Filter design for GOCE gravity gradients

The GOCE satellite observes gravity gradients with unprecedented accuracy and resolution. The GOCE observations are reliable within a well-defined measurement bandwidth. In this study, different finite and infinite impulse response filters have been designed to obtain the demanded pass. Exhaustive time and frequency domain investigations prove that the proposed infinite impulse response filter can be a real competitor of the existing solution of the filtering problem.     

GOCE引力梯度内部校准方法

GOCE卫星引力梯度仪的精确校准是反演高精度重力场的前提之一,本文利用GOCE卫星L1b数据中的引力梯度仪及恒星敏感器数据实现了卫星引力梯度的内部校准。以最小二乘联合多个恒星敏感器观测数据确定内部校准使用的角速度,有效避免了单个恒星敏感器低精度角速度分量对坐标转换过程的影响。考虑到恒星敏感器坐标系与梯度仪坐标系间旋转矩阵随时间的变化,本文在ESA官方内部校准方法的基础上,提出了顾及旋转矩阵校准参数的内部校准模型,并利用2009年11月的GOCE实测数据验证了该方法的效果。结果表明,该旋转矩阵校准参数数值约100″,且在该月存在3″~30″的漂移;与GOCE官方内部校准方法对比,从卫星引力梯度精度结果来看,在低于0.005 Hz频段内,同时解算旋转矩阵的校准参数与梯度仪内3个加速度计对的校准参数的内部校准模型优于仅考虑加速度计对校准参数的模型;除此之外,本文讨论了以该模型为基础的GOCE梯度仪数据校准的可能方法,为GOCE及后续重力卫星的数据处理工作提供参考。     

Preprocessing of gravity gradients at the GOCE high-level processing facility

One of the products derived from the gravity field and steady-state ocean circulation explorer (GOCE) observations are the gravity gradients. These gravity gradients are provided in the gradiometer reference frame (GRF) and are calibrated in-flight using satellite shaking and star sensor data. To use these gravity gradients for application in Earth scienes and gravity field analysis, additional preprocessing needs to be done, including corrections for temporal gravity field signals to isolate the static gravity field part, screening for outliers, calibration by comparison with existing external gravity field information and error assessment. The temporal gravity gradient corrections consist of tidal and nontidal corrections. These are all generally below the gravity gradient error level, which is predicted to show a 1/f behaviour for low frequencies. In the outlier detection, the 1/f error is compensated for by subtracting a local median from the data, while the data error is assessed using the median absolute deviation. The local median acts as a high-pass filter and it is robust as is the median absolute deviation. Three different methods have been implemented for the calibration of the gravity gradients. All three methods use a high-pass filter to compensate for the 1/f gravity gradient error. The baseline method uses state-of-the-art global gravity field models and the most accurate results are obtained if star sensor misalignments are estimated along with the calibration parameters. A second calibration method uses GOCE GPS data to estimate a low-degree gravity field model as well as gravity gradient scale factors. Both methods allow to estimate gravity gradient scale factors down to the 10⁻³ level. The third calibration method uses high accurate terrestrial gravity data in selected regions to validate the gravity gradient scale factors, focussing on the measurement band. Gravity gradient scale factors may be estimated down to the 10⁻² level with this method.     

GOCE gravitational gradients along the orbit

GOCE is ESA’s gravity field mission and the first satellite ever that measures gravitational gradients in space, that is, the second spatial derivatives of the Earth’s gravitational potential. The goal is to determine the Earth’s mean gravitational field with unprecedented accuracy at spatial resolutions down to 100 km. GOCE carries a gravity gradiometer that allows deriving the gravitational gradients with very high precision to achieve this goal. There are two types of GOCE Level 2 gravitational gradients (GGs) along the orbit: the gravitational gradients in the gradiometer reference frame (GRF) and the gravitational gradients in the local north oriented frame (LNOF) derived from the GGs in the GRF by point-wise rotation. Because the V _XX , V _YY , V _ZZ and V _XZ are much more accurate than V _XY and V _YZ , and because the error of the accurate GGs increases for low frequencies, the rotation requires that part of the measured GG signal is replaced by model signal. However, the actual quality of the gradients in GRF and LNOF needs to be assessed. We analysed the outliers in the GGs, validated the GGs in the GRF using independent gravity field information and compared their assessed error with the requirements. In addition, we compared the GGs in the LNOF with state-of-the-art global gravity field models and determined the model contribution to the rotated GGs. We found that the percentage of detected outliers is below 0.1% for all GGs, and external gravity data confirm that the GG scale factors do not differ from one down to the 10⁻³ level. Furthermore, we found that the error of V _XX and V _YY is approximately at the level of the requirement on the gravitational gradient trace, whereas the V _ZZ error is a factor of 2–3 above the requirement for higher frequencies. We show that the model contribution in the rotated GGs is 2–35% dependent on the gravitational gradient. Finally, we found that GOCE gravitational gradients and gradients derived from EIGEN-5C and EGM2008 are consistent over the oceans, but that over the continents the consistency may be less, especially in areas with poor terrestrial gravity data. All in all, our analyses show that the quality of the GOCE gravitational gradients is good and that with this type of data valuable new gravity field information is obtained.     

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     

Regularization of gravity field estimation from satellite gravity gradients

 The performance of the L-curve criterion and of the generalized cross-validation (GCV) method for the Tikhonov regularization of the ill-conditioned normal equations associated with the determination of the gravity field from satellite gravity gradiometry is investigated. Special attention is devoted to the computation of the corner point of the L-curve, to the numerically efficient computation of the trace term in the GCV target function, and to the choice of the norm of the residuals, which is important for the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) in the presence of colored observation noise. The trace term in the GCV target function is estimated using an unbiased minimum-variance stochastic estimator. The performance analysis is based on a simulation of gravity gradients along a 60-day repeat circular orbit and a gravity field recovery complete up to degree and order 300. Randomized GCV yields the optimal regularization parameter in all the simulations if the colored noise is properly taken into account. Moreover, it seems to be quite robust against the choice of the norm of the residuals. It performs much better than the L-curve criterion, which always yields over-smooth solutions. The numerical costs for randomized GCV are limited provided that a reasonable first guess of the regularization parameter can be found. Received: 17 May 2001 / Accepted: 17 January 2002     

First GOCE gravity field models derived by three different approaches

Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analysed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared with combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITG-Grace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-month GOCE gravity field models of 10?cm in terms of geoid heights, and 3?mGal in terms of gravity anomalies, globally at a resolution of 100?km (degree/order 200).     

利用重力梯度估计海底地形

利用重力异常估计海底地形目前是获取全球海底地形的主要方法.论文对利用重力梯度(在短波波段它比重力异常对地形更加敏感)估计海底地形的方法进行了研究.论文设计了频域和空间域两种估计方法,并利用卫星测高推导的垂向重力梯度数据对这两种方法进行了测试.     

Assessment of observing time-variable gravity from GOCE GPS and accelerometer observations

An assessment has been made of the possibility to estimate time-variable gravity from GPS-derived orbit perturbations and common-mode accelerometer observations of ESA’s GOCE Earth Explorer. A number of 20-day time series of Earth’s global long-wavelength gravity field have been derived for the period November 2009 to November 2012 using different parameter setups and estimation techniques. These techniques include a conventional approach where for each period, one set of gravity coefficients is estimated, either excluding or including empirical accelerations, and the so-called Wiese approach where higher frequency coefficients are estimated for the very long wavelengths. A principal component analysis of especially the time series of gravity field coefficients obtained by the Wiese approach and the conventional approach with empirical accelerations reveals an annual signal. When fitting this annual signal directly through the time series, the sine component (maximum in spring) displays features that are similar to well-known continental hydrological mass changes for the low latitude areas, such as mass variations in the Amazon basin, Africa and Australia for spatial scales down to 1,500 km. The cosine component (maximum in winter), however, displays large signals that can not be attributed to actual mass variations in the Earth system. The estimated gravity field changes from GOCE orbit perturbations are likely affected by missing GPS observations in case of high ionospheric perturbations during periods of increased solar activity, which is minimal in Summer and maximal towards the end of autumn.     

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.     

GPS-based precise orbit determination of the very low Earth-orbiting gravity mission GOCE

 A prerequisite for the success of future gravity missions like the European Gravity field and steady-state Ocean Circulation Explorer (GOCE) is a precise orbit determination (POD). A detailed simulation study has been carried out to assess the achievable orbit accuracy based on satellite-to-satellite tracking (SST) by the US global positioning system (GPS) and in conjunction the implications for gravity field determination. An orbit accuracy at the few centimeter level seems possible, sufficient to support the GOCE gravity mission and in particular its gravity gradiometer. Received: 21 January 2000 / Accepted: 4 July 2000     

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.     

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.     

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...     

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).     

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