Understanding data noise in gravity field recovery on the basis of inter-satellite ranging measurements acquired by the satellite gravimetry mission GRACE

Spectral analysis of data noise is performed in the context of gravity field recovery from inter-satellite ranging measurements acquired by the satellite gravimetry mission GRACE. The motivation of the study is two-fold: (i) to promote a further improvement of GRACE data processing techniques and (ii) to assist designing GRACE follow-on missions. The analyzed noise realizations are produced as the difference between the actual GRACE inter-satellite range measurements and the predictions based on state-of-the-art force models. The exploited functional model is based on the so-called “range combinations,” which can be understood as a finite-difference analog of inter-satellite accelerations projected onto the line-of-sight connecting the satellites. It is shown that low-frequency noise is caused by limited accuracy of the computed GRACE orbits. In the first instance, it leads to an inaccurate estimation of the radial component of the inter-satellite velocities. A large impact of this component stems from the fact that it is directly related to centrifugal accelerations, which have to be taken into account when the measured range-accelerations are linked with inter-satellite accelerations. Another effect of orbit inaccuracies is a miscalculation of forces acting on the satellites (particularly, the one described by the zero-degree term of the Earth’s gravitational field). The major contributors to the noise budget at high frequencies (above 9?mHz) are (i) ranging sensor errors and (ii) limited knowledge of the Earth’s static gravity field at high degrees. Importantly, we show that updating the model of the static field on the basis of the available data must be performed with a caution as the result may not be physical due to a non-unique recovery of high-degree coefficients. The source of noise in the range of intermediate frequencies (1–9?mHz), which is particularly critical for an accurate gravity field recovery, is not fully understood yet. We show, however, that it cannot be explained by inaccuracies in background models of time-varying gravity field. It is stressed that most of the obtained results can be treated as sufficiently general (i.e., applicable in the context of a statistically optimal estimation based on any functional model).     

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.     

Simulation study of a follow-on gravity mission to GRACE

The gravity recovery and climate experiment (GRACE) has been providing monthly estimates of the Earth’s time-variable gravity field since its launch in March 2002. The GRACE gravity estimates are used to study temporal mass variations on global and regional scales, which are largely caused by a redistribution of water mass in the Earth system. The accuracy of the GRACE gravity fields are primarily limited by the satellite-to-satellite range-rate measurement noise, accelerometer errors, attitude errors, orbit errors, and temporal aliasing caused by un-modeled high-frequency variations in the gravity signal. Recent work by Ball Aerospace & Technologies Corp., Boulder, CO has resulted in the successful development of an interferometric laser ranging system to specifically address the limitations of the K-band microwave ranging system that provides the satellite-to-satellite measurements for the GRACE mission. Full numerical simulations are performed for several possible configurations of a GRACE Follow-On (GFO) mission to determine if a future satellite gravity recovery mission equipped with a laser ranging system will provide better estimates of time-variable gravity, thus benefiting many areas of Earth systems research. The laser ranging system improves the range-rate measurement precision to ~0.6 nm/s as compared to ~0.2 μm/s for the GRACE K-band microwave ranging instrument. Four different mission scenarios are simulated to investigate the effect of the better instrument at two different altitudes. The first pair of simulated missions is flown at GRACE altitude (~480 km) assuming on-board accelerometers with the same noise characteristics as those currently used for GRACE. The second pair of missions is flown at an altitude of ~250 km which requires a drag-free system to prevent satellite re-entry. In addition to allowing a lower satellite altitude, the drag-free system also reduces the errors associated with the accelerometer. All simulated mission scenarios assume a two satellite co-orbiting pair similar to GRACE in a near-polar, near-circular orbit. A method for local time variable gravity recovery through mass concentration blocks (mascons) is used to form simulated gravity estimates for Greenland and the Amazon region for three GFO configurations and GRACE. Simulation results show that the increased precision of the laser does not improve gravity estimation when flown with on-board accelerometers at the same altitude and spacecraft separation as GRACE, even when time-varying background models are not included. This study also shows that only modest improvement is realized for the best-case scenario (laser, low-altitude, drag-free) as compared to GRACE due to temporal aliasing errors. These errors are caused by high-frequency variations in the hydrology signal and imperfections in the atmospheric, oceanographic, and tidal models which are used to remove unwanted signal. This work concludes that applying the updated technologies alone will not immediately advance the accuracy of the gravity estimates. If the scientific objectives of a GFO mission require more accurate gravity estimates, then future work should focus on improvements in the geophysical models, and ways in which the mission design or data processing could reduce the effects of temporal aliasing.     

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.     

Design considerations for a dedicated gravity recovery satellite mission consisting of two pairs of satellites

Future satellite missions dedicated to measuring time-variable gravity will need to address the concern of temporal aliasing errors; i.e., errors due to high-frequency mass variations. These errors have been shown to be a limiting error source for future missions with improved sensors. One method of reducing them is to fly multiple satellite pairs, thus increasing the sampling frequency of the mission. While one could imagine a system architecture consisting of dozens of satellite pairs, this paper explores the more economically feasible option of optimizing the orbits of two pairs of satellites. While the search space for this problem is infinite by nature, steps have been made to reduce it via proper assumptions regarding some parameters and a large number of numerical simulations exploring appropriate ranges for other parameters. A search space originally consisting of 15 variables is reduced to two variables with the utmost impact on mission performance: the repeat period of both pairs of satellites (shown to be near-optimal when they are equal to each other), as well as the inclination of one of the satellite pairs (the other pair is assumed to be in a polar orbit). To arrive at this conclusion, we assume circular orbits, repeat groundtracks for both pairs of satellites, a 100-km inter-satellite separation distance, and a minimum allowable operational satellite altitude of 290 km based on a projected 10-year mission lifetime. Given the scientific objectives of determining time-variable hydrology, ice mass variations, and ocean bottom pressure signals with higher spatial resolution, we find that an optimal architecture consists of a polar pair of satellites coupled with a pair inclined at 72°, both in 13-day repeating orbits. This architecture provides a 67% reduction in error over one pair of satellites, in addition to reducing the longitudinal striping to such a level that minimal post-processing is required, permitting a substantial increase in the spatial resolution of the gravity field products. It should be emphasized that given different sets of scientific objectives for the mission, or a different minimum allowable satellite altitude, different architectures might be selected.     

Comparing seven candidate mission configurations for temporal gravity field retrieval through full-scale numerical simulation

The goal of this contribution is to focus on improving the quality of gravity field models in the form of spherical harmonic representation via alternative configuration scenarios applied in future gravimetric satellite missions. We performed full-scale simulations of various mission scenarios within the frame work of the German joint research project “Concepts for future gravity field satellite missions” as part of the Geotechnologies Program, funded by the German Federal Ministry of Education and Research and the German Research Foundation. In contrast to most previous simulation studies including our own previous work, we extended the simulated time span from one to three consecutive months to improve the robustness of the assessed performance. New is that we performed simulations for seven dedicated satellite configurations in addition to the GRACE scenario, serving as a reference baseline. These scenarios include a “GRACE Follow-on” mission (with some modifications to the currently implemented GRACE-FO mission), and an in-line “Bender” mission, in addition to five mission scenarios that include additional cross-track and radial information. Our results clearly confirm the benefit of radial and cross-track measurement information compared to the GRACE along-track observable: the gravity fields recovered from the related alternative mission scenarios are superior in terms of error level and error isotropy. In fact, one of our main findings is that although the noise levels achievable with the particular configurations do vary between the simulated months, their order of performance remains the same. Our findings show also that the advanced pendulums provide the best performance of the investigated single formations, however an accuracy reduced by about 2–4 times in the important long-wavelength part of the spectrum (for spherical harmonic degrees ${<}50$ ), compared to the Bender mission, can be observed. Concerning state-of-the-art mission constraints, in particular the severe restriction of heterodyne lasers on maximum range-rates, only the moderate Pendulum and the Bender-mission are beneficial options, of course in addition to GRACE and GRACE-FO. Furthermore, a Bender-type constellation would result in the most accurate gravity field solution by a factor of about 12 at long wavelengths (up to degree/order 40) and by a factor of about 200 at short wavelengths (up to degree/order 120) compared to the present GRACE solution. Finally, we suggest the Pendulum and the Bender missions as candidate mission configurations depending on the available budget and technological progress.     

基于能量守恒的星间距离变率与地球重力场频谱关系的建立与分析(英文)

The spectral relationship between range-rate measurements and the gravity potential for low-low satellite-to-satellite tracking mission was established based on the energy conservation theory. Then the performances of satellite separation, the orbital altitude, and the accuracy of range-rate measurements in recovering the earth’s gravity field were simulated and analyzed by this method. Finally, the cumulative geoid errors of the reference mode were obtained by using the configuration parameters of the GRAC...     

基于GRACE卫星数据的高精度全球静态重力场模型

应用GRACE卫星数据反演高精度静态地球重力场是大地测量学界的热点之一。考虑到经典动力学法线性化误差随弧长拉长而迅速增长,本文以GRACE卫星轨道观测值为初值的线性化方法,建立了应用GRACE卫星轨道和星间距离变率反演地球重力场的改进动力学法理论模型。利用2003年1月至2010年12月的GRACE卫星姿态、轨道、星间距离变率和非保守力加速度等观测数据,解算了一个180阶次的无约束全球静态重力场模型Tongji-Dyn01s和一个采用Kaula规则约束的全球重力场模型Tongji-Dyn01k。与国际不同机构最新发布的纯GRACE数据解算的重力场模型(包括AIUB-GRACE03S、GGM05S、ITSG-Grace2014k和Tongji-GRACE01)进行比较,并利用DTU13海洋重力异常和GPS/水准高程异常进行外部检核,结果表明,Tongji-Dyn01s与国际最新模型精度处于同一水平,然而Tongji-Dyn01k模型总体上更加靠近EIGEN6C2重力场模型。     

卫星重力研究:21世纪大地测量研究的新热点

卫星重力发射将大大改善人们对地球重力场的了解 ,最近一些年已经和将要发射的 CHAMP、GRACE及GOCE卫星将把现有静态中长波长部分重力场的精度提高 1- 2个量级 ,并提供长波部分重力场随时间变化的信息。本文对这一大地测量的新进展作了简单叙述     

卫星重力研究:21世纪大地测量研究的新热点

卫星重力发射将大大改善人们对地球重力场的了解 ,最近一些年已经和将要发射的 CHAMP、GRACE及GOCE卫星将把现有静态中长波长部分重力场的精度提高 1- 2个量级 ,并提供长波部分重力场随时间变化的信息。本文对这一大地测量的新进展作了简单叙述     

EIGEN-GL04C重力场模型的精度分析

本文对GFZ发布的GRACE EIGEN-GL04C重力场模型从球谐系数分析、误差阶方差分析两方面进行了精度评价。研究表明,相较于以往的重力场模型,EIGEN-GL04C重力场模型精度对120阶以下(未包含J2项)的地球中长波部分具有明显的改善。GRACE重力卫星测量已经成为获取地球重力场信息的重要手段。     

Improved GRACE science results after adjustment of geometric biases in the Level-1B K-band ranging data

The GRACE (Gravity Recovery and Climate Experiment) satellite mission relies on the inter-satellite K-band microwave ranging (KBR) observations. We investigate systematic errors that are present in the Level-1B KBR data, namely in the geometric correction. This correction converts the original ranging observation (between the two KBR antennas phase centers) into an observation between the two satellites’ centers of mass. It is computed from data on the precise alignment between both satellites, that is, between the lines joining the center of mass and the antenna phase center of either satellite. The Level-1B data used to determine this alignment exhibit constant biases as large as 1–2 mrad in terms of pitch and yaw alignment angles. These biases induce non-constant errors in the Level-1B geometric correction. While the precise origin of the biases remains to be identified, we are able to estimate and reduce them in a re-calibration approach. This significantly improves time-variable gravity field solutions based on the CNES/GRGS processing strategy. Empirical assessments indicate that the systematic KBR data errors have previously induced gravity field errors on the level of 6–11 times the so-called GRACE baseline error level. The zonal coefficients (from degree 14) are particularly affected. The re-calibration reduces their rms errors by about 50%. As examples for geophysical inferences, the improvement enhances agreement between mass variations observed by GRACE and in-situ ocean bottom pressure observations. The improvement also importantly affects estimates of inter-annual mass variations of the Antarctic ice sheet.     

A simulation study of the errors of omission and commission for GRACE RL01 gravity fields

This paper examines the influence that certain omission and commission errors can have on the gravity field models estimated from the initial release of data (RL01) from the Gravity Recovery And Recovery Experiment (GRACE) satellite mission. The effects of omission errors were analyzed by limiting the degree and order to which the GPS and K-band range-rate (KBR) measurement partials were extended in the solution process. The commission error studies focused on the impact of an imperfect mean reference gravity field model on the solution. Combinations of both of these error sources were also explored. The nature of these errors makes them difficult to distinguish from the true gravity signal, so the exploration of these error sources was performed using simulations; however, comparisons to real-data solutions are provided. The results show how each of the specific error sources investigated influences the gravity field solution. The simulations also show how all of the errors examined can be sufficiently mitigated through the appropriate choice of processing parameters.     

Precise relative orbit determination of twin GRACE satellites

When formation flying spacecrafts are used as platform to gain earth oriented observation, precise baselines between these spacecrafts are always essential. Gravity recovery and climate experiment (GRACE) mission is aimed at mapping the global gravity field and its variation. Accurate baseline of GRACE satellites is necessary for the gravity field modeling. The determination of kinematic and reduced dynamic relative orbits of twin satellites has been studied in this paper, and an accuracy of 2 mm for dynamic relative orbits and 5 mm for kinematic ones can be obtained, whereby most of the double difference onboard GPS ambiguities are resolved.     

国际重力卫星研究进展和我国将来卫星重力测量计划

本文首先分别介绍了国际已经成功发射的专用地球重力测量卫星CHAMP、GRACE以及即将发射的GOCE、GRACE Follow-On和专用月球重力探测卫星GRAIL的研制机构、轨道参数、关键载荷、跟踪模式、测量原理、科学目标和技术特征;其次,阐述了当前相关学科对地球重力场测量精度的需求;最后,建议我国在将来实施的卫星重力测量计划中首选卫星跟踪卫星高低\低低模式,尽快开展轨道参数优化选取的定量系统研究论证和重力卫星系统的误差分析,依据匹配精度指标先期开展重力卫星各关键载荷的研制以及尽早启动卫星重力测量系统的虚拟仿真研究。     

各向异性组合滤波法反演陆地水储量变化

地球时变重力场模型反演陆地水储量变化已为全球气候变化研究作出巨大贡献,考虑到时变重力场模型球谐系数中存在相关性,其高阶次项具有较大的误差,需采用最优的滤波方法进行空间平滑。本文提出一种新的各向异性组合滤波方法,其基本思想是将改进的高斯滤波法与均方根(root mean square,RMS)滤波法组合,即对球谐系数的低阶次采用改进的高斯滤波法,而高阶次采用RMS滤波法。首先分析了最新的GRACE RL05系列时变重力场模型系数误差特性,基于全球水储量变化反演结果,分析比较了高斯滤波、改进的高斯滤波、RMS滤波和DDK滤波与本文提出的组合滤波法的有效性及精度,并利用模型结果进行了验证,计算结果表明,组合滤波法的中误差最小。研究结果表明,本文提出的组合法相比于先前的滤波方法,可有效地过滤高阶次的噪声,消除南北条带误差,同时减少信号泄漏,提高信噪比,保留更多有效的地球物理信号,进而提高反演精度。     

Short-arc analysis of intersatellite tracking data in a gravity mapping mission

 A technique for the analysis of low–low intersatellite range-rate data in a gravity mapping mission is explored. The technique is based on standard tracking data analysis for orbit determination but uses a spherical coordinate representation of the 12 epoch state parameters describing the baseline between the two satellites. This representation of the state parameters is exploited to allow the intersatellite range-rate analysis to benefit from information provided by other tracking data types without large simultaneous multiple-data-type solutions. The technique appears especially valuable for estimating gravity from short arcs (e.g. less than 15 minutes) of data. Gravity recovery simulations which use short arcs are compared with those using arcs a day in length. For a high-inclination orbit, the short-arc analysis recovers low-order gravity coefficients remarkably well, although higher-order terms, especially sectorial terms, are less accurate. Simulations suggest that either long or short arcs of the Gravity Recovery and Climate Experiment (GRACE) data are likely to improve parts of the geopotential spectrum by orders of magnitude. Received: 26 June 2001 / Accepted: 21 January 2002     

Precise GRACE baseline determination using GPS

Precision relative navigation is an essential aspect of spacecraft formation flying missions, both from an operational and a scientific point of view. When using GPS as a relative distance sensor, dual-frequency receivers are required for high accuracy at large inter-satellite separations. This allows for a correction of the relative ionospheric path delay and enables double difference integer ambiguity resolution. Although kinematic relative positioning techniques demonstrate promising results for hardware-in-the-loop simulations, they were found to lack an adequate robustness in real-world applications. To overcome this limitation, an extended Kalman Filter modeling the relative spacecraft dynamics has been developed. The filter processes single difference GPS pseudorange and carrier phase observations to estimate the relative position and velocity along with empirical accelerations and carrier phase ambiguities. In parallel, double difference carrier phase ambiguities are resolved on both frequencies using the least square ambiguity decorrelation adjustment (LAMBDA) method in order to fully exploit the inherent measurement accuracy. The combination of reduced dynamic filtering with the LAMBDA method results in smooth relative position estimates as well as fast and reliable ambiguity resolution. The proposed method has been validated with data from the gravity recovery and climate experiment (GRACE) mission. For an 11-day data arc, the resulting solution matches the GRACE K-Band Ranging System measurements with an accuracy of 1 mm, whereby 83% of the double difference ambiguities are resolved.     

Improved GRACE kinematic orbit determination using GPS receiver clock modeling

Kinematic positions of Low Earth Orbiters based on GPS tracking are frequently used as pseudo-observations for single satellite gravity field determination. Unfortunately, the accuracy of the satellite trajectory is partly limited because the receiver synchronization error has to be estimated along with the kinematic coordinates at every observation epoch. We review the requirements for GPS receiver clock modeling in Precise Point Positioning (PPP) and analyze its impact on kinematic orbit determination for the two satellites of the Gravity Recovery and Climate Experiment (GRACE) mission using both simulated and real data. We demonstrate that a piecewise linear parameterization can be used to model the ultra-stable oscillators that drive the GPS receivers on board of the GRACE satellites. Using such a continuous clock model allows position estimation even if the number of usable GPS satellites drops to three and improves the robustness of the solution with respect to outliers. Furthermore, simulations indicate a potential accuracy improvement of the satellite trajectory of at least 40 % in the radial direction and up to 7 % in the along-track and cross-track directions when a 60-s piecewise linear clock model is estimated instead of epoch-wise independent receiver clock offsets. For PPP with real GRACE data, the accuracy evaluation is hampered by the lack of a reference orbit of significantly higher accuracy. However, comparisons with a smooth reduced-dynamic orbit indicate a significant reduction of the high-frequency noise in the radial component of the kinematic orbit.     

基线法在卫星重力数据处理中的应用

提出了一种恢复地球重力场并同时改善部分轨道初始参数的方法——基线法。给出了基线法的基本原理,推导了基线参数与直角坐标形式参数的相互转换公式,分析了星间距离和距离变率对基线参数的敏感性。分别用基线法和经典动力学法处理了一组GRACE实际观测数据,结果表明,采用基线法较经典动力学法得到了一个精度更高的地球重力场模型,其大地水准面累积误差(最高60阶)减少了3 cm。