High Resolution Constituents of the Earth’s Gravitational Field

During the General Assembly of the European Geosciences Union in April 2008, the new Earth Gravitational Model 2008 (EGM08) was released with fully-normalized coefficients in the spherical harmonic expansion of the Earth’s gravitational potential complete to degree and order 2159 (for selected degrees up to 2190). EGM08 was derived through combination of a satellite-based geopotential model and 5 arcmin mean ground gravity data. Spherical harmonic coefficients of the global height function, that describes the surface of the solid Earth with the same angular resolution as EGM08, became available at the same time. This global topographical model can be used for estimation of selected constituents of EGM08, namely the gravitational potentials of the Earth’s atmosphere, ocean water (fluid masses below the geoid) and topographical masses (solid masses above the geoid), which can be evaluated numerically through spherical harmonic expansions. The spectral properties of the respective potential coefficients are studied in terms of power spectra and their relation to the EGM08 potential coefficients is analyzed by using correlation coefficients. The power spectra of the topographical and sea water potentials exceed the power of the EGM08 potential over substantial parts of the considered spectrum indicating large effects of global isostasy. The correlation analysis reveals significant correlations of all three potentials with the EGM08 potential. The potential constituents (namely their functionals such as directional derivatives) can be used for a step known in geodesy and geophysics as the gravity field reduction or stripping. Removing from EGM08 known constituents will help to analyze the internal structure of the Earth (geophysics) as well as to derive the Earth’s gravitational field harmonic outside the geoid (geodesy).     

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.     

Local geoid determination based on airborne gravity data

A mathematical model used for determination of a local geoid model by combining airborne gravity disturbances and the Earth Gravitational Model 2008 (EGM08) is shortly reviewed. The precision of the estimated local geoid model of Taiwan is tested by its comparison with the “real” geoid at Global Satellite Navigation Systems (GNSS)/levelling points. The same comparison at GNSS/levelling points is done for the geoid evaluated only by using EGM08. Conclusions concerning a rate of improvement of the “global” geoid from EGM08 using the “local” geoid from airborne gravity data are presented.     

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.     

Ability of the EGM2008 high degree geopotential model to calculate a local geoid model in Valencia,Eastern Spain

A new generation of global geopotential models (GGM) is being developed. These solutions offer a file with fully-normalized spherical harmonic coefficients of the Earth’s gravitational potential up to a degree greater than 2000 with very low commission errors. This paper analyses the recent Earth Gravitational Model EGM2008, developed up to degree and order 2159 with additional coefficients to degree 2190 and order 2159, which means recovering the gravitational field up to approximately 20 km wavelengths. 223 GPS/levelling points of the new Spanish High Precision Levelling Network in the Valencia region (Eastern Spain) are used as external tool for evaluation in that particular region. The same evaluation has been performed to other different global (EGM96 and EIGENCG03C), continental (EGG97), regional (IGG2005 and IBERGEO2006) and local (GCV07) geoid models for comparison purposes only. These comparisons show that EGM2008 is the geoid model that best fits to the GPS/levelling data in that region.     

Use of EGM08 model and shuttle radar topography mission data for geoid computation in the State of Rio de Janeiro,Brazil: A case study with Voronoi/Delaunay discretisations

The EGM08 geopotential model complete to degree and order 2159 was used in a remove-compute-restore (RCR) method for the geoid computation in the State of Rio de Janeiro, Brazil. Terrain and indirect effect corrections were computed using a 6-arcsec resolution DTE, derived from the TOPODATA Project (Shuttle Radar Topography Mission data) raised by the National Institute for Space Research. INPE, Brazil. We applied Voronoi/Delaunay discretisations for discrete Stokes integration. In these schemes, target area is partitioned into polygons/triangles, respectively, and the computation is carried out by point-wise numerical integration and no gridding is mandatory. For both procedures, the cells were produced using either observed gravity data combined with gridded Bouguer derived information. Particularly in Delaunay scheme, as the gravity anomalies are interpolated into the triangular cells, and geoid undulations are computed for their vertices, Stokes function singularity was gone. Externally estimated errors resulting from a comparison with GPS/leveling data were presented for both the schemes and classical ones, as well as for the EGM08 undulations. They yielded RMS differences equal to 0.105 m, 0.110 m, 0.110 m, 0.115 m and 0.228 m, respectively, for Voronoi, Delaunay, Voronoi/Delaunay with gridded-data alone and EGM08, computed between 32 GPS/leveling points.     

地形-均衡补偿重力、大地水准面异常频谱分析

将地形高、地球内部质量异常以及重力、大地水准面展开成球谐级数,依据岩石圈弹性挠曲均衡补偿理论建立地形—均衡补偿重力、大地水准面异常的球谐级数表达式.由此我们可以研究地形—均衡补偿重力、大地水准面异常与球谐级数阶次的关系,以及不同波长地形荷载与岩石圈挠曲补偿的关系,探讨地形—均衡补偿重力、大地水准面的频谱特性.从观测大地水准面异常和自由空气重力异常扣除地形—均衡补偿大地水准面、重力异常,可以得到均衡大地水准面异常和均衡重力异常.均衡大地水准面异常已经消除了浅层物质不均匀的影响,反映的是地球深部物质密度不均匀分布.均衡重力异常显示出中短波长特性,反映的是地壳上地幔物质分布的失衡和物质调整的动力学特征.     

One technique for refining the global Earth gravity models

The results of the theoretical and experimental research on the technique for refining the global Earth geopotential models such as EGM2008 in the continental regions are presented. The discussed technique is based on the high-resolution satellite data for the Earth’s surface topography which enables the allowance for the fine structure of the Earth’s gravitational field without the additional gravimetry data. The experimental studies are conducted by the example of the new GGMplus global gravity model of the Earth with a resolution about 0.5 km, which is obtained by expanding the EGM2008 model to degree 2190 with the corrections for the topograohy calculated from the SRTM data. The GGMplus and EGM2008 models are compared with the regional geoid models in 21 regions of North America, Australia, Africa, and Europe. The obtained estimates largely support the possibility of refining the global geopotential models such as EGM2008 by the procedure implemented in GGMplus, particularly in the regions with relatively high elevation difference.     

Geostatistical prediction of a local geometric geoid - kriging and cokriging with the use of EGM2008 geopotential model

We investigate prediction abilities of different variants of kriging and different combinations of data in a local geometric (GNSS/leveling based) geoid modeling. In order to generate local geoid models, we have used GNSS/leveling data and EGM2008 geopotential model. EGM2008 has been used twofold. Firstly, it was used as a basic long wave-length trend to be removed from geoid undulation data to generate a residual field of geoid heights modeled later by kriging (remove-restore technique). Secondly, EGM2008-based undulations were used as a secondary variable in a cokriging prediction procedure (as pseudo-observations). Besides the use of EGM2008, the kriging-based local geometric geoid models were generated only on the basis of raw undulations data. Kriging itself was used in two variants, i.e. ordinary kriging and universal kriging for univariate and bivariate cases (cokriging). The quality of kriging-based prediction for all its variants and all data combinations have been investigated on one fixed validation dataset consisting of 86 points and three training data sets characterized by a different density of sampling. Results of this study indicate that incorporation of EGM08 as a long wave-length trend in kriging prediction procedure outperforms cokriging strategy based on incorporation of EGM08 as a secondary spatially correlated variable.     

Update of the precision geoid determination in Korea

From the late 1990s, many studies on local geoid construction have been made in South Korea. However, the precision of the previous geoid has remained about 15 cm due to distribution and quality problems of gravity and GPS/levelling data. Since 2007, new land gravity data and GPS/levelling data have been obtained through many projects such as the Korean Land Spatilaization, Unified Control Point and Gravity survey on the Benchmark. The newly obtained data are regularly distributed to a certain degree and show much better improvement in their quality. In addition, an airborne gravity survey was conducted in 2008 to cover the Korean peninsula (South Korea only). Therefore, it is expected that the precision of the geoid could be improved. In this study, the new South Korean gravimetric geoid and hybrid geoid are presented based on land, airborne, ship‐borne, altimeter gravity data, geopotential model and topographic data. As for the methodology, the general remove‐restore approach was applied with the best chosen parameters in order to produce a precise local geoid. The global geopotential model EGM08 was used to remove the low‐frequency components using degree and order up to 360 and the short wavelength part of the gravity signal was dealt with by using the Shuttle Radar Topography Mission data. The parameters determined empirically in this study include for Stokes’ integral 0.5° and for Wong‐Gore kernel 110–120°, respectively and 10 km for both the Bjerhammar sphere depth and attenuation factor. The final gravimetric geoid in South Korea ranges from 20–31 m with a precision of 5.45 cm overall compared to 1096 GPS/levelling data. In addition, the South Korean hybrid geoid produces 3.46 cm and 3.92 cm for degrees of fitness and precision, respectively and a better statistic of 2.37 cm for plain and urban areas was achieved. The gravimetric and hybrid geoids are expected to improve further when the refined land gravity data are included in the near future.     

An estimation of the height system bias parameter <Emphasis Type="Italic">N</Emphasis><Subscript>0</Subscript> using least squares collocation from observed gravity and GPS-levelling data

This paper deals with the analysis of gravity anomaly and precise levelling in conjunction with GPS-Levelling data for the computation of a gravimetric geoid and an estimate of the height system bias parameter N_o for the vertical datum in Pakistan by means of least squares collocation technique. The long term objective is to obtain a regional geoid (or quasi-geoid) modeling using a combination of local data with a high degree and order Earth gravity model (EGM) and to determine a bias (if there is one) with respect to a global mean sea surface. An application of collocation with the optimal covariance parameters has facilitated to achieve gravimetric height anomalies in a global geocentric datum. Residual terrain modeling (RTM) technique has been used in combination with the EGM96 for the reduction and smoothing of the gravity data. A value for the bias parameter N_o has been estimated with reference to the local GPS-Levelling datum that appears to be 0.705 m with 0.07 m mean square error. The gravimetric height anomalies were compared with height anomalies obtained from GPS-Levelling stations using least square collocation with and without bias adjustment. The bias adjustment minimizes the difference between the gravimetric height anomalies with respect to residual GPS-Levelling data and the standard deviation of the differences drops from 35 cm to 2.6 cm. The results of this study suggest that N_o adjustment may be a good alternative for the fitting of the final gravimetric geoid as is generally done when using FFT methods.     

Geoid determination by spectral combination of an Earth gravitational model with airborne and terrestrial gravimetry — a theoretical study

Today air-gravimetry is a versatile technique to quickly collect gravity data over large regions, where terrestrial gravity data are sparse and/or of poor quality. The method requires the data to be downward continued to sea level for use in geoid determination, an inverse problem operation that calls for smoothing of the data and/or the kernel function involved (in either spectral or space domain). In this purely theoretical study we avoid this separate computational step by performing direct geoid estimation by so-called spectral combination/filtering of the data, which includes terrestrial gravimetry, airgravimetry, an Earth Gravitational Model (EGM) as well as their signal and error degree variances. Each derived geoid estimator is presented as the sum of one or two integral formulas and the harmonic series of the EGM together with the expected mean square error of the estimator. The article is limited to a theoretical study, leaving its practical tests for future investigation.     

Regional gravity field modeling based on rectangular harmonic analysis

Regional gravity field modeling with high-precision and high-resolution is one of the most important scientific objectives in geodesy,and can provide fundamental information for geophysics,geodynamics,seismology,and mineral exploration.Rectangular harmonic analysis(RHA)is proposed for regional gravity field modeling in this paper.By solving the Laplace’s equation of gravitational potential in local Cartesian coordinate system,the rectangular harmonic expansions of disturbing potential,gravity anomaly,gravity disturbance,geoid undulation and deflection of the vertical are derived,and so are the formula for signal degree variance and error degree variance of the rectangular harmonic coefficients(RHC).We also present the mathematical model and detailed algorithm for the solution of RHC using RHA from gravity observations.In order to reduce the edge effects caused by periodic continuation in RHA,we propose the strategy of extending the size of computation domain.The RHA-based modeling method is validated by conducting numerical experiments based on simulated ground and airborne gravity data that are generated from geopotential model EGM2008 and contaminated by Gauss white noise with standard deviation of 2 mGal.The accuracy of the 2.5′×2.5′geoid undulations computed from ground and airborne gravity data is 1 and 1.4cm,respectively.The standard error of the gravity disturbances that downward continued from the flight height of 4 km to the geoid is only 3.1 mGal.Numerical results confirm that RHA is able to provide a reliable and accurate regional gravity field model,which may be a new option for the representation of the fine structure of regional gravity field.     

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.     

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.     

A comparison of GOCE gravitational models with EGM2008

Four new gravity field models from GOCE, two of them combined with GRACE, are compared here with EGM2008. The objectives are to look into the differences in consecutive ranges of the spherical harmonic expansion globally as well as in selected geographical regions and in the regions of the various data sources used for EGM2008. In general, GOCE is able to contribute to improved global gravity models in the spherical harmonic range between 120 and 200 (and above). The agreement between EGM2008 and the GOCE models is very good in well-surveyed regions such as North America, Europe and Australia, with geoid RMS-differences on the order of 4–6 cm. In other regions, where the surface gravity data available for the development of EGM2008 were poor, such as South America, Africa, South-East Asia or China the RMS-differences are on a level of 30 cm. Here GOCE leads to a significant improvement. These findings are confirmed by the analysis of the areas of the various EGM2008 data sources. In the regions of the so-called “fill-in” data of EGM2008 RMS-geoid height differences are high. In Antarctica GOCE also gives important improvements in terms of spatial resolution and accuracy. In general, the agreement between EGM2008 and the GOCE-models up to degree and order (d/o) 200 is good, with a global (excluding the polar gaps of GOCE orbits, throughout) geoid difference RMS of 11 cm, in the ocean areas 8 cm and 20 cm in the continental areas. GOCE models are better suited for ocean circulation studies because no prior ocean information enters into the data reduction process, as it is the case when deducing gravity anomalies from an altimetric mean sea surface. On the other hand, the good consistency between GOCE-models and EGM2008 in ocean areas very likely indicates that the influence of ocean circulation information on EGM2008 is rather small. The four tested GOCE models behave similarly except at the highest latitudes where GOCE lacks data due to its orbit inclination of 96.5° and some form of regularization which has to be applied.     

海域航空重力快速构建区域大地水准面

我国在海域开展了大规模的航空重力勘探,这些资料对构建高精度大地水准面具有重要价值.基于此,本文提出一种利用海域航空重力测量数据快速构建大地水准面的方法.该方法基于移去-恢复法思想,利用位场最小曲率方法对航空重力数据进行高精度向下延拓并获取相应的扰动位,实现航空重力测量快速构建海域大地水准面.与斯托克斯积分计算相比,采用...     

中国及邻区大地水准面异常的场源深度探讨

利用中国及邻区地形、地震层析成像、沉积层底面、Moho面及岩石层底面资料,正演计算出中国及邻区岩石圈大地水准面异常;再从全阶大地水准面异常中扣除正演模拟得到的岩石圈大地水准面异常与不同阶次波段的大地水准面进行比较,寻求表示中国及邻区地幔物质不均匀的大地水准面异常的最佳阶次为2-60阶. 结果表明,对应于岩石圈的大地水准面异常的重力位球谐函数阶数为61-20阶;下地幔重力位球谐函数阶数为2-6阶;而-60阶重力位球谐函数则表示中国及邻近区域上地幔大地水准面异常.     

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.