in near-real-time, integrity monitoring (IM) is an essential operation of the POD process. In this contribution, the IM strategy for LEO POD in both the kinematic and reduced-dynamic modes is investigated. The overbounding parameters of the signal-in-space range error are investigated for the GPS products provided by the International GNSS Service’s Real-Time Service and the Multi-GNSS Advanced Demonstration of Orbit and Clock Analysis service. Benefiting from the dynamic models used and the improved model strength, the test results based on the data of the LEO satellite GRACE FO-1 show that the average-case mean protection levels (PLs) can be reduced from about 3–4 m in the kinematic mode to about 1 m in the reduced-dynamic mode in the radial, along-track and cross-track directions. The overbounding mean values of the SISRE play the dominant role in the final PLs. In the reduced-dynamic mode and average-case projection, the IM availabilities reach above 99% in the radial, along-track and cross-track directions with the alert limit (AL) set to 2 m. The values are still above 98% with the AL set to 4 m, when the duty cycle of tracking is reduced to 40%, e.g., in the case of power limits for miniature satellites such as CubeSats.

The Chinese Area Positioning System (CAPS) is a new type of satellite navigation systems. The geostationary satellites for normal communication with 2 or 3 sets of inclined geostationary orbit (GEO) satellites are employed to construct a satellite navigation system. The navigation signals are constructed and disseminated on the ground, and the satellites only transfer the signals to the ground by transponders. Such a system is easy to construct. But for such a positioning system, it needs precise and accurate orbit determination of geostationary satellites. A GEO satellite is placed at an altitude of 36000km above the equator and its movement with respect to the station on the ground is very small, so it is very difficult to determine the orbit precisely. An original method for determination of satellite orbits based on pairs of combined observations is developed at the National Time Service Center, Chinese Academy of Sciences. A special ranging observation with the distance between two stations via a satellite, called pairs of combined observations, is proposed. The satellite behaves with such an observation is not a ball, but an ellipse ball. The Dilution of Precision (DOP) is much less than those methods such as the SLR observation. The advantage of pairs of combined observations is that clock offsets for all observing stations and on-bound are cancelled by the method itself, so, the accuracy and precision of orbit determination is not affected by the clock offsets and the DOP is greatly improved. The ranging signals are generated and measured on the ground, and the instrumentation errors can be determined in real time with a high accuracy. Therefore, such a system is very stable. A very high rate of the PN code is employed, a disseminated power is very low, and it can observe signals from remote stations with a lower power less than 1w. With the spread-spectrum technique, multi-stations can work at the same time and same carrier frequency. It can reach the accuracy of raging about 1 cm. The accuracy and precision of the orbit determination are very high, up to a few centimeters. The observing system with the model proposed above for the GEO satellites including 5 stations (one main station and 4 slave stations) placed in China has been set up. The combined observations were made between the main station and the slave stations. The number of combined observations for the 5 stations is 9. The observations are made per second. The observations for 8 sets of satellites are continuous, lasting for about 5 years. The results for the GEO observations show that the observation residuals of the satellite orbit determination for a 3-day solution are 4.0 cm. With the development of the method to determine the orbit based on the discussion above, it is very easy to correct the orbit with observations nearly in real time. Obviously, the method proposed by the National Time Service Center is suitable for precise orbit determination of GEO satellites in a satellite navigation system, and also it is suitable for satellite management and deep space tracking.

Precise orbit and Earth parameter determination supported by LEO satellites, inter-satellite links and synchronized clocks of a future GNSS

Orbit determination and gravity field recovery from Doppler tracking data to the Lunar Reconnaissance Orbiter

Topex/Jason combined GPS/DORIS orbit determination in the tandem phase

Since the beginning of satellite altimetry missions, the ocean surface topography community requires precise and accurate satellite orbits. With start of the satellite Sentinel-3A on February 16, 2016, two radio- and one altimeter onboard the satellite accomplish the Copernicus program with a ocean- and land monitoring mission, planned for a nominal mission lifetime of 7 years. The satellite is orbiting the Earth on a polar, Sun-synchronous trajectory at an altitude of 815 km. For the purpose of precise orbit determination, the satellite is equipped with a geodetic-grade dual-frequency Global Positioning System (GPS) receiver. The GPS measurements are employed together with a set of gravitational and non-gravitational models in a Reduced-Dynamic Orbit Determination (RDOD) approach, which combines the advantages of a dynamic and a kinematic positioning for deriving precise satellite orbits. However, especially the non-gravitational force models require sophisticated modeling techniques. Therefore, a satellite macro model is introduced, which allows a proper modeling of accelerations due to Solar Radiation Pressure (SRP), Earth Radiation Pressure (ERP), and atmospheric drag. Especially the Sun-synchronous orbit, and the huge solar array, which is loosely coupled to the satellite body, makes the precise orbit determination challenging. As members of the Copernicus Quality Working Group, the Astronomical Institute of the University of Bern (AIUB), and the German Aerospace Center (DLR) are, among others, responsible for the orbit validation of Sentinel-3A. Both groups make use of a satellite macro model within a reduced-dynamic approach but differ in the employed software solutions and the pseudo-stochastic modeling. Basically, the pseudo-stochastic parameters allow to compensate potential deficits in the employed force models. Within this poster, the modeling aspects are briefly introduced, followed by comparing the results of both groups. The results include the estimated empirical accelerations, and the estimated satellite orbits of Sentinel-3A. Furthermore, the orbit quality is assessed by Satellite Laser Ranging (SLR), an external and independent tool for orbit validation.

Enhanced real-time onboard orbit determination for low-Earth-orbit satellites is essential for autonomous spacecraft operations. However, the accuracy of such systems is often limited by signal propagation delays between GPS satellites and the user spacecraft. These delays, primarily due to Earth’s rotation and ionospheric effects become particularly significant in high-dynamic LEO environments, leading to considerable errors in range and range rate measurements, and consequently, in position and velocity estimation. To mitigate these issues, this paper proposes a real-time orbit determination algorithm that applies Earth rotation correction and dual-frequency (L1 and L2) ionospheric compensation to raw GPS measurements. The enhanced orbit determination method is processed directly in the Earth-centered Earth-fixed frame, eliminating repeated coordinate transformations and improving integration with ground-based systems. The proposed method employs a reduced-dynamic orbit determination strategy to balance model fidelity and computational efficiency. A predictive correction model is also incorporated to compensate for GPS signal delays under dynamic motion, thereby enhancing positional accuracy. The overall algorithm is embedded within an extended Kalman filter framework, which assimilates the corrected GPS observations with a stochastic process noise model to account for dynamic modeling uncertainties. Simulation results using synthetic GPS measurements, including pseudoranges and pseudorange rates from a dual-frequency spaceborne receiver, demonstrate that the proposed method provides a significant improvement in orbit determination accuracy compared to conventional techniques that neglect signal propagation effects. These findings highlight the importance of performing orbit estimation directly in the Earth-centered, Earth-fixed reference frame, utilizing pseudoranges that are corrected for ionospheric errors, applying reduced-dynamic filtering methods, and compensating for signal delays. Together, these enhancements contribute to more reliable and precise satellite orbit determination for missions operating in low Earth orbit.

For precise orbit determination (POD) and precise applications with POD products, one of the critical issues is the modeling of non-conservative forces acting on satellites. Since the official publication of Galileo satellite metadata in 2017, analytical models including the box-wing model and thermal thrust models have been established to absorb a substantial amount of solar radiation pressure (SRP) and thermal thrust. These models serve as the foundation for the best overall modeling approach, combining the analytical box-wing model and thermal thrust model with parameterization of the remaining non-conservative perturbing forces using various optimized Empirical CODE Orbit Models (ECOMs) of the Center for Orbit Determination in Europe (CODE). Firstly, we have demonstrated the significance of the second-order signals in the D direction and the first-order signals in the B direction through spectral analyses of the pure box-wing model, which are consistent with the currently recommended 7-parameter Empirical CODE Orbit Model 2 (ECOM2). In spite of this, we still found that degradation in orbit accuracy frequently occurs during deep eclipse seasons when using the ECOM2 model. We confirm a high-frequency signal existing in the fluctuating orbit overlap differences through the spectral analysis. Considering this, the ECOM2 force model should be extended to higher order and adapted to absorb the remaining effects of potential perturbing forces. After extending the ECOM2 force model to the sixth order in the Sun direction, we demonstrated the significance of fourth- and sixth-order sine terms for deep eclipses. Due to the higher-order periodic terms, the averaged RMS values of orbit overlap difference over deep eclipses can be reduced from 5.3, 10.8, and 23.8 cm to 3.2, 3.9, and 9.9 cm for in-orbit validation (IOV) satellites, from 5.0, 8.6, and 17.7 cm to 3.0, 3.0, and 7.1 cm for the first generation of full operational capability (FOC-1) satellites, and from 5.4, 8.6, and 19.0 cm to 3.6, 3.6, and 7.4 cm for the second generation of FOC (FOC-2) satellites, in the radial, cross-track, and along-track directions, respectively. Fluctuations with a peak amplitude of approximately 0.4 nm/s2 in the bias in the solar panel axis (Y) direction (Y-bias) are effectively mitigated by the higher-order terms. Due to the higher-order terms, the vertical positioning errors during kinematic precise point positioning (PPP) convergence can be improved from 42.3 to 37.1 cm at the 95.5% confidence level. Meanwhile, a low correlation level of up to 0.02 is found between the newly introduced higher-order parameters and earth rotation parameters (ERPs).

Geostationary satellites (GEOs) play a significant role in the regional satellite navigation system. Simulation experiments show that the clock corrections could be mitigated through a single strategy or double differencing strategies for a navigation constellation, but for the mode of individual GEO orbit determination, high precision orbit and clock correction could not be obtained in the orbit determination based on the pseudorange data. A new GEO combined precise orbit determination (POD) strategy is studied in this paper, which combines pseudorange data and C-band transfer ranging data. This strategy overcomes the deficiency of C-band transfer ranging caused by limited stations and tracking time available. With the combination of transfer ranging and pseudorange data, clock corrections between the GEO and the stations can be estimated simultaneously along with orbital parameters, maintaining self-consistency between the satellite ephemeris and clock correction parameters. The error covariance analysis is conducted to illuminate the contributions from the transfer ranging data and the psudoranging data. Using data collected for a Chinese GEO satellite with 3 C-band transfer ranging stations and 4 L-band pseudorange tracking stations, POD experiments indicate that a meter-level accuracy is achievable. The root-mean-square (RMS) of the post-fit C-band ranging data is about 0.203 m, and the RMS of the post-fit pseudorange is 0.408 m. Radial component errors of the POD experiments are independently evaluated with the satellite laser ranging (SLR) data from a station in Beijing, with the residual RMS of 0.076 m. The SLR evaluation also suggests that for 2-h orbital predication, the predicted radial error is about 0.404 m, and the clock correction error is about 1.38 ns. Even for the combination of one C-band transfer ranging station and 4 pseudorange stations, POD is able to achieve a reasonable accuracy with the radial error of 0.280 m and the 2-h predicted radial error of 0.888 m. Clock synchronization between the GEO and tracking stations is achieved with an estimated accuracy of about 1.55 ns, meeting the navigation service requirements.

A novel method, which is based on the triple-frequency combination and Space-Based Telemetry, Tracking, and Command (STT&C) stations, is proposed in this paper. Considering BeiDou Navigation Satellite System (BDS) Geostationary Orbit (GEO) and Inclined Geostationary Orbit (IGSO) satellites as the STT&C facilities, firstly, we presented the BDS Medium Earth Orbit (MEO) satellites’ precise orbit determination scheme based on triple-frequency combination. Then, we gave the sufficient and necessary conditions about the visibility and the coverage rate calculation model of STT&C to BDS MEO satellite. And then we deduced the model of BDS MEO satellites precise orbit determination based on triple-frequency combination observations. At last, we designed the simulation calculation. The simulation results show that orbit determination of BDS MEO satellite based on STT&C station can be realized at all times. And most of the simulation period time, under the condition of the dm level orbit determination for GEO/IGSO satellites, the position accuracy of the relative orbit determination is better than 4 m, the horizontal accuracy of the relative orbit determination is within 2.5 m, and the vertical accuracy of the relative orbit determination is less than 3.5 m.

The inter-satellite link (ISL) equipment of the third-generation BeiDou Navigation Satellite System (BDS-3) can enhance the accuracy of precise orbit determination (POD) for BDS-3 satellites. However, the hardware delays within ISL observations can impact the measurement accuracy of ISL observations. This study evaluates the characteristics of hardware delays in ISL observations for BDS-3 medium Earth orbit (MEO) satellites and investigates their effects on POD. Analysis shows that most links exhibit slow variations in hardware delays, with standard deviations (STD) of 6.4 cm and 3.6 cm over 45 days for the C19-C25 and C20-C21 links, respectively. Conversely, some links like C30-C23 and C30-C33 show more pronounced variations, with STD values reaching 17.7 cm and 14.1 cm within three days. Links involving satellites C23 and C30 display higher variability and significant jumps in hardware delays compared with other links. The larger variations in hardware delays of these links can be absorbed by other parameters, thereby affecting the accuracy of parameter estimation. To mitigate their impact on POD, a segmented estimation strategy for hardware delays is proposed. This strategy divides a single link into multiple segments based on a set interval length and estimates the hardware delay parameters for each segment separately, thereby absorbing the variations in hardware delays or unmodeled errors. The effectiveness of this strategy is demonstrated through ISL POD residual analysis, comparison with L-band orbit determination results, and validation using Satellite Laser Ranging (SLR) residuals. The root mean square (RMS) of ISL observation residuals decreased by approximately 44.3%, from 6.1 cm to 3.4 cm. Comparisons with L-band orbits confirmed an approximate 28.2% reduction in the 3D RMS of orbit discrepancies, from 26.5 cm to 19.0 cm. Additionally, the RMS of the overall SLR residuals showed a slight decrease. Overall, the improved segmented estimation strategy effectively reduces the influence of hardware delay variations on BDS-3 orbit determination, notably enhancing the orbit determination accuracy of BDS-3 satellites, especially for C23 and C30, which experience larger hardware delay fluctuations.

Precise Orbit Determination (POD) is an integral part for analyzing measurements from space geodetic techniques such as Satellite Laser Ranging (SLR) and Global Navigation Satellite Systems (GNSS) such as the Global Positioning System (GPS). In the last two decades, POD based on GPS data has furthermore been established as one of the standard techniques to derive trajectories of satellites in the low Earth orbit (LEO) with highest accuracy. Since the launch of dedicated gravity missions, GPS sensors are not only used as a key tracking system for LEO POD, but also for extracting the long wavelength part of the Earth’s gravity field (together with SLR to spherical satellites). This chapter introduces SLR and GNSS measurements collected by the terrestrial networks of the International Laser Ranging Service (ILRS) and the International GNSS Service (IGS) as the observational basis for the realization of a terrestrial reference frame from satellite data. On this foundation, the basic equations and mathematical methods of orbit determination are introduced and extensively discussed. Pseudo-stochastic orbit modeling techniques are eventually presented as a general and efficient concept to determine satellite trajectories of highest quality even in presence of deficient force models, covering the full range between dynamic and purely kinematic solutions. Selected results from the application of the discussed orbit determination techniques are highlighted for GPS LEO data. Special emphasis is also put to present orbit determination in the context of more general orbit determination problems, where satellite trajectories are simultaneously determined with other parameters encompassing (at maximum) all pillars of geodesy, i.e., the shape, rotation, and gravity field of the Earth.

For real-time precise orbit determination (POD) of low earth orbit (LEO) satellites, high-accuracy global navigation satellite system (GNSS) orbit and clock products are necessary in real time. Recently, the Japanese multi-GNSS advanced demonstration of orbit and clock analysis precise point positioning (PPP) service and the new generation of the Australian/New Zealand satellite-based augmentation system (SBAS)-aided PPP service provide free and precise GNSS products that are directly broadcast through the navigation and geostationary earth orbit satellites, respectively. With the high quality of both products shown in this study, a 3D accuracy of centimeters can be achieved in the post-processing mode for the reduced-dynamic orbits of small LEO satellites having a duty cycle down to 40% and at sub-dm to dm level for the kinematic orbits. The results show a promising future for high-accuracy real-time POD onboard LEO satellites benefiting from the precise free-of-charge PPP corrections broadcast by navigation systems or SBAS.