Construction of Green's function for the Stokes boundary-value problem with ellipsoidal corrections in the boundary condition

Green's function for the boundary-value problem of Stokes's type with ellipsoidal corrections in the boundary condition for anomalous gravity is constructed in a closed form. The `spherical-ellipsoidal' Stokes function describing the effect of two ellipsoidal correcting terms occurring in the boundary condition for anomalous gravity is expressed in O(e <sup>2</sup> <sub>0</sub>)-approximation as a finite sum of elementary functions analytically representing the behaviour of the integration kernel at the singular point ψ=0. We show that the `spherical-ellipsoidal' Stokes function has only a logarithmic singularity in the vicinity of its singular point. The constructed Green function enables us to avoid applying an iterative approach to solve Stokes's boundary-value problem with ellipsoidal correction terms involved in the boundary condition for anomalous gravity. A new Green-function approach is more convenient from the numerical point of view since the solution of the boundary-value problem is determined in one step by computing a Stokes-type integral. The question of the convergence of an iterative scheme recommended so far to solve this boundary-value problem is thus irrelevant. Received: 5 June 1997 / Accepted: 20 February 1998     

The ellipsoidal correction to the Stokes kernel for precise geoid determination

Solving the geodetic boundary-value problem (GBVP) for the precise determination of the geoid requires proper use of the fundamental equation of physical geodesy as the boundary condition given on the geoid. The Stokes formula and kernel are the result of spherical approximation of this fundamental equation, which is a violation of the proper relation between the observed quantity (gravity anomaly) and the sought function (geoid). The violation is interpreted here as the improper formulation of the boundary condition, which implies the spherical Stokes kernel to be in error compared with the proper kernel of integral transformation. To remedy this error, two correction kernels to the Stokes kernel were derived: the first in both closed and spectral forms and the second only in spectral form. Contributions from the first correction kernel to the geoid across the globe were [−0.867 m, +1.002 m] in the low-frequency domain implied by the GRIM4-S4 purely satellite-derived geopotential model. It is a few centimeters, on average, in the high-frequency domain with some exceptions of a few meters in places of high topographical relief and sizable geological features in accordance with the EGM96 combined geopotential model. The contributions from the second correction kernel to the geoid are [−0.259 m, +0.217 m] and [−0.024 m, +0.023 m] in the low- and high-frequency domains, respectively.     

A new method for computing the ellipsoidal correction for Stokes's formula

 This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N ₀ evaluated from Stokes's formula and the ellipsoidal correction N ₁, makes the relative geoidal height error decrease from O(e ²) to O(e ⁴), which can be neglected for most practical purposes. The ellipsoidal correction N ₁ is expressed as a sum of an integral about the spherical geoidal height N ₀ and a simple analytical function of N ₀ and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N ₁ is done in an area where the spherical geoidal height N ₀ has already been evaluated, it is also more effective than the solution of Martinec and Grafarend. Received: 27 January 1999 / Accepted: 4 October 1999     

On a scalar fixed altimetry—gravimetry boundary value problem

For some years existence and uniqueness of the solution of a mixed altimetry — gravimetry boundary value problem is under investigation. Usually the size and the shape of the continental part of the Earth's surface is considered as unknown. This leads to a partly free and partly fixed mixed boundary value problem. A solution exists if the area of the continental part is sufficiently small. Obviously, this condition is not fulfilled in reality. Because the GPS is now fully operational, nowadays the size and the shape of the continental part of the Earth's surface can be considered as known, too. The aim of this article is to study the resulting fixed mixed boundary value problem and to prove the existence and uniqueness of its solution for an arbitrary distribution of land and sea at the Earth's surface.     

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。     

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.     

The geopotential from gravity measurements,levelling data and satellite results

The geodetic boundary value problem is formulated which uses as boundary values the differences between the geopotential of points at the surface of the continents and the potential of the geoid. These differences are computed by gravity measurements and levelling data. In addition, the shape of the geoid over the oceans is assumed to be known from satellite altimetry and the shape of the continents from satellite results together with three-dimensional triangulation. The boundary value problem thus formulated is equivalent to Dirichlet's exterior problem except for the unknown potential of the geoid. This constant is determined by an integral equation for the normal derivative of the gravitational potential which results from the first derivative of Green's fundamental formula. The general solution, which exists, of the integral equation gives besides the potential of the geoid the solution of the geodetic boundary value problem. In addition approximate solutions for a spherical surface of the earth are derived.     

Performance of three types of Stokes's kernel in the combined solution for the geoid

When regional gravity data are used to compute a gravimetric geoid in conjunction with a geopotential model, it is sometimes implied that the terrestrial gravity data correct any erroneous wavelengths present in the geopotential model. This assertion is investigated. The propagation of errors from the low-frequency terrestrial gravity field into the geoid is derived for the spherical Stokes integral, the spheroidal Stokes integral and the Molodensky-modified spheroidal Stokes integral. It is shown that error-free terrestrial gravity data, if used in a spherical cap of limited extent, cannot completely correct the geopotential model. Using a standard norm, it is shown that the spheroidal and Molodensky-modified integration kernels offer a preferable approach. This is because they can filter out a large amount of the low-frequency errors expected to exist in terrestrial gravity anomalies and thus rely more on the low-frequency geopotential model, which currently offers the best source of this information. Received: 11 August 1997 / Accepted: 18 August 1998     

Two different methods of geoidal height determinations using a spherical harmonic representation of the geopotential, topographic corrections and the height anomaly–geoidal height difference

 It is suggested that a spherical harmonic representation of the geoidal heights using global Earth gravity models (EGM) might be accurate enough for many applications, although we know that some short-wavelength signals are missing in a potential coefficient model. A `direct' method of geoidal height determination from a global Earth gravity model coefficient alone and an `indirect' approach of geoidal height determination through height anomaly computed from a global gravity model are investigated. In both methods, suitable correction terms are applied. The results of computations in two test areas show that the direct and indirect approaches of geoid height determination yield good agreement with the classical gravimetric geoidal heights which are determined from Stokes' formula. Surprisingly, the results of the indirect method of geoidal height determination yield better agreement with the global positioning system (GPS)-levelling derived geoid heights, which are used to demonstrate such improvements, than the results of gravimetric geoid heights at to the same GPS stations. It has been demonstrated that the application of correction terms in both methods improves the agreement of geoidal heights at GPS-levelling stations. It is also found that the correction terms in the direct method of geoidal height determination are mostly similar to the correction terms used for the indirect determination of geoidal heights from height anomalies. Received: 26 July 2001 / Accepted: 21 February 2002     

Construction of Green's function to the external Dirichlet boundary-value problem for the Laplace equation on an ellipsoid of revolution

Green's function to the external Dirichlet boundary-value problem for the Laplace equation with data distributed on an ellipsoid of revolution has been constructed in a closed form. The ellipsoidal Poisson kernel describing the effect of the ellipticity of the boundary on the solution of the investigated boundary-value problem has been expressed as a finite sum of elementary functions which describe analytically the behaviour of the ellipsoidal Poisson kernel at the singular point ψ = 0. We have shown that the degree of singularity of the ellipsoidal Poisson kernel in the vicinity of its singular point is of the same degree as that of the original spherical Poisson kernel. Received: 4 June 1996 / Accepted: 7 April 1997     

利用改进的Stokes-Helmert边值理论解算高精度似大地水准面

利用改进的Stokes-Helmert边值问题实现了临沂市厘米级似大地水准面精化。首先,基于Stokes理论和Molodensky理论,联合精密确定地表及其外部扰动引力位的严密解算理论,给出Stokes-Helmert边值问题的数学描述,以及直接地形影响和间接地形影响的严密理论表达式;然后,利用多源观测资料,根据"移去-恢复"技术构建临沂高精度的重力似大地水准面模型;最后,利用GPS/水准高程异常对重力似大地水准面模型进行控制拟合,求得最终的大地水准面模型,其外符合精度达到1.6 cm。     

Poisson重力边值问题

提出了Poisson重力边值问题,即关于扰动位的Poisson方程的Stokes问题和Neumann问题。作为导引,先研究Poisson方程的Dirichlet问题．再分别引入一种辅助函数,将Stokes问题和Neumann问题改化为Dirichlet问题,从而立即得到它们的积分解。最终解式表现为两部分叠加：一部分仅与边界观测相关,另一部分为对地形测量的响应,为研究地形测量对外部重力场和大地水准面的精化提供新的途径。     

论确定全球大地水准面的斯托克斯方法

确定全球大地水准面最常用的方法是斯托克司方法。然而,除了人们熟知的缺陷之外,斯托克司方法还存在人们没有意识到的理论困难：当大地水准面位于参考椭球（WGS84椭球）内部时,在大地水准面上及其与参考椭球面界定的区域中扰动位没有定义,当然在这部分区域也不调和。为了解决这一困难,可以选取一个包含在大地水准面内部的由四个基本参数唯一确定其外部正常重力位的参考椭球（简称内部椭球）,其中心与 WGS84 椭球的中心重合,其中的两个基本参数,旋转角速度和地心引力常数,与 WGS84 椭球面的相同,另外两个参数,半长轴和扁率,如此选取,使得内部椭球产生的新的正常重力位在 WGS84 椭球面上与大地水准面上的重力位 相等。这样,传统的斯托克司方法中存在的理论困难不复存在。     

Terrain corrections to power 3 in gravimetric geoid determination

In precise geoid determination by Stokes formula, direct and primary and secondary indirect terrain effects are applied for removing and restoring the terrain masses. We use Helmert's second condensation method to derive the sum of these effects, together called the total terrain effect for geoid. We develop the total terrain effect to third power of elevation H in the original Stokes formula, Earth gravity model and modified Stokes formula. It is shown that the original Stokes formula, Earth gravity model and modified Stokes formula all theoretically experience different total terrain effects. Numerical results indicate that the total terrain effect is very significant for moderate topographies and mountainous regions. Absolute global mean values of 5–10 cm can be reached for harmonic expansions of the terrain to degree and order 360. In another experiment, we conclude that the most important part of the total terrain effect is the contribution from the second power of H, while the contribution from the third power term is within 9 cm. Received: 2 September 1996 / Accepted: 4 August 1997     

Solutions of the linearized geodetic boundary value problem for an ellipsoidal boundary to order e 3

The geodetic boundary value problem (GBVP) was originally formulated for the topographic surface of the Earth. It degenerates to an ellipsoidal problem, for example when topographic and downward continuation reductions have been applied. Although these ellipsoidal GBVPs possess a simpler structure than the original ones, they cannot be solved analytically, since the boundary condition still contains disturbing terms due to anisotropy, ellipticity and centrifugal components in the reference potential. Solutions of the so-called scalar-free version of the GBVP, upon which most recent practical calculations of geoidal and quasigeoidal heights are based, are considered. Starting at the linearized boundary condition and presupposing a normal field of Somigliana–Pizzetti type, the boundary condition described in spherical coordinates is expanded into a series with respect to the flattening f of the Earth. This series is truncated after the linear terms in f, and first-order solutions of the corresponding GBVP are developed in closed form on the basis of spherical integral formulae, modified by suitable reduction terms. Three alternative representations of the solution are discussed, implying corrections by adding a first-order non-spherical term to the solution, by reducing the boundary data, or by modifying the integration kernel. A numerically efficient procedure for the evaluation of ellipsoidal effects, in the case of the linearized scalar-free version of the GBVP, involving first-order ellipsoidal terms in the boundary condition, is derived, utilizing geopotential models such as EGM96.     

Analysis of some systematic errors affecting altimeter-derived sea surface gradient with application to geoid determination over Taiwan

This paper analyzes several systematic errors affecting sea surface gradients derived from Seasat, Geosat/ERM, Geosat/GM, ERS-1/35d, ERS-1/GM and TOPEX/POSEIDON altimetry. Considering the data noises, the conclusion is: (1) only Seasat needs to correct for the non-geocentricity induced error, (2) only Seasat and Geosat/GM need to correct for the one cycle per revolution error, (3) only Seasat, ERS-1/GM and Geosat/GM need to correct for the tide model error; over shallow waters it is suggested to use a local tide model not solely from altimetry. The effects of the sea surface topography on gravity and geoid computations from altimetry are significant over areas with major oceanographic phenomena. In conclusion, sea surface gradient is a better data type than sea surface height. Sea surface gradients from altimetry, land gravity anomalies, ship gravity anomalies and elevation data were then used to calculate the geoid over Taiwan by least-squares collocation. The inclusion of sea surface gradients improves the geoid prediction by 27% when comparing the GPS-derived and the predicted geoidal heights, and by 30% when comparing the observed and the geoid-derived deflections of the vertical. The predicted geoid along coastal areas is accurate to 2 cm and can help GPS to do the third-order leveling. Received 22 January 1996; Accepted 13 September 1996     

The nullfield method for the ellipsoidal Stokes problem

The ellipsoidal Stokes problem is one of the basic boundary-value problems for the Laplace equation which arises in physical geodesy. Up to now, geodecists have treated this and related problems with high-order series expansions of spherical and spheroidal (ellipsoidal) harmonics. In view of increasing computational power and modern numerical techniques, boundary element methods have become more and more popular in the last decade. This article demonstrates and investigates the nullfield method for a class of Robin boundary-value problems. The ellipsoidal Stokes problem belongs to this class. An integral equation formulation is achieved, and existence and uniqueness conditions are attained in view of the Fredholm alternative. Explicit expressions for the eigenvalues and eigenfunctions for the boundary integral operator are provided. Received: 22 October 1996 / Accepted: 4 August 1997     

利用低阶大地水准面异常反演大尺度核幔边界起伏

核幔边界（core-mantle boundary,CMB）是地球内部最重要的物理化学界面之一,地核和地幔通过核幔边界发生多种相互作用,这对地球重力场、地球自转及地磁场等都能产生重要影响。大地水准面异常是地球重力场的重要观测量,反映了地球内部的物质密度异常及界面变化等重要信息。推导了通过大地水准面异常反演核幔边界起伏的公式,利用2~4阶大地水准面异常反演了大尺度核幔边界起伏形态。结果显示,核幔边界起伏的径向幅度达±5 km、与Morelli的地震层析成像结果的幅度接近,但在形态上略有差异。以高为5 km、底边长为1 000 km的棱柱体模型模拟计算了核幔边界密度异常引起的大地水准面异常响应,结果与观测大地水准面异常比较接近。     

不同扰动位泛函间的积分变换广义核函数

基于物理大地测量边值问题的解,利用一阶边界算子定义,推导重力异常Δg、单层密度μ、大地水准面高N,垂线偏差ε、扰动重力δg等扰动场元的解。利用球谐函数的正交特性,通过对核函数的算子运算,可以得到上述扰动场元的有关逆变换公式。相对经典物理大地测量公式应用的边界面条件,笔者将含有因子r的对应扰动场元反演关系的公式称为广义积分公式。针对常用的重力异常Δg、大地水准面高N,垂线偏差ε、扰动重力δg计算,重点分析它们之间的变换关系,给出利用某个选定扰动场元计算其他扰动场元的广义积分公式。同时,通过对积分边界面的讨论,分析经典公式与广义积分公式的差异和联系。最后,给出所有外部扰动场元与核函数映射的关系表。     

高阶地球重力场模型的评价及其优选

采用"移去-恢复"技术确定大地水准面,需要一个全球重力场模型作为参考场。本文采用基于模糊集合理论的最大隶属度原则,通过某试验区的33个GPS水准点的实测大地水准面差距与模型大地水准面差距的比较分析,从国际重力场服务提供的EGM96、EIGEN-CG01C、EIGEN-CG03C、GFZ93A、GFZ96、OSU91A、PGM2000A等七个高阶全球重力场模型中,选择OSU91A作为该地区最优的参考重力场模型。