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.     

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.     

GPS-based precise orbit determination of the very low Earth-orbiting gravity mission GOCE

 A prerequisite for the success of future gravity missions like the European Gravity field and steady-state Ocean Circulation Explorer (GOCE) is a precise orbit determination (POD). A detailed simulation study has been carried out to assess the achievable orbit accuracy based on satellite-to-satellite tracking (SST) by the US global positioning system (GPS) and in conjunction the implications for gravity field determination. An orbit accuracy at the few centimeter level seems possible, sufficient to support the GOCE gravity mission and in particular its gravity gradiometer. Received: 21 January 2000 / Accepted: 4 July 2000     

Precise ERS-2 orbit determination using SLR, PRARE, and RA observations

The second European Remote Sensing satellite ERS-2, launched in 1995, is supported by satellite laser ranging (SLR), the German precise range and range-rate equipment (PRARE) microwave tracking system, and radar altimetry (RA). SLR, PRARE and crossover RA observations (XO/RA) from May, June, and July 1996 have been analyzed using the GEOSAT software developed at Forsvarets forskningsinstitutt (FFI, The Norwegian defence research establishment). Orbits computed with the JGM3 gravity model and the more recent EGM96 gravity model have been compared with orbits from other analysis centers. Based on these comparisons in addition to the post-fit observation residuals, and results from internal orbital overlaps, we conclude that the radial component of the ERS-2 orbit can be determined with a precision of 6 cm. Received: 30 June 1997 / Accepted: 4 February 1998     

Exploring gravity field determination from orbit perturbations of the European Gravity Mission GOCE

 A comparison was made between two methods for gravity field recovery from orbit perturbations that can be derived from global positioning system satellite-to-satellite tracking observations of the future European gravity field mission GOCE (Gravity Field and Steady-State Ocean Circulation Explorer). The first method is based on the analytical linear orbit perturbation theory that leads under certain conditions to a block-diagonal normal matrix for the gravity unknowns, significantly reducing the required computation time. The second method makes use of numerical integration to derive the observation equations, leading to a full set of normal equations requiring powerful computer facilities. Simulations were carried out for gravity field recovery experiments up to spherical harmonic degree and order 80 from 10 days of observation. It was found that the first method leads to large approximation errors as soon as the maximum degree surpasses the first resonance orders and great care has to be taken with modeling resonance orbit perturbations, thereby loosing the block-diagonal structure. The second method proved to be successful, provided a proper division of the data period into orbital arcs that are not too long. Received: 28 April 2000 / Accepted: 6 November 2000     

Validation of GOCE gravity field models by means of orbit residuals and geoid comparisons

Three GOCE-based gravity field solutions have been computed by ESA’s high-level processing facility and were released to the user community. All models are accompanied by variance-covariance information resulting either from the least squares procedure or a Monte-Carlo approach. In order to obtain independent external quality parameters and to assess the current performance of these models, a set of independent tests based on satellite orbit determination and geoid comparisons is applied. Both test methods can be regarded as complementary because they either investigate the performance in the long wavelength spectral domain (orbit determination) or in the spatial domain (geoid comparisons). The test procedure was applied to the three GOCE gravity field solutions and to a number of selected pre-launch models for comparison. Orbit determination results suggest, that a pure GOCE gravity field model does not outperform the multi-year GRACE gravity field solutions. This was expected as GOCE is designed to improve the determination of the medium to high frequencies of the Earth gravity field (in the range of degree and order 50 to 200). Nevertheless, in case of an optimal combination of GOCE and GRACE data, orbit determination results should not deteriorate. So this validation procedure can also be used for testing the optimality of the approach adopted for producing combined GOCE and GRACE models. Results from geoid comparisons indicate that with the 2 months of GOCE data a significant improvement in the determination of the spherical harmonic spectrum of the global gravity field between degree 50 and 200 can be reached. Even though the ultimate mission goal has not yet been reached, especially due to the limited time span of used GOCE data (only 2 months), it was found that existing satellite-only gravity field models, which are based on 7 years of GRACE data, can already be enhanced in terms of spatial resolution. It is expected that with the accumulation of more GOCE data the gravity field model resolution and quality can be further enhanced, and the GOCE mission goal of 1–2 cm geoid accuracy with 100 km spatial resolution can be achieved.     

Filter design for GOCE gravity gradients

The GOCE satellite observes gravity gradients with unprecedented accuracy and resolution. The GOCE observations are reliable within a well-defined measurement bandwidth. In this study, different finite and infinite impulse response filters have been designed to obtain the demanded pass. Exhaustive time and frequency domain investigations prove that the proposed infinite impulse response filter can be a real competitor of the existing solution of the filtering problem.     

SVLBI用于大地测量的卫星轨道及其跟踪网设计

SVLBI （space very long baseline interferometry） has some important potential applications in geodesy and geodynamics, for which one of the most difficult tasks is to precisely determine the orbit of an SVLBI satellite. This work studies several technologies that will possibly be able to determine the orbit of a space VLBI satellite. Then, according to the types and charac- teristics of the satellite and the requirements for geodetic study and the geometry of the GNSS （GPS, GALILEO） satellite to track the space VLBI satellite, the six Keplerian elements of the SVLBI satellite （TEST-SVLBI） are determined. A program is designed to analyze the coverage area of space of different altitudes by the stations of the network, with which the tracking network of TEST-SVLBI is designed. The efficiency of tracking TEST-SVLBI by the network is studied, and the results are presented.     

模糊聚类定权法对SLR定轨精度的影响

针对卫星激光测距(satellite laser ranging,SLR)精密定轨过程中存在的测站观测数据合理定权问题,将一种改进的模糊聚类算法引入到SLR观测数据定权中。基于国际激光测距服务(International Laser Ranging Service,ILRS)提供的全球SLR测站性能报告,对测站进行近实时滑动分类定权,改变SLR数据处理中权重的经验或者随意性选取模式。经过LAGEOS1卫星2014年1月至2016年12月3年全球SLR实测数据处理的测试。结果表明,当考虑LAGEOS标准点总数、LAGEOS标准点RMS值以及LAGEOS标准点合格率这3项测站质量控制因素确定的测站权值能最大限度地提高卫星定轨精度和观测数据的使用效率,对参与计算的365个3d弧段数据,91.46%弧段精度得到提高,平均提高约3.7mm,且每个测站的定轨残差RMS也得到了降低。这对于正在迈向毫米级测量精度的SLR技术至关重要。     

GOCE gradiometer: estimation of biases and scale factors of all six individual accelerometers by precise orbit determination

A method has been implemented and tested for estimating bias and scale factor parameters for all six individual accelerometers that will fly on-board of GOCE and together form the so-called gradiometer. The method is based on inclusion of the individual accelerometer observations in precise orbit determinations, opposed to the baseline method where so-called common-mode accelerometer observations are used. The method was tested using simulated data from a detailed GOCE system simulator. It was found that the observations taken by individual accelerometers need to be corrected for (1) local satellite gravity gradient (SGG), and (2) rotational terms caused by centrifugal and angular accelerations, due to the fact that they are not located in the satellite’s center of mass. For these corrections, use is made of a reference gravity field model. In addition, the rotational terms are derived from on-board star tracker observations. With a perfect a priori gravity field model and with the estimation of not only accelerometer biases but also accelerometer drifts, scale factors can be determined with an accuracy and stability better than 0.01 for two of the three axes of each accelerometer, the exception being the axis pointing along the long axis of the satellite (more or less coinciding with the flight direction) for which the scale factor estimates are unreliable. This axis coincides with the axis of drag-free control, which results in a small variance of the signal to be calibrated and thus an inaccurate determination of its scale factor in the presence of relatively large (colored) accelerometer observation errors. In the presence of gravity field model errors, it was found that still an accuracy and stability of about 0.015 can be obtained for the accelerometer scale factors by simultaneously estimating empirical accelerations.     

GPS-derived orbits for the GOCE satellite

The first ESA (European Space Agency) Earth explorer core mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) was launched on 17 March 2009 into a sun-synchronous dusk–dawn orbit with an exceptionally low initial altitude of about 280 km. The onboard 12-channel dual-frequency GPS (Global Positioning System) receiver delivers 1 Hz data, which provides the basis for precise orbit determination (POD) for such a very low orbiting satellite. As part of the European GOCE Gravity Consortium the Astronomical Institute of the University of Bern and the Department of Earth Observation and Space Systems are responsible for the orbit determination of the GOCE satellite within the GOCE High-level Processing Facility. Both quick-look (rapid) and very precise orbit solutions are produced with typical latencies of 1 day and 2 weeks, respectively. This article summarizes the special characteristics of the GOCE GPS data, presents POD results for about 2 months of data, and shows that both latency and accuracy requirements are met. Satellite Laser Ranging validation shows that an accuracy of 4 and 7 cm is achieved for the reduced-dynamic and kinematic Rapid Science Orbit solutions, respectively. The validation of the reduced-dynamic and kinematic Precise Science Orbit solutions is at a level of about 2 cm.     

Atmospheric range correction for two-frequency SLR measurements

It has been widely known that the use of two-frequency Satellite Laser Ranging (SLR) system is limited by stringent precision requirements of the range measurements and the proper atmospheric model. Owing to the stringent requirements, this SLR system is impractical for the current requirement of SLR measurements within the framework of global geodetic observing system (GGOS). If in the future this stringent requirement could be met, this SLR system would be an attractive tool to reduce atmospheric propagation effects of SLR and would be of great benefit for the next generation of GGOS design. To anticipate possible future developments of the two-frequency SLR systems, we have developed a new atmospheric correction formula for the two-frequency SLR measurements. The new formula eliminates the total atmospheric density effect including its gradient and provides two terms to calculate the curvature effect and the water vapor distribution effect. While the curvature effect can be calculated by an accurate model, the required information about the water vapor distribution along the propagation path can be calculated using previous developments of optical delay modeling or alternatively using results from microwave measurements. Theoretical simulations using the two-frequency systems of the Graz and TIGO-Concepción stations shows that the new formula completely reduces all propagation effects at any elevation angle above 3° with an accuracy better than 1 mm. However, the required precision for the difference of the two-frequency SLR measurements, i.e. better than 45 μm for a single epoch, exceeds the capability of the current state of the art SLR systems.     

星地跟踪网的导航卫星精密轨道确定及预报

针对区域跟踪网不能覆盖导航卫星全弧段从而导致卫星定轨精度低的问题,简述了导航卫星和低轨卫星联合定轨模型,然后利用星地跟踪网观测数据同时确定了低轨卫星和导航卫星精密轨道,并根据实验结果详细分析了低轨卫星在联合定轨中所起到的作用。计算结果表明,引入低轨卫星之后,全球网和区域网定轨精度分别平均提高了20.0%和44.3%,区域网6h和24h的轨道预报精度分别优于10cm、13cm,利用星地跟踪网观测数据联合定轨方案是一种提高定轨精度并削弱对地面站依赖性的有效方法。     

Jason-2卫星的激光测距观测值噪声估计

针对Jason-2卫星定轨中关于多普勒无线电定轨定位系统(DORIS)和卫星激光测距(SLR)随机模型研究的不足,该文提出了3类k阶差分算子通用递推公式及相应的差分算子观测噪声估计方法。利用电离层平静和异常两个时段的Jason-2卫星DORIS和SLR多站数据进行了应用研究,计算结果验证了所提方法的正确性。同时分析了DORIS和SLR观测噪声随卫星高度角的变化规律以及观测噪声时的相关特性,并指出了3类差分算子的适用性。     

GOCE卫星引力梯度数据滤波方法研究

针对GOCE卫星引力梯度观测值中低精度分量和低频有色噪声的处理策略问题,该文采用模型模拟值代替低精度分量Vxy和Vyz,以减弱低精度分量在坐标系转换中对高精度分量的影响。深入分析比较了多种滤波方法处理GOCE卫星引力梯度观测值中有色噪声的效果,提出采用Butterworth零相移滤波方法加移去-恢复技术的思路,实测数据的处理效果验证了该方法的有效性。     

Improving GOCE cross-track gravity gradients

The GOCE gravity gradiometer measured highly accurate gravity gradients along the orbit during GOCE’s mission lifetime from March 17, 2009, to November 11, 2013. These measurements contain unique information on the gravity field at a spatial resolution of 80 km half wavelength, which is not provided to the same accuracy level by any other satellite mission now and in the foreseeable future. Unfortunately, the gravity gradient in cross-track direction is heavily perturbed in the regions around the geomagnetic poles. We show in this paper that the perturbing effect can be modeled accurately as a quadratic function of the non-gravitational acceleration of the satellite in cross-track direction. Most importantly, we can remove the perturbation from the cross-track gravity gradient to a great extent, which significantly improves the accuracy of the latter and offers opportunities for better scientific exploitation of the GOCE gravity gradient data set.     

Low-low satellite-to-satellite tracking: a comparison between analytical linear orbit perturbation theory and numerical integration

Low-low satellite-to-satellite tracking (ll-SST) range-rate observations have been predicted by two methods: one based on a linear perturbation theory in combination with the Hill equations, and one based on solving the equations of motion of two low-flying satellites by numerical integration. The two methods produce almost equivalent Fourier spectra of the range-rate observations after properly taking into account a few resonant terms. For a typical GRACE-type configuration, where the two satellites trail each other at a distance of 300 km at an altitude of 460 km, and in the presence of the EGM96 gravity field model, complete to spherical harmonic degree and order 70, the agreement between the Fourier spectra is about 1 mm/s compared to a root-mean-square (RMS) value of more than 220 mm/s for the range-rate signal. The discrepancy of 1 mm/s can be reduced significantly when not taking into account perturbations caused by the J₂ term. Excluding the J₂ term, the agreement between the two methods improves to 0.4 mm/s compared to a RMS value of 6 mm/s for the range-rate signal. These values are 0.01 and 2.3 mm/s when ignoring the spectrum for frequencies below two cycles per orbital revolution, reducing the discrepancy even further to about 0.5% of the signal. The selected linear perturbation theory is thus capable of modeling gravity field induced range-rate observations with very high precision for a large part of the spectrum.