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     

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     

Absolute gravity measurements on the canadian gravity standardization network

Summary Absolute measurements of gravity have been made at 6 locations ranging from Ottawa, Ont., in southern Canada, to Alert, N.W.T., the world's most northerly permanent settlement, as part of a program to provide scale and level for the Canadian Gravity Standardization Network (CGSN). Except at Resolute, N.W.T., CGSN-74 gravity values, upon which our gravity reductions are currently based, agree with the absolute gravity meter results to within about .25µm/s². The scale of our CGSN-80 gravity network, upon which our spring-balance type gravity meter scale constants are derived, agrees with the scale defined by the absolute gravity measurements to within about one part in ten thousand.     

融合多源重力数据构建局部高阶重力场模型

针对利用带限型径向基函数,融合航空和地面重力数据构建高阶地球重力场模型时,航空、地面重力数据的频谱信息不一致问题,该文提出残差与先验精度比较分析法,确定出了研究区域航空数据的最佳频谱范围.结果 表明:航空重力数据系统性偏差及其径向基函数展开阶次不当可分别导致±0.019m和±0.007 m的建模误差.基于此,构建了局部区域6000阶的径向基函数融合地球重力场模型CBFM2020.通过与GPS/水准数据比较,CBFM2020的精度比EGM2008、USGG2012以及仅由地面数据得到的重力场模型均有所提高.模型建立过程中航空重力数据未进行向下延拓,而是直接在原始高度与地面数据进行融合,可为局部地区高阶重力场模型的构建提供一定参考.     

卫星重力梯度数据重力异常的精度分析

针对GOCE卫星确定的地球重力场模型精度的不确定性,对比分析GOCE位模型与多个不同重力场模型确定的重力异常,并将其分别与船测重力数据、南极航空重力数据、北极重力数据以及美国和中国台湾地面重力数据比较研究。结果表明:GOCE位模型的内符合精度最高,与地面重力观测数据符合最优;与船测以及航空重力测量符合相对较差、精度较低。研究表明,在一定精度前提下,GOCE卫星确定的重力数据可用于无人区,从而提高重力观测数据的覆盖率。     

Comparison of undulation difference accuracies using gravity anomalies and gravity disturbances

Errors are considered in the outer zone contribution to oceanic undulation differences as obtained from a set of potential coefficients complete to degree 180. It is assumed that the gravity data of the inner zone (a spherical cap), consisting of either gravity anomalies or gravity disturbances, has negligible error. This implies that error estimates of the total undulation difference are analyzed. If the potential coefficients are derived from a global field of 1°×1° mean anomalies accurate to ε_Δg=10 mgal, then for a cap radius of 10°, the undulation difference error (for separations between 100 km and 2000 km) ranges from 13 cm to 55 cm in the gravity anomaly case and from 6 cm to 36 cm in the gravity disturbance case. If ε_Δg is reduced to 1 mgal, these errors in both cases are less than 10 cm. In the absence of a spherical cap, both cases yield identical error estimates: about 68 cm if ε_Δg=1 mgal (for most separations) and ranging from 93 cm to 160 cm if ε_Δg=10 mgal. Introducing a perfect 30-degree reference field, the latter errors are reduced to about 110 cm for most separations.     

Superconducting tensor gravity gradiometer

A sensitive gravity gradiometer can provide much needed gravity data of the earth and improve the accuracy of inertial navigation. A complete determination of all five independent components of the gravity gradient tensor is possible in principle by rotating a single in-line component gradiometer. In order to avoid dynamically induced noise sources arising from rotation, a hard-mounted assembly of component gradiometers may nevertheless be advantageous in an ultra-sensitive device. Superconductivity and other properties of materials at low temperatures can be used to obtain a sensitive, low-drift, gravity gradiometer. By differencing the outputs of accelerometer pairs using superconducting circuits, it is possible to construct a non-rotating tensor gravity gradiometer. Additional superconducting circuits can be provided to determine the linear and angular acceleration vectors. A three-axis in-line gravity gradiometer (a “vector” gradiometer) is being developed for satellite geodesy. A two-dimensional spring concept for a three-axis cross component gradiometer is discussed. The superconducting tensor gravity gradiometer constitutes a complete package of inertial navigation instruments with angular and linear acceleration readouts as well as gravity signals. Accuracy of inertial navigation could be improved by use of such a gradiometeraided inertial navigator.

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Sommaire Un gradiomètre gravimétrique sensible peut foumir une grande partie des données gravimétriques terrestres et augmenter la précision de la navigation inertielle. Une détermination complète des cinq composantes indépendantes du tenseur de gradient de gravité est possible en principe par la rotation d’un seul axe d’un gradiomètre. Dans le but d’éviter les sources de bruit provenant de la rotation, un assemblage complexe de gradiomètres peut être néanmoins avantageux dans un appareil ultra-sensible. La superconductivité et d’autres propriétés des matériaux à basse température peuvent être utilisées pour obtenir un gradiomètre gravimétrique sensible et à faible dérive. Il est possible, en différentiant les sorties de paires d’accéléromètres utilisant des circuits superconducteurs, de construire un gradimètre non-rotatif qui mesure le tenseur de gravité. Des circuits superconducteurs additionnels peuvent être ajoutés pour déterminer les vecteurs d’accélération linéaires et angulaires. Un gradiomètre gravimétrique à trois exes est présentement développé pour la géodésie spatiale. Le concept d’un ressort à deux dimensions comme composante d’un gradiomètre à trois axes est discuté. Le gradiomètre de mesure du tenseur de gravité à l’aide de circuits superconducteurs constitue l’ensemble complet pour un instrument de navigation inertielle avec lecture des accélérations angulaires et linéaires et des données de gravité. La précision de la navigation inertielle pourrait être améliorée par l’emploi d’un tel gradiomètre avec les systèmes inertiels.

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Presented at the Second International Symposium on Inertial Technology of Surveying and Geodesy, Banff, Canada, June 1–5, 1981.     

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.     

Botswana gravity reference net

Gravity reference stations for the National Gravity Survey of Botswana have been established at twenty-three sites throughout the country in a net linked to existing bases in South Africa, Kenya and Zambia with an internal accuracy of better than 0.5 gravity units (one gravity unit, gu, equals an acceleration of 10⁻⁶ m.s⁻²). The field procedure and reduction of data are explained and a list is given of the gravity values.     

重力辅助惯性导航中重力归算的垂直梯度计算

重力辅助惯性导航技术是利用地球物理特征信息数据—重力来完成水下运动载体的辅助导航与定位。为了实现水下运动载体上重力传感器输出的实测重力信息与重力数据库中存储的重力信息之间的匹配,首先必须将这两类数据归算到一个平面。本文研究分析了重力归算中重力垂直梯度求解的各种方法;探讨了不同数值积分区域对扰动重力垂直梯度精度的影响;并通过计算分析,提出了可以直接以重力异常垂直梯度代替扰动重力垂直梯度来求取重力垂直梯度。     

国际重力卫星研究进展和我国将来卫星重力测量计划

本文首先分别介绍了国际已经成功发射的专用地球重力测量卫星CHAMP、GRACE以及即将发射的GOCE、GRACE Follow-On和专用月球重力探测卫星GRAIL的研制机构、轨道参数、关键载荷、跟踪模式、测量原理、科学目标和技术特征;其次,阐述了当前相关学科对地球重力场测量精度的需求;最后,建议我国在将来实施的卫星重力测量计划中首选卫星跟踪卫星高低\低低模式,尽快开展轨道参数优化选取的定量系统研究论证和重力卫星系统的误差分析,依据匹配精度指标先期开展重力卫星各关键载荷的研制以及尽早启动卫星重力测量系统的虚拟仿真研究。     

Somigliana–Pizzetti gravity: the international gravity formula accurate to the sub-nanoGal level

 The Somigliana–Pizzetti gravity field (the International gravity formula), namely the gravity field of the level ellipsoid (the International Reference Ellipsoid), is derived to the sub-nanoGal accuracy level in order to fulfil the demands of modern gravimetry (absolute gravimeters, super conducting gravimeters, atomic gravimeters). Equations (53), (54) and (59) summarise Somigliana–Pizzetti gravity Γ(φ,u) as a function of Jacobi spheroidal latitude φ and height u to the order ?(10⁻¹⁰ Gal), and Γ(B,H) as a function of Gauss (surface normal) ellipsoidal latitude B and height H to the order ?(10⁻¹⁰ Gal) as determined by GPS (`global problem solver'). Within the test area of the state of Baden-Württemberg, Somigliana–Pizzetti gravity disturbances of an average of 25.452 mGal were produced. Computer programs for an operational application of the new international gravity formula with (L,B,H) or (λ,φ,u) coordinate inputs to a sub-nanoGal level of accuracy are available on the Internet. Received: 23 June 2000 / Accepted: 2 January 2001     

Downward continuation of Helmert's gravity

. The aim of this contribution is to show that mean Helmert's gravity anomalies obtained at the earth surface on a grid of a `reasonable' step can be transferred to corresponding mean Helmert's anomalies on the geoid. To demonstrate this, we take the by mean Helmert's anomalies from a very rugged region, the south-western corner of Canada which contains the two main chains of the Canadian Rocky Mountains, and formulate the problem of downward continuation of Helmert's anomalies for this region. This can be done exactly because Helmert's disturbing potential is harmonic everywhere outside the geoid, therefore even within the topography. Then we solve the problem numerically by transforming the Poisson integral to a system of 53,856 linear algebraic equations. Since the matrix of this system is well conditioned, there is no theoretical obstacle to the solution. The correctness of the solution is then checked by back substitution and by evaluating the contribution of the downward continuation term to Helmert's co-geoid. This contribution comes out positive for all the points. We thus claim that the determination of the downward continuation of mean Helmert's gravity anomalies on a grid of a `reasonable' step is a well posed problem with a unique solution and can be done routinely to any accuracy desired in the geoid computaion. Received 27 October 1995; Accepted 9 July 1996     

Improving GOCE cross-track gravity gradients

The GOCE gravity gradiometer measured highly accurate gravity gradients along the orbit during GOCE’s mission lifetime from March 17, 2009, to November 11, 2013. These measurements contain unique information on the gravity field at a spatial resolution of 80 km half wavelength, which is not provided to the same accuracy level by any other satellite mission now and in the foreseeable future. Unfortunately, the gravity gradient in cross-track direction is heavily perturbed in the regions around the geomagnetic poles. We show in this paper that the perturbing effect can be modeled accurately as a quadratic function of the non-gravitational acceleration of the satellite in cross-track direction. Most importantly, we can remove the perturbation from the cross-track gravity gradient to a great extent, which significantly improves the accuracy of the latter and offers opportunities for better scientific exploitation of the GOCE gravity gradient data set.     

On the combination of gravity anomalies and gravity disturbances for geoid determination in Western Australia

The geoid gradient over the Darling Fault in Western Australia is extremely high, rising by as much as 38 cm over only 2 km. This poses problems for gravimetric-only geoid models of the area, whose frequency content is limited by the spatial distribution of the gravity data. The gravimetric-only version of AUSGeoid98, for instance, is only able to resolve 46% of the gradient across the fault. Hence, the ability of GPS surveys to obtain accurate orthometric heights is reduced. It is described how further gravity data were collected over the Darling Fault, augmenting the existing gravity observations at key locations so as to obtain a more representative geoid gradient. As many of the gravity observations were collected at stations with a well-known GRS80 ellipsoidal height, the opportunity arose to compute a geoid model via both the Stokes and the Hotine approaches. A scheme was devised to convert free-air anomaly data to gravity disturbances using existing geoid models, followed by a Hotine integration to geoid heights. Interestingly, these results depended very weakly upon the choice of input geoid model. The extra gravity data did indeed improve the fit of the computed geoid to local GPS/Australian Height Datum (AHD) observations by 58% over the gravimetric-only AUSGeoid98. While the conventional Stokesian approach to geoid determination proved to be slightly better than the Hotine method, the latter still improved upon the gravimetric-only AUSGeoid98 solution, supporting the viability of conducting gravity surveys with GPS control for the purposes of geoid determination. AcknowledgementsThe author would like to thank Will Featherstone, Ron Gower, Ron Hackney, Linda Morgan, Geoscience Australia, Scripps Oceanographic Institute and the three anonymous reviewers of this paper. This research was funded by the Australian Research Council.     

Simulation of regional gravity field recovery from satellite gravity gradiometer data using collocation and FFT

When planning a satellite gravity gradiometer (SGG) mission, it is important to know the quality of the quantities to be recovered at ground level as a function of e.g. satellite altitude, data type and sampling rate, and signal variance and noise. This kind of knowledge may be provided either using the formal error estimates of wanted quantities using least-squares collocation (LSC) or by comparing simulated data at ground level with results computed by methods like LSC or Fast Fourier Transform (FFT). Results of a regional gravity field recovery in a 10^o×20^o area surrounding the Alps using LSC and FFT are reported. Data used as observations in satellite altitude (202 or161 km) and for comparison at ground level were generated using theOSU86F coefficient set, complete to degree 360. These observations are referred to points across simulated orbits. The simulated quantities were computed for a 45 days mission period and 4 s sampling. A covariance function which also included terms above degree 360 was used for prediction and error estimation. This had the effect that the formal error standard deviation for gravity anomalies were considerably larger than the standard deviations of predicted minus simulated quantities. This shows the importance of using data with frequency content above degree 360 in simulation studies. Using data at202 km altitude the standard deviation of the predicted minus simulated data was equal to8.3 mgal for gravity and0.33 m for geoid heights.