Study of geoid–quasigeoid separation obtained from terrestrial gravity data and two geopotential models

In this article, separation between the geoid and the quasigeoid was calculated using ground gravity data and the data extracted from two Global Geopotential Models (GGMs). The calculated results were compared together. To do so, the authors used the terrestrial gravity data in a vast region of Iran, comprising 8,245 stations which are kindly put in our disposal by the National Cartographic Center of Iran, as well as two GGMs, namely EGM96 and EGM2008 for comparison. The calculation of the separation for GGMs was performed by iteration method. The results showed that the geoid–quasigeoid separations obtained from the terrestrial data versus the orthometric heights are nonlinear in mountainous areas, whereas they are almost linear in flat regions due to decreasing the values of the topographic potential of the masses between the earth surface and the geoid. On the other hand, in case of GGMs, a positive correlation was observed between the separations and the orthometric heights in both mountainous and flat areas. As the difference between the separations extracted by two methods in mountainous areas—especially in the regions with ragged topography—differs strongly, it is recommended to use the dense gravity and height networks for accurate determination of the geoid–quasigeoid separation in these regions. Finally, we can conclude that the mean values of separation by two global geopotential models (EGM96 and EGM2008) are 21.87 and 21.23 cm, respectively, values which did not differ strongly, whereas this mean value obtained from ground gravity data is 16.10 cm, which differs from the GGMs’ results with approximately 5 cm.     

A new high-resolution gravimetric geoid for South Spain and the Gibraltar Strait area: SOSGIS

A new gravimetric geoid is computed for South Spain and the Gibraltar Strait area. This geoid is located just in the junction between two tectonic plates (Euro-Asiatic and African plates) and in the junction of two gravimetric geoids: IGG2005 (the Iberian Gravimetric Geoid obtained in 2005) and MORGEO (the MORoccan GEOid). IGG2005 is the Iberian geoid and MORGEO is the Moroccan geoid, both geoids have been previously obtained. The new geoid is the gravimetric geoid solution that connects the two above-mentioned geoids, getting a more accurate and reliable picture of this area than the other previous geoids. The method used is the Stokes integral in convolution form, which shows to be an efficient method to reach the proposed objective. The terrain correction and the indirect effect have been taken into account. The new geoid is obtained as a regular grid (with a mesh size of 1.5′ × 1.5′) in the GRS80 reference system, covering the study area from 34° to 40° of latitude and from −8° to 0° of longitude. This gravimetric geoid and the previous geoids: IGG2005 and MORGEO; are compared to the geoid undulations derived at the validation points located on the study area (four GPS/levelling points measured on Morocco and five points of the European vertical reference network (EUVN) measured on Iberia). As it is expected, the new geoid is a more precise and reliable model, fitting the geoidal heights of these validation points with more accuracy than the other previous geoids. This new model will be useful for orthometric height determination by GPS in the mountains and remote areas, where levelling has many logistic problems. Also, it can be interesting for other geophysical purposes different to the height measurements, because it can provide a constraint for the density distribution, the thermal state of Lithosphere and the viscosity in the mantle. Such details can be inferred from a geoid model and the seismic velocity structure.     

Quasigeoid-to-geoid determination by EGM08

We present a method to estimate the difference between quasigeoid and geoid heights globally from the Earth Gravitational Model EGM08 and a related topographic model. The numerical computations with the standard topographic density of 2.67 g/cm³ show that the maximum and minimum of the separations are estimated to 5.47 m and −0.11 m on the Tibet plateau and in the Indian Ocean, respectively. These estimates do not consider possible topographic density variations, which result in topographic bias changes proportional to the topographic elevation squared. Assuming such density changes of 10% from the standard value, the separation may change up to 5 dm.     

地面重力数据结合GPS/水准数据精化兴城物探实验区大地水准面

区域大地水准面的确定是GPS测量常需解决的问题。目前确定大地水准面的方法主要包括重力法、GPS水准几何法及组合法,其中组合法因其精度和可靠性都较高,常用于计算高精度区域大地水准面。高精度的大地水准面模型是组合法确定区域大地水准面的关键。在我国,EGM2008全球重力场模型精度和分辨率均高于此前的所有模型,研究基于该模型的组合法大地水准面精化具有重要的实践意义。笔者以吉林大学兴城教学实习基地物探实验区为例,基于实测重力数据、EGM2008重力场模型和GPS水准数据,采用组合法精化了区域大地水准面,比较了组合法大地水准面模型和无重力实测数据的几何法大地水准面模型的精度差异,分析了该方法在物探测量中的适用性。结果表明,实验区组合法大地水准面模型精度最高达到1.2 cm,并且误差分布区间较小,总体上精度和可靠性高于对比的几何方法,并且组合法和几何法获取的两种大地水准面模型均能满足大比例尺物探测量要求。EGM2008模型精度较高,故平坦地区使用组合法时,高密度的实测重力数据可能带来高频扰动,有可能降低EGM2008重力场模型本身的精度,所以重力数据采集过程中要顾及重力点的密度和空间分布。本文方法更适用于地形复杂的地区。     

The high‐resolution gravimetric geoid of North Iberia: NIBGEO

The gravimetric geoid computed in the northern part of Iberia, is presented in this paper. This computation has been performed considering two study windows fitted to the areas with higher density of gravity data, to reduce the computation errors associated to the scarcity of gravity data, as much as possible. The bad influence of a bathymetry with poorer resolution than the topography is also reduced considering the smallest marine area possible. Moreover, the computation of this gravimetric model is based on the most recent geopotential model: EIGEN‐GL04C (obtained in 2006). The method used in the computation of the new gravimetric geoid has been the Stokes integral in convolution form. The terrain correction has been applied to the gridded gravity anomalies, to obtain the corresponding reduced anomalies. Also the indirect effect has been taken into account. Thus, a new geoid model has been calculated and it is provided as a data grid in the Geodetic Reference System of 1980, distributed for the northern part of Iberia from 40 to 44 degrees of latitude and ?10 to 4 degrees of longitude, on a 161 × 561 regular grid with a mesh size of 1.5′ × 1.5′. This new geoid and the previous geoid Iberian Gravimetric Geoid 2005, are compared with the geoid undulations measured for eight points of the European Vertical Reference Network (EUVN) on Iberia. The new geoid shows an improvement in precision and reliability, fitting the geoidal heights of these EUVN points with more accuracy than the previous geoid. Moreover, this new geoid has a smaller standard deviation (12.6 cm) than that obtained by any previous geoid developed for the Iberian area up to date. This geoid obtained for the northern part of Iberia will complement the previously obtained geoid for South Spain and the Gibraltar Strait area; both geoids jointly will give a complete picture of the geoid for Spain and the Gibraltar Strait area. This new model will be useful for orthometric height determination by GPS over this study area, because it will allow orthometric height determination in the mountains and remote areas, in which levelling has many logistic problems. This new model contributes to our knowledge of the geoid, but the surrounding areas must be better known to constrain the lithospheric and mantle models.     

Utilisation of Fast Fourier Transform and Least-squares Modification of Stokes formula to compile a gravimetric geoid model over Khartoum State: Sudan

We use Fast Fourier Transform (FFT) and least-squares modification (LSM) of Stokes formula to compute the geoid over Khartoum State in Sudan. The two methods (FFT and LSM) have been utilised to test their efficiency with respect to EGM08 and the local GPS-levelling data. The FFT method has many advantages, it is fast and it reduces the computational complexity. The modification of Stokes formula is widely used in geoid modelling; however, its implementation based on point-wise summation requires a considerable amount of time. In FFT, we combine the terrestrial gravity data and the global geopotential model (GGM) by means of a remove-compute-restore procedure and we successfully apply the modification of the Stokes formula in the least-squares sense. FFT and LSM geoid solutions are evaluated against EGM2008 and the GPS-levelling data. The analysis of the undulation differences shows that the LSM solution is more compatible with EGM08 and GPS-levelling data. The discrepancies of the differences are removed using a 4-parameter model, the standard deviation (STD) of the undulation differences of LSM decreased from 0.41 to 0.37 m and from 0.48 to 0.39 m for FFT solution. There is no significant impact to the LSM geoid when adding the additive corrections, while the FFT geoid solution is slightly improved when terrain correction is applied.     

EGY-HGM2016: an improved hybrid local geoid model for Egypt based on the combination of GOCE-based geopotential model with gravimetric and GNSS/levelling measurements

An improved hybrid gravimetric geoid model for Egypt, EGY-HGM2016, has been recently computed implementing the least-squares collocation (LSC) method through the remove-compute-restore (RCR) procedure. The computation of EGY-HGM2016 involves different datasets in terms of gravity anomalies determined from the GOCE (gravity field and steady-state ocean circulation explorer)-based global geopotential model (SPW-R4) up to d/o 200 and EGM2008 from d/o 201 to 720 combined with terrestrial gravity datasets in terms of 2140 gravity field anomalies and about 121,480 marine surface gravity anomalies. In addition, orthometric heights from 17 GPS/levelling measurements have been considered during the modelling process to improve the determination of the hybrid gravimetric geoid over the Egyptian region. The EGY-HGM2016 model estimated over Egypt provides geoid heights that are ranging from 7.677 to 21.095 m with a standard deviation (st. dev.) of about 2.534 m in the northwest of the country excluding the involvement of the orthometric heights from GPS/levelling measurements. When the later dataset is considered during the implementation of LSC process, hybrid residual height anomalies ranging from ?1.5 to +0.9 m, with a mean of 0.22 m and a st. dev. of 0.17 m, are obtained. Comparison of the predicted hybrid gravimetric geoid with the corresponding ones obtained from EGM2008, GOCE-based SPW R4 model, and GPS/levelling reveals considerable improvements of our EGY-HGM2016 model over Egypt.     

Precision assessment of the orthometric heights determination in northern part of Algeria by combining the GPS data and the local geoid model

The main purpose of this article is to discuss the use of GPS positioning together with a gravimetrically determined geoid, for deriving orthometric heights in the North of Algeria, for which a limited number of GPS stations with known orthometric heights are available, and to check, by the same opportunity, the possibility of substituting the classical spirit levelling. For this work, 247 GPS stations which are homogeneously distributed and collected from the international TYRGEONET project, as well as the local GPS/Levelling surveys, have been used. The GPS/Levelling geoidal heights are obtained by connecting the points to the levelling network while gravimetric geoidal heights were interpolated from the geoid model computed by the Geodetic Laboratory of the National Centre of Spatial Techniques from gravity data supplied by BGI. However, and in order to minimise the discordances, systematic errors and datum inconsistencies between the available height data sets, we have tested two parametric models of corrector surface: a four parameter transformation and a third polynomial model are used to find the adequate functional representation of the correction that should be applied to the gravimetric geoid. The comparisons based on these GPS campaigns prove that a good fit between the geoid model and GPS/levelling data has been reached when the third order polynomial was used as corrector surface and that the orthometric heights can be deducted from GPS observations with an accuracy acceptable for the low order levelling network densification. In addition, the adopted methodology has been also applied for the altimetric auscultation of a storage reservoir situated at 40 km from the town of Oran. The comparison between the computed orthometric heights and observed ones allowed us to affirm that the alternative of levelling by GPS is attractive for this auscultation.     

Recent geoid computations for the Hellenic area

New geoid computations for the Hellenic area are carried out using (a) gravity anomalies for the land area available from old and new data bases, and gravity data for the sea area derived from altimetry and a recent digitization of sea gravity maps, and (b) a 1km × 1km digital terrain model. The EGM96 geopotential model is used as the reference field. In order to assess the quality of the computed geoid heights in the continental area comparisons were carried out with GPS/leveling heights and the recently available European Gravimetric Geoid EGG97. In the sea area the geoid heights were compared with sea surface heights of the recent and more accurate TOPEX/POSEIDON (T/P) altimetry mission. At the end of this article the improvement of the data bases is discussed and some plans for further development in the methodological schedule are pointed out.     

GPS高程拟合方法精度分析

近年来人们往往利用GPS数据来确定大地高,但大地高不同于正常高,为此,利用多项式拟合与地球重力场模型相结合的数学方法,使GPS所测大地高通过这些数学模型直接转换为具有厘米精度的正常高,将该方法得到的正常高与单独利用多项式拟合和地球重力场模型得到的结果进行了比较,其差值的标准差为±38 cm。     

Description and release of Australian gravity field model testing data

Gravimetric geoid and/or quasigeoid models are routinely evaluated using co-located GPS-levelling and/or astrogeodetic vertical deflections, globally and regionally. This short note describes these ground-truth data for Australia as of August 2017, which are provided as Electronic Supplementary Material. We provide ～7500 GPS-derived ellipsoidal heights, normal-orthometric heights from the 1971 adjustment of the Australian Height Datum, normal heights from a readjustment of levelling constrained to a model of the ocean's mean dynamic topography, and ～1000 historical astrogeodetic vertical deflections. Updates to these data will be posted on the Intergovernmental Committee on Surveying and Mapping GitHub repository (https://github.com/icsm-au), together with a readme.txt file describing them.     

Modelling the geoid and sea-surface topography in coastal areas

This paper looks at the relation between the time-averaged level of the sea surface and a gravimertic geoid, as determined in coastal areas. Measurements in local regions can now be accurate enough to demonstrate that the geoid and mean sea level are not even parallel to each other, let alone identical. The accuracy and pattern structure of surface gravity data in some shelf seas is comparable with those on land, so that a marine geoid can be derived from surface data without using satellite altimetry. The geodetic objective is then to combine the two to determine sea surface topography. In principle, gravimetric studies provide the absolute datum so that local oceanographic models on the shelf can be combined with sea surface topography models related to the global ocean circulation. In contrast, sea surface topography information near deep ocean coasts must come from external sources and satellite altimetry used to give the gravity data needed to offset the less good coverage by ship-borne gravimetry.Marine Bouguer anomalies enable two specific problems of gravity anomaly patterns near the continent ocean transition to be overcome. The necessary extension of Stokes' condensation reduction is developed and illustrated along a north-south profile from the Mediterranean across the Cote d'Azur. The effect on gravity of deep ocean water introduces a geoid correction in the form of a dipolar ridge whose amplitude at the shore is about 11 cm. In addition to geostrophic currents, a semi-quantitative model for the thermohaline effects on sea surface topography is discussed in relation to sea level differences between the Atlantic and Mediterranean.In considering appropriate algorithms for local geoid computation, Kirby's Iterative Fourier Combination routine for combining altimetry and surface gravity is extended to account for global sea surface topography. The impact of very fast spherical harmonic analysis algorithms is discussed and a simple physical model is given which explains the short coherence lengths found for the global gravity field. This necessary assumption for any local geoid computation was hitherto purely empirical.Finally, the use of land data such as tide gauges, ellipsoidal heights from GPS, and orthometric heights from first order levelling are reviewed as ways of corroborating geodetic estimates of sea surface topography and its relation to levelling datums. Successful examples are given from southern England.     

Using Newton's law and geophysical bounds on mass density contrast to ensure consistency between gravity and height data

In this paper we advocate the use of Newton's law of gravitational attraction to ensure perfect consistency between gravity and height data. Starting with the absolute gravity on the topography we decompose this signal into a number of quantities associated with physics of the system. To model gravitational attraction from topography we use DTM and Newton's law of gravitational attraction. A residual part of the gravity signal is interpreted as inconsistency between gravity and heights. In the paper we discuss a method by which such inconsistency (at least in principle) can be decomposed into a “gravity error” and a “terrain error”. In practice such separation is not possible because the two types of error are nearly 100% correlated. The inconsistency can be interpreted as a measure of ambiguity of the gravity-terrain models which are consistent with a set of measured/interpolated data. We discuss the influence of such ambiguity on the accuracy of the geoid for the investigated area of Jutland, Denmark.     

Toward a 10' resolution geoid for South America: a comparison study

Since the creation of the Sub-Commission for the Geoid in South America (SCGSA) in 1993, many efforts have been carried out in the different countries in order to improve the geoid computations. The validation of the gravity data in Brazil, Uruguay, Argentina and Chile has improved many of the gravity surveys in those countries. GPS observations carried out on benchmarks of the geometric levelling have been facilitated by the SIRGAS (Geocentric Reference System for South America) project and can contribute for testing the gravimetric determination of the geoid. Several countries made available GPS data for SCGSA like Brazil, Argentina, Venezuela and Chile. The Digital Terrain Model (DTM) has been improved considerably in Brazil and Argentina. A great number of topographic maps has been digitized to generate a DTM grid of 3′ resolution (DTM3). New gravity surveys in the Amazonas region have been in progress along Rio Negro and its tributaries. Many different organizations in most of the countries in South America have been involved with local or national geoid computations. This fact has brought attention to the data in several countries facilitating the efforts for a continental geoid. All these activities are strongly supported by Geophysical Exploration Technology (GETECH) — University of Leeds. The objective envisaged at the moment is to produce a 10′ resolution geoid for South America using FFT and to compare the result with that of the numerical integration of the modified Stokes integral.     

Improving the geoid accuracy by adapting the reference field

In the frame work of the Austrian Geoid 2000 project, the accuracy of the geoid has to be enhanced. One of the possibilities is to use an adapted reference field. The traditional remove/restore technique has the disadvantage of removing a part of the effect of the topography and its compensation twice. In order to avoid this double consideration the effect of the topographic-isostatic masses for the used fixed data window has been subtracted from the reference field yielding an adapted reference field. This adapted reference field has been used for the remove/restore technique. The paper shows a comparison between the geoid computed by the adapted technique using the enhanced reference field, the geoid computed by the traditional remove/restore technique and the geoid based on the RTM-method for the remove/restore process. All the models are checked against a geoid derived by GPS and levelling points. The geoid based on the adapted remove/restore technique shows the best results.     

Local geoid computation and evaluation

A local geoid solution for the northern part of Greece is presented based on a recent processing of newly available gravity data in the area 40.25 ≤ /o ≤ 41.00, 22.5 ≤λ ≤ 24.25. The derived gravimetric geoid heights are compared with geoid heights computed at recently measured GPS/ leveling benchmarks. A 4-parameter transformation model is applied to the differences between the two aforementioned geoid height sets, and a discussion is given on the current state of the leveling datum in the test area and the Greek territory. Regional and local transformation parameters are computed and some numerical tests are performed. A common adjustment of gravimetric geoid heights and corresponding GPS/leveling heights will be carried out in another study following an integrated procedure in order to study problems arising from the combination of different height data sets for geoid determination. Finally, some conclusions are drawn on the problems related to the optimization of a local geoid solution.     

GPS测量中的坐标转换

GPS测量中坐标系变换与基准变换：同一基准面下的坐标转换,不同基准面下的坐标转换;GPS数据坐标转换实现。     

青藏高原及邻区大地水准面异常场源结构

以青藏高原及邻区地形、地震层析成像、沉积层底面、Moho面及岩石层底面资料为基础 ,讨论了由地形起伏、地球内部各界面以及物质密度不均匀引起的大地水准面异常的计算方法 ,正演计算出青藏高原及邻区岩石圈内部物质不均匀产生的大地水准面异常 ,并把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 ,以寻求表示青藏高原及邻区地幔物质不均匀的大地水准面异常球谐函数的最佳阶次。结合地震层析成像资料分析结果得出 ,表示青藏高原及邻区下地幔、上地幔及岩石圈物质不均匀的大地水准面异常球谐函数的阶次范围分别为 2 6阶 ,7 60阶和 61 3 60阶。     

Tectonic implied results from geoid and topography signals

The topographic surface is a measure of static equilibrium from the actual density distribution within the outmost Earth's lithosphere. The natural height reference of this surface, known as geoid, reflects the mixed mass-density effects, caused by the same sources, without the contribution of topographic mass. Geoid undulation and topography are output signals, which carry in common a large part of the contribution from the causal “sources”. This contribution appears in both types of signal. Comparisons between the signals depict the geographical location and an estimation of the depth occurrence of areas with geophysical and tectonic formations depending on their correlation rate. We present results from the Greek region, known for its complex diversity in topography, tectonics and dynamics. The tests are in point and “surface” concept, from local and global signals of geoid and topography. Local geoid is represented at 91 GPS points and EGM 96 coefficients compute its global representation. The topography is point values within the area, and the ETOPO5 5′X5′ data within the geographical frame.     

萨吾尔山木斯岛冰川厚度特征及冰储量估算

冰下地形与冰川体积的估算对冰川水资源研究具有重要意义.以萨吾尔山木斯岛冰川为研究对象,利用Landsat影像数据、探地雷达(ground penetrating radar,简称GPR)冰川厚度数据以及差分GPS数据,分析模拟了萨吾尔山木斯岛冰川横纵剖面的厚度分布特征,采用多种插值方法比较分析,得到木斯岛冰川冰舌区的厚度分布图,初步估算了该冰川的冰储量.结合数字高程模型数据及冰川厚度分布图,绘制了木斯岛冰川冰舌区冰床地形图.研究表明,两个横剖面的冰川槽谷形态存在较大的差异.横剖面B1-B2有典型的“U”型地形发育,冰川厚度可达116.4 m;C1-C2横剖面底部地形比较平缓,冰川厚度分布较均匀,平均在70~90 m.纵测线A1-A2冰下地形成阶梯状分布,纵剖面冰体平均厚度约为80.89 m,最大冰体厚度为122.67 m.木斯岛冰川的冰床地形图与该冰川的冰厚度等值线图形成明显对比.在海拔3 240 m和3 280 m处存在明显的冰斗地形地貌.初步估算木斯岛冰川冰舌区的平均厚度和冰储量分别为60.5 m和0.195 km3.与传统计算冰储量的方法相比,利用GPR测量得到的冰川厚度数据来插值计算冰储量的方法,具有更高的准确性.