利用最小二乘直接法反演卫星重力场模型的MPI并行算法

针对海量卫星重力数据反演高阶次地球重力场模型的密集型计算任务与高内存耗用问题,基于MPI实现了最小二乘直接法恢复高阶次位系数的并行算法。引入并行读写、分块存储与分块计算等方式完成了设计矩阵的构建、法方程的形成与求解等密集型计算任务的并行算法,数值计算结果表明三者的并行相对效率峰值可分别达到95%、68%、63%。利用GOCE轨道跟踪和径向扰动重力梯度数据(共518 400个历元)分别反演了120、240阶次地球重力场模型,计算时间仅为40 min、7 h,内存耗用峰值仅为290 MB、1.57 GB;采用与GOCE同等噪声水平的观测数据恢复的重力场模型精度与GOCE已发布模型的解算精度相一致,联合GRACE和GOCE的解算模型能够实现二者独立信息的频谱互补,表明本文方法可高效稳定地恢复高阶次地球重力场模型。     

Efficient gravity field recovery from GOCE gravity gradient observations

 An efficient algorithm is proposed for gravity field recovery from Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite gravity gradient observations. The mathematical model is formulated in the time domain, which allows the inclusion of realistic observational noise models. The algorithm combines the iterative solution of the normal equations, using a Richardson-type iteration scheme, with the fast computation of the right-hand side of the normal equations in each iteration step by a suitable approximation of the design matrix. The convergence of the iteration is investigated, error estimates are provided, and the unbiasedness of the method is proved. It is also shown that the method does not converge to the solution of the normal equations. The performance of the approach for white noise and coloured noise is demonstrated along a simulated GOCE orbit up to spherical harmonic degree and order 180. The results also indicate that the approximation error may be neglected. Received: 30 November 1999 / Accepted: 31 May 2000     

Adaptive filtering of GOCE-derived gravity gradients of the disturbing potential in the context of the space-wise approach

Filtering and signal processing techniques have been widely used in the processing of satellite gravity observations to reduce measurement noise and correlation errors. The parameters and types of filters used depend on the statistical and spectral properties of the signal under investigation. Filtering is usually applied in a non-real-time environment. The present work focuses on the implementation of an adaptive filtering technique to process satellite gravity gradiometry data for gravity field modeling. Adaptive filtering algorithms are commonly used in communication systems, noise and echo cancellation, and biomedical applications. Two independent studies have been performed to introduce adaptive signal processing techniques and test the performance of the least mean-squared (LMS) adaptive algorithm for filtering satellite measurements obtained by the gravity field and steady-state ocean circulation explorer (GOCE) mission. In the first study, a Monte Carlo simulation is performed in order to gain insights about the implementation of the LMS algorithm on data with spectral behavior close to that of real GOCE data. In the second study, the LMS algorithm is implemented on real GOCE data. Experiments are also performed to determine suitable filtering parameters. Only the four accurate components of the full GOCE gravity gradient tensor of the disturbing potential are used. The characteristics of the filtered gravity gradients are examined in the time and spectral domain. The obtained filtered GOCE gravity gradients show an agreement of 63–84 mEötvös (depending on the gravity gradient component), in terms of RMS error, when compared to the gravity gradients derived from the EGM2008 geopotential model. Spectral-domain analysis of the filtered gradients shows that the adaptive filters slightly suppress frequencies in the bandwidth of approximately 10–30 mHz. The limitations of the adaptive LMS algorithm are also discussed. The tested filtering algorithm can be connected to and employed in the first computational steps of the space-wise approach, where a time-wise Wiener filter is applied at the first stage of GOCE gravity gradient filtering. The results of this work can be extended to using other adaptive filtering algorithms, such as the recursive least-squares and recursive least-squares lattice filters.     

GOCE实测数据反演高阶重力场模型的Torus方法

不同于当前广泛使用的空域法、时域法、直接解法,本文尝试采用Torus方法处理GOCE实测数据,利用71 d的GOCE卫星引力梯度数据反演了200阶次GOCE地球重力场模型,实现了对参考模型的精化。首先,采用Butterworth零相移滤波方法加移去—恢复技术,处理引力梯度观测值中的有色噪声,并利用泰勒级数展开和Kriging方法对GOCE卫星引力梯度数据进行归算和格网化,计算得到了名义轨道上格网点处的引力梯度数据。然后,利用2D-FFT技术和块对角最小二乘方法处理名义轨道上数据,获得了200阶次的GOCE地球重力场模型GOCE_Torus。利用中国和美国的GPS/水准数据进行外部检核结果说明,GOCE_Torus与ESA发布的同期模型的精度相当;GOCE_Torus模型与200阶次的EGM2008模型相比,在美国区域精度相当,但在中国区域精度提高了4.6 cm,这充分体现了GOCE卫星观测数据对地面重力稀疏区的贡献。Torus方法拥有快速高精度反演卫星重力场模型的优势,可以在重力梯度卫星的设计、误差分析及在轨快速评估等方面得到充分应用。     

Methodology and use of tensor invariants for satellite gravity gradiometry

Although its use is widespread in several other scientific disciplines, the theory of tensor invariants is only marginally adopted in gravity field modeling. We aim to close this gap by developing and applying the invariants approach for geopotential recovery. Gravitational tensor invariants are deduced from products of second-order derivatives of the gravitational potential. The benefit of the method presented arises from its independence of the gradiometer instrument’s orientation in space. Thus, we refrain from the classical methods for satellite gravity gradiometry analysis, i.e., in terms of individual gravity gradients, in favor of the alternative invariants approach. The invariants approach requires a tailored processing strategy. Firstly, the non-linear functionals with regard to the potential series expansion in spherical harmonics necessitates the linearization and iterative solution of the resulting least-squares problem. From the computational point of view, efficient linearization by means of perturbation theory has been adopted. It only requires the computation of reference gravity gradients. Secondly, the deduced pseudo-observations are composed of all the gravitational tensor elements, all of which require a comparable level of accuracy. Additionally, implementation of the invariants method for large data sets is a challenging task. We show the fundamentals of tensor invariants theory adapted to satellite gradiometry. With regard to the GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite gradiometry mission, we demonstrate that the iterative parameter estimation process converges within only two iterations. Additionally, for the GOCE configuration, we show the invariants approach to be insensitive to the synthesis of unobserved gravity gradients.     

面向精密位置服务的低轨卫星轨道预报精度分析EI北大核心CSCD

LEO卫星精密轨道预报是LEO导航增强系统中重要的技术环节之一,本文使用多种算法来实现不同任务需求下的轨道预报。对于在地面处理系统实现的LEO轨道预报,算法1采用定轨预报同时处理的策略,算法2将离散轨道点进行动力学拟合再进行积分外推。GRACE-C卫星预报5、10、15 min的URE平均精度分别为5.25、5.67、6.25 cm;HY2A卫星为7.83、8.69、9.66 cm;SWARM-A卫星为8.88、9.22、9.63 cm;SWARM-B卫星为8.49、8.98、9.63 cm。对于计算条件受限的LEO星上轨道预报,本文利用单个轨道点及简单动力学模型进行轨道积分外推的算法。该算法主要考虑地球中心引力及非球形引力摄动,因此地球重力场阶次对轨道预报精度产生较大影响。平均高度为500 km的LEO卫星选取60阶重力场,高度为1000 km的LEO卫星选取30阶重力场,可实现预报10 min轨道优于10 cm的预报精度。     

卫星重力梯度向下延拓的谱方法

本文提出在平面近似下解算卫星重力梯度向下延拓问题的谱方法，并采用模拟数据进行了试算，结果表明该方法是有效的。这为利用卫星重力梯度数据精化局部重力场提供了可供参考的方法。     

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.     

The ITG-Goce02 gravity field model from GOCE orbit and gradiometer data based on the short arc approach

In this contribution, we describe the global GOCE-only gravity field model ITG-Goce02 derived from 7.5 months of gradiometer and orbit data. This model represents an alternative to the official ESA products as it is computed completely independently, using a different processing strategy and a separate software package. Our model is derived using the short arc approach, which allows a very effective decorrelation of the highly correlated GOCE gradiometer and orbit data noise by introducing a full empirical covariance matrix for each arc, and gives the possibility to downweight ‘bad’ arcs. For the processing of the orbit data we rely on the integral equation approach instead of the energy integral method, which has been applied in several other GOCE models. An evaluation against high-resolution global gravity field models shows very similar differences of our model compared to the official GOCE results published by ESA (release 2), especially to the model derived by the time-wise approach. This conclusion is confirmed by comparison of the GOCE models to GPS/levelling and altimetry data.     

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.     

Using radial basis functions in airborne gravimetry for local geoid improvement

Radial basis functions (RBFs) have been used extensively in satellite geodetic applications. However, to the author’s knowledge, their role in processing and modeling airborne gravity data has not yet been fully advocated or extensively investigated in detail. Compared with satellite missions, the airborne data are more suitable for these kinds of localized basis functions especially considering the following facts: (1) Unlike the satellite missions that can provide global or near global data coverage, airborne gravity data are usually geographically limited. (2) It is also band limited in the frequency domain. (3) It is straightforward to formulate the RBF observation equations from an airborne gravimetric system. In this study, a set of band-limited RBF is developed to model and downward continue the airborne gravity data for local geoid improvement. First, EIGEN6c4 coefficients are used to simulate a harmonic field to test the performances of RBF on various sampling, noise, and flight height levels, in order to gain certain guidelines for processing the real data. Here, the RBF method not only successfully recovers the harmonic field but also presents filtering properties due to its particular design in the frequency domain. Next, the software was tested for the GSVS14 (Geoid Slope Validation Survey 2014) area in Iowa as well as for the area around Puerto Rico and the US Virgin Islands by use of the real airborne gravity data from the Gravity for the Redefinition of the American Vertical Datum (GRAV-D) project. By fully utilizing the three-dimensional correlation information among the flight tracks, the RBF can also be used as a data cleaning tool for airborne gravity data adjustment and cleaning. This property is further extended to surface gravity data cleaning, where conventional approaches have various limitations. All the related numerical results clearly show the importance and contribution of the use of the RBF for high- resolution local gravity field modeling.     

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.     

‘DEOS_CHAMP-01C_70’: a model of the Earth’s gravity field computed from accelerations of the CHAMP satellite

Performance of a recently proposed technique for gravity field modeling has been assessed with data from the CHAMP satellite. The modeling technique is a variant of the acceleration approach. It makes use of the satellite accelerations that are derived from the kinematic orbit with the 3-point numerical differentiation scheme. A 322-day data set with 30-s sampling has been used. Based on this, a new gravity field model – DEOS_CHAMP-01C_70 - is derived. The model is complete up to degree and order 70. The geoid height difference between the DEOS_CHAMP-01C_70 and EIGEN-GRACE01S models is 14 cm. This is less than for two other recently published models EIGEN-CHAMP03Sp and ITG-CHAMP01E. Furthermore, we analyze the sensitivity of the model to some empirically determined parameters (regularization parameter and the parameter that controls the frequency-dependent data weighting). We also show that inaccuracies related to non-gravitational accelerations, which are measured by the on-board accelerometer, have a minor influence on the computed gravity field model.     

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.     

GOCE重力卫星在军事上的应用前景分析

对GOCE重力卫星进行了详细的介绍;分析了星上载荷的配置和功能.结合GOCE任务的科学目标,详细分析了其在军事上的应用前景,并列出了军事上重要的应用方向.希望广大测量界人士对GoCE重力卫星应用领域进行更多的利用.     

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。     

GOCE卫星重力梯度数据反演重力场的滤波器设计与比较分析

地球重力场和海洋环流探测（gravity field and steady-state ocean circulation explorer,GOCE）卫星重力梯度数据有色噪声和低频系统误差的滤波处理是反演高精度地球重力场的一个关键问题。针对GOCE卫星重力梯度数据的滤波处理,基于移动平均（moving average,MA）方法和CPR(circle per revolution)经验参数方法设计了两类低频系统误差滤波器,并分别将这两类滤波器与基于自回归移动平均（auto-regressive and moving average,ARMA）模型设计的有色噪声滤波器组合起来形成级联滤波器。为了分析滤波器处理的实际效果,基于空域最小二乘法采用70 d的GOCE观测数据,并联合重力恢复与气候实验（gravity recovery and climate experiment,GRACE）数据分别反演了224阶次的重力场模型GOGR-MA(MA+ARMA级联滤波)和GOGR-CPR(CPR+ARMA级联滤波)。将反演模型与采用同期数据求解的第一代GOCE系列模型及GOCE和GRACE联合模...     

卫星重力测量

确定高精度和高分辨率地球重力场模型是现代大地测量的基本目标之一,卫星重力计划就是基于这一目标实施的。文章简单地评述了卫星重力的发展现状,介绍了三颗专用的重力卫星,给出了卫星重力测量的基本原理,最后比较了几种由重力卫星资料得到的地球重力场模型。     

低轨卫星初始状态误差对反演地球重力场的影响

基于动力学法反演地球重力场的基本理论,研究了卫星初始状态向量误差对应用低轨卫星精密轨道数据反演地球重力场的影响。在仅考虑低轨卫星初始状态误差的情况下进行了模拟计算,结果表明:在利用低轨卫星精密轨道数据反演地球重力场时,卫星初始状态向量误差需要重新进行估计;在目前的轨道精度水平下,若不顾及误差方程二次项的影响,反演弧长不宜过长;卫星初始状态速度误差(约1.5mm/s)的影响要大于位置误差(约10 cm)的影响。     

卫星重力梯度向下延拓的频域最小二乘配置法

本文深入研究频域最小二乘配置法的基本原理及其在求解卫星重力梯度向下延拓问题中的应用。与一般空域最小二乘配置法相比，该方法具有高效稳定的特点，特别适合于大规模重力场数据处理，并为利用卫星重力梯度数据精化局部重力场提供了可供参考的方法。