Demonstration of sub-meter GPS orbit determination and 1.5 parts in 108 three-dimensional baseline accuracy

Differential tracking of theGPS satellites in high-earth orbit provides a powerful relative positioning capability, even when a relatively small continental U.S. fiducial tracking network is used with less than one-third of the fullGPS constellation. To demonstrate this capability, we have determined baselines of up to2000 km in North America by estimating high-accuracyGPS orbits and ground receiver positions simultaneously. The2000 km baselines agree with very long baseline interferometry(VLBI) solutions at the level of1.5 parts in10 ⁸ and showrms daily repeatability of0.3–2 parts in10 ⁸. The orbits determined for the most thoroughly trackedGPS satellites are accurate to better than1 m. GPS orbit accuracy was assessed from orbit predictions, comparisons with independent data sets, and the accuracy of the continental baselines determined along with the orbits. The bestGPS orbit strategies included data arcs of at least one week, process noise models for tropospheric fluctuations, estimation ofGPS solar pressure coefficients, and combined processing ofGPS carrier phase and pseudorange data. For data arcs of two weeks, constrained process noise models forGPS dynamic parameters significantly improved the solutions.     

Calibration of satellite gradiometer data aided by ground gravity data

Parametric least squares collocation was used in order to study the detection of systematic errors of satellite gradiometer data. For this purpose, simulated data sets with a priori known systematic errors were produced using ground gravity data in the very smooth gravity field of the Canadian plains. Experiments carried out at different satellite altitudes showed that the recovery of bias parameters from the gradiometer “measurements” is possible with high accuracy, especially in the case of crossing tracks. The mean value of the differences (original minus estimated bias parameters) was relatively large compared to the standard deviation of the corresponding second-order derivative component at the corresponding height. This mean value almost vanished when gravity data at ground level were combined with the second-order derivative data set at satellite altitude. In the case of simultaneous estimation of bias and tilt parameters from ∂² T/∂z ²“measurements”, the recovery of both parameters agreed very well with the collocation error estimation. Received: 10 October 1996 / Accepted 25 May 1998     

Fast integration of low orbiter's trajectory perturbed by the earth's non-sphericity

A fast algorithm is proposed to integrate the trajectory of a low obiter perturbed by the earth's non-sphericity. The algorithm uses a separation degree to define the low-degree and the high-degree acceleration components, the former computed rigorously, and the latter interpolated from gridded accelerations. An FFT method is used to grid the accelerations. An optimal grid type for the algorithm depends on the trajectory's permissible error, speed, and memory capacity. Using the non-spherical accelerations computed from EGM96 to harmonic degree 360, orbit integrations were performed for a low orbiter at an altitude of 170 km. For a separation degree of 50, the new algorithm, together with the predict-pseudo correct method, speeds up the integration by 145 times compared to the conventional algorithm while keeping the errors in position and velocity below 10⁻⁴ m and 10⁻⁷ m/s for a 3-day arc. Received: 28 July 1997 / Accepted: 1 April 1998     

Calibrating the GOCE accelerations with star sensor data and a global gravity field model

A reliable and accurate gradiometer calibration is essential for the scientific return of the gravity field and steady-state ocean circulation explorer (GOCE) mission. This paper describes a new method for external calibration of the GOCE gradiometer accelerations. A global gravity field model in combination with star sensor quaternions is used to compute reference differential accelerations, which may be used to estimate various combinations of gradiometer scale factors, internal gradiometer misalignments and misalignments between star sensor and gradiometer. In many aspects, the new method is complementary to the GOCE in-flight calibration. In contrast to the in-flight calibration, which requires a satellite-shaking phase, the new method uses data from the nominal measurement phases. The results of a simulation study show that gradiometer scale factors can be estimated on a weekly basis with accuracies better than 2 × 10⁻³ for the ultrasensitive and 10⁻² for the less sensitive axes, which is compatible with the requirements of the gravity gradient error. Based on a 58-day data set, scale factors are found that can reduce the errors of the in-flight-calibrated measurements. The elements of the complete inverse calibration matrix, representing both the internal gradiometer misalignments and scale factors, can be estimated with accuracies in general better than 10⁻³.     

GOCE gravitational gradients along the orbit

GOCE is ESA’s gravity field mission and the first satellite ever that measures gravitational gradients in space, that is, the second spatial derivatives of the Earth’s gravitational potential. The goal is to determine the Earth’s mean gravitational field with unprecedented accuracy at spatial resolutions down to 100 km. GOCE carries a gravity gradiometer that allows deriving the gravitational gradients with very high precision to achieve this goal. There are two types of GOCE Level 2 gravitational gradients (GGs) along the orbit: the gravitational gradients in the gradiometer reference frame (GRF) and the gravitational gradients in the local north oriented frame (LNOF) derived from the GGs in the GRF by point-wise rotation. Because the V _XX , V _YY , V _ZZ and V _XZ are much more accurate than V _XY and V _YZ , and because the error of the accurate GGs increases for low frequencies, the rotation requires that part of the measured GG signal is replaced by model signal. However, the actual quality of the gradients in GRF and LNOF needs to be assessed. We analysed the outliers in the GGs, validated the GGs in the GRF using independent gravity field information and compared their assessed error with the requirements. In addition, we compared the GGs in the LNOF with state-of-the-art global gravity field models and determined the model contribution to the rotated GGs. We found that the percentage of detected outliers is below 0.1% for all GGs, and external gravity data confirm that the GG scale factors do not differ from one down to the 10⁻³ level. Furthermore, we found that the error of V _XX and V _YY is approximately at the level of the requirement on the gravitational gradient trace, whereas the V _ZZ error is a factor of 2–3 above the requirement for higher frequencies. We show that the model contribution in the rotated GGs is 2–35% dependent on the gravitational gradient. Finally, we found that GOCE gravitational gradients and gradients derived from EIGEN-5C and EGM2008 are consistent over the oceans, but that over the continents the consistency may be less, especially in areas with poor terrestrial gravity data. All in all, our analyses show that the quality of the GOCE gravitational gradients is good and that with this type of data valuable new gravity field information is obtained.     

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.     

Simulation of regional gravity field recovery from satellite gravity gradiometer data using collocation and FFT

When planning a satellite gravity gradiometer (SGG) mission, it is important to know the quality of the quantities to be recovered at ground level as a function of e.g. satellite altitude, data type and sampling rate, and signal variance and noise. This kind of knowledge may be provided either using the formal error estimates of wanted quantities using least-squares collocation (LSC) or by comparing simulated data at ground level with results computed by methods like LSC or Fast Fourier Transform (FFT). Results of a regional gravity field recovery in a 10^o×20^o area surrounding the Alps using LSC and FFT are reported. Data used as observations in satellite altitude (202 or161 km) and for comparison at ground level were generated using theOSU86F coefficient set, complete to degree 360. These observations are referred to points across simulated orbits. The simulated quantities were computed for a 45 days mission period and 4 s sampling. A covariance function which also included terms above degree 360 was used for prediction and error estimation. This had the effect that the formal error standard deviation for gravity anomalies were considerably larger than the standard deviations of predicted minus simulated quantities. This shows the importance of using data with frequency content above degree 360 in simulation studies. Using data at202 km altitude the standard deviation of the predicted minus simulated data was equal to8.3 mgal for gravity and0.33 m for geoid heights.     

重力场模型对Swarm卫星精密定轨的影响分析

采用星载GPS观测数据与简化动力学定轨方法,在方程中引入伪随机脉冲参数,从而实现对Swarm卫星的精密定轨. 详细分析了不同阶次的GOCO06s地球重力场模型对Swarm卫星简化动力学定轨精度的影响,对比了PGM2000a、EIGEN-2、EGM2008以及GECO重力场模型展开到100阶次时Swarm卫星解算的轨道精度. 结果表明:当GOCO06s地球重力场模型阶次处于30～100阶次时,Swarm-A、Swarm-B和Swarm-C卫星在径向、切向、法向上的定轨精度随着GOCO06s阶次的不断增加而越来越高,而在高于100阶次时,定轨精度基本稳定,且在各方向定轨精度优于3 cm. 此外,采用100阶次GECO、EGM2008和GOCO06s模型对三颗Swarm卫星进行定轨,解算的轨道精度相当,且要高于同阶次其他重力场模型的定轨结果.      

On the precision and accuracy of IGS orbits

In order to explore the precision and accuracy of International GNSS Service (IGS) orbits, we difference geocentric satellite positions midway between successive daily Final orbits for the period starting 5 November 2006, when the IGS switched its method of antenna calibration, through 31 December 2007. This yields a time series of orbit repeatabilities analogous to the classical geodetic test for position determinations. If we compare our average positional discontinuities to the official IGS accuracy codes, root-sum-squared (RSS) for each pair of days, we find the discontinuities are not well correlated with the predicted performance values. If instead the IGS weighted root-mean-square (WRMS) values from the Final combination long-arc analyses are taken as the measure of IGS accuracy, we find the position differences and long-arc values are correlated, but the long-arc values are exaggerated, particularly around eclipses, despite the fact that our day-boundary position differences apply to a single epoch each day and the long-arc analyses consider variations over a week. Our method is not well suited to probe the extent to which systematic effects dominate over random orbit errors, as indicated by satellite laser ranging residuals, but eclipsing satellites often display the most problematic behavior. A better metric than the current IGS orbit accuracy codes would probably be one based on the orbit discontinuities between successive days.     

The geopotential value <Emphasis Type="Italic">W</Emphasis><Subscript>0</Subscript> for specifying the relativistic atomic time scale and a global vertical reference system

The TOPEX/Poseidon (T/P) satellite alti- meter mission marked a new era in determining the geopotential constant W ₀. On the basis of T/P data during 1993–2003 (cycles 11–414), long-term variations in W ₀ have been investigated. The rounded value W ₀ = 62636856.0 ± 0.5) m ² s ⁻² has already been adopted by the International Astronomical Union for the definition of the constant L _G = W ₀/c ² = 6.969290134 × 10⁻¹⁰ (where c is the speed of light), which is required for the realization of the relativistic atomic time scale. The constant L _G , based on the above value of W ₀, is also included in the 2003 International Earth Rotation and Reference Frames Service conventions. It has also been suggested that W ₀ is used to specify a global vertical reference system (GVRS). W ₀ ensures the consistency with the International Terrestrial Reference System, i.e. after adopting W ₀, along with the geocentric gravitational constant (GM), the Earth’s rotational velocity (ω) and the second zonal geopotential coefficient (J ₂) as primary constants (parameters), then the ellipsoidal parameters (a,α) can be computed and adopted as derived parameters. The scale of the International Terrestrial Reference Frame 2000 (ITRF2000) has also been specified with the use of W ₀ to be consistent with the geocentric coordinate time. As an example of using W ₀ for a GVRS realization, the geopotential difference between the adopted W ₀ and the geopotential at the Rimouski tide-gauge point, specifying the North American Vertical Datum 1988 (NAVD88), has been estimated.     

GPS relative positioning at a precision level of one part per billion

Six days of data from the GIG'91 experiment have been analyzed with a fiducial strategy. The results obtained with our orbital software GEOSAT, show daily horizontal and length repeatabilities at the level of 1 part in 10⁹ plus 2 mm for baseline lengths up to 4000 km. A direct comparison with results from the GIPSY software shows, with some exceptions, mean differences at the sub-cm level. After transformation to ITRF'90 the rms of the coordinate differences is 14.8 mm. Studies of orbital predictions and comparisons with external high precision orbits indicate a mean orbit precision and accuracy of around 35 cm in each cartesian coordinate. Correlations between the GEOSAT and GIPSY solutions indicate some common model deficiencies.     

Precise orbit determination for the FORMOSAT-3/COSMIC satellite mission using GPS

The joint Taiwan–US mission FORMOSAT-3/ COSMIC (COSMIC) was launched on April 17, 2006. Each of the six satellites is equipped with two POD antennas. The orbits of the six satellites are determined from GPS data using zero-difference carrier-phase measurements by the reduced dynamic and kinematic methods. The effects of satellite center of mass (COM) variation, satellite attitude, GPS antenna phase center variation (PCV), and cable delay difference on the COSMIC orbit determination are studied. Nominal attitudes estimated from satellite state vectors deliver a better orbit accuracy when compared to observed attitude. Numerical tests show that the COSMIC COM must be precisely calibrated in order not to corrupt orbit determination. Based on the analyses of the 5 and 6-h orbit overlaps of two 30-h arcs, orbit accuracies from the reduced dynamic and kinematic solutions are nearly identical and are at the 2–3 cm level. The mean RMS difference between the orbits from this paper and those from UCAR (near real-time) and WHU (post-processed) is about 10 cm, which is largely due to different uses of GPS ephemerides, high-rate GPS clocks and force models. The kinematic orbits of COSMIC are expected to be used for recovery of temporal variations in the gravity field.     

The impact of accelerometry on CHAMP orbit determination

 The contribution of the STAR accelerometer to the CHAMP orbit precision is evaluated and quantified by means of the following results: orbital fit to the satellite laser ranging (SLR) observations, GPS reduced-dynamic vs SLR dynamic orbit comparisons, and comparison of the measured to the modeled non-gravitational accelerations (atmospheric drag in particular). In each of the four test periods in 2001, five CHAMP arcs of 2 days' length were analyzed. The mean RMS-of-fit of the SLR observations of the orbits computed with STAR data or the non-gravitational force model were 11 and 24 cm, respectively. If the accelerometer calibration parameters are not known at least at the few percent level, the SLR orbit fit deteriorates. This was tested by applying a 10% error to the along-track scale factor of the accelerometer, which increased the SLR RMS-of-fit on average to 17 cm. Reference orbits were computed employing the reduced-dynamic technique with GPS tracking data. This technique yields the most accurate orbit positions thanks to the estimation of a large number of empirical accelerations, which compensate for dynamic modeling errors. Comparison of the SLR orbits, computed with STAR data or the non-gravitational force model, to the GPS-based orbits showed that the SLR orbits employing accelerometer observations are twice as accurate. Finally, comparison of measured to modeled accelerations showed that the level of geomagnetic activity is highly correlated with the atmospheric drag model error, and that the largest errors occur around the geomagnetic poles. Received: 7 May 2002 / Accepted: 18 November 2002 Correspondence to: S. Bruinsma Acknowledgments. The TIGCM results were obtained from the CEDAR database. This study was supported by the Centre National d'Etudes Spatiales (CNES). The referees are thanked for their helpful remarks and suggestions.     

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.     

Improvement in global gravity field recovery using GFZ-1 satellite laser tracking data

The passive satellite GFZ-1 has been orbiting the Earth since April 1995. The purpose of this mission is to improve the current knowledge of the Earth's gravity field by analysing gravitational orbit perturbations observed at unique low altitudes, below 400 km. GFZ-1 is one target of the international satellite laser ranging ground network. An evaluation of the first 30 months of GFZ-1 laser tracking data led to a new version of the global GRIM4-S4 satellite-only gravity field model: GRIM4-S4G. Information was obtained from GFZ-1 data for spherical harmonic coefficients up to degree 100, which was not possible in any earlier satellite-only gravity field solution. GFZ-1's contribution to a global 5 × 5° geoid and gravity field representations is moderate but visible with a 1 cm and 0.1 mGal gain in accuracy on a level of 75 cm and 5 mGal, respectively. Received: 10 November 1998 / Accepted: 19 April 1999     

Accuracy of GPS-derived relative positions as a function of interstation distance and observing-session duration

 Ten days of GPS data from 1998 were processed to determine how the accuracy of a derived three-dimensional relative position vector between GPS antennas depends on the chord distance (denoted L) between these antennas and on the duration of the GPS observing session (denoted T). It was found that the dependence of accuracy on L is negligibly small when (a) using the `final' GPS satellite orbits disseminated by the International GPS Service, (b) fixing integer ambiguities, (c) estimating appropriate neutral-atmosphere-delay parameters, (d) 26 km ≤ L ≤ 300 km, and (e) 4 h ≤T ≤ 24 h. Under these same conditions, the standard error for the relative position in the north–south dimension (denoted S _n and expressed in mm) is adequately approximated by the equation S _n =k _n /T ^0.5 with k _n =9.5 ± 2.1 mm · h^0.5 and T expressed in hours. Similarly, the standard errors for the relative position in the east–west and in the up-down dimensions are adequately approximated by the equations S _e =k _e /T ^0.5 and S _u =k _u /T ^0.5, respectively, with k _e =9.9 ± 3.1 mm · h^0.5 and k _u =36.5 ± 9.1 mm · h^0.5. Received: 5 February 2001 / Accepted: 14 May 2001     

Monitoring GOCE gradiometer calibration parameters using accelerometer and star sensor data: methodology and first results

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite, launched on 17 March 2009, is designed to measure the Earth’s mean gravity field with unprecedented accuracy at spatial resolutions down to 100?km. The accurate calibration of the gravity gradiometer on-board GOCE is of utmost importance for achieving the mission goals. ESA’s baseline method for the calibration uses star sensor and accelerometer data of a dedicated calibration procedure, which is executed every 2?months. In this paper, we describe a method for monitoring the evolution of calibration parameter during that time. The method works with star sensor and accelerometer data and does not require gravity field models, which distinguishes it from other existing methods. We present time series of calibration parameters estimated from GOCE data from 1 November 2009 to 17 May 2010. The time series confirm drifts in the calibration parameters that are present in the results of other methods, including ESA’s baseline method. Although these drifts are very small, they degrade the gravity gradients, leading to the conclusion that the calibration parameters of the ESA’s baseline method need to be linearly interpolated. Further, we find a correction of ?36 × 10^?6 for one calibration parameter (in-line differential scale factor of the cross-track gradiometer arm), which improves the gravity gradient performance. The results are validated by investigating the trace of the calibrated gravity gradients and comparing calibrated gravity gradients with reference gradients computed along the GOCE orbit using the ITG-Grace-2010s gravity field model.     

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.      

Precise orbit determination for the GRACE mission using only GPS data

The GRACE (gravity recovery and climate experiment) satellites, launched in March 2002, are each equipped with a BlackJack GPS onboard receiver for precise orbit determination and gravity field recovery. Since launch, there have been significant improvements in the background force models used for satellite orbit determination, most notably the model for the geopotential. This has resulted in significant improvements to orbit accuracy for very low altitude satellites. The purpose of this paper is to investigate how well the orbits of the GRACE satellites (about 470 km in altitude) can currently be determined using only GPS data and based on the current models and methods. The orbit accuracy is assessed using a number of tests, which include analysis of orbit fits, orbit overlaps, orbit connecting points, satellite Laser ranging residuals and K-band ranging (KBR) residuals. We show that 1-cm radial orbit accuracy for the GRACE satellites has probably been achieved. These precise GRACE orbits can be used for such purposes as improving gravity recovery from the GRACE KBR data and for atmospheric profiling, and they demonstrate the quality of the background force models being used.     

Precise alpine geoid determination without digital terrain models

The short wavelength geoid undulations, caused by topography, amount to several decimeters in mountainous areas. Up to now these effects are computed by means of digital terrain models in a grid of 100–500m. However, for many countries these data are not yet available or their collection is too expensive. This problem can be overcome by considering the special behaviour of the gravity potential along mountain slopes. It is shown that 90 per cent of the topographic effects are represented by a simple summation formula, based on the average height differences and distances between valleys and ridges along the geoid profiles, δN=[30.H.D.+16.(H−H′).D] in mm/km, (error<10%), whereH, H′, D are estimated in a map to the nearest 0.2km. The formula is valid for asymmetric sides of valleys (H, H′) and can easily be corrected for special shapes. It can be used for topographic refinement of low resolution geoids and for astrogeodetic projects. The “slope method” was tested in two alpine areas (heights up to 3800m, astrogeodetic deflection points every 170km ²) and resulted in a geoid accuracy of ±3cm. In first order triangulation networks (astro points every 1000km ²) or for gravimetric deflections the accuracy is about 10cm per 30km. Since a map scale of 1∶500.000 is sufficient, the method is suitable for developing countries, too.