Outlier detection algorithms and their performance in GOCE gravity field processing

The satellite missions CHAMP, GRACE, and GOCE mark the beginning of a new era in gravity field determination and modeling. They provide unique models of the global stationary gravity field and its variation in time. Due to inevitable measurement errors, sophisticated pre-processing steps have to be applied before further use of the satellite measurements. In the framework of the GOCE mission, this includes outlier detection, absolute calibration and validation of the SGG (satellite gravity gradiometry) measurements, and removal of temporal effects. In general, outliers are defined as observations that appear to be inconsistent with the remainder of the data set. One goal is to evaluate the effect of additive, innovative and bulk outliers on the estimates of the spherical harmonic coefficients. It can be shown that even a small number of undetected outliers (<0.2 of all data points) can have an adverse effect on the coefficient estimates. Consequently, concepts for the identification and removal of outliers have to be developed. Novel outlier detection algorithms are derived and statistical methods are presented that may be used for this purpose. The methods aim at high outlier identification rates as well as small failure rates. A combined algorithm, based on wavelets and a statistical method, shows best performance with an identification rate of about 99%. To further reduce the influence of undetected outliers, an outlier detection algorithm is implemented inside the gravity field solver (the Quick-Look Gravity Field Analysis tool was used). This results in spherical harmonic coefficient estimates that are of similar quality to those obtained without outliers in the input data.     

球近似下地球外空间任意类型场元的地形影响

传统的重力归算方法只适用于地球表面上的重力异常,不能用于扰动重力、垂线偏差、重力梯度等其他类型扰动重力场元,不适合处理除地面外其他高度上场元的地形影响问题。当前,地球重力场探测的场元类型越来越丰富,探测的高度也逐渐转向航空和卫星高度,精确处理地球外空间各种类型重力场元的地形影响已成为地球重力场领域面临的重要课题。本文通过直接分解由地形生成的具有调和性质的引力场,从而导出地球外空间任意高度、任意类型扰动重力场元的地形影响,在此基础上给出在球近似下地形影响的严密算法和高精度快速算法。利用本文推荐的地形影响计算方案,可以方便地处理各种类型地面重力、海洋重力、航空重力、卫星重力、卫星测高数据的地形影响,从而丰富重力场数据处理的内涵,改善地球重力场算法的性能。     

Triangulated spherical splines for geopotential reconstruction

We present an alternate mathematical technique than contemporary spherical harmonics to approximate the geopotential based on triangulated spherical spline functions, which are smooth piecewise spherical harmonic polynomials over spherical triangulations. The new method is capable of multi-spatial resolution modeling and could thus enhance spatial resolutions for regional gravity field inversion using data from space gravimetry missions such as CHAMP, GRACE or GOCE. First, we propose to use the minimal energy spherical spline interpolation to find a good approximation of the geopotential at the orbital altitude of the satellite. Then we explain how to solve Laplace’s equation on the Earth’s exterior to compute a spherical spline to approximate the geopotential at the Earth’s surface. We propose a domain decomposition technique, which can compute an approximation of the minimal energy spherical spline interpolation on the orbital altitude and a multiple star technique to compute the spherical spline approximation by the collocation method. We prove that the spherical spline constructed by means of the domain decomposition technique converges to the minimal energy spline interpolation. We also prove that the modeled spline geopotential is continuous from the satellite altitude down to the Earth’s surface. We have implemented the two computational algorithms and applied them in a numerical experiment using simulated CHAMP geopotential observations computed at satellite altitude (450 km) assuming EGM96 (n _max = 90) is the truth model. We then validate our approach by comparing the computed geopotential values using the resulting spherical spline model down to the Earth’s surface, with the truth EGM96 values over several study regions. Our numerical evidence demonstrates that the algorithms produce a viable alternative of regional gravity field solution potentially exploiting the full accuracy of data from space gravimetry missions. The major advantage of our method is that it allows us to compute the geopotential over the regions of interest as well as enhancing the spatial resolution commensurable with the characteristics of satellite coverage, which could not be done using a global spherical harmonic representation. The results in this paper are based on the research supported by the National Science Foundation under the grant no. 0327577.     

国际新一代卫星重力探测计划研究现状与进展

回顾了卫星重力探测技术的发展,讨论了当前卫星重力计划CHAMP、GRACE、GOCE研究现状和存在的主要问题,评述了国际下一代4个卫星重力计划E.motion、GRACE-FO、NGGM和GETRIS的最新研究进展,并介绍了SWARM卫星星群任务在地球重力场研究中的作用和地位,最后对我国未来卫星重力计划的实施提出了几点建议。     

An integrated wavelet concept of physical geodesy

For the determination of the earth's gravity field many types of observations are nowadays available, including terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gradio-metry, etc. The mathematical connection between these observables on the one hand and gravity field and shape of the earth on the other is called the integrated concept of physical geodesy. In this paper harmonic wavelets are introduced by which the gravitational part of the gravity field can be approximated progressively better and better, reflecting an increasing flow of observations. An integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are integration formulas relating an integral over an internal sphere to suitable linear combinations of observation functionals, i.e. linear functionals representing the geodetic observables. A scale discrete version of multiresolution is described for approximating the gravitational potential outside and on the earth's surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of band-limited wavelets. A method for combined global outer harmonic and local harmonic wavelet modelling is proposed corresponding to realistic earth's models. As examples, the role of wavelets is discussed for the classical Stokes problem, the oblique derivative problem, satellite-to-satellite tracking, satellite gravity gradiometry and combined satellite-to-satellite tracking and gradiometry. Received: 28 February 1997 / Accepted: 17 November 1997     

Regional gravity modeling in terms of spherical base functions

This article provides a survey on modern methods of regional gravity field modeling on the sphere. Starting with the classical theory of spherical harmonics, we outline the transition towards space-localizing methods such as spherical splines and wavelets. Special emphasis is given to the relations among these methods, which all involve radial base functions. Moreover, we provide extensive applications of these methods and numerical results from real space-borne data of recent satellite gravity missions, namely the Challenging Minisatellite Payload (CHAMP) and the Gravity Recovery and Climate Experiment (GRACE). We also derive high-resolution gravity field models by effectively combining space-borne and surface measurements using a new weighted level-combination concept. In addition, we outline and apply a strategy for constructing spatio-temporal fields from regional data sets spanning different observation periods.     

Downward continuation and geoid determination based on band-limited airborne gravity data

 The downward continuation of the harmonic disturbing gravity potential, derived at flight level from discrete observations of airborne gravity by the spherical Hotine integral, to the geoid is discussed. The initial-boundary-value approach, based on both the direct and inverse solution to Dirichlet's problem of potential theory, is used. Evaluation of the discretized Fredholm integral equation of the first kind and its inverse is numerically tested using synthetic airborne gravity data. Characteristics of the synthetic gravity data correspond to typical airborne data used for geoid determination today and in the foreseeable future: discrete gravity observations at a mean flight height of 2 to 6 km above mean sea level with minimum spatial resolution of 2.5 arcmin and a noise level of 1.5 mGal. Numerical results for both approaches are presented and discussed. The direct approach can successfully be used for the downward continuation of airborne potential without any numerical instabilities associated with the inverse approach. In addition to these two-step approaches, a one-step procedure is also discussed. This procedure is based on a direct relationship between gravity disturbances at flight level and the disturbing gravity potential at sea level. This procedure provided the best results in terms of accuracy, stability and numerical efficiency. As a general result, numerically stable downward continuation of airborne gravity data can be seen as another advantage of airborne gravimetry in the field of geoid determination. Received: 6 June 2001 / Accepted: 3 January 2002     

Evaluation of pre-CHAMP gravity models `GRIM5-S1' and `GRIM5-C1' with satellite crossover altimetry

 The new GFZ/GRGS gravity field models GRIM5-S1 and GRIM5-C1, currently used as initial models for the CHAMP mission, have been compared with other recent models (JGM 3, EGM 96) for radial orbit accuracy (by means of latitude lumped coefficients) in computations on altimetry satellite orbits. The bases for accuracy judgements are multi-year averages of crossover sea height differences from Geosat and ERS 1/2 missions. This radially sensitive data is fully independent of the data used to develop these gravity models. There is good agreement between the observed differences in all of the world's oceans and projections of the same errors from the scaled covariance matrix of their harmonic geopotential coefficients. It was found that the tentative scale factor of five for the formal standard deviations of the harmonic coefficients of the new GRIM fields is justified, i.e. the accuracy estimates, provided together with the GRIM geopotential coefficients, are realistic. Received: 20 February 2001 / Accepted: 24 October 2001     

A study on the combination of satellite, airborne, and terrestrial gravity data

Satellite gravity missions, such as CHAMP, GRACE and GOCE, and airborne gravity campaigns in areas without ground gravity will enhance the present knowledge of the Earths gravity field. Combining the new gravity information with the existing marine and ground gravity anomalies is a major task for which the mathematical tools have to be developed. In one way or another they will be based on the spectral information available for gravity data and noise. The integration of the additional gravity information from satellite and airborne campaigns with existing data has not been studied in sufficient detail and a number of open questions remain. A strategy for the combination of satellite, airborne and ground measurements is presented. It is based on ideas independently introduced by Sjöberg and Wenzel in the early 1980s and has been modified by using a quasi-deterministic approach for the determination of the weighting functions. In addition, the original approach of Sjöberg and Wenzel is extended to more than two measurement types, combining the Meissl scheme with the least-squares spectral combination. Satellite (or geopotential) harmonics, ground gravity anomalies and airborne gravity disturbances are used as measurement types, but other combinations are possible. Different error characteristics and measurement-type combinations and their impact on the final solution are studied. Using simulated data, the results show a geoid accuracy in the centimeter range for a local test area.     

扰动重力梯度的球冠谐分析建模

从球冠谐理论出发,详细推导了球冠坐标系下扰动重力梯度的无奇异性计算公式。基于Tikhonov正则化方法,利用GOCE卫星实际观测数据解算局部重力场球冠谐模型。数值计算表明,基于扰动重力梯度的球冠谐分析建模方法能够有效地恢复局部重力场中的短波信号,与GO_CONS_GCF_2_DIR_R5模型的差异在±0.3×10~(-5) m/s~2水平。     

Earth’s gravity field modelling based on satellite accelerations derived from onboard GPS phase measurements

GPS data collected by satellite gravity missions can be used for extracting the long-wavelength part of the Earth’s gravity field. We propose a new data processing method which makes use of the ‘average acceleration’ approach to gravity field modelling. In this method, satellite accelerations are directly derived from GPS carrier phase measurements with an epoch-differenced scheme. As a result, no ambiguity solutions are needed and the systematic errors that do not change much from epoch to epoch are largely eliminated. The GPS data collected by the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite mission are used to demonstrate the added value of the proposed method. An analysis of the residual accelerations shows that accelerations derived in this way are more precise, with noise being reduced by about 20 and 5% at the cross-track component and the other two components, respectively, as compared to those based on kinematic orbits. The accelerations obtained in this way allow the recovery of the gravity field to a slightly higher maximum degree compared to the solution based on kinematic orbits. Furthermore, the gravity field solution has an overall better performance. Errors in spherical harmonic coefficients are smaller, especially at low degrees. The cumulative geoid height error is reduced by about 15 and 5% up to degree 50 and 150, respectively. An analysis in the spatial domain shows that large errors along the geomagnetic equator, which are caused by a high electron density coupled with large short-term variations, are substantially reduced. Finally, the new method allows for a better observation of mass transport signals. In particular, sufficiently realistic signatures of regional mass anomalies in North America and south-west Africa are obtained.     

Frequency-dependent data weighting in global gravity field modeling from satellite data contaminated by non-stationary noise

Satellite data that are used to model the global gravity field of the Earth are typically corrupted by correlated noise, which can be related to a frequency dependence of the data accuracy. We show an opportunity to take such noise into account by using a proper noise covariance matrix in the estimation procedure. If the dependence of noise on frequency is not known a priori, it can be estimated on the basis of a posteriori residuals. The methodology can be applied to data with gaps. Non-stationarity of noise can also be dealt with, provided that the necessary a priori information exists. The proposed methodology is illustrated with CHAllenging Mini-satellite Payload (CHAMP) data processing. It is shown, in particular, that the usage of a proper noise model can make the measurements of non-gravitational satellite accelerations unnecessarily. This opens the door for high-quality modeling of the Earth’s gravity field on the basis of observed orbits of non-dedicated satellites (i.e., satellites without an on-board accelerometer). Furthermore, the processing of data from dedicated satellite missions – GRACE (Gravity Recovery and Climate Experiment) and GOCE (Gravity field and steady-state Ocean Circulation Explorer) – may also benefit from the proposed methodology.     

星载重力测量和固体地球的透视

为了进一步改进地球重力场模型 ,美国科学院 (NRC)于 1997年提出一个报告 ,号召美国和欧洲学者们组成一个由空间探测地球重力场委员会。他们评比了大约 10个计划的或想象的卫星重力任务 ,最后提出了 5种卫星重力任务的方案 ,称为专用重力任务。本文介绍这 5种任务的内容以及地球科学将由这些任务获得重大进展的研究领域。     

On the spectral consistency of the altimetric ocean and geoid surface: a one-dimensional example

Geoid models from the new generation of satellite gravity missions, such as GRACE and GOCE, in combination with sea surface from satellite altimetry allow to obtain absolute dynamic ocean topography with rather high spatial resolution and accuracy. However, this implies combination of data with fundamentally different characteristics and different spatial resolutions. Spectral consistency would imply the removal of the short-scale features of the altimetric sea surface height by filtering, to provide altimetric data consistent with the resolution of the geoid field. The goal must be to lose as little as possible from the high precision of the altimetric signal. Using a one-dimensional example we show how the spectrum is changing when a function defined only on a limited domain (ocean in the real case) is extended or not as to cover the complete domain (the whole sphere in the real case). The results depend on the spectral characteristics of the altimetric signal and of the applied filter. Referring to the periodicity condition, as it is requested in the case of Fourier analysis, the action of the two classical filters (Ideal Low Pass and Gauss filter) and of two alternative procedures (wavelets and Slepian) is studied.     

卫星跟踪卫星技术的进展及应用前景

卫星跟踪卫星技术被认为是 2 1世纪初最有价值和应用前景的高效重力探测技术 ,旨在测定中长波重力场的精细结构及时间相依变化。本文首先简要阐述卫星跟踪卫星技术的发展背景及概况 ,其次介绍目前已经实施和将要实施的卫星跟踪卫星计划 CHAMP和 GRACE的进展情况 ,最后讨论该技术在精化地球重力场和研究相关地学问题中的应用前景。     

Geoid and high resolution sea surface topography modelling in the mediterranean from gravimetry,altimetry and GOCE data: evaluation by simulation

The determination of local geoid models has traditionally been carried out on land and at sea using gravity anomaly and satellite altimetry data, while it will be aided by the data expected from satellite missions such as those from the Gravity field and steady-state ocean circulation explorer (GOCE). To assess the performance of heterogeneous data combination to local geoid determination, simulated data for the central Mediterranean Sea are analyzed. These data include marine and land gravity anomalies, altimetric sea surface heights, and GOCE observations processed with the space-wise approach. A spectral analysis of the aforementioned data shows their complementary character. GOCE data cover long wavelengths and account for the lack of such information from gravity anomalies. This is exploited for the estimation of local covariance function models, where it is seen that models computed with GOCE data and gravity anomaly empirical covariance functions perform better than models computed without GOCE data. The geoid is estimated by different data combinations and the results show that GOCE data improve the solutions for areas covered poorly with other data types, while also accounting for any long wavelength errors of the adopted reference model that exist even when the ground gravity data are dense. At sea, the altimetric data provide the dominant geoid information. However, the geoid accuracy is sensitive to orbit calibration errors and unmodeled sea surface topography (SST) effects. If such effects are present, the combination of GOCE and gravity anomaly data can improve the geoid accuracy. The present work also presents results from simulations for the recovery of the stationary SST, which show that the combination of geoid heights obtained from a spherical harmonic geopotential model derived from GOCE with satellite altimetry data can provide SST models with some centimeters of error. However, combining data from GOCE with gravity anomalies in a collocation approach can result in the estimation of a higher resolution geoid, more suitable for high resolution mean dynamic SST modeling. Such simulations can be performed toward the development and evaluation of SST recovery methods.     

A synthetic Earth for use in geodesy

 A synthetic Earth and its gravity field that can be represented at different resolutions for testing and comparing existing and new methods used for global gravity-field determination are created. Both the boundary and boundary values of the gravity potential can be generated. The approach chosen also allows observables to be generated at aircraft flight height or at satellite altitude. The generation of the synthetic Earth shape (SES) and gravity-field quantities is based upon spherical harmonic expansions of the isostatically compensated equivalent rock topography and the EGM96 global geopotential model. Spherical harmonic models are developed for both the synthetic Earth topography (SET) and the synthetic Earth potential (SEP) up to degree and order 2160 corresponding to a 5′×5′ resolution. Various sets of SET, SES and SEP with boundary geometry and boundary values at different resolutions can be generated using low-pass filters applied to the expansions. The representation is achieved in point sets based upon refined triangulation of a octahedral geometry projected onto the chosen reference ellipsoid. The filter cut-offs relate to the sampling pattern in order to avoid aliasing effects. Examples of the SET and its gravity field are shown for a resolution with a Nyquist sampling rate of 8.27 degrees. Received: 6 August 1999 / Accepted: 26 April 2000     

Integral transformations of deflections of the vertical onto satellite-to-satellite tracking and gradiometric data

With the advent of geodetic satellite missions mapping almost globally the Earth’s gravitational field, new methods and theoretical approaches have been developed and investigated to fully exploit the potential of their new observables. Besides estimating values of numerical coefficients in harmonic series of the gravitational potential, new applications emerged such as data validation and combination. In this contribution, new integral transformations are presented which transform principal components of the terrestrial deflection of the vertical onto disturbing satellite-to-satellite tracking and gradiometric data at altitude. Using spherical approximation, necessary integral kernel functions are derived in both spectral and closed forms. The behaviour of isotropic kernel functions is studied and the new integral transformations are tested in a closed-loop simulation using synthetic terrestrial and satellite data synthesized from a global gravitational model. New integral transformations can be used for data validation and combination purposes.     

联合EGM2008模型重力异常和GOCE观测数据构建超高阶地球重力场模型SGG-UGM-1

本文研究了联合卫星观测数据和重力异常数据确定超高阶重力场模型的理论方法,并使用EGM2008模型重力异常和GOCE（gravity field and ocean circulation explorer）观测数据构建了重力场模型SGG-UGM-1。重点研究了由球面格网重力异常快速构建超高阶重力场模型的块对角最小二乘方法,将OpenMP技术引入到块对角最小二乘中以提高计算效率,并基于模拟数据验证了方法及算法和软件模块的正确性。采用本文制定的联合解算策略,利用GOCE重力卫星观测数据构建的220阶次法方程和EGM2008模型重力异常构建的2159阶次块对角法方程,联合求解了2159阶次的重力场模型SGG-UGM-1。将SGG-UGM-1与EGM2008、EIGEN-6C2、EIGEN-6C4等超高阶模型在频谱域内进行了比较分析,结果表明SGG-UGM-1相对参考模型的系数误差较小,且在220阶次内的系数精度相比EGM2008模型有了提高。采用中国与美国的GPS/水准数据和毛乌素测区的航空重力观测数据对这些模型进行了外符合精度的检验。检核结果表明,在中国区域,SGG-UGM-1模型大地水准面的精度在EIGEN-6C2和EIGEN-6C4两个模型之间,优于GOSG-EGM模型和EGM2008模型,与美国区域几个模型的精度相当。利用毛乌素测区的航空重力数据对几个模型进行了检核,结果表明SGG-UGM-1模型计算的重力扰动精度与EGM2008、EIGEN-6C4模型相当,优于GOSG-EGM模型和EIGEN-6C2模型。