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.     

联合多代测高数据建立中国海域重力异常模型

本文联合T/P数据、T/P新轨道数据、ERS数据、GFO数据、GeosatGM数据和ERS-1/168数据,用测高卫星记录点的位置信息直接计算沿轨大地水准面的方向导数,结合测线轨迹方向的方位角在交叉点处推求垂线偏差,然后利用逆Vening-Meinesz公式计算了中国近海(0o～41oN,105o～132oN)2′×2′格网分辨率的海域重力异常模型。将其与CLS_SHOW99重力异常模型比较,统计结果表示与该模型差异的RMS为8.15mgal,在剔除差值大于20mgal的点(剔除3.3%)以后,RMS为4.72mgal;与某海区船测重力异常比较的RMS为8.91mgal。     

Inverse Vening Meinesz formula and deflection-geoid formula: applications to the predictions of gravity and geoid over the South China Sea

Using the spherical harmonic representations of the earth's disturbing potential and its functionals, we derive the inverse Vening Meinesz formula, which converts deflection of the vertical to gravity anomaly using the gradient of the H function. The deflection-geoid formula is also derived that converts deflection to geoidal undulation using the gradient of the C function. The two formulae are implemented by the 1D FFT and the 2D FFT methods. The innermost zone effect is derived. The inverse Vening Meinesz formula is employed to compute gravity anomalies and geoidal undulations over the South China Sea using deflections from Seasat, Geosat, ERS-1 and TOPEX//POSEIDON satellite altimetry. The 1D FFT yields the best result of 9.9-mgal rms difference with the shipborne gravity anomalies. Using the simulated deflections from EGM96, the deflection-geoid formula yields a 4-cm rms difference with the EGM96-generated geoid. The predicted gravity anomalies and geoidal undulations can be used to study the tectonic structure and the ocean circulations of the South China Sea. Received: 7 April 1997 / Accepted: 7 January 1998     

Improving gravity anomalies over China marginal sea from retracked Geosat and ERS-1 data

The quality of altimeter data and ocean tide model is critical to the recovery of coastal gravity anomalies. In this contribution, three retracking methods (threshold, improved threshold and Beta-5) are investigated with the aim of improving the altimeter data over a shallow water area. Comparison indicates that the improved threshold is the best retracking method over China Sea. Two ocean tide models, NAO99b and CSR4.0, are analyzed. Results show that different tide models used in the processing of altimeter data may result in differences more than 10 mGal in recovered coastal gravity anomalies. Also, NAO99b is more suitable than CSR4.0 over the shallow water area of China Sea. Finally, gravity anomalies over China Sea are calculated from retracked Geosat/GM and ERS-1/GM data by least squares collocation. Comparison with shipborne gravimetry data demonstrates that gravity anomalies from retracked data are significantly superior to those from non-retracked data. Our results have the same order as the other two altimeter-derived gravity models: Sandwell&Smith(V16) and DNSC08.     

Estimation of dynamic ocean topography in the Gulf Stream area using the Hotine formula and altimetry data

Two modifications of the Hotine formula using the truncation theory and marine gravity disturbances with altimetry data are developed and used to compute a marine gravimetric geoid in the Gulf Stream area. The purpose of the geoid computation from marine gravity information is to derive the absolute dynamic ocean topography based on the best estimate of the mean surface height from recent altimetry missions such as Geosat, ERS-1, and Topex. This paper also tries to overcome difficulties of using Fast Fourier Transformation (FFT) techniques to the geoid computation when the Hotine kernel is modified according to the truncation theory. The derived absolute dynamic ocean topography is compared with that from global circulation models such as POCM4B and POP96. The RMS difference between altimetry-derived and global circulation model dynamic ocean topography is at the level of 25cm. The corresponding mean difference for POCM4B and POP96 is only a few centimeters. This study also shows that the POP96 model is in slightly better agreement with the results derived from the Hotine formula and altimetry data than POCM4B in the Gulf Stream area. In addition, Hotine formula with modification (II) gives the better agreement with the results from the two global circulation models than the other techniques discussed in this paper. Received: 10 October 1996 / Accepted: 16 January 1998     

Coastal Gravity Anomalies from Retracked Geosat/GM Altimetry: Improvement, Limitation and the Role of Airborne Gravity Data

We process geophysical and waveform data records of the Geosat/GM (geodetic mission) satellite altimeter mission for waveform retracking and applications. An improved threshold retracker is developed. The performances of the Beta-5, threshold and improved threshold retrackers are assessed over waters around Taiwan. The improved threshold retracker outperforms the other two. The improvement in the accuracy of sea surface height (SSH) is investigated according to marine zone and the distance of waters to the shore. The improvement rate increases closer to the land, with the largest improvement rate of about 20% in waters within 10 km of the shore. Over waters around islands and coasts, there are still retracked SSHs with large errors. Least-squares collocation is used to compute gravity anomalies from the Geosat/GM altimeter data. Use of retracked SSHs improves the accuracy of gravity anomalies by about 11%. Adding airborne gravity data further improves the accuracy, especially in the immediate vicinity of the coasts. Tide model errors over coastal waters remain a problem in altimetry applications, even if the waveforms are properly retracked.     

卫星测高数据的沿轨迹重力异常反演法及其应用

本文给出了一套基于直角坐标系下的垂线偏差求解重力异常公式 ,并将之发展成为一套新的沿轨迹重力异常求解公式。与其他方法相比 ,本方法无须求解交叠点处沿轨迹和跨轨迹方向的海面高斜率 ,仅需计算沿轨迹方向的海面高斜率 ,因而更为简洁、有效 ,而且分辨率可以更高并可与真正的沿航迹实际船测重力相比较、验证。据此 ,利用 Geosat/GM、ERS-1 /35天及TOPEX/Poseidon三种测高数据 ,反演了南中国海域 (0°～ 2 5°N,1 0 5°～ 1 2 2°E)的 2′× 2′重力异常—— IGG-S。通过与实际船测资料和国际同行提供的重力模型相比 ,IGG-S总体精度达到1 0× 1 0 - 5ms- 2。     

Role of ocean variability and dynamic ocean topography in the recovery of the mean sea surface and gravity anomalies from satellite altimeter data

Procedures to calculate mean sea surface heights and gravity anomalies from altimeter-derived sea surface heights and along-track sea surface slopes using the least-squares collocation procedure are derived. The slope data is used when repeat track averaging is not possible to reduce ocean variability effects. Tests were carried out using Topex, Geosat, ERS-1 [35-day and 168-day (2 cycle)] data. Calculations of gravity anomalies in the Gulf Stream region were made using the sea surface height and slope data. Tests were also made correcting the sea surface heights for dynamic ocean topography calculated from a degree 360 expansion of data from the POCM-4B global ocean circulation model. Comparisons of the anomaly predictions were carried out with ship data using anomalies calculated for this paper as well as others. Received: 19 August 1996 / Accepted: 14 April 1997     

Global mean sea surface and marine gravity anomaly from multi-satellite altimetry: applications of deflection-geoid and inverse Vening Meinesz formulae

 Global mean sea surface heights (SSHs) and gravity anomalies on a 2^′×2^′ grid were determined from Seasat, Geosat (Exact Repeat Mission and Geodetic Mission), ERS-1 (1.5-year mean of 35-day, and GM), TOPEX/POSEIDON (T/P) (5.6-year mean) and ERS-2 (2-year mean) altimeter data over the region 0^∘–360^∘ longitude and –80^∘–80^∘ latitude. To reduce ocean variabilities and data noises, SSHs from non-repeat missions were filtered by Gaussian filters of various wavelengths. A Levitus oceanic dynamic topography was subtracted from the altimeter-derived SSHs, and the resulting heights were used to compute along-track deflection of the vertical (DOV). Geoidal heights and gravity anomalies were then computed from DOV using the deflection-geoid and inverse Vening Meinesz formulae. The Levitus oceanic dynamic topography was added back to the geoidal heights to obtain a preliminary sea surface grid. The difference between the T/P mean sea surface and the preliminary sea surface was computed on a grid by a minimum curvature method and then was added to the preliminary grid. The comparison of the NCTU01 mean sea surface height (MSSH) with the T/P and the ERS-1 MSSH result in overall root-mean-square (RMS) differences of 5.0 and 3.1 cm in SSH, respectively, and 7.1 and 3.2 μrad in SSH gradient, respectively. The RMS differences between the predicted and shipborne gravity anomalies range from 3.0 to 13.4 mGal in 12 areas of the world's oceans. Received: 26 September 2001 / Accepted: 3 April 2002 Correspondence to: C. Hwang Acknowledgements. This research is partly supported by the National Science Council of ROC, under grants NSC89-2611-M-009-003-OP2 and NSC89-2211-E-009-095. This is a contribution to the IAG Special Study Group 3.186. The Geosat and ERS1/2 data are from NOAA and CERSAT/France, respectively. The T/P data were provided by AVISO. The CLS and GSFC00 MSS models were kindly provided by NASA/GSFC and CLS, respectively. Drs. Levitus, Monterey, and Boyer are thanked for providing the SST model. Dr. T. Gruber and two anonymous reviewers provided very detailed reviews that improved the quality of this paper.     

利用卫星测高数据反演海洋重力异常研究

全面研究了利用卫得测高数据反演海洋重力异常3种主要方法（即Stokes数据解析反解以及逆Vening-Meinesz公式）的技术特点，建立了3种算法的数学模型及其谱计算式，在以1440阶次位模型定义的标准场中完成了3种算法的数值比较和内部检核，通过仿真试验实现了3种算法的可靠性和稳定性检验，最后，本文利用卫得测高实测对南中国海地区的海洋重力异常进行了实际反演，并将反演结果同船测数据进行了比较。     

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.     

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     

基于Laplace方程的垂线偏差法反演全球海域重力异常

海洋重力场模型反演的质量主要依赖于采用测高数据的精度、空间分辨率和数据分布密集程度。本文联合Geosat GM/ERM、ERS-1 GM/ERM、TOPEX/Poseidon、Envisat、Cryosat-2、Jason-1 ERM/GM和SARAL/AltiKa等多种测高观测数据集,深入比较了多种波形重跟踪算法的效果,回波数据重跟踪处理后的沿轨海面高标准差。统计表明,Sandwell算法优于MLE-4算法、Davis阈值法、改进阈值法和β参数拟合法;基于不同测高数据波形重采样的结果给出了沿轨海面梯度计算中低通滤波的参数选择方法,并采用Sandwell提出的垂线偏差法,反演了全球海域1′×1′的重力场模型。检核表明,反演结果与DTU13和SIO V23.1模型检核的差值均方根分别为3.4、1.8 mGal,与NGDC船测数据的检核精度为4~8 mGal,且本文模型在部分典型海区内精度更优。     

基于多源卫星测高数据的新一代全球平均海面高模型

高精度高分辨率平均海面高模型是研究海洋动力环境变化的重要基础。本文基于多源多代卫星测高数据,包括Geosat/GM、Geosat/ERM、ERS-1/168、ERS-2、Envisat-1、T/P和Jason-1数据,采用统一精密的模型或方法对各项地球物理改正和传播介质误差进行了改进或重新计算,然后以T/P和Jason-1两种数据的共线平差作为参考基准,通过发展的全组合交叉点整体平差方法大幅度削弱了径向轨道误差和时变海面高影响,经比较后优选了最小二乘配置格网化方法,以EGM2008高阶重力场模型由移去－恢复方法得到了全球纬度80°S～82°N范围内海域2′×2′平均海平面高模型WHU2009。与国际上较常用的CLS01平均海平面高模型及实测2年平均Jason-1比较表明WHU2009模型整体精度优于CLS01模型。     

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.     

Spectral accuracy of Jgm3 from satellite crossover altimetry

A detailed accuracy assessment of the geopotential model Jgm3 is made based on independent single- and dual-satellite sea-height differences at crossovers from altimetry with Jgm3-based orbits. These differences, averaged over long time spans and in latitude bands, are converted to spectra (latitude-lumped coefficients) by least-squares estimation. The observed error spectra so obtained are then compared directly to error projections for them from the Jgm3 variance–covariance matrix. It is found from these comparisons that Jgm3 is generally well calibrated with respect to the crossover altimetry of and between Geosat, TOPEX/Poseidon (T/P), and Ers 1. Some significant discrepancies at a few lower orders (namely m=1 and 3) indicate a need for further improvement of Jgm3. A companion calibration (by order) of the geopotential model Jgm2 shows its variance–covariance matrix also to be generally well calibrated for the same single- and dual-satellite altimeter data sets (but based on Jgm2 orbits), except that the error projections for Geosat are too pessimistic. The analysis of the dual-satellite crossovers reveals possible relative coordinate system offsets (particularly for Geosat with respect to T/P) which have been discussed previously. The long-term detailed seasonally averaged Geosat sea level with respect to T/P (covering 1985–1996) should be useful in gauging the relative change in sea level between different parts of the ocean over the single 4-year gap between these missions (1988–1992). Received: 16 February 1998 / Accepted: 25 November 1998     

Making the connection between Geosat and TOPEX/Poseidon

We can presently construct two independent time series of sea level, each at a precision of a few centimeters, from Geosat (1985–1988) and TOPEX/Poseidon (1992–1995) collinear altimetry. Both are based on precise satellite orbits computed using a common geopotential model, JGM-2 (Nerem et al. 1994). We have attempted to connect these series using Geosat-T/P crossover differences in order to assess long-term ocean changes between these missions. Unfortunately, the observed result are large-scale sea level differences which appear to be due to a combination of geodetic and geopotential error sources. The most significant geodetic component seems to be a coordinate system bias for Geosat sea level (relative to T/P) of −7 to −9 cm in the y-axis (towards the Eastern Pacific). The Geosat-T/P sea height differences at crossovers (with JGM-2 orbits) probably also contain stationary geopotential-orbit error of about the same magnitude which also distort any oceanographic interpretation of the observed changes. We also found JGM-3 Geosat orbits have not resolved the datum errors evident from the JGM-2 Geosat -T/P results. We conclude that the direct altimetric approach to accurate determination of sea level change using Geosat and T/P data still depends on further improvement in the Geosat orbits, including definition of the geocenter. Received: 11 March 1996; Accepted: 19 September 1996     

The GeoForschungsZentrum Potsdam/Groupe de Recherche de Gèodésie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGEN-GL04C

The recent improvements in the Gravity Recovery And Climate Experiment (GRACE) tracking data processing at GeoForschungsZentrum Potsdam (GFZ) and Groupe de Recherche de Géodésie Spatiale (GRGS) Toulouse, the availability of newer surface gravity data sets in the Arctic, Antarctica and North-America, and the availability of a new mean sea surface height model from altimetry processing at GFZ gave rise to the generation of two new global gravity field models. The first, EIGEN-GL04S1, a satellite-only model complete to degree and order 150 in terms of spherical harmonics, was derived by combination of the latest GFZ Potsdam GRACE-only (EIGEN-GRACE04S) and GRGS Toulouse GRACE/LAGEOS (EIGEN-GL04S) mean field solutions. The second, EIGEN-GL04S1 was combined with surface gravity data from altimetry over the oceans and gravimetry over the continents to derive a new high-resolution global gravity field model called EIGEN-GL04C. This model is complete to degree and order 360 and thus resolves geoid and gravity anomalies at half- wavelengths of 55 km at the equator. A degree-dependent combination method has been applied in order to preserve the high accuracy from the GRACE satellite data in the lower frequency band of the geopotential and to form a smooth transition to the high-frequency information coming from the surface data. Compared to pre-CHAMP global high-resolution models, the accuracy was improved at a spatial resolution of 200 km (half-wavelength) by one order of magnitude to 3 cm in terms of geoid heights. The accuracy of this model (i.e. the commission error) at its full spatial resolution is estimated to be 15 cm. The model shows a reduced artificial meridional striping and an increased correlation of EIGEN-GL04C-derived geostrophic meridional currents with World Ocean Atlas 2001 (WOA01) data. These improvements have led to select EIGEN-GL04C for JASON-1 satellite altimeter data reprocessing. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Accuracy assessment and refinement of the JGM-2 and JGM-3 gravity fields for radial positioning of ERS-1

ERS-1 radial positioning using the JGM-2 and JGM-3 gravity fields is assessed by analysing dual crossovers with TOPEX/Poseidon, neither field containing ERS-1 data. This method allows a more complete recovery of ERS-1 radial orbit error, specifically of the previously unattainable mean geographical error. The global analysis shows that the theoretical error derived from the JGM-2 covariance matrix is realistic and that JGM-3 represents a slight improvement, at least at the inclination of ERS-1. A latitudinal-based study in the southern ocean indicates possible weaknesses in both fields, notably for low and resonant geopotential orders m. A refinement of JGM-2, RGM-2, is undertaken through inclusion of ERS-1 and STELLA laser tracking and ERS-1 altimetry, reducing several of its deficiencies. Received: 14 May 1996 / Accepted: 17 February 1997     

利用卫星测高资料推求西北太平洋海域的海洋大地水准面

本文根据卫星测高数据和海面动力地形资料，绘制了不相上下北太平洋海域局部大地水准面的精细结构图，对解决卫星测高技术中大地水准面积和海面地形的可分性问题作了初步尝试。