Intersatellite laser ranging instrument for the GRACE follow-on mission

The Gravity Recovery and Climate Experiment (GRACE) has demonstrated that low–low satellite-to-satellite tracking enables monitoring the time variations of the Earth’s gravity field on a global scale, in particular those caused by mass-transport within the hydrosphere. Due to the importance of long-term continued monitoring of the variations of the Earth’s gravitational field and the limited lifetime of GRACE, a follow-on mission is currently planned to be launched in 2017. In order to minimise risk and the time to launch, the follow-on mission will be basically a rebuild of GRACE with microwave ranging as the primary instrument for measuring changes of the intersatellite distance. Laser interferometry has been proposed as a method to achieve improved ranging precision for future GRACE-like missions and is therefore foreseen to be included as demonstrator experiment in the follow-on mission now under development. This paper presents the top-level architecture of an interferometric laser ranging system designed to demonstrate the technology which can also operate in parallel with the microwave ranging system of the GRACE follow-on mission.     

Alternative mission architectures for a gravity recovery satellite mission

Since its launch in 2002, the Gravity Recovery and Climate Experiment (GRACE) mission has been providing measurements of the time-varying Earth gravity field. The GRACE mission architecture includes two satellites in near-circular, near-polar orbits separated in the along-track direction by approximately 220 km (e.g. collinear). A microwave ranging instrument measures changes in the distance between the spacecraft, while accelerometers on each spacecraft are used to measure changes in distance due to non-gravitational forces. The fact that the satellites are in near-polar orbits coupled with the fact that the inter-satellite range measurements are directed in the along-track direction, contributes to longitudinal striping in the estimated gravity fields. This paper examines four candidate mission architectures for a future gravity recovery satellite mission to assess their potential in measuring the gravity field more accurately than GRACE. All satellites were assumed to have an improved measurement system, with an inter-satellite laser ranging instrument and a drag-free system for removal of non-gravitational accelerations. Four formations were studied: a two-satellite collinear pair similar to GRACE; a four-satellite architecture with two collinear pairs; a two-satellite cartwheel formation; and a four-satellite cartwheel formation. A cartwheel formation consists of satellites performing in-plane, relative elliptical motion about their geometric center, so that inter-satellite measurements are, at times, directed radially (e.g. parallel to the direction towards the center of the Earth) rather than along-track. Radial measurements, unlike along-track measurements, have equal sensitivity to mass distribution in all directions along the Earth’s surface and can lead to higher spatial resolution in the derived gravity field. The ability of each architecture to recover the gravity field was evaluated using numerical simulations performed with JPL’s GIPSY-OASIS software package. Thirty days of data were used to estimate gravity fields complete to degree and order 60. Evaluations were done for 250 and 400 km nominal orbit altitudes. The sensitivity of the recovered gravity field to under-sampled effects was assessed using simulated errors in atmospheric/ocean dealiasing (AOD) models. Results showed the gravity field errors associated with the four-satellite cartwheel formation were approximately one order of magnitude lower than the collinear satellite pair when only measurement system errors were included. When short-period AOD model errors were introduced, the gravity field errors for each formation were approximately the same. The cartwheel formations eliminated most of the longitudinal striping seen in the gravity field errors. A covariance analysis showed the error spectrum of the cartwheel formations to be lower and more isotropic than that of the collinear formations.     

Simulation of gravity gradients: a comparison study

At different European institutes software has been developed for evaluation of the gravitational potential of the Earth using high degree spherical harmonic expansions. In this report the results of a comparison of a number of these software packages are presented. We compared the results for the second order derivatives (gravity gradients). It appeared that one of the most critical points in these computations is the definition of the coordinates, which should be as accurate as possible. Machine dependency and algorithm setup were of less importance, the former being only reflected in CPU timing results.     

GRACE时变重力场滤波方法

针对GRACE时变重力场模型高阶项误差较大导致的"南-北"条带噪声,该文利用模拟的GRACE数据分析了去相关滤波、Gaussian滤波、组合滤波和平滑先验信息滤波方法对噪声的滤除效果和对真实信号的衰减程度。实验表明:4种滤波算法均能有效降低条带噪声,但单独使用去相关滤波时效果较差,需与其他算法结合使用;Guass滤波和组合滤波在减小噪声条带的同时,也在一定程度上牺牲了空间分辨率;平滑先验信息滤波在移除噪声、保留有效信号方面比其他3种算法有较为明显的优势。     

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.     

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).     

GRACE时变重力场各向异性高斯组合滤波方法

受测量误差等因素影响,直接使用GRACE时变重力场模型的地表质量变化反演结果呈现严重条带噪声,必须采用滤波消除。本文对不同滤波方法进行了试验分析,以信噪比最大为准则,确定了不同滤波方法的最优滤波参数,并在此基础上提出了一种各向异性组合滤波方法。该方法根据时变重力场模型球谐系数误差特性,结合各向异性高斯滤波和均方根滤波特点,对精度较高的低次项系数采用较大权重以保留更多有效信号,而对精度较差的高次项系数采用较小权重以压制噪声。不同于传统的两步法组合滤波,该方法仅需进行一步滤波处理。试验结果表明,本文提出的各向异性组合滤波方法计算步骤简单,能够有效消除条带噪声;与单一滤波和传统两步法组合滤波方法相比,提高了反演结果信噪比,保留了更多真实信号。     

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 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.     

Reducing errors in the GRACE gravity solutions using regularization

The nature of the gravity field inverse problem amplifies the noise in the GRACE data, which creeps into the mid and high degree and order harmonic coefficients of the Earth’s monthly gravity fields provided by GRACE. Due to the use of imperfect background models and data noise, these errors are manifested as north-south striping in the monthly global maps of equivalent water heights. In order to reduce these errors, this study investigates the use of the L-curve method with Tikhonov regularization. L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems using Tikhonov regularization. However, the computational effort required to determine the L-curve is prohibitively high for a large-scale problem like GRACE. This study implements a parameter-choice method, using Lanczos bidiagonalization which is a computationally inexpensive approximation to L-curve. Lanczos bidiagonalization is implemented with orthogonal transformation in a parallel computing environment and projects a large estimation problem on a problem of the size of about 2 orders of magnitude smaller for computing the regularization parameter. Errors in the GRACE solution time series have certain characteristics that vary depending on the ground track coverage of the solutions. These errors increase with increasing degree and order. In addition, certain resonant and near-resonant harmonic coefficients have higher errors as compared with the other coefficients. Using the knowledge of these characteristics, this study designs a regularization matrix that provides a constraint on the geopotential coefficients as a function of its degree and order. This regularization matrix is then used to compute the appropriate regularization parameter for each monthly solution. A 7-year time-series of the candidate regularized solutions (Mar 2003–Feb 2010) show markedly reduced error stripes compared with the unconstrained GRACE release 4 solutions (RL04) from the Center for Space Research (CSR). Post-fit residual analysis shows that the regularized solutions fit the data to within the noise level of GRACE. A time series of filtered hydrological model is used to confirm that signal attenuation for basins in the Total Runoff Integrating Pathways (TRIP) database over 320 km radii is less than 1 cm equivalent water height RMS, which is within the noise level of GRACE.     

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.     

GRACE时变重力数据后处理方法评价

针对GRACE Level2卫星时变重力数据后处理方法如何评价的问题,该文以中国数字地震观测网络获得的青藏高原地区地面重力变化图像为参考,基于平均结构相似性等图像相似度指标,研究了与该区域地面重力观测同期、不同后处理方法得到的GRACE卫星重力变化图像的可靠性。结果显示,GRACE卫星重力和地面重力观测结果具有一定的可比性,滑动窗口去相关滤波和高斯400 km滤波的组合方法可以获得最优的处理效果。本文的方法和结论对GRACE及GRACE Follow-On卫星重力数据应用中后处理方法和参数的选取有一定的借鉴意义。     

Combination of GRACE monthly gravity field solutions from different processing strategies

We combine the publicly available GRACE monthly gravity field time series to produce gravity fields with reduced systematic errors. We first compare the monthly gravity fields in the spatial domain in terms of signal and noise. Then, we combine the individual gravity fields with comparable signal content, but diverse noise characteristics. We test five different weighting schemes: equal weights, non-iterative coefficient-wise, order-wise, or field-wise weights, and iterative field-wise weights applying variance component estimation (VCE). The combined solutions are evaluated in terms of signal and noise in the spectral and spatial domains. Compared to the individual contributions, they in general show lower noise. In case the noise characteristics of the individual solutions differ significantly, the weighted means are less noisy, compared to the arithmetic mean: The non-seasonal variability over the oceans is reduced by up to 7.7% and the root mean square (RMS) of the residuals of mass change estimates within Antarctic drainage basins is reduced by 18.1% on average. The field-wise weighting schemes in general show better performance, compared to the order- or coefficient-wise weighting schemes. The combination of the full set of considered time series results in lower noise levels, compared to the combination of a subset consisting of the official GRACE Science Data System gravity fields only: The RMS of coefficient-wise anomalies is smaller by up to 22.4% and the non-seasonal variability over the oceans by 25.4%. This study was performed in the frame of the European Gravity Service for Improved Emergency Management (EGSIEM; http://www.egsiem.eu) project. The gravity fields provided by the EGSIEM scientific combination service (ftp://ftp.aiub.unibe.ch/EGSIEM/) are combined, based on the weights derived by VCE as described in this article.     

GRACE和GOCE卫星监测的青藏高原重力场特征

文章阐述了对青藏高原重力场进行研究的意义,并进一步利用重力卫星GRACE和GOCE的数据对该区域的重力场特征进行了描述.通过对该区域的重力异常、径向引力梯度的计算和分析,可以得出:在青藏高原的西部,有明显的3条重力异常区,这与当地的地形有关,也与断层的位置有关;引力梯度比重力异常具有更高的空间分辨率;重力变化剧烈的区域与梯度的异常区有一定的对应关系,同时也是地球动力活动变化剧烈的区域.     

基于SLR的GRACE卫星定轨中重力场模型对轨道精度的影响

以GRACE卫星为例,分析比较利用SLR观测资料进行卫星定轨时,采用不同重力场模型对GRACE卫星定轨精度的影响;以及重力场截断阶引起的积分轨道差异;同时,将定轨结果与采用GPS确定的定轨结果进行比较,分析与GPS定轨结果的差异.实验证明.重力场模型选择GGM02C的定轨结果优于选择JGM-3的定轨结果,基于SLR的定...     

GGM02 – An improved Earth gravity field model from GRACE

A new generation of Earth gravity field models called GGM02 are derived using approximately 14 months of data spanning from April 2002 to December 2003 from the Gravity Recovery And Climate Experiment (GRACE). Relative to the preceding generation, GGM01, there have been improvements to the data products, the gravity estimation methods and the background models. Based on the calibrated covariances, GGM02 (both the GRACE-only model GGM02S and the combination model GGM02C) represents an improvement greater than a factor of two over the previous GGM01 models. Error estimates indicate a cumulative error less than 1 cm geoid height to spherical harmonic degree 70, which can be said to have met the GRACE minimum mission goals. Electronic Supplementary Material Supplementary material is available in the online version of this article at     

A preliminary error analysis of the gravity field recovery from a lunar Satellite-to-Satellite mission

Summary A low cost lunar Satellite-to-Satellite radio tracking mission in a low-low configuration could considerably improve the existing knowledge about the lunar gravity field. The impact of various mission parameters that may contribute to the recovery of the gravity field, such as satellite altitude, satellite separation, mission duration, measurement precision and sampling interval were quantified using the Jekeli-Rapp algorithm. Preliminary results indicate that the gravity field resolution up to harmonic degree 40 to 80 is feasible depending on various mission configurations. Radio tracking data from a six-month mission with a precision of 1 mm s^–1 every 10 s and 300 km satellite separation at 150 km altitude will permit the determination of 5^o×5^o mean gravity anomalies with an error of approximately 15 mgals. Consideration of other unaccounted error sources of instrumental, operational as well as environmental nature may lower this resolution.     

Decorrelated GRACE time-variable gravity solutions by GFZ,and their validation using a hydrological model

We have analyzed recent gravity recovery and climate experiment (GRACE) RL04 monthly gravity solutions, using a new decorrelating post-processing approach. We find very good agreement with mass anomalies derived from a global hydrological model. The post-processed GRACE solutions exhibit only little amplitude damping and an almost negligible phase shift and period distortion for relevant hydrological basins. Furthermore, these post-processed GRACE solutions have been inspected in terms of data fit with respect to the original inter-satellite ranging and to SLR and GPS observations. This kind of comparison is new. We find variations of the data fit due to solution post-processing only within very narrow limits. This confirms our suspicion that GRACE data do not firmly ‘pinpoint’ the standard unconstrained solutions. Regarding the original Kusche (J Geod 81:733–749, 2007) decorrelation and smoothing method, a simplified (order-convolution) approach has been developed. This simplified approach allows to realize a higher resolution—as necessary, e.g., for generating computed GRACE observations—and needs far less coefficients to be stored.     

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.     

青藏高原GRACE卫星重力长期变化研究

青藏高原隆升是地球上新生代以来最壮观的地质事件之一,对东亚乃至全球大陆动力学研究具有举足轻重的作用,一直是国际地球科学研究的热点区域.随着近年来现代大地测量技术(如GPS、卫星测高、绝对重力、雷达干涉测量等)的蓬勃发展,特别是2002年GRACE重力卫星的成功发射,为研究全球物质分布和季节性变化提供了重要参考依据,同时也为研究青藏高原动力学提供了可靠的观测数据和便捷的观测手段.GRACE卫星发射至今已超过13年,随着资料的累积和数据处理方法的改进与成熟,用GRACE卫星资料研究长期性变化成为新的热点.