Real-time precise orbit determination for BDS satellites using the square root information filter

Validation and evaluation on B1IB3I-based and B1CB2a-based BDS-3 precise orbits from iGMAS

Real-time satellite orbit and clock estimations are the prerequisite for Global Navigation Satellite System (GNSS) real-time precise positioning services. To meet the high-rate update requirement of satellite clock corrections, the computational efficiency is a key factor and a challenge due to the rapid development of multi-GNSS constellations. The Square Root Information Filter (SRIF) is widely used in real-time GNSS data processing thanks to its high numerical stability and computational efficiency. In real-time clock estimation, the outlier detection and elimination are critical to guarantee the precision and stability of the product but could be time-consuming. In this study, we developed a new quality control procedure including the three standard steps: i.e., detection, identification, and adaption, for real-time data processing of huge GNSS networks. Effort is made to improve the computational efficiency by optimizing the algorithm to provide only the essential information required in the processing, so that it can be applied in real-time and high-rate estimation of satellite clocks. The processing procedure is implemented in the PANDA (Positioning and Navigation Data Analyst) software package and evaluated in the operational generation of real-time GNSS orbit and clock products. We demonstrated that the new algorithm can efficiently eliminate outliers, and a clock precision of 0.06 ns, 0.24 ns, 0.06 ns, and 0.11 ns can be achieved for the GPS, GLONASS, Galileo, and BDS-2 IGSO/MEO satellites, respectively. The computation time per epoch is about 2 to 3 s depending on the number of existing outliers. Overall, the algorithm can satisfy the IGS real-time clock estimation in terms of both the computational efficiency and product quality.

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.

The precise orbit determination (POD) for BeiDou satellites mainly relies on a regional tracking network distributed in China area at the present time. The multi-satellite POD for BeiDou regional navigation satellite system via double-difference observations is presented. First, the double-difference phase measurements are modeled on ionosphere-free combination of B1 and B2 signals. The POD strategies are detailed, followed by a brief data-processing flow. Then, the POD tests based on regional network of domestic monitoring stations are developed, and the orbit quality assessed by arc overlaps and Satellite Laser Ranging (SLR) residuals shows an accuracy of about 20 cm in radial direction. Finally, contributions of the overseas monitoring stations to the POD accuracy of BeiDou satellites are investigated by introducing several IGS Multi-GNSS Experiment (IGS M-GEX) monitoring stations with BeiDou tracking capability into the multi-satellite POD test. The improved orbit accuracy indicates that more BeiDou observing sites should be established worldwide in future.

Global Navigation Satellite System (GNSS) precise orbit, clock and observable-specific bias (OSB) products are the prerequisite for precise point positioning with ambiguity solution (PPP-AR). Under the condition of sparse or regional ground network, integrating low Earth orbit (LEO) onboard observations is a potential approach to supplementing GNSS global tracking measurements. In this contribution, a method of integrating ground and LEO onboard observations to generate GNSS precise satellite clock and phase OSB based on square root information filter (SRIF) estimation is proposed. Particularly, a method of phase OSB estimation with a 1.5 h sampling rate is established under regional ground stations and 8 LEO satellites. The results show that integrating 30 global ground stations and 8 LEO satellites, the standard deviation (STD) and convergence time of satellite clocks reach to 0.093 ns and 4.93 h, which reduced by 41.5% and 35.4% compared to ground-only solutions, respectively. Phase OSB accuracy improves from 0.092 to 0.078 cycles, and the product availability increases from 42.2% to 95.1%. The kinematic PPP-AR accuracy of integrated solutions improved by 26%, 17% and 18% on the E, N and U component, and convergence time accelerated by 17%. In addition, phase OSB products generated by 20 European regional ground stations integrating 8 LEO satellites achieve an STD of 0.086 cycles, with the product availability of more than 98%, and kinematic PPP-AR accuracy is close to the solutions with phase OSB generated by 70 ground stations.

Regarding the problem of systematic multipath bias on BeiDou navigation system code observations, ten days BeiDou triple-frequency observations of multi-stations were collected, the relationships between the characteristic of multipath bias and the type of the satellite, the signal frequency and the satellite elevation angle were analyse d. In order to solve this problem, a modified sidereal filtering based on zero-phase filter was proposed to mitigate the multipath bias on GEO and IGSO satellites code observations, and elevation-angle correction method was introduced to decrease the multipath bias of IGSO and MEO satellite code observations. The experimental results show that the modified sidereal filtering can decrease the multipath bias of B1, B2 and B3 frequencies code observations for GEO and IGSO satellites over 17.0, 18.6 and 10.5%, respectively, and the elevation-angle correction method can reduce the multipath bias of IGSO and MEO satellite B1, B2 and B3 frequencies code measurements by 5.0, 6.8, and 4.1%, respectively. After mitigating the multipath bias of BeiDou code observations, the precision of single point position for east, north and up directions can be improved about 34, 45 and 51%, respectively.

The precise orbit determination (POD) accuracy of the Chinese BeiDou Navigation Satellite System (BDS) is still not comparable to that of the Global Positioning System because of the unfavorable geometry of the BDS and the uneven distribution of BDS ground monitoring stations. Fortunately, low Earth orbit (LEO) satellites, serving as fast moving stations, can efficiently improve BDS geometry. Nearly all studies on Global Navigation Satellite System POD enhancement using large LEO constellations are based on simulations and their results are usually overly optimistic. The receivers mounted on a spacecraft or an LEO satellite are usually different from geodetic receivers and the observation conditions in space are more challenging than those on the ground. The noise level of spaceborne observations needs to be carefully calibrated. Moreover, spaceborne observational errors caused by space weather events, i.e., solar geomagnetic storms, are usually ignored. Accordingly, in this study, the actual spaceborne observation noises are first analyzed and then used in subsequent observation simulations. Then, the observation residuals from the actual-processed LEO POD during a solar storm on 8 September 2017 are extracted and added to the simulated spaceborne observations. The effect of the observational errors on the BDS POD augmented with different LEO constellation configurations is analyzed. The results indicate that the noise levels from the Swarm-A, GRACE-A, and Sentinel-3A satellites are different and that the carrier-phase measurement noise ranges from 2 mm to 6 mm. Such different noise levels for LEO spaceborne observations cause considerable differences in the BDS POD solutions. Experiments calculating the augmented BDS POD for different LEO constellations considering spaceborne observational errors extracted from the solar storm indicate that these errors have a significant influence on the accuracy of the BDS POD. The 3D root mean squares of the BDS GEO, IGSO, and MEO satellite orbits are 1.30 m, 1.16 m, and 1.02 m, respectively, with a Walker 2/1/0 LEO constellation, and increase to 1.57 m, 1.72 m, and 1.32 m, respectively, with a Walker 12/3/1 constellation. When the number of LEO satellites increases to 60, the precision of the BDS POD improves significantly to 0.89 m, 0.77 m, and 0.69 m for the GEO, IGSO, and MEO satellites, respectively. While 12 satellites are sufficient to enhance the BDS POD to the sub-decimeter level, up to 60 satellites can effectively reduce the influence of large spaceborne observational errors, i.e., from solar storms.

Due to the continued development of the GLONASS satellites, precise orbit determination (POD) still poses a series of challenges. This study examines the impact of introducing the analytical tube-wing model for GLONASS-M and the box-wing model for GLONASS-K in a series of hybrid POD strategies that consider both the analytical model and a series of empirical parameters. We assess the perturbing accelerations acting on GLONASS spacecraft based on the analytical model. All GLONASS satellites are equipped with laser retroreflectors for satellite laser ranging (SLR). We apply the SLR observations for the GLONASS POD in a series of GNSS + SLR combined solutions. The application of the box-wing model significantly improves GLONASS orbits, especially for GLONASS-K, reducing the STD of SLR residuals from 92.6 to 27.6 mm. Although the metadata for all GLONASS-M satellites reveal similar construction characteristics, we found differences in empirical accelerations and SLR offsets not only between GLONASS-M and GLONASS-M+ but also within the GLONASS-M+ series. Moreover, we identify satellites with inferior orbit solutions and check if we can improve them using the analytical model and SLR observations. For GLONASS-M SVN730, the STD of the SLR residuals for orbits determined using the empirical solution is 48.7 mm. The STD diminishes to 41.2 and 37.8 mm when introducing the tube-wing model and SLR observations, respectively. As a result, both the application of the SLR observations and the analytical model significantly improve the orbit solution as well as reduce systematic errors affecting orbits of GLONASS satellites.

To meet the demands of research and precise point positioning (PPP) in a multi-GNSS environment, we developed a GNSS data processing software named multi-GNSS automatic precise positioning software (MG-APP). MG-APP is an open-source software that can be run on Windows/Linux/UNIX and other operating systems. It can simultaneously process GPS/GLONASS/BDS/Galileo observations using a Kalman filter or a square root information filter (SRIF). Compared to the Kalman filter, the SRIF has better numerical stability and maintains stable convergence even with a significant round-off error. MG-APP has a comprehensive and friendly graphical user interface that conveniently allows the user to select models and set parameters. It also contains several types of tropospheric and estimation models that make it easy to analyze the impact of different models and parameters on PPP data processing. After the data processing finishes, zenith tropospheric delays, receiver clock offsets, satellite ambiguity parameters, observation residuals, and other results will be saved into files. Users can further analyze the solution results and construct graphs easily.

<p>Numerous active low Earth orbiters (LEOs) and Global Navigation Satellite System (GNSS) satellites, including the Galileo constellation, are equipped with laser retroreflectors used for Satellite Laser Ranging (SLR). Moreover, most of LEOs are equipped with GNSS receivers for precise orbit determination. SLR measurements to LEOs, GNSS, and geodetic satellites vary in terms of the number of registered normal points (NPs) or registered satellite passes. In 2016-2018, SLR measurements to LEOs constituted 81% of all NPs and 59% of all registered satellite passes, whereas 10% of NPs and 30% of satellite passes, respectively, were assigned to GNSS. The remaining SLR measurements were completed by geodetic satellites, including LAGEOS-1/2, and LARES-1.</p><p>In this study, we show that the SLR observations to Galileo, passive geodetic and active LEO satellites together with precise GNSS-based orbits of LEOs and Galileo, can be used for the determination of global geodetic parameters, such as geocenter coordinates (GCC) and Earth rotation parameters (ERPs), i.e. pole coordinates, and length-of-day parameter.</p><p>GCC are typically determined using SLR observations to passive geodetic satellites, such as LAGEOS-1/2. Also, the SLR observations to LAGEOS-1/2 together with GNSS and Very Long Baseline Interferometry data are used for the determination of ERPs. Here, we use SLR observations to Galileo, LAGEOS-1/2, LARES-1, Sentinel-3A, SWARM-A/B/C, TerraSAR-X, Jason-2, GRACE-A/B satellites to investigate whether they can be applied for the reference frame realization and for deriving high-quality global geodetic parameters.</p><p>We present various types of solutions to investigate the best solution set-up. The studied solutions differ in terms of solution lengths, the combination of different sets of satellites and the relative weights for the variance scaling factors of technique and satellite-specific normal equations. We compare our results with the standard LAGEOS-based solutions, the combined EOP-14-C04 products and show the consistency of the results.</p>

Precise orbit determination and rapid orbit recovery supported by time synchronization

Satellite laser ranging (SLR) to the satellites of the global navigation satellite systems (GNSS) provides substantial and valuable information about the accuracy and quality of GNSS orbits and allows for the SLR-GNSS colocation in space. In the framework of the NAVSTAR-SLR experiment two GPS satellites of Block-IIA were equipped with laser retroreflector arrays (LRAs), whereas all satellites of the GLONASS system are equipped with LRAs in an operational mode. We summarize the outcome of the NAVSTAR-SLR experiment by processing 20 years of SLR observations to GPS and 12 years of SLR observations to GLONASS satellites using the reprocessedmicrowave orbits provided by the center for orbit determination in Europe (CODE). The dependency of the SLR residuals on the size, shape, and number of corner cubes in LRAs is studied. We show that the mean SLR residuals and the RMS of residuals
\ndepend on the coating of the LRAs and the block or type of GNSS satellites. The SLR mean residuals are also a function of the equipment used at SLR stations including the single-photon and multi-photon detection modes.We also showthat the SLR observations to GNSS satellites are important to validate GNSS orbits and to assess deficiencies in the solar radiation pressure models. We found that the satellite signature effect, which is defined as a spread of optical pulse signals due to reflection from multiple reflectors, causes the variations of mean SLR residuals of up to 15mmbetween the observations at nadir angles of 0◦ and 14◦. in case of multiphoton SLR stations. For single-photon SLR stations this effect does not exceed 1 mm. When using the new empirical
\nCODE orbit model (ECOM), the SLR mean residual falls into the range 0.1–1.8 mm for high-performing singlephoton SLR stations observing GLONASS-M satellites with uncoated corner cubes. For best-performing multi-photon stations the mean SLR residuals are between −12.2 and
\n−25.6 mm due to the satellite signature effect.

The differential code bias (DCB) in Global Navigation Satellite System (GNSS) satellites should be precisely determined using real ground GNSS data when designing certain applications, such as precise positioning, navigation and timing. This paper analyzed the relationship between satellite elevation angle with DSNR (difference signal to noise ratio) of BDS (BeiDou Navigation Satellite System) different orbital satellite. The experimental results showed that the DSNR of IGSO satellite was less than 5dBHz and more stable, much better than the other satellites; the change of GEO satellite DSNR was small, but the absolute value was large; MEO satellite DSNR changed frequently and big. A new algorithm to get the BDS observation weight is introduced by considering the distance between the satellite and the Earth and DSNR. The results indicated that the new weight algorithm improved the DCB accuracy and enhanced the stabilities, which improved GEO satellites 1–15 %, IGSO satellites 4.5–16 %, MEO satellites 20–22 %.

<p><span><span>Global Navigation Satellite Systems (GNSS) play a critical role for providing real-time positioning and navigation services, and the precise satellite orbit and clock products are essential for the high-precision GNSS applications. The International GNSS Service (IGS) and its Analysis Centers (ACs) have been working on the study on precise GNSS data processing and provision of the precise products. The German Research Center for Geosciences (GFZ), as one of the ACs, also provides the multi-GNSS rapid products: the GBM product. We introduce the GBM data processing strategy, analyze the precision of GBM multi-GNSS orbits from 2015 to 2021, and present the impact of applying the undifferenced ambiguity resolution on satellite orbits. The GPS orbits of GBM products agree with the IGS final orbits at the level of 11-13 mm in the three directions, and the GPS orbit 6-hour prediction precision is around 6 cm. The 6-hour prediction precision of GLONASS is around 12 cm, slightly worse than that of GALILEO, which hold an average value of 10 cm in the same period but shows a significant improvement to around 5 cm after end of 2016. The prediction precision of BDS MEO satellites are around 10 cm, and that of the BDS GEO satellites and QZSS satellites are at the level of 1 to 3 meter. The Satellite Laser Ranging (SLR) residuals show that the orbit precision of GALILEO, GLONASS, and BD3-MEO is 23 mm, 41 mm, and 47 mm, respectively. Moreover, comparing the double-differenced ambiguity resolution, adopting the undifferenced ambiguity resolution improves the 6-hour orbit prediction precision by 9-15%</span></span>，<span><span>15-18%</span></span>，<span><span>11-13%</span></span>，<span><span>6-17%</span></span>和<span><span>14-25% for the GPS, GLONASS, GALILEO, BDS-2 and BDS-3 MEO satellites, respectively.</span></span></p>

GPS triple-frequency undifferenced and uncombined precise orbit determination with the consideration of receiver time-variant bias