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.     

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.     

SGG重力场球谐系数正则解的误差估计方法

重力梯度为重力位的二阶导数,可以通过星载梯度仪进行观测。重力场球谐函数系数可以通过正则化方法由重力梯度算出。本文在对正则化方法分析的基础上提出了估计球谐函数系数正则解误差的方法,为我国今后发射重力梯度卫星提供技术准备。     

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     

虚拟压缩恢复法在物理大地测量学中的应用(英文)

The fictitious compress recovery approach is introduced, which could be applied to the establishment of the Rungerarup theorem, the determination of the Bjerhammar＇s fictitious gravity anomaly, the solution of the ＂downward con- tinuation＂ problem of the gravity field, the confirmation of the convergence of the spherical harmonic expansion series of the Earth＇s potential field, and the gravity field determination in three cases： gravitational potential case, gravitation case, and gravitational gradient case. Several tests using simulation experiments show that the fictitious compress recovery approach shows promise in physical geodesy applications.     

Applications of an orbiting gravity gradiometer

Considering present attempts to develop a gradiometer with an accuracy between 10⁻³ E and 10⁻⁴ E, two applications for such a device have been studied: (a) mapping the gravitational field of the Earth, and (b) estimating the geocentric distance of a satellite carrying the instrument. Given a certain power spectrum for the signal and 10⁻⁴ E (rms) of white measurement noise, the results of an error analysis indicate that a six-month mission in polar orbit at a height of 200 km, with samples taken every three seconds, should provide data for estimating the spherical harmonic potential coefficients up to degree and order 300 with less than 50% error, and improve the coefficients through degree 30 by up to four orders of magnitude compared to existing models. A simulation study based on numerical orbit integrations suggests that a simple adjustment of the initial conditions based on gradiometer data could produce orbits where the geocentric distance is accurate to 10 cm or better, provided the orbits are 2000 km high and some improvement in the gravity field up to degree 30 is first achieved. In this sense, the gravity-mapping capability of the gradiometer complements its use in orbit refinement. This idea can be of use in determining orbits for satellite altimetry. Furthermore, by tracking the gradiometer-carrying spacecraft when it passes nearly above a terrestrial station, the geocentric distance of this station can also be estimated to about one decimeter accuracy. This principle could be used in combination with VLBI and other modern methods to set up a world-wide 3-D network of high accuracy.     

利用卫星引力梯度确定地球重力场的张量不变方法研究

应用张量不变理论对利用卫星重力梯度数据确定地球重力场的方法进行了研究,对张量不变观测方程的线性化处理、非全张量观测值的数据处理策略以及采用白噪声特性下的梯度观测值恢复地球重力场的精度等进行了数值分析.结果表明,张量不变解实现了不同观测值的联合求解,基于先验重力场模型的线性化方法在实现张量不变观测模型线性化处理的同时,提...     

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).     

Harmonic analysis of the Earth's gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP

An algorithm for the determination of the spherical harmonic coefficients of the terrestrial gravitational field representation from the analysis of a kinematic orbit solution of a low earth orbiting GPS-tracked satellite is presented and examined. A gain in accuracy is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high precision, in the range of a few centimeters. In particular, advantage is taken of Newton's Law of Motion, which balances the acceleration vector with respect to an inertial frame of reference (IRF) and the gradient of the gravitational potential. By means of triple differences, and in particular higher-order differences (seven-point scheme, nine-point scheme), based upon Newton's interpolation formula, the local acceleration vector is estimated from relative GPS position time series. The gradient of the gravitational potential is conventionally given in a body-fixed frame of reference (BRF) where it is nearly time independent or stationary. Accordingly, the gradient of the gravitational potential has to be transformed from spherical BRF to Cartesian IRF. Such a transformation is possible by differentiating the gravitational potential, given as a spherical harmonics series expansion, with respect to Cartesian coordinates by means of the chain rule, and expressing zero- and first-order Ferrer's associated Legendre functions in terms of Cartesian coordinates. Subsequently, the BRF Cartesian coordinates are transformed into IRF Cartesian coordinates by means of the polar motion matrix, the precession–nutation matrices and the Greenwich sidereal time angle (GAST). In such a way a spherical harmonic representation of the terrestrial gravitational field intensity with respect to an IRF is achieved. Numerical tests of a resulting Gauss–Markov model document not only the quality and the high resolution of such a space gravity spectroscopy, but also the problems resulting from noise amplification in the acceleration determination process.     

国际卫星重力梯度测量计划研究进展

本文首先阐述了重力梯度测量原理、从20世纪初到21世纪初重力梯度仪的研究历程、卫星重力梯度仪(静电悬浮重力梯度仪、超导重力梯度仪和量子重力梯度仪)的技术特征以及卫星重力梯度测量的特点;其次,介绍了基于卫星重力梯度技术恢复250阶GOCE地球重力场以及论证首先开展一维径向重力梯度仪的研制进而恢复高精度和高空间解析度中高频地球重力场可行性方面的研究进展;最后,建议我国尽早开展基于时空域混合法解算中高频地球重力场和卫星重力梯度测量系统误差分析的预先研究。     

Energy integral method for gravity field determination from satellite orbit coordinates

A fast iterative method for gravity field determination from low Earth satellite orbit coordinates has been developed and implemented successfully. The method is based on energy conservation and avoids problems related to orbit dynamics and initial state. In addition, the particular geometry of a repeat orbit is exploited by using a very efficient iterative estimation scheme, in which a set of normal equations is approximated by a sparse block-diagonal equivalent. Recovery experiments for spherical harmonic gravity field models up to degree and order 80 and 120 were conducted based on a 29-day simulated data set of orbit coordinates. The method was found to be very flexible and could be easily adapted to include observations of non-conservative accelerations, such as (to be) provided by satellites like CHAMP, GRACE, and GOCE. A serious drawback of the method is its large sensitivity to satellite velocity errors. Existing orbit determination strategies need to be altered or augmented to include algorithms that focus on optimizing the accuracy of estimated velocities.     

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     

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.     

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.     

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.     

扰动重力梯度的球冠谐分析建模

从球冠谐理论出发,详细推导了球冠坐标系下扰动重力梯度的无奇异性计算公式。基于Tikhonov正则化方法,利用GOCE卫星实际观测数据解算局部重力场球冠谐模型。数值计算表明,基于扰动重力梯度的球冠谐分析建模方法能够有效地恢复局部重力场中的短波信号,与GO_CONS_GCF_2_DIR_R5模型的差异在±0.3×10~(-5) m/s~2水平。     

ITG-CHAMP01: a CHAMP gravity field model from short kinematic arcs over a one-year observation period

Global gravity field models have been determined based on kinematic orbits covering an observation period of one year beginning from March 2002. Three different models have been derived up to a maximum degree of n=90 of a spherical harmonic expansion of the gravitational potential. One version, ITG-CHAMP01E, has been regularized beginning from degree n=40 upwards, based on the potential coefficients of the gravity field model EGM96. A second model, ITG-CHAMP01K, has been determined based on Kaulas rule of thumb, also beginning from degree n=40. A third version, ITG-CHAMP01S, has been determined without any regularization. The physical model of the gravity field recovery technique is based on Newtons equation of motion, formulated as a boundary value problem in the form of a Fredholm-type integral equation. The observation equations are formulated in the space domain by dividing the one-year orbit into short sections of approximately 30-minute arcs. For every short arc, a variance factor has been determined by an iterative computation procedure. The three gravity field models have been validated based on various criteria, and demonstrate the quality of not only the gravity field recovery technique but also the kinematically determined orbits.     

Precise orbit determination for the GRACE mission using only GPS data

The GRACE (gravity recovery and climate experiment) satellites, launched in March 2002, are each equipped with a BlackJack GPS onboard receiver for precise orbit determination and gravity field recovery. Since launch, there have been significant improvements in the background force models used for satellite orbit determination, most notably the model for the geopotential. This has resulted in significant improvements to orbit accuracy for very low altitude satellites. The purpose of this paper is to investigate how well the orbits of the GRACE satellites (about 470 km in altitude) can currently be determined using only GPS data and based on the current models and methods. The orbit accuracy is assessed using a number of tests, which include analysis of orbit fits, orbit overlaps, orbit connecting points, satellite Laser ranging residuals and K-band ranging (KBR) residuals. We show that 1-cm radial orbit accuracy for the GRACE satellites has probably been achieved. These precise GRACE orbits can be used for such purposes as improving gravity recovery from the GRACE KBR data and for atmospheric profiling, and they demonstrate the quality of the background force models being used.     

多种跟踪组合卫星重力场恢复方法初探

将卫星地面跟踪、高低卫星跟踪和低低卫星跟踪恢复重力场的原理进行统一 ,导出了两种能充分综合多种跟踪观测的卫星重力场恢复基本观测方程。在建立基本观测量与重力场参数的函数传输关系的基础上 ,对目前存在的重力场恢复算法给出了直观的解释 ,并以距离二次变率为重力场恢复的基本观测量 ,提出了一种能组合各种卫星跟踪观测量的动态 /动力组合定轨和重力场恢复方法。