Comparison of GOCE-GPS gravity fields derived by different approaches

Several techniques have been proposed to exploit GNSS-derived kinematic orbit information for the determination of long-wavelength gravity field features. These methods include the (i) celestial mechanics approach, (ii) short-arc approach, (iii) point-wise acceleration approach, (iv) averaged acceleration approach, and (v) energy balance approach. Although there is a general consensus that—except for energy balance—these methods theoretically provide equivalent results, real data gravity field solutions from kinematic orbit analysis have never been evaluated against each other within a consistent data processing environment. This contribution strives to close this gap. Target consistency criteria for our study are the input data sets, period of investigation, spherical harmonic resolution, a priori gravity field information, etc. We compare GOCE gravity field estimates based on the aforementioned approaches as computed at the Graz University of Technology, the University of Bern, the University of Stuttgart/Austrian Academy of Sciences, and by RHEA Systems for the European Space Agency. The involved research groups complied with most of the consistency criterions. Deviations only occur where technical unfeasibility exists. Performance measures include formal errors, differences with respect to a state-of-the-art GRACE gravity field, (cumulative) geoid height differences, and SLR residuals from precise orbit determination of geodetic satellites. We found that for the approaches (i) to (iv), the cumulative geoid height differences at spherical harmonic degree 100 differ by only \({\approx }10~\%\) ; in the absence of the polar data gap, SLR residuals agree by \({\approx }96~\%\) . From our investigations, we conclude that real data analysis results are in agreement with the theoretical considerations concerning the (relative) performance of the different approaches.     

Validation of GOCE gravity field models by means of orbit residuals and geoid comparisons

Three GOCE-based gravity field solutions have been computed by ESA’s high-level processing facility and were released to the user community. All models are accompanied by variance-covariance information resulting either from the least squares procedure or a Monte-Carlo approach. In order to obtain independent external quality parameters and to assess the current performance of these models, a set of independent tests based on satellite orbit determination and geoid comparisons is applied. Both test methods can be regarded as complementary because they either investigate the performance in the long wavelength spectral domain (orbit determination) or in the spatial domain (geoid comparisons). The test procedure was applied to the three GOCE gravity field solutions and to a number of selected pre-launch models for comparison. Orbit determination results suggest, that a pure GOCE gravity field model does not outperform the multi-year GRACE gravity field solutions. This was expected as GOCE is designed to improve the determination of the medium to high frequencies of the Earth gravity field (in the range of degree and order 50 to 200). Nevertheless, in case of an optimal combination of GOCE and GRACE data, orbit determination results should not deteriorate. So this validation procedure can also be used for testing the optimality of the approach adopted for producing combined GOCE and GRACE models. Results from geoid comparisons indicate that with the 2 months of GOCE data a significant improvement in the determination of the spherical harmonic spectrum of the global gravity field between degree 50 and 200 can be reached. Even though the ultimate mission goal has not yet been reached, especially due to the limited time span of used GOCE data (only 2 months), it was found that existing satellite-only gravity field models, which are based on 7 years of GRACE data, can already be enhanced in terms of spatial resolution. It is expected that with the accumulation of more GOCE data the gravity field model resolution and quality can be further enhanced, and the GOCE mission goal of 1–2 cm geoid accuracy with 100 km spatial resolution can be achieved.     

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     

Evaluation of the third- and fourth-generation GOCE Earth gravity field models with Australian terrestrial gravity data in spherical harmonics

In March 2013, the fourth generation of European Space Agency’s (ESA) global gravity field models, DIR4 (Bruinsma et al. in Proceedings of the ESA living planet symposium, 28 June–2 July, Bergen, ESA, Publication SP-686, 2010b) and TIM4 (Migliaccio et al. in Proceedings of the ESA living planet symposium, 28 June–2 July, Bergen, ESA, Publication SP-686, 2010), generated from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gravity observation satellite was released. We evaluate the models using an independent ground truth data set of gravity anomalies over Australia. Combined with Gravity Recovery and Climate Experiment (GRACE) satellite gravity, a new gravity model is obtained that is used to perform comparisons with GOCE models in spherical harmonics. Over Australia, the new gravity model proves to have significantly higher accuracy in the degrees below 120 as compared to EGM2008 and seems to be at least comparable to the accuracy of this model between degree 150 and degree 260. Comparisons in terms of residual quasi-geoid heights, gravity disturbances, and radial gravity gradients evaluated on the ellipsoid and at approximate GOCE mean satellite altitude ( $h=250$  km) show both fourth generation models to improve significantly w.r.t. their predecessors. Relatively, we find a root-mean-square improvement of 39 % for the DIR4 and 23 % for TIM4 over the respective third release models at a spatial scale of 100 km (degree 200). In terms of absolute errors, TIM4 is found to perform slightly better in the bands from degree 120 up to degree 160 and DIR4 is found to perform slightly better than TIM4 from degree 170 up to degree 250. Our analyses cannot confirm the DIR4 formal error of 1 cm geoid height (0.35 mGal in terms of gravity) at degree 200. The formal errors of TIM4, with 3.2 cm geoid height (0.9 mGal in terms of gravity) at degree 200, seem to be realistic. Due to combination with GRACE and SLR data, the DIR models, at satellite altitude, clearly show lower RMS values compared to TIM models in the long wavelength part of the spectrum (below degree and order 120). Our study shows different spectral sensitivity of different functionals at ground level and at GOCE satellite altitude and establishes the link among these findings and the Meissl scheme (Rummel and van Gelderen in Manusrcipta Geodaetica 20:379–385, 1995).     

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.     

Assessment of three numerical solution strategies for gravity field recovery from GOCE satellite gravity gradiometry implemented on a parallel platform

 The recovery of a full set of gravity field parameters from satellite gravity gradiometry (SGG) is a huge numerical and computational task. In practice, parallel computing has to be applied to estimate the more than 90 000 harmonic coefficients parameterizing the Earth's gravity field up to a maximum spherical harmonic degree of 300. Three independent solution strategies (preconditioned conjugate gradient method, semi-analytic approach, and distributed non-approximative adjustment), which are based on different concepts, are assessed and compared both theoretically and on the basis of a realistic-as-possible numerical simulation regarding the accuracy of the results, as well as the computational effort. Special concern is given to the correct treatment of the coloured noise characteristics of the gradiometer. The numerical simulations show that the three methods deliver nearly identical results—even in the case of large data gaps in the observation time series. The newly proposed distributed non-approximative adjustment approach, which is the only one of the three methods that solves the inverse problem in a strict sense, also turns out to be a feasible method for practical applications. Received: 17 December 2001 / Accepted: 17 July 2002 Acknowledgments. We would like to thank Prof. W.-D. Schuh, Institute of Theoretical Geodesy, University of Bonn, for providing us with the serial version of the PCGMA algorithm, which forms the basis for the parallel PCGMA package developed at our institute. This study was partially performed in the course of the GOCE project `From E?tv?s to mGal+', funded by the European Space Agency (ESA) under contract No. 14287/00/NL/DC. Correspondence to: R. Pail     

Evaluation of the first GOCE static gravity field models using terrestrial gravity,vertical deflections and EGM2008 quasigeoid heights

Recently, four global geopotential models (GGMs) were computed and released based on the first 2 months of data collected by the Gravity field and steady-state Ocean Circulation Explorer (GOCE) dedicated satellite gravity field mission. Given that GOCE is a technologically complex mission and different processing strategies were applied to real space-collected GOCE data for the first time, evaluation of the new models is an important aspect. As a first assessment strategy, we use terrestrial gravity data over Switzerland and Australia and astrogeodetic vertical deflections over Europe and Australia as ground-truth data sets for GOCE model evaluation. We apply a spectral enhancement method (SEM) to the truncated GOCE GGMs to make their spectral content more comparable with the terrestrial data. The SEM utilises the high-degree bands of EGM2008 and residual terrain model data as a data source to widely bridge the spectral gap between the satellite and terrestrial data. Analysis of root mean square (RMS) errors is carried out as a function of (i) the GOCE GGM expansion degree and (ii) the four different GOCE GGMs. The RMS curves are also compared against those from EGM2008 and GRACE-based GGMs. As a second assessment strategy, we compare global grids of GOCE GGM and EGM2008 quasigeoid heights. In connection with EGM2008 error estimates, this allows location of regions where GOCE is likely to deliver improved knowledge on the Earth’s gravity field. Our ground truth data sets, together with the EGM2008 quasigeoid comparisons, signal clear improvements in the spectral band ~160–165 to ~180–185 in terms of spherical harmonic degrees for the GOCE-based GGMs, fairly independently of the individual GOCE model used. The results from both assessments together provide strong evidence that the first 2 months of GOCE observations improve the knowledge of the Earth’s static gravity field at spatial scales between ~125 and ~110 km, particularly over parts of Asia, Africa, South America and Antarctica, in comparison with the pre-GOCE-era.     

GOCE卫星重力梯度观测值的时变重力场变化改正及影响分析

GOCE卫星重力梯度观测值为高阶静态重力场反演提供了重要的数据支撑,但其在使用前需考虑扣除时变重力场变化的影响.本文研究了GOCE卫星重力梯度观测值的时变重力场变化改正方法,更新了ESA标准和背景模型,以更好地扣除时变重力场变化的影响,自主实现了由GOCE卫星Level1b重力梯度数据直接进行重力场反演.本文通过3种时...     

利用先验重力场模型求定GOCE卫星重力梯度仪校准参数

重力梯度仪校准参数的确定是GOCE重力梯度观测数据处理的关键环节。本文对GOCE卫星重力梯度观测值中的时变信号与粗差进行了分析,利用高精度全球重力场模型,确定了GOCE重力梯度观测值各分量的尺度因子与偏差,并对校准结果进行了精度评定。结果表明,在测量带宽内,海潮对重力梯度观测值影响在mE量级,与重力梯度仪的精度水平相当,陆地水等非潮汐重力场时变信号略小于海潮,量级约为10^-4E;各分量重力梯度观测值的粗差比例均大于0.2%;除EGM96模型外的其他模型对GOCE重力梯度仪进行校准后,V_xx、V_yy、V_zz、V_yz分量上尺度因子的稳定性均在10^-4量级,V_xz分量能达到10^-5量级,V_xy分量为10^-2量级,这与梯度观测值各分量的精度水平一致。     

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地球重力场模型在青藏地区的评价及分析

2009年GOCE卫星升空以后,卫星重力梯度数据参与解算的GOCE系列重力场模型已有多家研究机构相继公布。本文分别采用青藏地区的GPS/水准和重力异常实测数据对GOCE重力场模型进行了外部测试,并在重力异常验证过程中引入了一种新的滤波方法,验证结果表明在青藏地区GOCE重力场模型相比其它系列模型的优势在于中波段。同时,探讨了GOCE重力场模型与其他系列模型在青藏地区主要差异值的空间分布以及首次利用统计分析方法找出模型之间主要差异值的阶次分布,得出如下结论：模型之间的较大差异值在空间水平方向上主要分布在喜马拉雅山脉、天山等地形起伏较大的区域,在垂直方向上主要集中在岩石圈。     

Efficient GOCE satellite gravity field recovery based on least-squares using QR decomposition

We develop and apply an efficient strategy for Earth gravity field recovery from satellite gravity gradiometry data. Our approach is based upon the Paige-Saunders iterative least-squares method using QR decomposition (LSQR). We modify the original algorithm for space-geodetic applications: firstly, we investigate how convergence can be accelerated by means of both subspace and block-diagonal preconditioning. The efficiency of the latter dominates if the design matrix exhibits block-dominant structure. Secondly, we address Tikhonov-Phillips regularization in general. Thirdly, we demonstrate an effective implementation of the algorithm in a high-performance computing environment. In this context, an important issue is to avoid the twofold computation of the design matrix in each iteration. The computational platform is a 64-processor shared-memory supercomputer. The runtime results prove the successful parallelization of the LSQR solver. The numerical examples are chosen in view of the forthcoming satellite mission GOCE (Gravity field and steady-state Ocean Circulation Explorer). The closed-loop scenario covers 1 month of simulated data with 5 s sampling. We focus exclusively on the analysis of radial components of satellite accelerations and gravity gradients. Our extensions to the basic algorithm enable the method to be competitive with well-established inversion strategies in satellite geodesy, such as conjugate gradient methods or the brute-force approach. In its current development stage, the LSQR method appears ready to deal with real-data applications.     

Exploring gravity field determination from orbit perturbations of the European Gravity Mission GOCE

 A comparison was made between two methods for gravity field recovery from orbit perturbations that can be derived from global positioning system satellite-to-satellite tracking observations of the future European gravity field mission GOCE (Gravity Field and Steady-State Ocean Circulation Explorer). The first method is based on the analytical linear orbit perturbation theory that leads under certain conditions to a block-diagonal normal matrix for the gravity unknowns, significantly reducing the required computation time. The second method makes use of numerical integration to derive the observation equations, leading to a full set of normal equations requiring powerful computer facilities. Simulations were carried out for gravity field recovery experiments up to spherical harmonic degree and order 80 from 10 days of observation. It was found that the first method leads to large approximation errors as soon as the maximum degree surpasses the first resonance orders and great care has to be taken with modeling resonance orbit perturbations, thereby loosing the block-diagonal structure. The second method proved to be successful, provided a proper division of the data period into orbital arcs that are not too long. Received: 28 April 2000 / Accepted: 6 November 2000     

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。     

Calibrating the GOCE accelerations with star sensor data and a global gravity field model

A reliable and accurate gradiometer calibration is essential for the scientific return of the gravity field and steady-state ocean circulation explorer (GOCE) mission. This paper describes a new method for external calibration of the GOCE gradiometer accelerations. A global gravity field model in combination with star sensor quaternions is used to compute reference differential accelerations, which may be used to estimate various combinations of gradiometer scale factors, internal gradiometer misalignments and misalignments between star sensor and gradiometer. In many aspects, the new method is complementary to the GOCE in-flight calibration. In contrast to the in-flight calibration, which requires a satellite-shaking phase, the new method uses data from the nominal measurement phases. The results of a simulation study show that gradiometer scale factors can be estimated on a weekly basis with accuracies better than 2 × 10⁻³ for the ultrasensitive and 10⁻² for the less sensitive axes, which is compatible with the requirements of the gravity gradient error. Based on a 58-day data set, scale factors are found that can reduce the errors of the in-flight-calibrated measurements. The elements of the complete inverse calibration matrix, representing both the internal gradiometer misalignments and scale factors, can be estimated with accuracies in general better than 10⁻³.     

Comparison of present SST gravity field models

IntroductionSince 2000 , with the launch of CHAMP satellite,several series of high-accuracy and high-resolutionstatic Earth’s gravityfield models have been createdbased on aboundent SST data. With these models ,the research in solid geophysics ,oceangraphy,andgeodesy can be promoted greatly[1].In this paper ,the SSTgravity models’accura-cyin various frequently domainis studied.First-ly,the difference among these models is compu-ted and compared,and then their accuracyis an-alyzed. Fina…     

Comparison of present SST gravity field models

Taking the main land of Europe as the region to be studied, the potential of the new satellite gravity technique: satellite-to-satellite tracking (SST) and improving the accuracy of regional gravity field model with the SST models are investigated. The drawbacks of these models are discussed. With GPM98C as the reference, the gravity anomaly residuals of several other models, the latest SST global gravity field models (EIGEN series and GGM series), were computed and compared. The results of the comparison show that in the selected region, some systematic errors with periodical properties exist in the EIGEN and GGM's S series models in the high degree and order. Some information that was not shown in the classic gravity models is detected in the low and middle degree and order of EIGEN and GGM's S series models. At last, the effective maximum degrees and orders of SST models are suggested.     

A parametric study on the impact of satellite attitude errors on GOCE gravity field recovery

In the recent design of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite mission, the gravity gradients are defined in the gradiometer reference frame (GRF), which deviates from the actual flight direction (local orbit reference frame, LORF) by up to 3–4°. The main objective of this paper is to investigate the effect of uncertainties in the knowledge of the gradiometer orientation due to attitude reconstitution errors on the gravity field solution. In the framework of several numerical simulations, which are based on a realistic mission configuration, different scenarios are investigated, to provide the accuracy requirements of the orientation information. It turns out that orientation errors have to be seriously considered, because they may represent a significant error component of the gravity field solution. While in a realistic mission scenario (colored gradiometer noise) the gravity field solutions are quite insensitive to small orientation biases, random noise applied to the attitude information can have a considerable impact on the accuracy of the resolved gravity field models.