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     

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.     

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     

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.     

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.     

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     

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.     

Variations in the accuracy of gravity recovery due to ground track variability: GRACE,CHAMP, and GOCE

Following an earlier recognition of degraded monthly geopotential recovery from GRACE (Gravity Recovery And Climate Experiment) due to prolonged passage through a short repeat (low order resonant) orbit, we extend these insights also to CHAMP (CHAllenging Minisatellite Payload) and GOCE (Gravity field and steady state Ocean Circulation Explorer). We show wide track-density variations over time for these orbits in both latitude and longitude, and estimate that geopotential recovery will be as widely affected as well within all these regimes, with lesser track density leading to poorer recoveries. We then use recent models of atmospheric density to estimate the future orbit of GRACE and warn of degraded performance as other low order resonances are encountered in GRACE’s free fall. Finally implications for the GOCE orbit are discussed.     

Improvement in global gravity field recovery using GFZ-1 satellite laser tracking data

The passive satellite GFZ-1 has been orbiting the Earth since April 1995. The purpose of this mission is to improve the current knowledge of the Earth's gravity field by analysing gravitational orbit perturbations observed at unique low altitudes, below 400 km. GFZ-1 is one target of the international satellite laser ranging ground network. An evaluation of the first 30 months of GFZ-1 laser tracking data led to a new version of the global GRIM4-S4 satellite-only gravity field model: GRIM4-S4G. Information was obtained from GFZ-1 data for spherical harmonic coefficients up to degree 100, which was not possible in any earlier satellite-only gravity field solution. GFZ-1's contribution to a global 5 × 5° geoid and gravity field representations is moderate but visible with a 1 cm and 0.1 mGal gain in accuracy on a level of 75 cm and 5 mGal, respectively. Received: 10 November 1998 / Accepted: 19 April 1999     

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.     

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。     

Accuracy Analysis of Gravity Model Based on Fictitious Compress Recovery Approach

The potential field determined based on the fictitious compress recovery approach is influenced by the errors contained in the boundary (the Earth’s surface or the surface corresponding to the satellite altitude) values. Given the boundary value with definite accuracy, the accuracy of the field determined based on the fictitious compress recovery approach is estimated, and it is theoretically shown that the determined field has the same accuracy level as the given boundary value.     

基于SLR的GRACE卫星定轨中重力场模型对轨道精度的影响

以GRACE卫星为例,分析比较利用SLR观测资料进行卫星定轨时,采用不同重力场模型对GRACE卫星定轨精度的影响;以及重力场截断阶引起的积分轨道差异;同时,将定轨结果与采用GPS确定的定轨结果进行比较,分析与GPS定轨结果的差异.实验证明.重力场模型选择GGM02C的定轨结果优于选择JGM-3的定轨结果,基于SLR的定...     

Efficient paddy field mapping using Landsat-8 imagery and object-based image analysis based on advanced fractel net evolution approach

This paper proposes an efficient paddy field mapping method using object-based image analysis and a bitemporal data set acquired by Landsat-8 Operational Land Imager. In the proposed approach, image segmentation is the first step and its quality has a serious impact on the accuracy of paddy field classification. In order to improve segmentation quality, a new segmentation algorithm based on a frequently used method, fractal net evolution approach, is developed, with improvement mainly in merging criteria. In order to automate the process of scale parameter determination, an unsupervised scale selection method is utilized to determine the optimal scale parameter for the proposed image segmentation approach. After segmentation, four types of object-based features including geometric, spectral, textural, and contextual information are extracted and input into the subsequent classification procedure. By using a random forest classifier, paddy fields and nonpaddy fields are separated. The proposed image segmentation method and the final classification result are both quantitatively evaluated. Our segmentation method outperformed two popular algorithms according to three supervised evaluation criteria. The classification result with overall accuracy of 91.00% and kappa statistic of 0.82 validated the effectiveness of the proposed framework. Further analysis on feature importance indicated that spectral features made the most contribution as compared to the other three types of object-based features.