GOCE gravitational gradiometry

GOCE is the first gravitational gradiometry satellite mission. Gravitational gradiometry is the measurement of the second derivatives of the gravitational potential. The nine derivatives form a 3 × 3 matrix, which in geodesy is referred to as Marussi tensor. From the basic properties of the gravitational field, it follows that the matrix is symmetric and trace free. The latter property corresponds to Laplace equation, which gives the theoretical foundation of its representation in terms of spherical harmonic or Fourier series. At the same time, it provides the most powerful quality check of the actual measured gradients. GOCE gradiometry is based on the principle of differential accelerometry. As the satellite carries out a rotational motion in space, the accelerometer differences contain angular effects that must be removed. The GOCE gradiometer provides the components V _xx , V _yy , V _zz and V _xz with high precision, while the components V _xy and V _yz are of low precision, all expressed in the gradiometer reference frame. The best performance is achieved inside the measurement band from 5 × 10^–3 to 0.1 Hz. At lower frequencies, the noise increases with 1/f and is superimposed by cyclic distortions, which are modulated from the orbit and attitude motion into the gradient measurements. Global maps with the individual components show typical patterns related to topographic and tectonic features. The maps are separated into those for ascending and those for descending tracks as the components are expressed in the instrument frame. All results are derived from the measurements of the period from November to December 2009. While the components V _xx and V _yy reach a noise level of about \({10\;\rm{\frac{mE}{\sqrt{Hz}}}}\), that of V _zz and V _xz is about \({20\; \rm{\frac{mE}{\sqrt{Hz}}}}\). The cause of the latter’s higher noise is not yet understood. This is also the reason why the deviation from the Laplace condition is at the \({20 \;\rm{\frac{mE}{\sqrt{Hz}}}}\) level instead of the originally planned \({11\;\rm{\frac{mE}{\sqrt{Hz}}}}\). Each additional measurement cycle will improve the accuracy and to a smaller extent also the resolution of the spherical harmonic coefficients derived from the measured gradients.     

GOCE level 1b data processing

In this article, the processing steps applied to the raw GOCE science payload instrument data (level 0) in order to obtain input data for the gravity field determination (level 1b) are described. The raw gradiometer measurements, which are given at the level of control voltages, have to be transformed into accelerations and gradients. For the latter step, knowledge about the GOCE attitude is required, which is provided by the star trackers. In addition, the data of the satellite to satellite tracking instrument are used to date the measurements, after its clock error has been corrected. All intermediate steps of the processing flow are described. Together with the explanation of the processing flow, an overview of the main level 1b products is given. The final part of the article discusses the means of quality control of the L1b data currently used and gives an outlook on potential processor evolutions.     

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。     

GOCE卫星引力梯度数据滤波方法研究

针对GOCE卫星引力梯度观测值中低精度分量和低频有色噪声的处理策略问题,该文采用模型模拟值代替低精度分量Vxy和Vyz,以减弱低精度分量在坐标系转换中对高精度分量的影响。深入分析比较了多种滤波方法处理GOCE卫星引力梯度观测值中有色噪声的效果,提出采用Butterworth零相移滤波方法加移去-恢复技术的思路,实测数据的处理效果验证了该方法的有效性。     

Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues

The issue of optimal regularization is investigated in the context of the processing of satellite gravity gradiometry (SGG) data that will be acquired by the GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) satellite. These data are considered as the input for determination of the Earths gravity field in the form of a series of spherical harmonics. Exploitation of a recently developed fast processing algorithm allowed a very realistic setup of the numerical experiments to be specified, in particular: a non-repeat orbit; 1-s sampling rate; half-year duration of data series; and maximum degree and order set to 300. The first goal of the study is to compare different regularization techniques (regularization matrices). The conclusion is that the first-order Tikhonov regularization matrix (the elements are practically proportional to the degree squared) and the Kaula regularization matrix (the elements are proportional to the fourth power of the degree) are somewhat superior to other regularization techniques. The second goal is to assess the generalized cross-validation method for the selection of the regularization parameter. The inference is that the regularization parameter found this way is very reasonable. The time expenditure required by the generalized cross-validation method remains modest even when a half-year set of SGG data is considered. The numerical study also allows conclusions to be drawn regarding the quality of the Earths gravity field model that can be obtained from the GOCE SGG data. In particular, it is shown that the cumulative geoid height error between degrees 31 and 200 will not exceed 1 cm. AcknowledgmentsThe authors thank Dr. E. Schrama for valuable discussions and for computing the orbit used to generate the long data set. They are also grateful to Prof. Tscherning and two anonymous reviewers for numerous valuable remarks and suggestions. The orbit to generate the short data set was kindly provided by J. van den IJssel. Computing resources were provided by Stichting Nationale Computerfaciliteiten (NCF), grant SG-027.     

Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis

Collocation is widely used in physical geodesy. Its application requires to solve systems with a dimension equal to the number of observations, causing numerical problems when many observations are available. To overcome this drawback, tailored step-wise techniques are usually applied. An example of these step-wise techniques is the space-wise approach to the GOCE mission data processing. The original idea of this approach was to implement a two-step procedure, which consists of first predicting gridded values at satellite altitude by collocation and then deriving the geo-potential spherical harmonic coefficients by numerical integration. The idea was generalized to a multi-step iterative procedure by introducing a time-wise Wiener filter to reduce the highly correlated observation noise. Recent studies have shown how to optimize the original two-step procedure, while the theoretical optimization of the full multi-step procedure is investigated in this work. An iterative operator is derived so that the final estimated spherical harmonic coefficients are optimal with respect to the Wiener–Kolmogorov principle, as if they were estimated by a direct collocation. The logical scheme used to derive this optimal operator can be applied not only in the case of the space-wise approach but, in general, for any case of step-wise collocation. Several numerical tests based on simulated realistic GOCE data are performed. The results show that adding a pre-processing time-wise filter to the two-step procedure of data gridding and spherical harmonic analysis is useful, in the sense that the accuracy of the estimated geo-potential coefficients is improved. This happens because, in its practical implementation, the gridding is made by collocation over local patches of data, while the observation noise has a time-correlation so long that it cannot be treated inside the patch size. Therefore, the multi-step operator, which is in theory equivalent to the two-step operator and to the direct collocation, is in practice superior thanks to the time-wise filter that reduces the noise correlation before the gridding. The criteria for the choice of this filter are investigated numerically.     

Satellite gradiometry using a satellite pair

The GRACE mission has substantiated the low–low satellite-to-satellite tracking (LL-SST) concept. The LL-SST configuration can be combined with the previously realized high–low SST concept in the CHAMP mission to provide a much higher accuracy. The line of sight (LOS) acceleration difference between the GRACE satellite pair, the simplest form of the combined observable, is mostly used for mapping the global gravity field of the Earth in terms of spherical harmonic coefficients. As an alternative observable, a linear combination of the gravitational gradient tensor components is proposed. Being a one-point function and having a direct relation with the field geometry (curvature of the field at the point) are two noteworthy achievements of the alternative formulation. In addition, using an observation quantity that is related to the second-instead of the first-order derivatives of the gravitational potential amplifies the high-frequency part of the signal. Since the transition from the first- to the second-order derivatives includes the application of a finite-differences scheme, the high-frequency part of the noise is also amplified. Nevertheless, due to the different spectral behaviour of signal and noise, in the end the second-order approach leads to improved gravitational field resolution. Mathematical formulae for the gradiometry approach, for both linear and higher-degree approximations, are derived. The proposed approach is implemented for recovery of the global gravitational field and the results are compared with those of LOS acceleration differences. Moreover, LOS acceleration difference residuals are calculated, which are at the level of a few tenths of mGal. Error analysis shows that the residuals of the estimated degree variances are less than 10^–3. Furthermore, the gravity anomaly residuals are less than 2 mGal for most points on the Earth.     

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.     

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     

Alternative method for angular rate determination within the GOCE gradiometer processing

The most crucial part of the GOCE gradiometer processing is, besides the internal calibration of the gradiometer, the determination of the satellite’s inertial angular rate. This paper describes a new method for the angular rate determination. It is based on the stochastic properties of the GOCE star sensors and the gradiometer. The attitude information of both instrument types is combined at the level of angular rates. The combination is done in the spectral domain by Wiener filtering, and thus using an optimal relative weighting of the star sensor and gradiometer attitude information. Since the complete processing chain from raw measurements to gravity field solutions is performed, the results are not only analyzed at the level of gravity gradients, but also of gravity field solutions. Compared to the nominal method, already the resulting gravity gradients show a significantly improved performance for the frequencies (mainly) below the gradiometer measurement bandwidth. This can be verified by analysis of the gravity gradient trace. The improvement is propagated to the level of gravity field models, where a better accuracy can be observed for selected groups of coefficients at characteristic bands at orders k × 16, with integer k, up to high harmonic degrees.     

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.     

Monitoring GOCE gradiometer calibration parameters using accelerometer and star sensor data: methodology and first results

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite, launched on 17 March 2009, is designed to measure the Earth’s mean gravity field with unprecedented accuracy at spatial resolutions down to 100?km. The accurate calibration of the gravity gradiometer on-board GOCE is of utmost importance for achieving the mission goals. ESA’s baseline method for the calibration uses star sensor and accelerometer data of a dedicated calibration procedure, which is executed every 2?months. In this paper, we describe a method for monitoring the evolution of calibration parameter during that time. The method works with star sensor and accelerometer data and does not require gravity field models, which distinguishes it from other existing methods. We present time series of calibration parameters estimated from GOCE data from 1 November 2009 to 17 May 2010. The time series confirm drifts in the calibration parameters that are present in the results of other methods, including ESA’s baseline method. Although these drifts are very small, they degrade the gravity gradients, leading to the conclusion that the calibration parameters of the ESA’s baseline method need to be linearly interpolated. Further, we find a correction of ?36 × 10^?6 for one calibration parameter (in-line differential scale factor of the cross-track gradiometer arm), which improves the gravity gradient performance. The results are validated by investigating the trace of the calibrated gravity gradients and comparing calibrated gravity gradients with reference gradients computed along the GOCE orbit using the ITG-Grace-2010s gravity field model.     

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。     

A unified approach to post-mission processing of intertial data

The outpout of inertial survey systems is available to the user in two basic forms: as Kalman filtered information at updates or as integrated velocity and position information at regular time intervals. In case of the second data type, the post-mission processing starts with the approximation of the velocity error curve. This approximation is either based on a system error model as in Kalman filtering or uses curve fitting techniques. From there on, smoothing or adjustment procedures are used as further steps in the post-mission treatment of both data types. A unified treatment of the various post-mission approaches starts with the formulation of appropriate error models of the system outputs. It is then possible to present all existing methods as intermediate steps of a rigorous adjustment procedure. This unified approach gives insight into the limitations of individual methods and provides a means to detect inconsistencies in post-mission processing strategies. An analysis of existing approaches is made and a new method, spectral decomposition, is treated in detail. Compared to the existing procedures, it has advantages with respect to a rigorous covariance propagation and blunder detection. Paper presented at the 18^th. General Assembly of the IAG, Symposium d: The Future of Terrestrial and Space Methods for Positioning, Hamburg, August 15–27, 1983.     

A unified approach to post-mission processing of intertial data

The outpout of inertial survey systems is available to the user in two basic forms: as Kalman filtered information at updates or as integrated velocity and position information at regular time intervals. In case of the second data type, the post-mission processing starts with the approximation of the velocity error curve. This approximation is either based on a system error model as in Kalman filtering or uses curve fitting techniques. From there on, smoothing or adjustment procedures are used as further steps in the post-mission treatment of both data types. A unified treatment of the various post-mission approaches starts with the formulation of appropriate error models of the system outputs. It is then possible to present all existing methods as intermediate steps of a rigorous adjustment procedure. This unified approach gives insight into the limitations of individual methods and provides a means to detect inconsistencies in post-mission processing strategies. An analysis of existing approaches is made and a new method, spectral decomposition, is treated in detail. Compared to the existing procedures, it has advantages with respect to a rigorous covariance propagation and blunder detection.     

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.     

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.     

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.