A local geoid determination based on ellipsoid approximation in Western China

A new methodology for precise geoid determination with finest local details based on ellipsoidal approximation is presented. This methodology is formulated through the “fixed-free two-boundary value problem” based on the observable of the type modulus of gravity intensity, gravity acceleration and gravity potential at the GPS positioned stations, with support of the known geoid's potential value, W0.     

ITG-CHAMP01: a CHAMP gravity field model from short kinematic arcs over a one-year observation period

Global gravity field models have been determined based on kinematic orbits covering an observation period of one year beginning from March 2002. Three different models have been derived up to a maximum degree of n=90 of a spherical harmonic expansion of the gravitational potential. One version, ITG-CHAMP01E, has been regularized beginning from degree n=40 upwards, based on the potential coefficients of the gravity field model EGM96. A second model, ITG-CHAMP01K, has been determined based on Kaulas rule of thumb, also beginning from degree n=40. A third version, ITG-CHAMP01S, has been determined without any regularization. The physical model of the gravity field recovery technique is based on Newtons equation of motion, formulated as a boundary value problem in the form of a Fredholm-type integral equation. The observation equations are formulated in the space domain by dividing the one-year orbit into short sections of approximately 30-minute arcs. For every short arc, a variance factor has been determined by an iterative computation procedure. The three gravity field models have been validated based on various criteria, and demonstrate the quality of not only the gravity field recovery technique but also the kinematically determined orbits.     

Global height datum unification: a new approach in gravity potential space

The problem of “global height datum unification” is solved in the gravity potential space based on: (1) high-resolution local gravity field modeling, (2) geocentric coordinates of the reference benchmark, and (3) a known value of the geoid’s potential. The high-resolution local gravity field model is derived based on a solution of the fixed-free two-boundary-value problem of the Earth’s gravity field using (a) potential difference values (from precise leveling), (b) modulus of the gravity vector (from gravimetry), (c) astronomical longitude and latitude (from geodetic astronomy and/or combination of (GNSS) Global Navigation Satellite System observations with total station measurements), (d) and satellite altimetry. Knowing the height of the reference benchmark in the national height system and its geocentric GNSS coordinates, and using the derived high-resolution local gravity field model, the gravity potential value of the zero point of the height system is computed. The difference between the derived gravity potential value of the zero point of the height system and the geoid’s potential value is computed. This potential difference gives the offset of the zero point of the height system from geoid in the “potential space”, which is transferred into “geometry space” using the transformation formula derived in this paper. The method was applied to the computation of the offset of the zero point of the Iranian height datum from the geoid’s potential value W ₀=62636855.8 m²/s². According to the geometry space computations, the height datum of Iran is 0.09 m below the geoid.     

全球高程基准研究进展

建立统一的全球高程基准是国际大地测量科学界的核心目标之一,也是全球尺度地球科学研究、跨境工程应用等的必要基础设施。国际大地测量协会（international geodesy association,IAG）2015年发布了国际高程参考系统的定义,并于2019年提出了建立国际高程参考框架的目标。从全球高程参考系统的理论基础和定义出发,对国际高程参考系统与框架的理论、方法和实际问题开展论述与研究,主要包括全球大地水准面重力位$W_{\mathrm{0}}$的确定、基于高阶重力场模型的重力位确定、基于区域重力场建模的重力位确定,并重点论述和分析了IAG组织的科罗拉多大地水准面建模试验和中国2020珠峰高程测量实现国际高程参考系统2项典型案例研究。结果表明,在平坦地区和一般山区,重力大地水准面模型精度能达到1 cm（重力位0.1 m2/s2）,即使在珠穆朗玛峰这样的特大山区,也有望达到2~3 cm精度（重力位0.2~0.3 m2/s2）。综合典型案例研究结果、观测技术、数据资源和区域分布等因素,提出了建立国际高程参考框架的初步策略,包括IHRF参考站布设、重力位确定方法、数据要求、应遵循的标准/约定和预期精度指标等,展望了光学原子钟与相对论大地测量对于全球高程基准统一的潜在贡献。     

Geodetic methods to determine the relativistic redshift at the level of 10$$^{}$$ in the context of international timescales: a review and practical results

The frequency stability and uncertainty of the latest generation of optical atomic clocks is now approaching the one part in \(10^{18}\) level. Comparisons between earthbound clocks at rest must account for the relativistic redshift of the clock frequencies, which is proportional to the corresponding gravity (gravitational plus centrifugal) potential difference. For contributions to international timescales, the relativistic redshift correction must be computed with respect to a conventional zero potential value in order to be consistent with the definition of Terrestrial Time. To benefit fully from the uncertainty of the optical clocks, the gravity potential must be determined with an accuracy of about \(0.1\,\hbox {m}^{2}\,\hbox {s}^{-2}\), equivalent to about 0.01 m in height. This contribution focuses on the static part of the gravity field, assuming that temporal variations are accounted for separately by appropriate reductions. Two geodetic approaches are investigated for the derivation of gravity potential values: geometric levelling and the Global Navigation Satellite Systems (GNSS)/geoid approach. Geometric levelling gives potential differences with millimetre uncertainty over shorter distances (several kilometres), but is susceptible to systematic errors at the decimetre level over large distances. The GNSS/geoid approach gives absolute gravity potential values, but with an uncertainty corresponding to about 2 cm in height. For large distances, the GNSS/geoid approach should therefore be better than geometric levelling. This is demonstrated by the results from practical investigations related to three clock sites in Germany and one in France. The estimated uncertainty for the relativistic redshift correction at each site is about \(2 \times 10^{-18}\).     

利用重力异常数据构建重力梯度场的方法比较

为实现大范围、高精度基准重力梯度数据库的构建,考虑到重力梯度场对地形质量的敏感效应,一般利用恒密度数字高程模型来求取重力梯度值,从而忽略了地形密度变化以及水准面以下密度异常对重力梯度的影响。根据重力位理论中求解边值问题的数值应用方法,直接利用重力异常数据求取重力梯度场,弥补了密度变化和密度异常在重力梯度上的反映。根据模型算例和实测重力异常数据求取了剖面重力梯度值,结果表明,限于重力数据空间分辨率的影响,利用重力异常数据可恢复中长波段重力梯度场。该方法与地形数据求取重力梯度和卫星重力梯度测量等方法技术相结合,对重力梯度数据库的建设具有实际应用价值。     

A comparison of methods for the inversion of airborne gravity data

Four integral-based methods for the inversion of gravity disturbances, derived from airborne gravity measurements, into the disturbing potential on the Bjerhammar sphere and the Earths surface are investigated and compared with least-squares (LS) collocation. The performance of the methods is numerically investigated using noise-free and noisy observations, which have been generated using a synthetic gravity field model. It is found that advanced interpolation of gravity disturbances at the nodes of higher-order numerical integration formulas significantly improves the performance of the integral-based methods. This is preferable to the commonly used one-point composed Newton–Cotes integration formulas, which intrinsically imply a piecewise constant interpolation over a patch centered at the observation point. It is shown that the investigated methods behave similarly for noise-free observations, but differently for noisy observations. The best results in terms of root-mean-square (RMS) height-anomaly errors are obtained when the gravity disturbances are first downward continued (inverse Poisson integral) and then transformed into potential values (Hotine integral). The latter has a strong smoothing effect, which damps high-frequency errors inherent in the downward-continued gravity disturbances. An integral method based on the single-layer representation of the disturbing potential shows a similar performance. This representation has the advantage that it can be used directly on surfaces with non-spherical geometry, whereas classical integral-based methods require an additional step if gravity field functionals have to be computed on non-spherical geometries. It is shown that defining the single-layer density on the Bjerhammar sphere gives results with the same quality as obtained when using the Earths topography as support for the single-layer density. A comparison of the four integral-based methods with LS collocation shows that the latter method performs slightly better in terms of RMS height-anomaly errors.     

Geoid determination using one-step integration

A residual (high-frequency) gravimetric geoid is usually computed from geographically limited ground, sea and/or airborne gravimetric data. The mathematical model for its determination from ground gravity is based on the transformation of observed discrete values of gravity into gravity potential related to either the international ellipsoid or the geoid. The two reference surfaces are used depending on height information that accompanies ground gravity data: traditionally orthometric heights determined by geodetic levelling were used while GPS positioning nowadays allows for estimation of geodetic (ellipsoidal) heights. This transformation is usually performed in two steps: (1) observed values of gravity are downward continued to the ellipsoid or the geoid, and (2) gravity at the ellipsoid or the geoid is transformed into the corresponding potential. Each of these two steps represents the solution of one geodetic boundary-value problem of potential theory, namely the first and second or third problem. Thus two different geodetic boundary-value problems must be formulated and solved, which requires numerical evaluation of two surface integrals. In this contribution, a mathematical model in the form of a single Fredholm integral equation of the first kind is presented and numerically investigated. This model combines the solution of the first and second/third boundary-value problems and transforms ground gravity disturbances or anomalies into the harmonically downward continued disturbing potential at the ellipsoid or the geoid directly. Numerical tests show that the new approach offers an efficient and stable solution for the determination of the residual geoid from ground gravity data.     

利用大地边值问题方法统一全球高程基准

为解决世界各国高程基准差异的问题,提出联合卫星重力场模型、地面重力数据、GNSS大地高、局部高程基准的正高或正常高,按大地边值问题法确定局部高程基准重力位差的方法。首先推导了利用传统地面"有偏"重力异常确定高程基准重力位差的方法;接着利用改化Stokes核函数削弱"有偏"重力异常的影响,并联合卫星重力场模型和地面"有偏"重力数据,得到独立于任何局部高程基准的重力水准面,以此来确定局部高程基准重力位差;最后利用GNSS+水准数据和重力大地水准面确定了美国高程基准与全球高程基准W₀的重力位差为-4.82±0.05 m²s^-2。     

均质椭球体的引力位以及调和延拓在边值问题中的应用

本文作者首次推导均质椭球体的内部和外部引力位的数学表达式 ,由此应用调和延拓的思想引入外大地水准面和外部位函数的概念 ;建立了关于外扰动位的边值问题 ,该问题的界面规则而且具有 O( T2 )的精度 ;作为一个整体 ,讨论了外正高的计算和将地面点的重力测量值在 O( T2 )精度下归算到外大地水准面的方法     

联合多源重力数据反演菲律宾海域海底地形

对比分析重力-海深的“理论导纳”和实际数据的“观测导纳”,获得研究海域有效弹性厚度理论值为10 km。联合重力异常和重力异常垂直梯度数据,应用自适应赋权技术,采用导纳函数方法构建菲律宾海域1'×1'海底地形模型。试验发现,当重力异常垂直梯度反演海深结果与重力异常反演海深结果的权比为2∶3时,所构建的海深模型检核精度最高。同时,联合多源重力数据反演海深能够综合重力异常和重力异常垂直梯度在对待不同海底地形上的反演优势,生成精度优于单独使用重力异常数据和重力异常垂直梯度数据反演的海底地形模型。以船测数据作为外部检核条件,反演模型检核精度略低于V18.1海深模型,而相较于ETOPO1海深模型和DTU10海深模型检核精度分别提高了27.17%和39.02%左右;反演模型相对误差的绝对值在5%范围内的检核点大约占检核点总数的94.25%。     

High-resolution regional geoid computation without applying Stokes’s formula: a case study of the Iranian geoid

A new theory for high-resolution regional geoid computation without applying Stokess formula is presented. Operationally, it uses various types of gravity functionals, namely data of type gravity potential (gravimetric leveling), vertical derivatives of the gravity potential (modulus of gravity intensity from gravimetric surveys), horizontal derivatives of the gravity potential (vertical deflections from astrogeodetic observations) or higher-order derivatives such as gravity gradients. Its algorithmic version can be described as follows: (1) Remove the effect of a very high degree/order potential reference field at the point of measurement (POM), in particular GPS positioned, either on the Earths surface or in its external space. (2) Remove the centrifugal potential and its higher-order derivatives at the POM. (3) Remove the gravitational field of topographic masses (terrain effect) in a zone of influence of radius r. A proper choice of such a radius of influence is 2r=4×10⁴ km/n, where n is the highest degree of the harmonic expansion. (cf. Nyquist frequency). This third remove step aims at generating a harmonic gravitational field outside a reference ellipsoid, which is an equipotential surface of a reference potential field. (4) The residual gravitational functionals are downward continued to the reference ellipsoid by means of the inverse solution of the ellipsoidal Dirichlet boundary-value problem based upon the ellipsoidal Abel–Poisson kernel. As a discretized integral equation of the first kind, downward continuation is Phillips–Tikhonov regularized by an optimal choice of the regularization factor. (5) Restore the effect of a very high degree/order potential reference field at the corresponding point to the POM on the reference ellipsoid. (6) Restore the centrifugal potential and its higher-order derivatives at the ellipsoidal corresponding point to the POM. (7) Restore the gravitational field of topographic masses ( terrain effect) at the ellipsoidal corresponding point to the POM. (8) Convert the gravitational potential on the reference ellipsoid to geoidal undulations by means of the ellipsoidal Bruns formula. A large-scale application of the new concept of geoid computation is made for the Iran geoid. According to the numerical investigations based on the applied methodology, a new geoid solution for Iran with an accuracy of a few centimeters is achieved.Acknowledgments. The project of high-resolution geoid computation of Iran has been support by National Cartographic Center (NCC) of Iran. The University of Tehran, via grant number 621/3/602, supported the computation of a global geoid solution for Iran. Their support is gratefully acknowledged. A. Ardalan would like to thank Mr. Y. Hatam, and Mr. K. Ghazavi from NCC and Mr. M. Sharifi, Mr. A. Safari, and Mr. M. Motagh from the University of Tehran for their support in data gathering and computations. The authors would like to thank the comments and corrections made by the four reviewers and the editor of the paper, Professor Will Featherstone. Their comments helped us to correct the mistakes and improve the paper.     

On the feasibility of using satellite gravity observations for detecting large-scale solid mass transfer events

The main focus of this paper is to assess the feasibility of utilizing dedicated satellite gravity missions in order to detect large-scale solid mass transfer events (e.g. landslides). Specifically, a sensitivity analysis of Gravity Recovery and Climate Experiment (GRACE) gravity field solutions in conjunction with simulated case studies is employed to predict gravity changes due to past subaerial and submarine mass transfer events, namely the Agulhas slump in southeastern Africa and the Heart Mountain Landslide in northwestern Wyoming. The detectability of these events is evaluated by taking into account the expected noise level in the GRACE gravity field solutions and simulating their impact on the gravity field through forward modelling of the mass transfer. The spectral content of the estimated gravity changes induced by a simulated large-scale landslide event is estimated for the known spatial resolution of the GRACE observations using wavelet multiresolution analysis. The results indicate that both the Agulhas slump and the Heart Mountain Landslide could have been detected by GRACE, resulting in \({\vert }0.4{\vert }\) and \({\vert }0.18{\vert }\) mGal change on GRACE solutions, respectively. The suggested methodology is further extended to the case studies of the submarine landslide in Tohoku, Japan, and the Grand Banks landslide in Newfoundland, Canada. The detectability of these events using GRACE solutions is assessed through their impact on the gravity field.     

The spherical horizontal and spherical vertical boundary value problem – vertical deflections and geoidal undulations – the completed Meissl diagram

 In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere (IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced. Received: 10 May 2000 / Accepted: 26 February 2001     

Further remarks on the altimetry-gravimetry problems

The two Altimetry-Gravimetry linearized boundary value problems, both characterized by gravity anomaly boundary data on the continents, while on the oceans either the gravity potential or its radial derivative is assumed to be known, are investigated in their modified version, in the spherical boundary approximation. A constant unknown bias is added to the data set on the oceans, while a vanishing zerodegree harmonic component is assumed as additional constraint. Existence, uniqueness and regularity results are obtained, it is proved that, in order to obtain regularization of the solution, not only regular data must be imposed, on the continents and on the oceans, but suitable constraints on the coast line are needed.     

A study on the combination of satellite, airborne, and terrestrial gravity data

Satellite gravity missions, such as CHAMP, GRACE and GOCE, and airborne gravity campaigns in areas without ground gravity will enhance the present knowledge of the Earths gravity field. Combining the new gravity information with the existing marine and ground gravity anomalies is a major task for which the mathematical tools have to be developed. In one way or another they will be based on the spectral information available for gravity data and noise. The integration of the additional gravity information from satellite and airborne campaigns with existing data has not been studied in sufficient detail and a number of open questions remain. A strategy for the combination of satellite, airborne and ground measurements is presented. It is based on ideas independently introduced by Sjöberg and Wenzel in the early 1980s and has been modified by using a quasi-deterministic approach for the determination of the weighting functions. In addition, the original approach of Sjöberg and Wenzel is extended to more than two measurement types, combining the Meissl scheme with the least-squares spectral combination. Satellite (or geopotential) harmonics, ground gravity anomalies and airborne gravity disturbances are used as measurement types, but other combinations are possible. Different error characteristics and measurement-type combinations and their impact on the final solution are studied. Using simulated data, the results show a geoid accuracy in the centimeter range for a local test area.     

The free versus fixed geodetic boundary value problem for different combinations of geodetic observables

Various formulations of the geodetic fixed and free boundary value problem are presented, depending upon the type of boundary data. For the free problem, boundary data of type astronomical latitude, astronomical longitude and a pair of the triplet potential, zero and first-order vertical gradient of gravity are presupposed. For the fixed problem, either the potential or gravity or the vertical gradient of gravity is assumed to be given on the boundary. The potential and its derivatives on the boundary surface are linearized with respect to a reference potential and a reference surface by Taylor expansion. The Eulerian and Lagrangean concepts of a perturbation theory of the nonlinear geodetic boundary value problem are reviewed. Finally the boundary value problems are solved by Hilbert space techniques leading to new generalized Stokes and Hotine functions. Reduced Stokes and Hotine functions are recommended for numerical reasons. For the case of a boundary surface representing the topography a base representation of the solution is achieved by solving an infinite dimensional system of equations. This system of equations is obtained by means of the product-sum-formula for scalar surface spherical harmonics with Wigner 3j-coefficients.     

The free versus fixed geodetic boundary value problem for different combinations of geodetic observables

Various formulations of the geodetic fixed and free boundary value problem are presented, depending upon the type of boundary data. For the free problem, boundary data of type astronomical latitude, astronomical longitude and a pair of the triplet potential, zero and first-order vertical gradient of gravity are presupposed. For the fixed problem, either the potential or gravity or the vertical gradient of gravity is assumed to be given on the boundary. The potential and its derivatives on the boundary surface are linearized with respect to a reference potential and a reference surface by Taylor expansion. The Eulerian and Lagrangean concepts of a perturbation theory of the nonlinear geodetic boundary value problem are reviewed. Finally the boundary value problems are solved by Hilbert space techniques leading to new generalized Stokes and Hotine functions. Reduced Stokes and Hotine functions are recommended for numerical reasons. For the case of a boundary surface representing the topography a base representation of the solution is achieved by solving an infinite dimensional system of equations. This system of equations is obtained by means of the product-sum-formula for scalar surface spherical harmonics with Wigner 3j-coefficients.     

An isostatically compensated gravity model using spherical layer distributions

A new isostatic model for the Earths gravity field is presented based on a simple hypothesis of layers approximating constant density contrasts. The spherical layer distribution used to describe the hydrostatic equilibrium of the Earths masses leads to a new set of spherical harmonic coefficients for the gravitational potential. First attempts to quantify the information content of these coefficients led to the outcome that they seem to explain the observed gravity field for a certain wavelength band, while they are insufficient for short and very long wavelengths. A synthesis of the derived coefficients over specific degree ranges provided a computation of band-limited geoid undulations on a global scale. The association of these potential quantities with known tectonic structures, such as the topography of the core–mantle boundary, strengthens the belief that the interpretation of Earth gravity models, especially those arising from global digital elevation models, should be considered in close relation with deep-Earth structure.     

A methodology for least-squares local quasi-geoid modelling using a noisy satellite-only gravity field model

The paper is about a methodology to combine a noisy satellite-only global gravity field model (GGM) with other noisy datasets to estimate a local quasi-geoid model using weighted least-squares techniques. In this way, we attempt to improve the quality of the estimated quasi-geoid model and to complement it with a full noise covariance matrix for quality control and further data processing. The methodology goes beyond the classical remove–compute–restore approach, which does not account for the noise in the satellite-only GGM. We suggest and analyse three different approaches of data combination. Two of them are based on a local single-scale spherical radial basis function (SRBF) model of the disturbing potential, and one is based on a two-scale SRBF model. Using numerical experiments, we show that a single-scale SRBF model does not fully exploit the information in the satellite-only GGM. We explain this by a lack of flexibility of a single-scale SRBF model to deal with datasets of significantly different bandwidths. The two-scale SRBF model performs well in this respect, provided that the model coefficients representing the two scales are estimated separately. The corresponding methodology is developed in this paper. Using the statistics of the least-squares residuals and the statistics of the errors in the estimated two-scale quasi-geoid model, we demonstrate that the developed methodology provides a two-scale quasi-geoid model, which exploits the information in all datasets.