Pseudo-Stochastic Orbit Modeling Techniques for Low-Earth Orbiters

The Earth’s non-spherical mass distribution and atmospheric drag cause the strongest perturbations on very low-Earth orbiting satellites (LEOs). Models of gravitational and non-gravitational accelerations are utilized in dynamic precise orbit determination (POD) with GPS data, but it is also possible to derive LEO positions based on GPS precise point positioning without dynamical information. We use the reduced-dynamic technique for LEO POD, which combines the geometric strength of the GPS observations with the force models, and investigate the performance of different pseudo-stochastic orbit parametrizations, such as instantaneous velocity changes (pulses), piecewise constant accelerations, and continuous piecewise linear accelerations. The estimation of such empirical orbit parameters in a standard least-squares adjustment process of GPS observations, together with other relevant parameters, strives for the highest precision in the computation of LEO trajectories. We used the procedures for the CHAMP satellite and found that the orbits may be validated by means of independent SLR measurements at the level of 3.2 cm RMS. Validations with independent accelerometer data revealed correlations at the level of 95% in the along-track direction. As expected, the empirical parameters compensate to a certain extent for deficiencies in the dynamic models. We analyzed the capability of pseudo-stochastic parameters for deriving information about the mismodeled part of the force field and found evidence that the resulting orbits may be used to recover force field parameters, if the number of pseudo-stochastic parameters is large enough. Results based on simulations showed a significantly better performance of acceleration-based orbits for gravity field recovery than for pulse-based orbits, with a quality comparable to a direct estimation if unconstrained accelerations are set up every 30 s.     

GOCE: precise orbit determination for the entire mission

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) was the first Earth explorer core mission of the European Space Agency. It was launched on March 17, 2009 into a Sun-synchronous dusk-dawn orbit and re-entered into the Earth’s atmosphere on November 11, 2013. The satellite altitude was between 255 and 225 km for the measurement phases. The European GOCE Gravity consortium is responsible for the Level 1b to Level 2 data processing in the frame of the GOCE High-level processing facility (HPF). The Precise Science Orbit (PSO) is one Level 2 product, which was produced under the responsibility of the Astronomical Institute of the University of Bern within the HPF. This PSO product has been continuously delivered during the entire mission. Regular checks guaranteed a high consistency and quality of the orbits. A correlation between solar activity, GPS data availability and quality of the orbits was found. The accuracy of the kinematic orbit primarily suffers from this. Improvements in modeling the range corrections at the retro-reflector array for the SLR measurements were made and implemented in the independent SLR validation for the GOCE PSO products. The satellite laser ranging (SLR) validation finally states an orbit accuracy of 2.42 cm for the kinematic and 1.84 cm for the reduced-dynamic orbits over the entire mission. The common-mode accelerations from the GOCE gradiometer were not used for the official PSO product, but in addition to the operational HPF work a study was performed to investigate to which extent common-mode accelerations improve the reduced-dynamic orbit determination results. The accelerometer data may be used to derive realistic constraints for the empirical accelerations estimated for the reduced-dynamic orbit determination, which already improves the orbit quality. On top of that the accelerometer data may further improve the orbit quality if realistic constraints and state-of-the-art background models such as gravity field and ocean tide models are used for the reduced-dynamic orbit determination.     

Combined processing of the IGS and the IGEX network

 Since the beginning of the International Global Navigation Satellite System (GLONASS) Experiment, IGEX, in October 1998, the Center for Orbit Determination in Europe (CODE) has acted as an analysis center providing precise GLONASS orbits on a regular basis. In CODE's IGEX routine analysis the Global Positioning System (GPS) orbits and Earth rotation parameters are introduced as known quantities into the GLONASS processing. A new approach is studied, where data from the IGEX network are combined with GPS observations from the International GPS Service (IGS) network and all parameters (GPS and GLONASS orbits, Earth rotation parameters, and site coordinates) are estimated in one processing step. The influence of different solar radiation pressure parameterizations on the GLONASS orbits is studied using different parameter subsets of the extended CODE orbit model. Parameterization with three constant terms in the three orthogonal directions, D, Y, and X (D = direction satellite–Sun, Y = direction of the satellite's solar panel axis), and two periodic terms in the X-direction, proves to be adequate for GLONASS satellites. As a result of the processing it is found that the solar radiation pressure effect for the GLONASS satellites is significantly different in the Y-direction from that for the GPS satellites, and an extensive analysis is carried out to investigate the effect in detail. SLR observations from the ILRS network are used as an independent check on the quality of the GLONASS orbital solutions. Both processing aspects, combining the two networks and changing the orbit parameterization, significantly improve the quality of the determined GLONASS orbits compared to the orbits stemming from CODE's IGEX routine processing. Received: 10 May 2000 / Accepted: 9 October 2000     

Swarm系列卫星非差运动学厘米级精密定轨

采用2015年5月24日—30日的Swarm星载GPS双频观测数据,基于Melbourne-Wübbena(MW)和消电离层线性组合,在精密单点定位技术的基础上,采用批处理最小二乘估计法对不同轨道高度的Swarm系列卫星进行非差运动学精密定轨。利用星载GPS相位观测值残差、与欧空局发布的简化动力学轨道对比,以及SLR检核3种方法对Swarm系列卫星非差运动学定轨结果进行精度评估。结果表明:①Swarm系列卫星星载GPS相位观测值残差RMS为6~7 mm;②与欧空局发布的简化动力学轨道进行求差,径向、切向及法向轨道差值RMS为2~4 cm;③与欧空局发布的运动学轨道进行求差,径向、切向及法向轨道差值RMS为1~2 cm;④SLR检核结果表明Swarm-A/B/C卫星轨道精度为3~4 cm。因此,采用非差运动学定轨方法与本文提供的定轨策略进行Swarm系列卫星精密定轨是切实可行的,定轨精度为厘米级。     

基于星载GPS的CHAMP卫星精密简动力定轨

定轨是地球探测卫星任务顺利执行的关键。星载GPS技术提供了大量、连续的高低卫星跟踪观测,为低轨卫星精密定轨提供了技术支撑。为了确定CHAMP卫星的轨道,并分析定轨精度,利用CHAMP卫星星载GPS数据,运用零差简动力法进行精密定轨,给出了精密定轨流程。利用实际数据进行了精密定轨实验,结果与德国地学研究中心（GFZ）公布的CHAMP卫星快速轨道（RSO）进行了对比,结果显示：求解轨道可以达到厘米量级。     

Laser-based validation of GLONASS orbits by short-arc technique

 The International GLONASS Experiment (IGEX-98) was carried out between 19 October 1998 and 19 April 1999. Among several objectives was the precise orbit determination of GPS and GLONASS satellites and its validation by laser ranging observations. Local laser-based orbit corrections (radial, tangential and normal components in a rotating orbital local reference frame) are computed using a geometrical short-arc technique. The order of magnitude of these corrections is at the level of few decimeters, depending on the considered components. The orbit corrections are analyzed as a function of several parameters (date, orbital plane, geographical area). The mean corrections are at the level of several centimeters. However, when averaging over the entire campaign and for all the satellites, no mean radial, tangential and normal orbit corrections are found. The origin of the observed corrections is considered (errors due to the geocentric gravitational constant, the non-gravitational forces, the thermal equilibrium of on-board equipment, the reference systems, the location and the signature of the retroreflector array, and the precision of the satellite laser ranges). Some features are also due to errors in the radio-tracking GLONASS orbits. Further investigations will be needed to better understand the origin of various biases. Received: 17 February 2000 / Accepted: 31 January 2001     

TOPEX/Poseidon orbit error assessment

This paper discusses the accuracy of TOPEX/Poseidon orbits computed at Delft University, Section Space Research & Technology (DUT/SSR&T), from several types of tracking data,i.e. SLR, DORIS, and GPS. To quantify the orbit error, three schemes are presented. The first scheme relies on the direct altimeter observations and the covariance of the JGM-2 gravity field. The second scheme is based on crossover difference residuals while the third scheme uses the differences of dynamic orbit solutions with the GPS reduced-dynamic orbit. All three schemes give comparable results and indicate that the radial orbit error of TOPEX/Poseidon is 3–4 cm. From the orbit comparisons with GPS reduced dynamic, both the along-track and cross-track errors of the dynamic orbit solutions were found to be within 10–15 cm.     

On fine orbit selection for particular geodetic and oceanographic missions involving passage through resonances

 The identification of mean semi-major axes (suitably defined) for satellite orbits to satisfy a variety of requirements for geodesy, geophysics and oceanography, in terms of repeat orbits (with orbital resonances), is investigated. Various options for the definition of semi-major axis, from the viewpoint of satellite dynamics, are described. Simple simulations of the expected resonant changes in inclination are presented, and tools for the analysis of orbit resonances to extract certain lumped harmonic coefficients of the geopotential (e.g. from the very precise CHAMP orbit) are resurrected. Finally, a preliminary example of the 46th-order resonance analysis possible for CHAMP, based on the mean orbital elements produced by GFZ (GeoForschungs Zentrum) for ephemeris prediction, is presented. Received: 10 July 2001 / Accepted: 17 July 2002 Correspondence to: J. Klokočník at Ondřejov Observatory Acknowledgements. We thank Prof. Dr. Ch. Reigber, Dr. P. Schwintzer, Dr. T. Gruber and Dr. R. K?nig from GFZ Potsdam for various consultations and discussions, and for the CHAMP two-line mean elements. This investigation was performed under the aegis of CEDR (Center for Earth's Dynamics Research, Prague-Ondřejov); it has been supported by project LN00A005 (provided by the Ministry of Education of the Czech Republic) and by grant A 3004 of the Grant Agency of the Academy of Sciences of the Czech Republic.     

基于星载GPS数据的Jason-3卫星简化动力学和运动学法精密定轨

利用Jason-3星载GPS观测数据,采用简化动力学方法和运动学方法对Jason-3卫星进行精密定轨研究. 通过载波相位残差、重叠轨道对比、参考轨道对比和卫星激光测距(SLR)轨道检核四种方式评定轨道精度. 计算相位残差均方根(RMS)值,简化动力学轨道的RMS值在0.7～0.8 cm,运动学轨道的RMS值在0.50～0.55 cm;简化动力学轨道重叠部分径向RMS值达到0.32 cm,运动学轨道重叠部分径向RMS值达到1.12 cm;与国际DORIS服务(IDS)官方提供的参考轨道对比,简化动力学轨道径向精度达到1.47 cm,运动学轨道径向精度达到4.36 cm;利用SLR观测数据进行核验,简化动力学轨道精度整体优于2.1 cm,运动学轨道精度整体优于3.3 cm. 通过实验证明:Jason-3卫星的简化动力学轨道和运动学轨道的精度均达到cm级.      

附加Helmert变换参数的低轨卫星约化动力学精密定轨

在运动学精密定轨以及动力学轨道积分的基础上,提出基于Helmert变换的约化动力学精密定轨模型.该模型对动力积分轨道以及运动学轨道建立Helmert变换,进而修正轨道积分中的卫星初始轨道以及各种动力学参数.应用该模型,文章采用的约化动力学精密定轨包含两个部分:运动学精密定轨以及基于Helmert变换的动力学轨道平滑.对CHAMP、GRACE两个星期的观测数据进行计算,结果显示:在引入Helmert变换平移参数的参数设置下,相对于运动学轨道,约化动力学轨道的精度平均提高了约30%;对于CHAMP卫星,约化动力学轨道与参考轨道差值在XYZ 3个方向RMS的平均值分别为(0.14,0.14,0.16) m,差值3D RMS的平均值为0.26 m;对于GRACE-A卫星,约化动力学轨道与参考轨道差值在XYZ 3个方向RMS的平均值分别为(0.17,0.15,0.13) m,差值3D RMS的平均值为0.26 m.文中还详细讨论和分析了模型中不同参数设置下轨道精度的情况.     

Evaluation of CHAMP satellite orbit with SLR measurements

The technique of Evaluating CHAMP satellite orbit with SLR measurements is presented. As an independent evaluation of the orbit solution, SLR data observed from January 1 to 16, 2002 are processed to compute the residuals after fixing the GFZ's post science orbits solutions. The SLR residuals are computed as the differences of the SLR measurements minus the corresponding distances between the SLR station and the GPS-derived orbit positions. On the basis of the SLR residuals analysis, it is found that the accuracy of GFZ' s spost science orbits is better than 10 cm and that there is no systematic error in GFZ's post science orbits.     

Evaluation of CHAMP satellite orbit with SLR measurements

The technique of Evaluating CHAMP satellite orbit with SLR measurements iS presented. As an independent evaluation of the orbit solution, SLR data observed from January 1 to 16, 2002 are processed to compute the residuals after fixing the GFZ＇s post science orbits solutions. The SLR residuals are computed as the differences of the SLR measurements minus the corresponding distances between the SLR station and the GPS-derived orbit positions. On the basis of the SLR residuals analysis, it is found that the accuracy of GFZ＇ s post science orbits is better than 10 cm and that there is no systematic error in GFZ＇s post science orbits.     

Model improvements and validation of TerraSAR-X precise orbit determination

The radar imaging satellite mission TerraSAR-X requires precisely determined satellite orbits for validating geodetic remote sensing techniques. Since the achieved quality of the operationally derived, reduced-dynamic (RD) orbit solutions limits the capabilities of the synthetic aperture radar (SAR) validation, an effort is made to improve the estimated orbit solutions. This paper discusses the benefits of refined dynamical models on orbit accuracy as well as estimated empirical accelerations and compares different dynamic models in a RD orbit determination. Modeling aspects discussed in the paper include the use of a macro-model for drag and radiation pressure computation, the use of high-quality atmospheric density and wind models as well as the benefit of high-fidelity gravity and ocean tide models. The Sun-synchronous dusk–dawn orbit geometry of TerraSAR-X results in a particular high correlation of solar radiation pressure modeling and estimated normal-direction positions. Furthermore, this mission offers a unique suite of independent sensors for orbit validation. Several parameters serve as quality indicators for the estimated satellite orbit solutions. These include the magnitude of the estimated empirical accelerations, satellite laser ranging (SLR) residuals, and SLR-based orbit corrections. Moreover, the radargrammetric distance measurements of the SAR instrument are selected for assessing the quality of the orbit solutions and compared to the SLR analysis. The use of high-fidelity satellite dynamics models in the RD approach is shown to clearly improve the orbit quality compared to simplified models and loosely constrained empirical accelerations. The estimated empirical accelerations are substantially reduced by 30% in tangential direction when working with the refined dynamical models. Likewise the SLR residuals are reduced from \(-3\,\pm \,17\) to \(2\,\pm \,13\) mm, and the SLR-derived normal-direction position corrections are reduced from 15 to 6 mm, obtained from the 2012–2014 period. The radar range bias is reduced from \(-10.3\) to \(-6.1\) mm with the updated orbit solutions, which coincides with the reduced standard deviation of the SLR residuals. The improvements are mainly driven by the satellite macro-model for the purpose of solar radiation pressure modeling, improved atmospheric density models, and the use of state-of-the-art gravity field models.     

Regularization of geopotential determination from satellite data by variance components

 Different types of present or future satellite data have to be combined by applying appropriate weighting for the determination of the gravity field of the Earth, for instance GPS observations for CHAMP with satellite to satellite tracking for the coming mission GRACE as well as gradiometer measurements for GOCE. In addition, the estimate of the geopotential has to be smoothed or regularized because of the inversion problem. It is proposed to solve these two tasks by Bayesian inference on variance components. The estimates of the variance components are computed by a stochastic estimator of the traces of matrices connected with the inverse of the matrix of normal equations, thus leading to a new method for determining variance components for large linear systems. The posterior density function for the variance components, weighting factors and regularization parameters are given in order to compute the confidence intervals for these quantities. Test computations with simulated gradiometer observations for GOCE and satellite to satellite tracking for GRACE show the validity of the approach. Received: 5 June 2001 / Accepted: 28 November 2001     

Apropos laser tracking to GPS satellites

. Laser tracking to GPS satellites (PRN5 and 6) provides an opportunity to compare GPS and laser systems directly and to combine data of both in a single solution. A few examples of this are given in this study. The most important results of the analysis are that (1) daily SLR station coordinate solutions could be generated with a few cm accuracy; (2) coordinates of nine stations were determined in a 2.3-year-long arc solution; (3) the contribution of laser data on the `SLR-GPS' combined orbit, resulting from the simultaneous processing of SLR and GPS data, is significant and (4) laser-only orbits have an accuracy of 10–20 cm, 1-day predictions of SLR orbits differ from IGS orbits by about 20–40 cm, 2-day predictions by 50–60 cm. Received: 1 October 1996 / Accepted: 14 February 1997     

High precision orbit determination of CHAMP satellite

IntroductionCHAMP satellite was launched on July 15 ,2000 to study the earth gravity field, magneticfield, and at mosphere/ionosphere . CHAMPmission is managed by GeoForshungsZentrum(GFZ) Potsdam withinternational partners .In-cluding accelerometer ,star …     

Time transfer using GPS carrier phase: error propagation and results

 A joint time-transfer project between the Astronomical Institute of the University of Berne (AIUB) and the Swiss Federal Office of Metrology and Accreditation (METAS) was initiated to investigate the power of the time transfer using GPS carrier phase observations. Studies carried out in the context of this project are presented. The error propagation for the time-transfer solution using GPS carrier phase observations was investigated. To this purpose a simulation study was performed. Special interest was focussed on errors in the vertical component of the station position, antenna phase-center variations and orbit errors. A constant error in the vertical component introduces a drift in the time-transfer results for long baselines in east–west directions. The simulation study was completed by investigating the profit for time transfer when introducing the integer carrier phase ambiguities from a double-difference solution. This may reduce the drift in the time-transfer results caused by constant vertical error sources. The results from the present time-transfer solution are shown in comparison to results obtained with independent time-transfer techniques. The interpretation of the comparison benefits from the investigations of the error propagation study. Two types of solutions are produced on a regular basis at AIUB: one based on the rapid orbits from CODE, the other on the CODE final orbits. The rapid solution is available the day after the observations and has nearly the same quality as the final solution, which has a latency of about one week. The differences between these two solutions are below the nanosecond level. The differences from independent time-transfer techniques such as TWSTFT (two-way satellite time and frequency transfer) are a few nanoseconds for both products. Received: 15 November 2001 / Accepted: 6 September 2002 Correspondence to:R. Dach     

GOCE orbit predictions for SLR tracking

After a descent phase of about half a year, the Gravity field and steady-state Ocean Circulation Explorer (GOCE) reached the final orbital altitude of the first measurement and operational phase (MOP-1) in September 2009. Due to this very low orbital altitude and the inactive drag compensation during descent, the generation of reliable predictions of the GOCE trajectory turned out to be a major challenge even for short prediction intervals. As predictions of good quality are a prerequisite for frequent ranging from the tracking network of the International Laser Ranging Service (ILRS), Satellite Laser Ranging (SLR) data of GOCE was very sparse at mission start and made it difficult to independently calibrate and optimize the orbit determination based on data of the Global Positioning System (GPS). In addition to the GOCE orbit predictions provided by the European Space Agency (ESA), the Astronomical Institute of the University of Bern (AIUB) started providing predictions on July 22, 2009, as part of the Level 1b to Level 2 data processing performed at AIUB. The predictions based on the 12-h ultra-rapid products of the International GNSS Service (IGS) were originally intended to primarily serve the daylight passes in the early evening hours over Europe. The corresponding along-track prediction errors were often kept below 50 m during the descent phase and allowed for the first successful SLR tracking of GOCE over Europe on July 29, 2009, by the Zimmerwald observatory. Additional predictions based on the IGS 18-h ultra-rapid products are provided by AIUB since September 20, 2009, to further optimize the GOCE SLR tracking. In this article, the development of the GOCE prediction service at AIUB is presented, and the quality of the orbit predictions is assessed for periods with and without active drag compensation. The prediction quality is discussed as a function of the prediction interval, the quality of the input products for the GPS satellite orbits and clocks, and the availability of the GOCE GPS data. From the methodological point of view, different approaches for the treatment of the non-gravitational accelerations acting on the GOCE satellite are discussed and their impact on the prediction quality is assessed, in particular during the descent phase. Eventually, an outlook is given on the significance of GOCE SLR tracking to identify systematic errors in the GPS-based orbit determination, e.g., cross-track errors induced by mismodeled GOCE GPS phase center variations (PCVs).     

GRACE卫星非差简化动力学定轨研究

本文基于卫星精密定轨的基本理论,研究了GRACE卫星非差简化动力学定轨的方法;并用自行研制的定轨软件CASMORD对实测的星载GPS数据进行非差数据的简化动力学定轨,通过比较GRACE卫星解算的轨道与JPL事后轨道及SLR测距信息,结果表明:利用非差观测值进行CRACE卫星的简化动力学定轨,三维位置精度(3D-RMS)...     

The celestial mechanics approach: application to data of the GRACE mission

The celestial mechanics approach (CMA) has its roots in the Bernese GPS software and was extensively used for determining the orbits of high-orbiting satellites. The CMA was extended to determine the orbits of Low Earth Orbiting satellites (LEOs) equipped with GPS receivers and of constellations of LEOs equipped in addition with inter-satellite links. In recent years the CMA was further developed and used for gravity field determination. The CMA was developed by the Astronomical Institute of the University of Bern (AIUB). The CMA is presented from the theoretical perspective in (Beutler et al. 2010). The key elements of the CMA are illustrated here using data from 50 days of GPS, K-Band, and accelerometer observations gathered by the Gravity Recovery And Climate Experiment (GRACE) mission in 2007. We study in particular the impact of (1) analyzing different observables [Global Positioning System (GPS) observations only, inter-satellite measurements only], (2) analyzing a combination of observations of different types on the level of the normal equation systems (NEQs), (3) using accelerometer data, (4) different orbit parametrizations (short-arc, reduced-dynamic) by imposing different constraints on the stochastic orbit parameters, and (5) using either the inter-satellite ranges or their time derivatives. The so-called GRACE baseline, i.e., the achievable accuracy of the GRACE gravity field for a particular solution strategy, is established for the CMA.