Accuracy Analysis of GPT2/GPT2w Models for SLR-Based Satellite Orbits Validation

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.

Satellite laser ranging (SLR) constitutes a fundamental space geodetic technique providing global geodetic parameters, such as geocenter coordinates, Earth rotation parameters, and low-degree gravity field coefficients. The tropospheric delay correction is one of the crucial corrections that have to be taken into account when processing SLR data. Current conventional models of the troposphere delays assume a full symmetry of the atmosphere above SLR stations. Neglecting horizontal gradients in SLR solutions introduces a systematic error in SLR products, especially for the observations at low elevation angles, and leads to a deterioration of the consistency between SLR and other space geodetic techniques, such as global navigational satellite systems and very-long-baseline interferometry. We derive new mapping function coefficients, as well as first- and second-order horizontal gradients, all of which are based on numerical weather models, in order to properly consider the azimuthal asymmetry in SLR solutions. We test the enhanced mapping function and horizontal gradients on the solutions based on 11 years of SLR observations to LAGEOS-1/2 satellites and 1 year of SLR observations to Sentinel-3A. The consideration of azimuthal asymmetry of the atmosphere above the SLR stations has a systematic effect on SLR-derived products, such as station and geocenter coordinates and pole coordinates. Horizontal gradients in SLR solutions improve the consistency between SLR-derived pole coordinates and the combined IERS-C04 series by means of reducing the offset for the X and Y pole coordinates by 20 {upmu }mathrm{as}. The second-order horizontal gradients are negligible in SLR solutions; thus, including first-order gradients is sufficient for SLR solutions.

In the last decade, we have been witnessing a rapid development of the constellations of Global and Regional Navigation Satellite Systems (GNSS/RNSS). Besides the well-known GPS and GLONASS, newly developed systems such as Galileo, BeiDou, QZSS and NAVIC have become increasingly important. All satellites of new GNSS are equipped with laser retroreflector arrays (LRA) dedicated to Satellite Laser Ranging (SLR). SLR allows, e.g., an independent validation of microwave-based orbit products. Therefore, a fully operational online service called the multi-GNSS Orbit Validation Visualizer Using SLR (GOVUS) has been developed allowing for near real-time analysis of the quality of multi-GNSS orbits. The mean offsets of SLR residuals for Center for Orbit Determination in Europe (CODE) orbits in 2016 are at the level of −8, −38, −14, and −107 mm, for BeiDou, Galileo, GLONASS, and QZSS, respectively, with the standard deviations of 66, 36, 29, and 100 mm. Moreover, GOVUS can be used as a database containing information on equipment used at SLR stations and multi-GNSS satellite parameters. This paper includes a comprehensive description of the functionality and the structure of the developed service with exemplary analyses. The paper points out the most critical issues, limitations and challenges of multi-GNSS and SLR tracking network in the context of the SLR orbit validation. The goal of the paper and GOVUS itself is to determine: (1) what is the current quality of multi-GNSS orbits validated using SLR results; (2) what kinds of systematic errors can affect GNSS orbits and SLR observations; and (3) how to provide the online analysis tools to the broadest possible multi-GNSS community. The service has been officially operating since March 2017 as the Associate Analysis Center of the International Laser Ranging Service (ILRS ACC).

This study aims to evaluate and analyze the orbit predictions of selected satellites: geodetic, Global Navigational Satellite Systems (GNSS), and scientific low-orbiting, which are tracked by laser stations. The possibility of conducting satellite laser ranging (SLR) to artificial satellites depends on the access to high-quality predictions of satellite orbits. The predictions provide information to laser stations where to aim the telescope in search of a satellite to get the returns from the retroreflectors installed onboard. If the orbit predictions are very imprecise, SLR stations must spend more time to correct the telescope pointing, and thus the number of collected observations is small or, in an extreme case, there are none of them at all. Currently, there are about 120 satellites equipped with laser retroreflectors orbiting the Earth. Therefore, the necessity to determine the quality of predictions provided by various analysis centers is important in the context of the increasing number of satellites tracked by SLR stations. We compare the orbit predictions to final GNSS orbits, precise orbits of geodetic satellites based on SLR measurements determined in postprocessing, and kinematic orbits of low-orbiting satellites based on GPS data. We assess the quality degradation of the orbit predictions over time depending on the type of orbit and the satellite being analyzed. We estimate the time of usefulness of prediction files, and indicate those centers which publish most accurate predictions of the satellites’ trajectories. The best-quality predictions for geodetic satellites and Galileo reach the mean error of 0.5–1 m for the whole 5-day prediction file (for all three components), while the worst ones can reach values of up to several thousand meters during the first day of the prediction.

The Russian Global Navigation Satellite System (GLONASS) satellites have a stretched body shape and take a specific attitude mode inside the eclipse. Based on previous studies, the new Empirical CODE orbit model (ECOM2) performs better than the classical ECOM model if a satellite has elongated shape or does not maintain yaw-steering mode, and the use of an a priori box-wing (BW) model improves the orbits significantly when employing the ECOM model. However, we find that the ECOM model performs better than the ECOM2 model for GLONASS satellites outside eclipse seasons, while it performs two times worse in eclipse seasons. The use of the conventional box-wing model results in very little improvement. By assessing the ECOM Y _{0} estimates, we conclude that there are potential radiators on the -x surface of GLONASS satellites causing orbit perturbations also inside the eclipse. The higher-order Fourier terms of the ECOM2 model can compensate for such effects better than the ECOM model. Based on this finding, we first confirm that GLONASS-K satellites take a similar attitude mode as GLONASS-M satellites inside the eclipse. Then, we adjust optical parameters of GLONASS satellites as part of precise orbit determination (POD) considering the potential radiator and thermal radiation effects. Finally, the adjusted parameters are introduced into a new box-wing model and jointly used with the ECOM and ECOM2 model, respectively. Results show that the amplitude and the dependency of the empirical parameters on the beta angle are greatly reduced for both ECOM and ECOM2 models. Rather than the conventional box-wing model, the new box-wing model reduces the orbit misclosure between two consecutive arcs for both GLONASS-M and GLONASS-K satellites. In particular, the improvement in GLONASS-M satellites is more than 30% for the ECOM model during eclipse seasons. Further evaluation from 24-h predicted orbits demonstrates that the improvement during eclipse seasons is mainly in along- and cross-track directions. Finally, we validate GLONASS satellite orbits using Satellite Laser Ranging (SLR) observations. The use of the new box-wing model reduces the spurious pattern of the SLR residuals as a function of beta and Delta u significantly, and the linear dependency of the SLR residuals on the elongation drops from as large as -0.760 mm/deg to almost zero for both ECOM and ECOM2 models. In general, GLONASS-M satellites benefit more from the new a priori box-wing model and the BW+ECOM model results in the best SLR residuals, with an improvement of about 50% and 20%, respectively, for the mean and standard deviation (STD) values with respect to the orbit products without a priori model.

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.

The challenge of precise orbit determination for STSAT-2C using extremely sparse SLR data

Atmospheric pressure loading displacement of SLR stations

The quality of BeiDou Navigation Satellite System (BDS) Geostationary Earth Orbit (GEO) ultra-rapid products is unsatisfactory because GEO satellites are nearly stationary relative to ground stations. To optimize the quality of these ultra-rapid orbit products, we investigated the effects of the fitting arc length, an a priori Solar-Radiation Pressure (SRP) model, and the along-track empirical acceleration on the prediction of BDS GEO satellite orbits. The predicted orbit arcs of 24-h were evaluated through comparisons with the corresponding observed orbit arc and Satellite Laser Ranging (SLR) observations. In both eclipse and non-eclipse seasons, accuracy of the orbit predictions obtained using a 48-h fitting arc length were better than those obtained using 24-h and 72-h fitting arc lengths. Although the overlapping precision of predicted orbits exhibited no obvious improvement when an a priori SRP model was employed, the systematic bias in the SLR residuals was significantly reduced. Specifically, the mean value of SLR residuals decreased from −0.248 m to −0.024 m during non-eclipse seasons and from −0.333 m to −0.041 m during eclipse seasons, respectively. In addition, when an empirical acceleration in the along-track direction was introduced, the three-Dimensional Root-Mean-Square (3D RMS) of overlapping orbits during eclipse seasons decreased from 2.964 to 1.080 m, which is comparable to that during non-eclipse seasons. Furthermore, the Standard Deviation (STD) of SLR residuals decreased from 0.419 to 0.221 m during eclipse seasons. The analysis of SRP estimates shows that the stability of SRP parameters was significantly enhanced after the introduction of along-track empirical acceleration in eclipse seasons. The optimal BDS GEO ultra-rapid orbit prediction products were yielded by using a 48-h fitting arc length, an a priori SRP model and an along-track empirical acceleration.

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.

LAser RElativity Satellite 2 (LARES-2) successfully joined the constellation of geodetic satellites tracked by Satellite Laser Ranging (SLR) stations on July 13, 2022. LARES-2 has a spherical shape and a very favorable area-to-mass ratio that minimizes the non-gravitational orbit perturbations. Due to very small retroreflectors, the spread of center-of-mass corrections for different detectors installed at SLR sites is much smaller than for LAGEOS satellites. LARES-2 orbits at a similar height as LAGEOS-1, however, with a complementary inclination angle of 70° forming a butterfly configuration together with LAGEOS-1.Although the primary objective of LARES-2 is verification of the Lense-Thirring effect emerging from general relativity, the satellite also has a substantial impact on the geodetic parameters derived from SLR observations. We process 18 months of LARES-2 data and compare the LAGEOS-1/2 solutions with the combined LAGEOS-1/2+LARES-2 solutions. We show the impact of LARES-2 on the (1) SLR station coordinates, (2) pole coordinates, (3) length-of-day excess, (4) low-degree gravity field parameters focusing on C20 and C30 coefficients, (5) scale of the reference frame, (6) geocenter motion. We show that LARES-2 can especially improve the Z component of the geocenter coordinates and de-correlate C20 from the length-of-day parameter. The secular drifts of the ascending nodes for LARES-1 and LAGEOS-1 caused by C20 are the same in terms of absolute values but with opposite signs. This allows us to successfully separate the measurements of length-of-day excess (or the UT rate) from the C20-induced changes. We also analyze the empirical accelerations acting on LARES-2 which result from unmodeled non-gravitational orbit perturbations, such as thermal effects, and compare them to those observed for LAGEOS satellites. The observation geometry of LARES-2 is especially beneficial for stations located at high and medium latitudes, which allows it to improve the estimation of station coordinates provided by LAGEOS-1/2. Therefore, LARES-2 substantially contributes not only to general relativity and fundamental physics but also to space geodesy improving the future realizations of the international terrestrial reference frames.

Atmospheric pressure loading plays a crucial role in precise space geodesy, thus its omission, especially the nontidal loading (ANTL) part, leads to the inconsistency between solutions based on the microwave [Global Navigation Satellite Systems (GNSS)] and optical [satellite laser ranging (SLR)] observations. SLR observations are performed only during the cloudless conditions coincident with high values of air pressure that deforms the earth's crust downward. The systematic shift of the estimated SLR station coordinates, which arises from the ANTL omission, is called the Blue-Sky effect. The offset is related to the long-term averaging of ANTL for SLR observations which are provided in sparse intervals, unlike GNSS, which observes continuously. Based on the ANTL model applied on SLR observations to GNSS, we determined the values of the Blue-Sky effect. The largest values of the Blue-Sky effect are observed for the inland stations, i.e., 2.3, 2.1, 2.0, and 1.9 mm for Svetloe, Potsdam, Baikonur, and Altay, respectively. We also investigate the impact of the omission of ANTL corrections on geodetic parameters, i.e., earth rotation parameters, geocenter motion, precise GNSS orbits, as well as on the global SLR network. The negligence of ANTL causes a systematic effect on geocenter coordinates with the amplitude of the annual signal at the level of 1.9 mm for the Z-component. The omission of ANTL corrections causes also a systematic shift of the multi-GNSS orbit constellation with the amplitude of the annual signal at the level of 2.7 mm for the Z-component.

The National Geographic Institute of Spain (IGN Spain) develops, operates, and exploits the Spanish national geodetic networks and their associated infrastructure. This includes the permanent networks of reference GNSS (Global Navigation Satellite System) receivers, VLBI (Very Long Baseline Interferometry) telescopes, and a new SLR (Satellite Laser Ranging) station under construction at the Yebes Observatory. In order to fully exploit the opportunities offered by the availability of these space geodetic techniques, IGN has been operating Analysis Centres of GNSS and VLBI for a number of years, with the recent addition of an Associated Analysis Centre of the International Laser Ranging Service. IGN Spain is a EUREF Analysis Centre since 2001, contributing with their weekly and daily coordinate solutions to the realisation of the European Terrestrial Reference System. The activities of the AC include projects such as IBERRED, for geodynamic purposes, and the participation in the European E-GVAP programme for meteorological applications. Over the last years, IGN Spain has expanded its contribution to geodetic VLBI analysis, starting in 2019 the operational analysis of VLBI sessions and the submission of the respective solutions (containing consistently estimated Earth Orientation Parameters, station coordinates and source positions) to the International VLBI Service for Geodesy and Astrometry (IVS). Additionally, the reprocessing of the historical VLBI data since 1979 is ongoing, which is the basis for future contributions to the IVS combination series that will be provided for future realisations of the international terrestrial reference frame (ITRF). The latest addition to IGN Spain analysis capabilities is the IGN-Yebes Associated Analysis Centre. Highlights of its activities are the participation in the ITRF2020 reprocessing in collaboration with the Analysis Centre NSGF in the UK, and the computation of SLR centre-of-mass corrections for spherical geodetic satellites. IGN strives to further exploit the synergies between these groups and work towards the combined analysis of the data. A description of the analysis activities of IGN Spain, along with its future prospects, is presented.

Precise orbit determination for geostationary satellites with multiple tracking techniques

GipsyX/RTGx is the Jet Propulsion Laboratory’s (JPL) next generation software package for positioning, navigation, timing, and Earth science using measurements from three geodetic techniques: Global Navigation Satellite Systems (GNSS), Satellite Laser Ranging (SLR), and Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS); with Very Long Baseline Interferometry (VLBI) under development. The software facilitates combined estimation of geodetic and geophysical parameters using a Kalman filter approach on real or simulated data in both post-processing and in real-time. The estimated parameters include station coordinates and velocities, satellite orbits and clocks, Earth orientation, ionospheric and tropospheric delays. The software is also capable of full realization of a dynamic terrestrial reference through analysis and combination of time series of ground station coordinates.Applying lessons learned from its predecessors, GIPSY-OASIS and Real Time GIPSY (RTG), GipsyX/RTGx was re-designed from the ground up to offer improved precision, accuracy, usability, and operational flexibility. We present some key aspects of its new architecture, and describe some of its major applications, including Real-time orbit determination and ephemeris predictions in the U.S. Air Force Next Generation GPS Operational Control Segment (OCX), as well as in JPL’s Global Differential GPS (GDGPS) System, supporting User Range Error (URE) of <5 cm RMS; precision post-processing GNSS orbit determination, including JPL’s contributions to the International GNSS Service (IGS) with URE in the 2 cm RMS range; Precise point positioning (PPP) with ambiguity resolution, both statically and kinematically, for geodetic applications with 2 mm horizontal, and 6.5 mm vertical repeatability for static positioning; Operational orbit and clock determination for Low Earth Orbiting (LEO) satellites, such as NASA’s Gravity Recovery and Climate Experiment (GRACE) mission with GRACE relative clock alignment at the 20 ps level; calibration of radio occultation data from LEO satellites for weather forecasting and climate studies; Satellite Laser Ranging (SLR) to GNSS and LEO satellites, DORIS-based and multi-technique orbit determination for LEO; production of terrestrial reference frames and Earth rotation parameters in support of JPL’s contribution to the International Terrestrial Reference Frame (ITRF).