Estimating low resolution gravity fields at short time intervals to reduce temporal aliasing errors

The Gravity Recovery and Climate Experiment (GRACE) satellite mission has been estimating temporal changes in the Earth’s gravitational field since its launch in 2002. While it is not yet fully resolved what the limiting source of error is for GRACE, studies on future missions have shown that temporal aliasing errors due to undersampling signals of interest (such as hydrological variations) and errors in atmospheric, ocean, and tide models will be a limiting source of error for missions taking advantage of improved technologies (flying drag-free with a laser interferometer). This paper explores the option of reducing the effects of temporal aliasing errors by directly estimating low degree and order gravity fields at short time intervals, ultimately resulting in data products with improved spatial resolution. Three potential architectures are considered: a single pair of polar orbiting satellites, two pairs of polar orbiting satellites, and a polar orbiting pair of satellites coupled with a lower inclined pair of satellites. Results show that improvements in spatial resolution are obtained when one estimates a low resolution gravity field every two days for the case of a single pair of satellites, and every day for the case of two polar pairs of satellites. However, the spatial resolution for these cases is still lower than that provided by simply destriping and smoothing the solutions via standard GRACE post-processing techniques. Alternately, estimating daily gravity fields for the case of a polar pair of satellites coupled with a lower inclined pair results in solutions with superior spatial resolution than that offered by simply destriping and smoothing the solutions.     

Improved kHz-SLR tracking techniques and orbit quality analysis for LEO missions

Satellite gravity field missions such as CHAMP, GRACE and GOCE are designed as low Earth orbiting spacecraft (LEO) with orbit heights of about 250–500 km. The challenging mission objectives require a very precise knowledge of the satellite orbit position in space. For these missions precise orbit information is typically provided by GPS satellite-to-satellite tracking (SST) observations supported by satellite laser ranging (SLR).     

重力卫星星间高精度测距技术研究

卫卫跟踪(SST)技术是当前地球重力场测量最有价值和应用前景的方法之一.高精度星间测距系统是低低卫卫跟踪(SST-ll)重力卫星的关键有效载荷.GRACE卫星携带的K波段测距系统(KBR K Band Ranging System)是一微米量级的测距系统,通过处理高精度的星间距离和距离变化率数据,可以恢复出地球重力场.GRACE后续计划又提出了一种更高精度的激光干涉测距系统.在研究KBR及激光干涉测距系统测量原理的基础上,提出了一种KBR系统的基本结构,详细分析了两种测距系统的关键技术及国内目前的研究水平,提出了我国开展星间测距系统研究的一些建议.      

Characteristics and accuracies of the GRACE inter-satellite pointing

For almost 10 years, the Gravity Recovery and Climate Experiment (GRACE) has provided information about the Earth gravity field with unprecedented accuracy. Efforts are ongoing to approach the GRACE baseline accuracy as there still remains an order of magnitude between the present error level of the gravity field solutions and the GRACE baseline. At the current level of accuracy, thorough investigation of sensor related effects is necessary as they are one of the potential contributors to the error budget. In the science mode operations, the twin satellites are kept precisely pointed with their KBR antennas towards each other. It is the task of the onboard attitude and orbit control system (AOCS) to keep the satellites in the required formation. We analyzed long time series of the inter-satellite pointing variations as they reflect the AOCS performance and characteristics. We present significant systematic effects in the inter-satellite pointing and discuss their possible sources. Prominent features are especially related to the magnetic torquer characteristics, star cameras’ performance and KBR antenna calibration parameters. The relation between the magnetic torquer attitude control and the Earth magnetic field, impact of the different performance of the two star camera heads on the attitude control and the features due to uncertainties in the calibration parameters relating the star camera frame to K-frame are discussed in detail. Proper understanding of these effects will help to reduce their impact on the science data and subsequently increase the accuracy of the gravity field solutions. Moreover, understanding the complexity of the onboard system is essential not only for increasing the accuracy of the GRACE data but also for the development of the future gravity field satellite missions.     

Impact of a novel hybrid accelerometer on satellite gravimetry performance

The state-of-the-art electrostatic accelerometers (EA) used for the retrieval of non-gravitational forces acting on a satellite constitute a core component of every dedicated gravity field mission. However, due to their difficult-to-control thermal drift in the low observation frequencies, they are also one of the most limiting factors of the achievable performance of gravity recovery. Recently, a hybrid accelerometer consisting of a regular EA and a novel cold atom interferometer (CAI) that features a time-invariant observation stability and constantly recalibrates the EA has been developed in order to remedy this major drawback. In this paper we aim to assess the value of the hybrid accelerometer for gravity field retrieval in the context of GRACE-type and Bender-type missions by means of numerical closed-loop simulations where possible noise specifications of the novel instrument are considered and different components of the Earth’s gravity field signal are added subsequently. It is shown that the quality of the gravity field solutions is mainly dependent on the CAI’s measurement accuracy. While a low CAI performance (10^?8 to 10^?9?m/s²/Hz^1/2) does not lead to any gains compared to a stand-alone EA, a sufficiently high one (10^?11?m/s²/Hz^1/2) may improve the retrieval performance by over one order of magnitude. We also show that improvements which are limited to low-frequency observations may even propagate into high spherical harmonic degrees. Further, the accelerometer performance seems to play a less prominent role if the overall observation geometry is improved as it is the case for a Bender-type mission. The impact of the accelerometer measurements diminishes further when temporal variations of the gravity field are introduced, pointing out the need for proper de-aliasing techniques. An additional study reveals that the hybrid accelerometer is – contrary to a stand-alone EA – widely unaffected by scale factor instabilities.     

Performance of GPS-based accelerometry: A simulation experiment

Recent studies have shown that with the availability of high-quality CHAMP and GRACE gravity field models, it is feasible to determine accurate non-gravitational accelerations for low Earth orbiting satellites indirectly from precise GPS satellite-to-satellite observations. Possible applications of this so-called GPS-based accelerometry approach consist of accelerometer calibration and atmospheric density and wind computations. With the growing number of high-quality space-borne GPS receivers, this method could be applied to a large range of satellites. In this paper an extensive simulation study has been carried out, based on real accelerometer data from the GRACE mission, in order to determine the optimal processing strategy and the resulting accuracy of the estimated non-gravitational accelerations. It is shown that the optimal processing strategy consists of a piecewise linear parameterization of the estimated empirical accelerations, together with short 6-h orbit arcs. The GPS-based accelerometry approach makes use of triple-differenced GPS observations and the impact of considering the correlated observation noise was found to be marginal in the presence of other error sources such as GPS ephemeris errors. Using a priori non-gravitational force models improves the recovery of low temporal resolution accelerations, except during huge geomagnetic storms. With this strategy, non-gravitational accelerations can be recovered during high solar activity with an accuracy of better than 10% of the total signal in along-track direction and around 25–40% in cross-track direction, at time resolutions of around 8–20 min. During solar minimum conditions, the relative recovery error will increase to approximately 50% in along-track direction and around 60–70% in cross-track direction, due to the reduced atmospheric drag signal. Unfortunately, GPS-based accelerometry is hardly sensitive in the radial direction.     

Regional gravity field model in Pakistan area from the combination of CHAMP,GRACE and ground data using least squares collocation: A case study

This study describes a methodology of recovery of the Earth’s gravity field from CHAMP and GRACE satellites data in Pakistan using least squares collocation (LSC) based downward continuation technique. The CHAMP height anomalies and GRACE gravity disturbances derived from the observed satellite data have been used in combination solution using LSC with observed gravity values at the Earth surface. The combined covariance functions of height anomalies and/or gravity disturbances at satellite altitudes and observed gravity anomalies at Earth surface have been used as the basis for combination and downward continuation solution. The variance of predicted gravity anomalies from GRACE gravity disturbances is relatively lower than the corresponding results of gravity anomalies from CHAMP height anomalies. This fact may be attributed partly to the amplification of noise and partly to the unstable inverse transformation process of height anomalies to gravity anomalies. The impact of data error variance has been studied in the context of smoothing and noise reduction in the final solution of downward continuation using least squares collocation. The raising of data error suppresses the noise and as a result a smooth final solution is obtained. The prediction results appear to be dependent on the quality of data and goodness of combined covariance function, which are fairly comparable for the CHAMP and GRACE data. The recovered gravity field from satellite data appears to contribute mainly to medium and long wavelength parts of total gravity field spectrum. Due to flexibility of data handling in least squares collocation, this procedure is applicable to any observable of gravity field being at different altitudes and with different data spacing.     

Numerical simulation of time delay interferometry for a LISA-like mission with the simplification of having only one interferometer

In order to attain the requisite sensitivity for LISA, laser frequency noise must be suppressed below the secondary noises such as the optical path noise, acceleration noise etc. In a previous paper (Dhurandhar, S.V., Nayak, K.R., Vinet, J.-Y. Time delay interferometry for LISA with one arm dysfunctional. Class. Quantum Grav. 27, 135013, 2010), we have found a large family of second-generation analytic solutions of time delay interferometry with one arm dysfunctional, and we also estimated the laser noise due to residual time-delay semi-analytically from orbit perturbations due to Earth. Since other planets and solar-system bodies also perturb the orbits of LISA spacecraft and affect the time delay interferometry (TDI), we simulate the time delay numerically in this paper for all solutions with the generation number n ? 3. We have worked out a set of 3-year optimized mission orbits of LISA spacecraft starting at January 1, 2021 using the CGC2.7 ephemeris framework. We then use this numerical solution to calculate the residual optical path differences in the second-generation solutions of our previous paper, and compare with the semi-analytic error estimate. The accuracy of this calculation is better than 1 cm (or 30 ps). The maximum path length difference, for all configuration calculated, is below 1 m (3 ns). This is well below the limit under which the laser frequency noise is required to be suppressed. The numerical simulation in this paper can be applied to other space-borne interferometers for gravitational wave detection with the simplification of having only one interferometer.     

Validation of the in-flight calibration procedures for the MICROSCOPE space mission

The MICROSCOPE space mission aims to test the Equivalence Principle with an accuracy of 10^-15${10}^{-15}$. The drag-free micro-satellite will orbit around the Earth and embark a differential electrostatic accelerometer including two cylindrical test masses submitted to the same gravitational field and made of different materials. The experience consists in testing the equality of the electrostatic acceleration applied to the masses to maintain them relatively motionless. The accuracy of the measurements exploited for the test of the Equivalence Principle is limited by our a priori knowledge of several physical parameters of the instrument. These parameters are partially estimated on-ground, but with an insufficient accuracy, and an in-orbit calibration is therefore required to correct the measurements. The calibration procedures have been defined and their analytical performances have been evaluated. In addition, a simulator software including the dynamics model of the instrument, the satellite drag-free system and the perturbing environment has been developed to numerically validate the analytical results. After an overall presentation of the MICROSCOPE mission, this paper will describe the calibration procedures and focus on the simulator. Such an in-flight calibration is mandatory for similar space missions taking advantage of a drag-free system.     

Recovery of the Earth’s gravity field from formation-flying satellites: Temporal aliasing issues

Temporal and mean gravity field models derived from the twin-satellite, leader–follower mission GRACE have provided a new type of information for Earth sciences. In this contribution, we study the potential of various alternative satellite formations for gravity field determination in the post-GRACE era in a simulation environment. In particular, the effects of spherical harmonic truncation and of temporal aliasing in the processing of gravity products from such future formations are investigated.     

GRACE gravity field modeling with an investigation on correlation between nuisance parameters and gravity field coefficients

The GRACE (Gravity Recovery And Climate Experiment) monthly gravity models have been independently produced and published by several research institutions, such as Center for Space Research (CSR), GeoForschungsZentrum (GFZ), Jet Propulsion Laboratory (JPL), Centre National d’Etudes Spatiales (CNES) and Delft Institute of Earth Observation and Space Systems (DEOS). According to their processing standards, above institutions use the traditional variational approach except that the DEOS exploits the acceleration approach. The background force models employed are rather similar. The produced gravity field models generally agree with one another in the spatial pattern. However, there are some discrepancies in the gravity signal amplitude between solutions produced by different institutions. In particular, 10%–30% signal amplitude differences in some river basins can be observed. In this paper, we implemented a variant of the traditional variational approach and computed two sets of monthly gravity field solutions using the data from January 2005 to December 2006. The input data are K-band range-rates (KBRR) and kinematic orbits of GRACE satellites. The main difference in the production of our two types of models is how to deal with nuisance parameters. This type of parameters is necessary to absorb low-frequency errors in the data, which are mainly the aliasing and instrument errors. One way is to remove the nuisance parameters before estimating the geopotential coefficients, called NPARB approach in the paper. The other way is to estimate the nuisance parameters and geopotential coefficients simultaneously, called NPESS approach. These two types of solutions mainly differ in geopotential coefficients from degree 2 to 5. This can be explained by the fact that the nuisance parameters and the gravity field coefficients are highly correlated, particularly at low degrees. We compare these solutions with the official and published ones by means of spectral analysis. It is found that our solutions are, in general, consistent with others in the spatial pattern. The water storage variations of the Amazon, Chari and Ganges river basins have also been computed. The variations computed with the NPARB approach are closer to those produced by JPL and DEOS solutions, while the variations produced with the NPESS approach are in good agreement with those produced by the CSR and GFZ solutions. A simulation study is implemented with considering realistic noise and low-frequency error. The two approaches are used to recover the true model. The NPESS solution appears closer to the true one. Therefore we are inclined to estimate the nuisance parameters simultaneously with the geopential coefficients.     

重力卫星高精度测距系统观测量的降速率滤波算法

在卫星时频传递、矢量测量技术领域，有效观测数据的高精度滤波处理十分重要，其基本要求是在保留有效观测信息的同时能够降低噪声干扰，实现原始信息的高效、高精度传递。以中国地球重力场测量卫星高精度测距系统的工程研制为背景，为实现高频噪声抑制、保留低频重力场信息，基于对FIR滤波器结构的优化和窗函数的改进，提出了一种对10 Hz采样率的原始测距数据降速率、降噪滤波的算法。相比其他类似算法，所提方法同时具备降速滤波和差分运算两种功能，噪声抑制抑制度优于44%，且频率截止特性陡峭，第一旁瓣抑制度达到–94 dB。该算法在提高测距精度、充分保留重力场信息的同时，实现了地面数据后处理流程的简化，为微波测量系统、激光测量系统的数据处理提供了一种优选方案。     

Magnetic torquer induced disturbing signals within GRACE accelerometer data

The GRACE (Gravity Recovery And Climate Experiment) gravity field satellite mission was launched in 2002. Although many investigations have been carried out, not all disturbances and perturbations upon satellite instruments and sensors are resolved yet. In this work the issue of acceleration disturbances onboard of GRACE due to magnetic torquers is investigated and discussed. Each of the GRACE satellites is equipped with a three-axes capacitive accelerometer to measure non-gravitational forces acting on the spacecraft. We used 10 Hz Level 1a raw accelerometer data in order to determine the impact of electric current changes on the accelerometer. After reducing signals which are induced by highly dominating processes in the low frequency range, such as thermospheric drag and solar radiation pressure, which can easily be done by applying a high-pass filter, disturbing signals from onboard instruments such as thruster firing events or heater switch events need to be removed from the previously filtered data. Afterwards the spikes which are induced by the torquers can be very well observed. Spikes vary in amplitude with respect to an increasing or decreasing current used for magnetic torquers, and can be as large as 20 nm/s². Furthermore, we were able to set up a model for the spikes of each scenario with which we were able to compute model spike time series. With these time series the spikes can successfully be removed from the 10 Hz raw accelerometer data. Spectral analysis of the time series reveal that an influence onto gravity field determination due to these effects is very unlikely, but can theoretically not be excluded.     

GOCE orbit analysis: Long-wavelength gravity field determination using the acceleration approach

The restricted sensitivity of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gradiometer instrument requires satellite gravity gradiometry to be supplemented by orbit analysis in order to resolve long-wavelength features of the geopotential. For the hitherto published releases of the GOCE time-wise (TIM) and GOCE space-wise gravity field series—two of the official ESA products—the energy conservation method has been adopted to exploit GPS-based satellite-to-satellite tracking information. On the other hand, gravity field recovery from data collected by the CHAllenging Mini-satellite Payload (CHAMP) satellite showed the energy conservation principle to be a sub-optimal choice. For this reason, we propose to estimate the low-frequency part of the gravity field by the point-wise solution of Newton’s equation of motion, also known as the acceleration approach. This approach balances the gravitational vector with satellite accelerations, and hence is characterized by (second-order) numerical differentiation of the kinematic orbit. In order to apply the method to GOCE, we present tailored processing strategies with regard to low-pass filtering, variance–covariance information handling, and robust parameter estimation. By comparison of our GIWF solutions (initials GI for “Geodätisches Institut” and IWF for “Institut für WeltraumForschung”) and the GOCE-TIM estimates with a state-of-the-art gravity field solution derived from GRACE (Gravity Recovery And Climate Experiment), we conclude that the acceleration approach is better suited for GOCE-only gravity field determination as opposed to the energy conservation method.     

GPS-only gravity field recovery with GOCE,CHAMP, and GRACE

Gravity missions such as the Gravity field and steady-state Ocean Circulation Explorer (GOCE) are equipped with onboard Global Positioning System (GPS) receivers for precise orbit determination (POD), instrument time-tagging, and the extraction of the long wavelength part of the Earth’s gravity field. The very low orbital altitude of the GOCE satellite and the availability of dense 1 s GPS tracking data are ideal characteristics to exploit the contribution of GPS high-low Satellite-to-Satellite Tracking (hl-SST) to gravity field determination. We present gravity field solutions based on about 8 months of GOCE GPS hl-SST data from 2009 and compare the results with those obtained from the CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) missions. The very low orbital altitude of GOCE significantly improves gravity field recovery from GPS hl-SST data above degree 20, but not for the degrees below 20, where the quality of the spherical harmonic coefficients remains essentially unchanged. Despite the limited time span of GOCE data used, the gravity field of the Earth can be resolved up to about degree 115 using GPS data only. Empirically determined phase center variations (PCVs) of the GOCE onboard GPS helix antenna are, however, mandatory to achieve this performance.     

DORIS-based point mascons for the long term stability of precise orbit solutions

In recent years non-tidal Time Varying Gravity (TVG) has emerged as the most important contributor in the error budget of Precision Orbit Determination (POD) solutions for altimeter satellites’ orbits. The Gravity Recovery And Climate Experiment (GRACE) mission has provided POD analysts with static and time-varying gravity models that are very accurate over the 2002–2012 time interval, but whose linear rates cannot be safely extrapolated before and after the GRACE lifespan. One such model based on a combination of data from GRACE and Lageos from 2002–2010, is used in the dynamic POD solutions developed for the Geophysical Data Records (GDRs) of the Jason series of altimeter missions and the equivalent products from lower altitude missions such as Envisat, Cryosat-2, and HY-2A. In order to accommodate long-term time-variable gravity variations not included in the background geopotential model, we assess the feasibility of using DORIS data to observe local mass variations using point mascons. In particular, we show that the point-mascon approach can stabilize the geographically correlated orbit errors which are of fundamental interest for the analysis of regional Mean Sea Level trends based on altimeter data, and can therefore provide an interim solution in the event of GRACE data loss. The time series of point-mass solutions for Greenland and Antarctica show good agreement with independent series derived from GRACE data, indicating a mass loss at rate of 210 Gt/year and 110 Gt/year respectively.     

深空卫星重力测量计划研究综述

地球卫星重力测量计划CHAMP(CHAllenging Minisatellite Payload)、GRACE(Gravity Recovery and Climate Experiment)、GOCE(Gravity field and steady-state Ocean Circulation Explorer)和月球卫星重力测量计划(Gravity Recovery and Interior Laboratory,GRAIL)的成功实施,以及下一代地球重力卫星(GRACE Follow-On)的即将发射昭示着我们将迎来一个前所未有的高精度和高空间分辨的深空卫星重力探测时代。围绕深空卫星重力测量的研究背景、必要性、可行性、卫星重力反演软件平台构建、轨道摄动和未来研究方向开展了研究论证。研究表明:深空卫星重力测量作为新世纪重力探测技术,在精化量体重力场、提高惯性导航精度、天体动力学、天体物理学和军事技术的研究,以及促进国民经济发展和提高社会效益等方面具有广泛的应用前景。     

Influence of time variable geopotential models on precise orbits of altimetry satellites,global and regional mean sea level trends

During the last decade a significant progress has been reached in the investigation of the gravity field of the Earth. Besides static, also time variable geopotential models have been recently created. In this paper we investigate the impact of the recent time variable geopotential models on altimetry satellite orbits and such altimetry products based on these orbits, as global and regional mean sea level trends. We show that the modeling of time variable gravity improves the orbit solutions, at least for the GRACE period where time variable gravity is sufficiently accurately observed by this mission. Our analysis includes six geopotential models jointly developed by GFZ German Research Centre for Geosciences and Space Geodesy Research Group (CNES/GRGS) Toulouse: the stationary model EIGEN-GL04S, a stationary version of EIGEN-6S (EIGEN-6S_stat), a corrected version of EIGEN-6S and three enhanced versions of EIGEN-6S called EIGEN-6S2, EIGEN-6S2A and EIGEN-6S2B. By “stationary” we mean “containing periodic parameters such as annual and semi-annual variations, but no secular (drift) terms”. We computed precise orbits for the radar altimetry satellites ERS-1, ERS-2, TOPEX/Poseidon, and Envisat over 20 years between 1991 and 2011. The orbit, single-mission and multi-mission altimetry crossover analyses show that the time variable models EIGEN-6S_corrected, EIGEN-6S2 and its two precursors EIGEN-6S2A/B perform notably better than the stationary models for the GRACE period from 2003 onwards. Thus, using EIGEN-6S2 and EIGEN-6S2A/B we have got 3.6% smaller root mean square fits of satellite laser ranging observations for Envisat, as when using EIGEN-GL04S. However, for the pre-GRACE period 1991–2003, the stationary geopotential models EIGEN-GL04S and EIGEN-6S_stat as well as EIGEN-6S2 having no drift terms for degree 3–50 at this time interval perform superior compared to EIGEN-6S_correct and EIGEN-6S2A/B which contain drifts for this period. We found, that the time variable geopotential models have a low (0.1–0.2 mm/yr) impact on our results for the global mean sea level trend. However, we found strong East/West differences up to 3 mm/yr in the regional mean sea level trends when using orbits of all four satellites based on time variable and stationary geopotential models. We show that these differences are related to the relative drifts of the centers-of-origin between the orbit solutions based on the time variable and stationary geopotential models. From the results of our detailed study, we conclude that the final version of the time variable gravity field model EIGEN-6S2 performs best for the four satellites tested. This model provides the most reliable and mission-consistent sea level estimates for the whole time period from 1992 to 2010. This model is of maximum spherical harmonic degree and order 260 and contains time series for drifts as well as annual and semiannual variations of the spherical harmonic coefficients for degree 2–50.     

应用双向时间传递的重力卫星时标误差自主测量

星间时标误差的自主测量对于中国自主研发地球重力卫星具有重要意义. 提出利用双向时间传递法实现重力卫星时标误差的自主测量, 设计了测量方案, 建立了包含卫星运动导致的链路非对称、电离层效应、设备零值以及随机测时误差在内的测量模型. 结合地球重力卫星相关特性, 分析了时标误差测量中各误差源的影响及相应的误差校正方法. 以GRACE重力卫星为例, 利用提出的方法和校正措施, 星间时标误差自主测量精度可以达到0.62ns, 其中误差主要来自系统零值标定误差和随机测时误差.