Recovery of marine gravity anomalies from ERS1, ERS2 and ENVISAT satellite altimetry data for geoid computations over Norway

Free-Air Anomalies (FAA) for the Norwegian marine area including some parts of the North Sea, the Norwegian Sea and the Barents Sea are computed from satellite altimetry data. A total of 84 cycles of ERS2 along-track data, 25 cycles of ENVISAT along-track data and high density ERS1 data during its geodetic mission are used. The new geopotential model from the Gravity Recovery and Climate Experiment (GRACE) mission, GGM02S (Tapely et al., 2005) is used to compute the long wavelength contributions of the geoid and the FAA. To correct data for mean dynamic topography, the available Levitus climatology model (Levitus and Boyer, 1994) is used. Corrected data are then used to compute along-track gradients in each cycle-pass to suppress the orbital and the atmospheric errors below the noise level of the altimeter. Resulted gradients are then stacked and the east-west and the north-south components of the deflection of verticals are computed where ascending and descending tracks meet each other. Finally, the inverse Vening-Meinesz formula is implemented on the gridded deflections to compute FAA. Results are then compared with available marine and airborne data. Standard deviations of ± 4.301 and ± 6.159 mGal in comparison with airborne and marine FAA were achieved. Thereafter, the derived anomalies are combined with marine and airborne FAA together with the land FAA to compute a fine resolution geoid for Norway and the surrounding marine areas. This geoid is evaluated over sea and land with the synthetic geoid (the geoid derived from the mean sea surface by subtracting the mean dynamic topography) and Global Positioning System (GPS)-levelling and the standard deviations of the differences are ± 20.9 and ± 12.8 cm respectively. ali.soltanpour@ntnu.no, hossein.nahavandchi@ntnu.no, kourosh.ghazavi@ntnu.no     

Regional gravimetric quasi-geoid model and transformation surface to national height system for Turkey (THG-09)

Turkish regional geoid models have been developed by employing a reference earth gravitational model, surface gravity observations and digital terrain models. The gravimetric geoid models provide a ready transformation from ellipsoidal heights to the orthometric heights through the use of GPS/leveling geoid heights determined through the national geodetic networks. The recent gravimetric models for Turkish territory were computed depending on OSU91 (TG-91) and EGM96 (TG-03) earth gravitational models. The release of the Earth Gravitational Model 2008 (EGM08), the collection of new surface gravity observations, the advanced satellite altimetry-derived gravity over the sea, and the availability of the high resolution digital terrain model have encouraged us to compute a new geoid model for Turkey. We used the Remove-Restore procedure based on EGM08 and applied Residual Terrain Model (RTM) reduction of the surface gravity data. Fast Fourier Transformation (FFT) was then used to obtain the residual quasigeoid from the reduced gravity. We restored the individual contributions of EGM08 and RTM to the whole quasi-geoid height (TQG-09). Since the Helmert orthometric height system is adopted in Turkey, the quasi-geoid model (TQG-09) was then converted to the geoid model (TG-09) by making use of Bouguer gravity anomalies and digital terrain model. After all we combined a gravimetric geoid model with GPS/leveling geoid heights in order to obtain a hybrid geoid model (THG-09) (or a transformation surface) to be used in GPS applications. The RMS of the post-fit residuals after the combination was found to be ± 0.95 cm, which represents the internal precision of the final combination. And finally, we tested the hybrid geoid model with GPS/leveling data, which were not used in the combination, to assess the external accuracy. Results show that the external accuracy of the THG-09 model is ± 8.4 cm, a precision previously not achieved in Turkey until this study.     

Contribution of satellite altimetry data in the geophysical investigation of the Red Sea Region, Egypt

Successful development of geodetic satellite missions has aroused new interest in determining global and regional gravity field based on satellite data. Satellite altimetry data enable direct determination of the geoid over sea regions. In Egypt, where land and marine geophysical data are inadequate because of rough topography and economic reasons, the use of satellite altimetry data is of special importance. The northern Red Sea region has been selected as a site for case study of the current research, after applying spectral analysis to reveal near-surface structure, the residual geoid of the studied region shows a good correlation with the known geologic features. Moreover, satellite-based gravity data enhance small-scale features and agrees well with land and marine gravity data. Thus, geoid undulation and satellite gravity data can be a complementary source of data to determine near-surface and deep structures.     

A study of regional gravity field recovery from GOCE vertical gravity gradient data in the Auvergne test area using collocation

Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the first satellite mission that observes gravity gradients from the space, to be primarily used for the determination of high precision global gravity field models. However, the GOCE gradients, having a dense data distribution, may potentially provide better predictions of the regional gravity field than those obtained using a spherical harmonic Earth Geopotential Model (EGM). This is investigated in Auvergne test area using Least Squares Collocation (LSC) with GOCE vertical gravity gradient anomalies (T_zz), removing the long wavelength part from EGM2008 and the short wavelength part by residual terrain modelling (RTM). The results show that terrain effects on the vertical gravity gradient are significant at satellite altitude, reaching a level of 0.11 E?tv?s unit (E.U.) in the mountainous areas. Removing the RTM effects from GOCE T_zz leads to significant improvements on the LSC predictions of surface gravity anomalies and quasigeoid heights. Comparison with ground truth data shows that using LSC surface free air gravity anomalies and quasi-geoid heights are recovered from GOCE T_zz with standard deviations of 11 mGal and 18 cm, which is better than those obtained by using GOCE EGMs, demonstrating that information beyond the maximal degree of the GOCE EGMs is present. Investigation of using covariance functions created separately from GOCE T_zz and terrestrial free air gravity anomalies, suggests that both covariance functions give almost identical predictions. However, using covariance function obtained from GOCE T_zz has the effect that the predicted formal average error estimates are considerably larger than the standard deviations of predicted minus observed gravity anomalies. Therefore, GOCE T_zz should be used with caution to determine the covariance functions in areas where surface gravity anomalies are not available, if error estimates are needed.     

用GRACE卫星跟踪数据反演地球重力场

利用141天GRACE卫星观测资料,包括K波段、星载加速度和卫星轨道数据,反演了80阶地球重力场模型IGGGRACE01S,该模型在半波长为500km的空间分辨率上,确定大地水准面的精度约为0012m,中长波(<80阶)精度优于重力卫星发射以前研制的重力场模型. 与EIGEN_GRACE02S、EIGEN_CHAMP03S和EGM96模型的位系数相比,该模型系数最接近于EIGEN_GRACE02S,与另两个模型差异较大. 比较几种模型确定的全球重力异常和大地水准面起伏,结果发现IGGGRACE01S与EIGEN_GRACE02S模型的计算结果比较接近,与EGM96模型结果差异较大,差别较大地区主要在南极等地区. 对于中国大陆,比较IGGGRACE01S模型（前72阶）计算的重力异常和NIMA重力异常数据（25°×25°网格）,两者之间的标准偏差为48mGal.     

地形/均衡重力场模型构制及其拟稳分析

重力空白区数据填补的一个主要方法是基于地壳均衡理论进行的,该方法亦用于EGM系列模型的构建中.本文研究了地形数据在构制地形/均衡重力场模型中的应用,分析了补偿深度对Airy位模型和面凝聚位模型的影响,给出二者的最佳补偿深度分别为50 km和40 km.以纯卫星重力模型为参考,后者在前120阶的精度要高于前者,但在121~250阶的精度较低,组合模型精度高于单一模型精度.对地形/均衡地球重力场模型进行了EGM2008拟稳分析,研究了不同分辨率基准的拟稳效果,分析表明：30'分辨率的拟稳基准所得拟稳模型对应的阶方差与参考阶方差曲线直到360阶都有较好的一致性,以EGM2008为基准,其相对累计大地水准面高误差在140阶时为6.83cm,相对累计重力异常误差在220阶时为1.10 mGal.     

应用多种来源重力异常编制中国海陆及邻区空间重力异常图及重力场解读

本文通过分析陆地实测空间重力异常数据、海洋船载测量空间重力异常数据、卫星测高重力异常,布格重力异常数据、EGM2008地球重力模型数据等多种来源数据的性质和精度,并对相关数据进行对比,研究了编制1:500万中国海陆空间重力异常图的数据使用方案和技术方法.在地形较为平坦、实测数据分布均匀的陆区,使用实测数据,在地形复杂,实测数据稀少以及没有实测数据的陆区或岛屿,利用布格重力异常反推空间异常的方法合成平均空间重力数据,西藏地区的数据对比实验证明合成平均空间重力异常数据是一种有效的数据补充.利用三观测列方差分解法在南海地区对船载测量空间重力数据和美国SS系列及丹麦DNSC08GRA卫星重力数据进行了方差分解计算,结果表明不同来源的卫星测高重力数据具有很大的一致性,数据精度较以往有了很大的提高.海区空间重力数据使用原则是在船载重力测量数据校准下,全面使用卫星测高重力数据进行编图.海陆过渡区的异常处理应以EGM2008地球重力模型重力场为基准参考场,实现海陆异常平缓过渡,无缝连接.对中国海陆空间重力异常场进行了小波变换处理,对空间重力异常场进行了解读,勾画出三横四竖的一级重力梯级带及其所围限的8个一级重力异常区,并划分了二级重力异常区和梯级带,为块体构造学体系中大地构造格架的建立提供了地球物理证据.     

我国海域大地水准面的计算及其与大陆大地水准面拼接的研究和实施

研究和实施了由卫星测高数据计算垂线偏差，用莫洛 金斯基（Molodensky）公式反演 大地水准面高，由此求得我国海域大地水准面高. 为了检核，将测高垂线偏差利用逆维宁迈 纳斯（Vening Meinesz）公式反演重力异常，与海上船测重力值进行了外部检核；同时还用 司托克斯（Stokes）公式，将上述反演的重力异常计算大地水准面高，与莫洛金斯基公式直 接解得的相应结果进行比较作为内部检核. 在积分计算中充分应用了FFT的严格公式.由重力和GPS水准数据确定的陆地大地水准面，和主要由卫星测高数据确定的海洋大地水准 面，二者之间一般都存在以系统误差为主的拼接差，本文分析了产生这一现象的主要原因, 并结合我国在陆海大地水准面拼接区重力资料稀疏的实际，提出了新的拼接技术，最后将拟 合参数校正中国全部海域的重 力大地水准面，以最大限度地削弱拼接点和制约测高海洋大地水准面可能存在的系统误差.     

联合多代卫星测高和多源重力数据的局部大地水准面精化方法

本文研究了基于泊松小波径向基函数融合多代卫星测高及多源重力数据精化大地水准面模型的方法.分别以沿轨垂线偏差和大地水准面高高差作为卫星测高观测量,研究了使用不同类型测高数据对于大地水准面建模精度的影响.针对全球潮汐模型在浅水区域及部分开阔海域精度较低的问题,引入局部潮汐模型研究了不同潮汐模型对于大地水准面的影响.数值分析表明:相比于使用沿轨垂线偏差作为测高观测量,基于沿轨大地水准面高高差解算得到的大地水准面模型的精度更高,特别是在海域区域,其精度提高了2.3cm.由于使用沿轨大地水准面高高差作为测高观测量削弱了潮汐模型长波误差的影响,采用不同潮汐模型对大地水准面解算的影响较小.总体而言,船载重力及测高观测数据在海洋重力场的确定中呈现互补性关系,联合两类重力场观测量可以提高局部重力场的建模精度.     

Geoid and Sea Surface Topography Derived from ERS-1 Altimeter Data by the Adjoint Method

A method for splitting sea surface height measurements from satellite altimetry into geoid undulations and sea surface topography is presented. The method is based on a combination of the information from altimeter data and a dynamic sea surface height model. The model consists of geoid undulations and a quasi-geostrophic model for expressing the sea surface topography. The goal is the estimation of those values of the parameters of the sea surface height model that provide a least-squares fit of the model to the data. The solution is accomplished by the adjoint method which makes use of the adjoint model for computing the gradient of the cost function of the least-squares adjustment and an optimization algorithm for obtaining improved parameters. The estimation is applied to the North Atlantic. ERS-1 altimeter data of the year 1993 are used. The resulting geoid agrees well with the geoid of the EGM96 gravity model.     

Comparison of global geopotential models from the champ and grace missions for regional geoid modelling in Turkey

The continuous efforts on establishment and modernization of the geodetic control in Turkey include a number of regional geoid models that have been determined since 1976. The recently released gravimetric Geoid of Turkey, TG03, is used in geodetic applications where GPS-heights need to be converted to the local vertical datum. To reach a regional geoid model with improved accuracy, the selection of the appropriate global geopotential model is of primary importance. This study assesses the performance of a number of recent satellite-only and combined global geopotential models (GGMs) derived from CHAMP and GRACE missions’ data in comparison to the older EGM96 model, which is the underlying reference model for TG03. In this respect, gravity anomalies and geoid heights from the global geopotential models were compared with terrestrial gravity data and low-pass filtered GPS/levelling data, respectively. Also, five new gravimetric geoid models, computed by the Fast Fourier Transform technique using terrestrial gravity data and the geopotential models, were validated at the GPS/levelling benchmarks. The findings were also compared with the validation results of the TG03 model. The tests showed that as it was expected any of the high-degree combined models (EIGEN-CG03C, EIGEN-GL04C, EGM96) can be employed for determining the gravity anomalies over Turkey. In the west of Turkey, EGM96 and EIGEN-CHAMP03S fit the GPS/levelling surface better. However, all the tested GGMs revealed equal performance when they were employed in gravimetric geoid modelling after de-trending the gravimetric geoid model with corrector surface fitting. The new geoid models have improved accuracy (after fit) compared to TG03.     

Collinear and cross-over adjustment of Geosat ERM and Seasat altimeter data in the Mediterranean Sea

The computation of mean sea surface heights from a set of collinear Geosat ERM altimeter data tracks was carried out in a collinear adjustment, where 1 cy/rev cosine and sine coefficients for each track are estimated, so the differences between the collinear tracks are minimized. Then bias/tilt cross-over adjustments of stacked Geosat and Seasat altimetry were carried out. The problems with the free surface in the cross-over adjusted altimetric surface were treated using the absolute sea surface heights relative to the geoid model OSU89B. In a combined adjustment of the two altimeter data sets using cross-over and height informations simultaneously a RMS of the cross-overs of 0.080 m and a RMS of the sea surface heights relative to OSU89B of 0.611 m were obtained.     

Global detailed geoid computation and model analysis

This paper presents a survey of recent work on the gravimetric geoid. The gravity models considered are those published in the past few years by the Goddard Space Flight Center (GSFC), the Smithsonian Astrophysical Observatory (SAO) and the Ohio State University (OSU). Comparisons and analyses have been carried out through the ose of detailed gravimetric geoids which we have computed by combining the above-mentioned models with a set of 26 000, 1^ox1^o mean free air gravity anomalies. The accuracy of the detailed gravimetric geoid computed using the most recent Goddard Earth Model (GEM-6) in conjunction with the set 1^ox1^o mean free air gravity anomalies is assessed at 2 m on the continents of North America, Europe And Australia, 2 to 5 m in the North-East Pacific and North Atlantic areas and 5 to 10 m in other areas where surface gravity data are sparse. Rms differences between this detailed geoid and the detailed geoids computed using the other satellite gravity fields in conjunction with same set of surface data range from 3 to 7 m. The maximum differences in all cases occurred in the Southern Hemisphere where surface data and satellite observations are sparse. These differences exhibited wavelengths of approximately 30^o to 50^o in longitude. Detailed geoidal heights were also computed with models truncated to 12th degree and order as well as 8th degree and order. This truncation resulted in a reduction of the rms differences to a maximum of 5 m. Comparisons have been made with the astrogeodetic data of Rice (United States), Bomford (Europe), and Mather (Australia) and also with geoidal heights from satellite solutions for geocentric station coordinates in North America and the Caribbean.     

Merging of Airborne Gravity and Gravity Derived from Satellite Altimetry: Test Cases Along the Coast of Greenland

The National Survey and Cadastre - Denmark (KMS) has for several years produced gravity anomaly maps over the oceans derived from satellite altimetry. During the last four years, KMS has also conducted airborne gravity surveys along the coast of Greenland dedicated to complement the existing onshore gravity coverage and fill in new data in the very-near coastal area, where altimetry data may contain gross errors. The airborne surveys extend from the coastline to approximately 100 km offshore, along 6000 km of coastline. An adequate merging of these different data sources is important for the use of gravity data especially, when computing geoid models in coastal regions.The presence of reliable marine gravity data for independent control offers an opportunity to study procedures for the merging of airborne and satellite data around Greenland. Two different merging techniques, both based on collocation, are investigated in this paper. Collocation offers a way of combining the individual airborne gravity observation with either the residual geoid observations derived from satellite altimetry or with gravity derived from these data using the inverse Stokes method implemented by Fast Fourier Transform (FFT).     

Transformation between gravimetric and GPS/levelling-derived geoids using additional gravity information

The transformation from the gravimetric to the GPS/levelling-derived geoid using additional gravity information for the covariance function of geoid height differences has been investigated in a test area in south-western Canada. A “corrector surface” model, which accounts for datum inconsistencies, long-wavelength geoid errors, vertical network distortions and GPS errors, has been constructed using least-squares collocation. The local covariance function of geoid height differences is usually obtained from residual values between the GPS/levelling and gravimetric geoid heights after the elimination of all known systematic distortions. If additional gravity data (in the form of gravity anomalies) are available, the covariance function of geoid height differences can be determined by the following steps: (1) transforming the GPS/levelling-derived geoid heights into gravity anomalies; (2) forming differences between the computed in step 1 and given gravity anomalies; (3) determining the parameters of the local covariance function of the gravity anomaly differences; (4) constructing an analytical covariance model for the geoid height differences from the covariance function of the gravity anomaly differences using the parameters derived in step 3. The advantage of the proposed method stems from the great number of gravity data used to derive the empirical covariance function. A comparison with the least-squares adjustment shows that the standard deviation of the residuals of the predicted geoid height differences with respect to the control point values decreases by 2.4 cm.     

由多卫星测高大地水准面推断西太平洋边缘海构造动力格局

由于卫星观测技术的发展，现在已能利用测高大地水准面简捷可靠地研究海底构造与动力问题. 根据Geosat T2/ERM、Topex/Poseidon 和ERS 1/2测高数据给定的0°N ～45°N、100°E～150°E范围内4′×4′大地水准面，采用全球地形/均衡的重力位效应改正，计算布格、格莱尼和均衡大地水准面. 由格莱尼大地水准面反演Moho面埋深，再从均衡大地水准面起伏推算小尺度地幔流应力场. 结果表明，菲律宾海和南海显示了与磁条带、扩张脊对应或斜交的高频成分大地水准面起伏条带. 各边缘海盆的Moho面埋深有往南变浅的趋势，与菲律宾海各海盆的Moho面埋深大致相当，说明琉球—台湾—菲律宾岛弧两侧的构造动力强度基本相近. 从各种构造特征和大、中、小尺度的地幔流应力场的驱动机制，可以证明岛弧内侧的边缘海具有不同于大洋、大陆的独特构造动力格局和特征.     

Error sources and data limitations for the prediction of surface gravity: a case study using benchmarks

Gravity-based heights require gravity values at levelled benchmarks (BMs), which sometimes have to be predicted from surrounding observations. We use the Earth Gravitational Model 2008 (EGM2008) and the Australian National Gravity Database (ANGD) as examples of model and terrestrial observed data respectively to predict gravity at Australian National Levelling Network (ANLN) BMs. The aim is to quantify errors that may propagate into the predicted BM gravity values and then into gravimetric height corrections (HCs). Our results indicate that an approximate ±1 arc-min horizontal position error of the BMs causes maximum errors in EGM2008 BM gravity of ~22 mGal (~55 mm in the HC at ~2200 m elevation) and ~18 mGal for ANGD BM gravity because the values are not computed at the true location of the BM. We use RTM (residual terrain modelling) techniques to show that ~50% of EGM2008 BM gravity error in a moderately mountainous region can be accounted for by signal omission. Non-representative sampling of ANGD gravity in this region may cause errors of up to 50 mGals (~120 mm for the Helmert orthometric correction at ~2200 m elevation). For modelled gravity at BMs to be viable, levelling networks need horizontal BM positions accurate to a few metres, while RTM techniques can be used to reduce signal omission error. Unrepresentative gravity sampling in mountains can be remedied by denser and more representative re-surveys, and/or gravity can be forward modelled into regions of sparser gravity.     

A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran

A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.

- Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.

- The removed effects at the application points are restored on the surface of reference ellipsoid.

- Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.

- Computation of the geoid of Iran has successfully tested this new methodology.

融合多源数据构建区域航空重力梯度扰动全张量

通过联合全球重力位模型(EGM2008)、航空重力扰动数据和剩余地形模型(RTM)数据,基于频谱域(二维FFT变换)和空间域(Stokes数值积分)算法对毛乌素测区GT-2A航空重力测量系统采集的空中测线后处理重力扰动数据进行解算,构建了该地区的航空重力梯度扰动全张量.(1)残余航空重力扰动延拓结果表明:残余航空重力扰动经向下延拓至大地水准面,再向上延拓至航空高度后与原数据差值的标准差为1.0078 mGal,考虑边缘效应后,内缩计算范围得到的差值标准差减小至0.1269 mGal.(2)基于残余重力扰动数据(原航空高度数据及向下延拓数据),通过不同方案解算得到的梯度扰动结果表明:两种方案得到的研究区域重力梯度扰动各分量之差的最大标准差为6.4798E(Γ_(yz)分量),最小标准差为2.6968E(Γ_(xy)分量),内缩计算范围后得到的差值标准差最大值为1.8307E(Γ_(zz)分量),最小值为0.7223E(Γ_(yz)分量).本文的思路和方法可为未来我国自主构建航空重力梯度标定场提供参考.