Marine gravity and geoid determination by optimal combination of satellite altimetry and shipborne gravimetry data

. Satellite altimetry derived geoid heights and marine gravity anomalies can be combined to determine a detailed gravity field over the oceans using the least-squares collocation method and spectral combination techniques. Least-squares collocation, least-squares adjustment in the frequency domain and input-output system theory are employed to determine the gravity field (both geoid and anomalies) and its errors. This paper intercompares these three techniques using simulated data. Simulation studies show that best results are obtained by the input-output system theory among the three prediction methods. The least-squares collocation method gives results which are very close to but a little bit worse than those obtained using input-output system theory. This slightly poorer performance of the least-squares collocation method can be explained by the fact that it uses isotropic structured covariance (thus approximate signal PSD information) while the system theory method uses detailed signal PSD information. The method of least-squares adjustment in the frequency domain gives the poorest results among these three methods because it uses less information than the other two methods (it ignores the signal PSDs). The computations also show that the least-squares collocation and input-output system theory methods are not as sensitive to noise levels as the least-squares adjustment in the frequency domain method is. Received 19 January 1996; Accepted 17 July 1996     

利用重力异常数据构建重力梯度场的方法比较

为实现大范围、高精度基准重力梯度数据库的构建,考虑到重力梯度场对地形质量的敏感效应,一般利用恒密度数字高程模型来求取重力梯度值,从而忽略了地形密度变化以及水准面以下密度异常对重力梯度的影响。根据重力位理论中求解边值问题的数值应用方法,直接利用重力异常数据求取重力梯度场,弥补了密度变化和密度异常在重力梯度上的反映。根据模型算例和实测重力异常数据求取了剖面重力梯度值,结果表明,限于重力数据空间分辨率的影响,利用重力异常数据可恢复中长波段重力梯度场。该方法与地形数据求取重力梯度和卫星重力梯度测量等方法技术相结合,对重力梯度数据库的建设具有实际应用价值。     

On the combination of high-resolution and satellite-only global gravity models

The issue of combining high-resolution gravity models, based on observations taken on the Earth surface, with those derived from satellite-only observations is of increasing importance, due to the new data provided by gravity satellite missions, CHAMP, GRACE and GOCE. The paper addresses this issue with a twofold purpose. On the one hand, it is an attempt to discuss and assess general concepts, well known in literature, such as achievable resolution, regularization in the least-squares sense or in an infinite dimensional setup, combination criteria, symmetry and block diagonal structures. In particular, as for the symmetry question, a well-defined result, generalizing known facts, is derived. On the other hand, the outcomes of the general discussion are specifically applied to the combination of a high-resolution model (e.g. EGM08) with a GOCE gravity model estimated by the so-called space-wise approach. Small numerical examples are developed to clarify the property of the proposed solution.     

The computation of aliasing effects in local gravity field approximation

The aliasing effects in local gravity field computations are presented in this study. First the relation between the power spectral density of a 2-D continuous signal and its corresponding sampled version is derived. Then the power spectral density of the aliasing errors related to non band-limited signals is derived. Finally the variance of these aliasing errors is computed using gravity anomalies at different grid spacings. This computation prerequires some known gravity anomaly power spectral density model. The model used in this study corresponds to a second-order Gauss-Markov covariance function for the anomalous potential. Editor’s notice: Comments on this paper will follow in the next issue of Bulletin Géodésique.     

On the error of analytical continuation in physical geodesy

Analytical continuation of gravity anomalies and height anomalies is compared with Helmert's second condensation method. Assuming that the density of the terrain is constant and known the latter method can be regarded as correct. All solutions are limited to the second power of H/R, where H is the orthometric height of the terrain and R is mean sea-level radius. We conclude that the prediction of free-air anomalies and height anomalies by analytical continuation with Poisson's formula and Stokes's formula goes without error. Applying the same technique for geoid determination yields an error of the order of H², stemming from the failure of analytical continuation inside the masses of the Earth.     

Quasi-stationary sea surface topography estimation by the multiple input/output method

 Multiple input/multiple output system theory (MIMOST) is briefly presented, and the application of the method to the quasi-stationary sea surface topography (QSST) estimation and the filtering of the input observations are discussed. The repeat character of satellite altimetry missions provides more than one sample of the measured sea surface height (SSH) field, and an approximation of the input signal and error power spectral densities can be determined using this successive information. A case study in the Labrador Sea is considered using SSHs from ERS1 phases C and G, ERS1-GM, ERS2 phase A and TOPEX/POSEIDON altimetric missions in combination with shipborne gravity anomalies. The time period of the observations in this study is from 1993 to 1998. Some comparisons between the techniques used for the power spectral density approximation are carried out and some remarks on the properties of the estimated QSST are presented. Received: 19 October 1999 / Accepted: 23 October 2000     

Simulation study of a follow-on gravity mission to GRACE

The gravity recovery and climate experiment (GRACE) has been providing monthly estimates of the Earth’s time-variable gravity field since its launch in March 2002. The GRACE gravity estimates are used to study temporal mass variations on global and regional scales, which are largely caused by a redistribution of water mass in the Earth system. The accuracy of the GRACE gravity fields are primarily limited by the satellite-to-satellite range-rate measurement noise, accelerometer errors, attitude errors, orbit errors, and temporal aliasing caused by un-modeled high-frequency variations in the gravity signal. Recent work by Ball Aerospace & Technologies Corp., Boulder, CO has resulted in the successful development of an interferometric laser ranging system to specifically address the limitations of the K-band microwave ranging system that provides the satellite-to-satellite measurements for the GRACE mission. Full numerical simulations are performed for several possible configurations of a GRACE Follow-On (GFO) mission to determine if a future satellite gravity recovery mission equipped with a laser ranging system will provide better estimates of time-variable gravity, thus benefiting many areas of Earth systems research. The laser ranging system improves the range-rate measurement precision to ~0.6 nm/s as compared to ~0.2 μm/s for the GRACE K-band microwave ranging instrument. Four different mission scenarios are simulated to investigate the effect of the better instrument at two different altitudes. The first pair of simulated missions is flown at GRACE altitude (~480 km) assuming on-board accelerometers with the same noise characteristics as those currently used for GRACE. The second pair of missions is flown at an altitude of ~250 km which requires a drag-free system to prevent satellite re-entry. In addition to allowing a lower satellite altitude, the drag-free system also reduces the errors associated with the accelerometer. All simulated mission scenarios assume a two satellite co-orbiting pair similar to GRACE in a near-polar, near-circular orbit. A method for local time variable gravity recovery through mass concentration blocks (mascons) is used to form simulated gravity estimates for Greenland and the Amazon region for three GFO configurations and GRACE. Simulation results show that the increased precision of the laser does not improve gravity estimation when flown with on-board accelerometers at the same altitude and spacecraft separation as GRACE, even when time-varying background models are not included. This study also shows that only modest improvement is realized for the best-case scenario (laser, low-altitude, drag-free) as compared to GRACE due to temporal aliasing errors. These errors are caused by high-frequency variations in the hydrology signal and imperfections in the atmospheric, oceanographic, and tidal models which are used to remove unwanted signal. This work concludes that applying the updated technologies alone will not immediately advance the accuracy of the gravity estimates. If the scientific objectives of a GFO mission require more accurate gravity estimates, then future work should focus on improvements in the geophysical models, and ways in which the mission design or data processing could reduce the effects of temporal aliasing.     

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.     

利用GOCE模拟观测反演重力场的Torus法

在介绍Torus方法反演地球重力场模型的基本原理和方法的基础上,基于圆环面上均匀分布的卫星引力梯度模拟观测值解算了200阶次的地球重力场模型,在无误差情况下,Torus方法解算模型的阶误差RMS小于10-16,验证了该方法的严密性。利用61dGOCE卫星轨道上无误差的模拟引力梯度观测值解算了200阶次的地球重力场模型,分析了格网化误差、极空白对解算精度的影响,迭代3次后,在不考虑低次系数情况下,模型的大地水准面阶误差和累积误差均较小,最大值仅为0.022mm和0.099mm。在沿轨卫星引力梯度模拟数据中加入5mE/Hz1/2的白噪声,基于Torus方法和空域最小二乘法解算了200阶次的地球重力场模型,Torus方法的精度略低于空域最小二乘法的精度,在不考虑低次项的情况下,两种方法解算模型的大地水准面阶误差最大值分别为1.58cm和1.45cm,累积误差最大值分别为6.37cm和5.55cm。但由于采用了二维快速傅里叶技术和块对角最小二乘法,极大地提高了计算效率。本文数值结果说明Torus方法是一种独立有效的方法,可用于GOCE任务海量卫星引力梯度观测值反演重力场的快速解算。     

Integral transformations of deflections of the vertical onto satellite-to-satellite tracking and gradiometric data

With the advent of geodetic satellite missions mapping almost globally the Earth’s gravitational field, new methods and theoretical approaches have been developed and investigated to fully exploit the potential of their new observables. Besides estimating values of numerical coefficients in harmonic series of the gravitational potential, new applications emerged such as data validation and combination. In this contribution, new integral transformations are presented which transform principal components of the terrestrial deflection of the vertical onto disturbing satellite-to-satellite tracking and gradiometric data at altitude. Using spherical approximation, necessary integral kernel functions are derived in both spectral and closed forms. The behaviour of isotropic kernel functions is studied and the new integral transformations are tested in a closed-loop simulation using synthetic terrestrial and satellite data synthesized from a global gravitational model. New integral transformations can be used for data validation and combination purposes.     

Adaptive filtering of GOCE-derived gravity gradients of the disturbing potential in the context of the space-wise approach

Filtering and signal processing techniques have been widely used in the processing of satellite gravity observations to reduce measurement noise and correlation errors. The parameters and types of filters used depend on the statistical and spectral properties of the signal under investigation. Filtering is usually applied in a non-real-time environment. The present work focuses on the implementation of an adaptive filtering technique to process satellite gravity gradiometry data for gravity field modeling. Adaptive filtering algorithms are commonly used in communication systems, noise and echo cancellation, and biomedical applications. Two independent studies have been performed to introduce adaptive signal processing techniques and test the performance of the least mean-squared (LMS) adaptive algorithm for filtering satellite measurements obtained by the gravity field and steady-state ocean circulation explorer (GOCE) mission. In the first study, a Monte Carlo simulation is performed in order to gain insights about the implementation of the LMS algorithm on data with spectral behavior close to that of real GOCE data. In the second study, the LMS algorithm is implemented on real GOCE data. Experiments are also performed to determine suitable filtering parameters. Only the four accurate components of the full GOCE gravity gradient tensor of the disturbing potential are used. The characteristics of the filtered gravity gradients are examined in the time and spectral domain. The obtained filtered GOCE gravity gradients show an agreement of 63–84 mEötvös (depending on the gravity gradient component), in terms of RMS error, when compared to the gravity gradients derived from the EGM2008 geopotential model. Spectral-domain analysis of the filtered gradients shows that the adaptive filters slightly suppress frequencies in the bandwidth of approximately 10–30 mHz. The limitations of the adaptive LMS algorithm are also discussed. The tested filtering algorithm can be connected to and employed in the first computational steps of the space-wise approach, where a time-wise Wiener filter is applied at the first stage of GOCE gravity gradient filtering. The results of this work can be extended to using other adaptive filtering algorithms, such as the recursive least-squares and recursive least-squares lattice filters.     

The height datum problem and the role of satellite gravity models

Regional height systems do not refer to a common equipotential surface, such as the geoid. They are usually referred to the mean sea level at a reference tide gauge. As mean sea level varies (by ±1 to 2 m) from place to place and from continent to continent each tide gauge has an unknown bias with respect to a common reference surface, whose determination is what the height datum problem is concerned with. This paper deals with this problem, in connection to the availability of satellite gravity missions data. Since biased heights enter into the computation of terrestrial gravity anomalies, which in turn are used for geoid determination, the biases enter as secondary or indirect effect also in such a geoid model. In contrast to terrestrial gravity anomalies, gravity and geoid models derived from satellite gravity missions, and in particular GRACE and GOCE, do not suffer from those inconsistencies. Those models can be regarded as unbiased. After a review of the mathematical formulation of the problem, the paper examines two alternative approaches to its solution. The first one compares the gravity potential coefficients in the range of degrees from 100 to 200 of an unbiased gravity field from GOCE with those of the combined model EGM2008, that in this range is affected by the height biases. This first proposal yields a solution too inaccurate to be useful. The second approach compares height anomalies derived from GNSS ellipsoidal heights and biased normal heights, with anomalies derived from an anomalous potential which combines a satellite-only model up to degree 200 and a high-resolution global model above 200. The point is to show that in this last combination the indirect effects of the height biases are negligible. To this aim, an error budget analysis is performed. The biases of the high frequency part are proved to be irrelevant, so that an accuracy of 5 cm per individual GNSS station is found. This seems to be a promising practical method to solve the problem.     

联合高低卫-卫跟踪和卫星重力梯度数据恢复地球重力场的谱组合法

GOCE采用的高低卫-卫跟踪和卫星重力梯度测量技术在恢复重力场方面各有所长并互为补充,如何有效利用这两类观测数据最优确定地球重力场是GOCE重力场反演的关键问题。本文研究了联合高低卫-卫跟踪和卫星重力梯度数据恢复地球重力场的最小二乘谱组合法,基于球谐分析方法推导并建立了卫星轨道面扰动位T和径向重力梯度Tzz、以及扰动位T和重力梯度分量组合{Tzz-Txx-Tyy}的谱组合计算模型与误差估计公式。数值模拟结果表明,谱组合计算模型可以有效顾及各类数据的精度和频谱特性进行最优联合求解。采用61天GOCE实测数据反演的两个180阶次地球重力场模型WHU_GOCE_SC01S（扰动位和径向重力梯度数据求解）和WHU_GOCE_SC02S（扰动位和重力梯度分量组合数据求解）,结果显示后者精度优于前者,并且它们的整体精度优于GOCE时域解,而与GOCE空域解的精度接近,验证了谱组合法的可行性与有效性。     

Gravity-field improvement in the Mediterranean Sea by estimating the bottom topography using collocation

The contribution of bathymetry to the prediction of quantities related to the gravity field (e.g., gravity anomalies, geoid heights) is discussed in an extended test area of the central Mediterranean Sea. Sea gravity anomalies and a priori statistical characteristics of depths are used in a least-squares collocation procedure in order to produce new depths, giving a better smoothing of the gravity field when using a remove-restore procedure. The effect of the bottom topography on gravity-field modeling is studied using both the original and the new depths through a residual terrain modeling reduction. The numerical tests show a considerable smoothing of the sea gravity anomalies and the available altimeter heights when the new depth information is taken into account according to the covariance analysis performed. Moreover, geoid heights are computed by combining the sea gravity anomalies either with the original depths or with the new ones, using as a reference surface the OSU91A geopotential model. Comparing the computed geoid heights with adjusted altimeter sea-surface heights (SSHs), better results are obtained when subtracting the attraction of the new depth information. Similar results are obtained when predicting gravity anomalies from altimeter SSHs where the terrain effect on altimetry is based on the new bottom topography. Received: 10 September 1996 / Accepted: 4 August 1997     

星载重力测量和固体地球的透视

为了进一步改进地球重力场模型 ,美国科学院 (NRC)于 1997年提出一个报告 ,号召美国和欧洲学者们组成一个由空间探测地球重力场委员会。他们评比了大约 10个计划的或想象的卫星重力任务 ,最后提出了 5种卫星重力任务的方案 ,称为专用重力任务。本文介绍这 5种任务的内容以及地球科学将由这些任务获得重大进展的研究领域。     

Integral formulas for geopotential coefficient determination from gravity anomalies versus gravity disturbances

Integral formulas are derived for the determination of geopotential coefficients from gravity anomalies and gravity disturbances over the surface of the Earth. First order topographic corrections to spherical formulas are presented. In addition new integral formulas are derived for the determination of the external gravity field from surface gravity. Taking advantage of modern satellite positioning techniques, it is suggested that, in general, the external gravity field as well as individual coefficients are better determined from gravity disturbances than from gravity anomalies.     

Gravity anomalies from satellite altimetry: comparison between computation via geoid heights and via deflections of the vertical

The accumulation of good quality satellite altimetry missions allows us to have a precise geoid with fair resolution and to compute free air gravity anomalies easily by fast Fourier transform (FFT) techniques.In this study we are comparing two methods to get gravity anomalies. The first one is to establish a geoid grid and transform it into anomalies using inverse Stokes formula in the spectral domain via FFT. The second one computes deflection of the vertical grids and transforms them into anomalies.The comparison is made using different data sets: Geosat, ERS-1 and Topex-Poseidon exact repeat misions (ERMs) north of 30°S and Geosat geodetic mission (GM) south of 30°S. The second method which transforms the geoid gradients converted into deflection of the vertical values is much better and the results have been favourably evaluated by comparison with marine gravity data.     

Geoid and high resolution sea surface topography modelling in the mediterranean from gravimetry,altimetry and GOCE data: evaluation by simulation

The determination of local geoid models has traditionally been carried out on land and at sea using gravity anomaly and satellite altimetry data, while it will be aided by the data expected from satellite missions such as those from the Gravity field and steady-state ocean circulation explorer (GOCE). To assess the performance of heterogeneous data combination to local geoid determination, simulated data for the central Mediterranean Sea are analyzed. These data include marine and land gravity anomalies, altimetric sea surface heights, and GOCE observations processed with the space-wise approach. A spectral analysis of the aforementioned data shows their complementary character. GOCE data cover long wavelengths and account for the lack of such information from gravity anomalies. This is exploited for the estimation of local covariance function models, where it is seen that models computed with GOCE data and gravity anomaly empirical covariance functions perform better than models computed without GOCE data. The geoid is estimated by different data combinations and the results show that GOCE data improve the solutions for areas covered poorly with other data types, while also accounting for any long wavelength errors of the adopted reference model that exist even when the ground gravity data are dense. At sea, the altimetric data provide the dominant geoid information. However, the geoid accuracy is sensitive to orbit calibration errors and unmodeled sea surface topography (SST) effects. If such effects are present, the combination of GOCE and gravity anomaly data can improve the geoid accuracy. The present work also presents results from simulations for the recovery of the stationary SST, which show that the combination of geoid heights obtained from a spherical harmonic geopotential model derived from GOCE with satellite altimetry data can provide SST models with some centimeters of error. However, combining data from GOCE with gravity anomalies in a collocation approach can result in the estimation of a higher resolution geoid, more suitable for high resolution mean dynamic SST modeling. Such simulations can be performed toward the development and evaluation of SST recovery methods.     

Mission design,operation and exploitation of the gravity field and steady-state ocean circulation explorer mission

The European Space Agency’s Gravity field and steady-state ocean circulation explorer mission (GOCE) was launched on 17 March 2009. As the first of the Earth Explorer family of satellites within the Agency’s Living Planet Programme, it is aiming at a better understanding of the Earth system. The mission objective of GOCE is the determination of the Earth’s gravity field and geoid with high accuracy and maximum spatial resolution. The geoid, combined with the de facto mean ocean surface derived from twenty-odd years of satellite radar altimetry, yields the global dynamic ocean topography. It serves ocean circulation and ocean transport studies and sea level research. GOCE geoid heights allow the conversion of global positioning system (GPS) heights to high precision heights above sea level. Gravity anomalies and also gravity gradients from GOCE are used for gravity-to-density inversion and in particular for studies of the Earth’s lithosphere and upper mantle. GOCE is the first-ever satellite to carry a gravitational gradiometer, and in order to achieve its challenging mission objectives the satellite embarks a number of world-first technologies. In essence the spacecraft together with its sensors can be regarded as a spaceborne gravimeter. In this work, we describe the mission and the way it is operated and exploited in order to make available the best-possible measurements of the Earth gravity field. The main lessons learned from the first 19 months in orbit are also provided, in as far as they affect the quality of the science data products and therefore are of specific interest for GOCE data users.