The solution of mixed boundary value problems with the reference ellipsoid as boundary

In this paper we study the following mixed boundary value problem

随机Dirichlet问题解的调和展开

针对重力学随机Dirichlet问题,通过适当地对边界检验函数的分解,并在随机边界样本空间中提取确定性部分的对偶基,本文将随机Dirichlet问题的一般解展开为一随机系数的调和级数形式。     

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     

Explicit expression and regularization of the harmonic reproducing kernels for the earth's ellipsoid

The reproducing kernel functions and the integral formulae involving the solutions of the fixed gravimetric boundary value problems for the earth's ellipsoid are investigated in a suitable polar coordinate system defined on the boundary ellipsoid. The infinite series expressions of the reproducing kernel functions are represented explicitly up to the square of the first excentricity of the boundary surface. The obtained results show that the reproducing kernel functions for the earth's ellipsoid are, in contrast to the case of the spherical boundary, inhomogeneous and anisotropic. Moreover, the anisotropy is stronger near the pole. Subsequently the kernel functions are regularized in order to overcome the weak singularity of the integrals.     

尺度比稳健估计及参考椭球参数确定

短程精密测距精度一般优于GNSS测量精度,工程中常用GNSS测量与精密测距方法联合建立高精度工程测量控制网,对两者所存在的尺度差异,常采用尺度比进行统一,其关键是尺度比的合理选择。文中参照精密测距边长确定的实测尺度比相对于理论尺度比的残差,建立空间分布与尺度残差线性关系,对尺度差异进行估计并确定合理的尺度比,再在投影层面将尺度比等价转换成参考椭球参数进行数据归算,从而实现GNSS与精密测距成果的高度融合,得到高精度工程测量控制网的可靠成果。     

试论选择地方参考椭球体长半径的合理公式

在建立地方独立坐标系时需要匹配一个与该坐标系投影面吻合的地方参考椭球体,在保证地方椭球体的中心、轴向和扁率与国家参考椭球体相同的前提下,采用何种计算公式来确定长半轴的改正量,以使新椭球体面与地方坐标系投影面吻合得最好,本文对现今个别规范和教科书中采用的计算公式作了分析,指出了公式错误,同时导出了新的计算公式,并得出了现用公式对实际测量工作无影响的结论。     

Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric Earth-fixed reference frame

The derivatives of the Earth gravitational potential are considered in the global Cartesian Earth-fixed reference frame. Spherical harmonic series are constructed for the potential derivatives of the first and second orders on the basis of a general expression of Cunningham (Celest Mech 2:207–216, 1970) for arbitrary order derivatives of a spherical harmonic. A common structure of the series for the potential and its first- and second-order derivatives allows to develop a general procedure for constructing similar series for the potential derivatives of arbitrary orders. The coefficients of the derivatives are defined by means of recurrence relations in which a coefficient of a certain order derivative is a linear function of two coefficients of a preceding order derivative. The coefficients of the second-order derivatives are also presented as explicit functions of three coefficients of the potential. On the basis of the geopotential model EGM2008, the spherical harmonic coefficients are calculated for the first-, second-, and some third-order derivatives of the disturbing potential T, representing the full potential V, after eliminating from it the zero- and first-degree harmonics. The coefficients of two lowest degrees in the series for the derivatives of T are presented. The corresponding degree variances are estimated. The obtained results can be applied for solving various problems of satellite geodesy and celestial mechanics.     

The direct and inverse solutions for the great elliptic line on the reference ellipsoid

New formulae are given for the line of the great elliptic on the reference ellipsoid providing solutions to both the forward and the inverse problems of exceptional accuracy. The solution incorporates a closed equation for the great elliptic azimuth, and the derivation of this equation is presented and illustrated.     

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     

Ellipsoidal terrain correction based on multi-cylindrical equal-area map projection of the reference ellipsoid

An operational algorithm for computation of terrain correction (or local gravity field modeling) based on application of closed-form solution of the Newton integral in terms of Cartesian coordinates in multi-cylindrical equal-area map projection of the reference ellipsoid is presented. Multi-cylindrical equal-area map projection of the reference ellipsoid has been derived and is described in detail for the first time. Ellipsoidal mass elements with various sizes on the surface of the reference ellipsoid are selected and the gravitational potential and vector of gravitational intensity (i.e. gravitational acceleration) of the mass elements are computed via numerical solution of the Newton integral in terms of geodetic coordinates {,,h}. Four base- edge points of the ellipsoidal mass elements are transformed into a multi-cylindrical equal-area map projection surface to build Cartesian mass elements by associating the height of the corresponding ellipsoidal mass elements to the transformed area elements. Using the closed-form solution of the Newton integral in terms of Cartesian coordinates, the gravitational potential and vector of gravitational intensity of the transformed Cartesian mass elements are computed and compared with those of the numerical solution of the Newton integral for the ellipsoidal mass elements in terms of geodetic coordinates. Numerical tests indicate that the difference between the two computations, i.e. numerical solution of the Newton integral for ellipsoidal mass elements in terms of geodetic coordinates and closed-form solution of the Newton integral in terms of Cartesian coordinates, in a multi-cylindrical equal-area map projection, is less than 1.6×10^–8 m²/s² for a mass element with a cross section area of 10×10 m and a height of 10,000 m. For a mass element with a cross section area of 1×1 km and a height of 10,000 m the difference is less than 1.5×10^–4m²/s². Since 1.5× 10^–4 m²/s² is equivalent to 1.5×10^–5m in the vertical direction, it can be concluded that a method for terrain correction (or local gravity field modeling) based on closed-form solution of the Newton integral in terms of Cartesian coordinates of a multi-cylindrical equal-area map projection of the reference ellipsoid has been developed which has the accuracy of terrain correction (or local gravity field modeling) based on the Newton integral in terms of ellipsoidal coordinates.Acknowledgments. This research has been financially supported by the University of Tehran based on grant number 621/4/859. This support is gratefully acknowledged. The authors are also grateful for the comments and corrections made to the initial version of the paper by Dr. S. Petrovic from GFZ Potsdam and the other two anonymous reviewers. Their comments helped to improve the structure of the paper significantly.     

On the computation and approximation of ultra-high-degree spherical harmonic series

Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics. This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series also offers a computational savings of at least one third.     

Anholonomity when using the development method for the reduction of observations to the reference ellipsoid

For computing the geodetic coordinates ϕ and γ on the ellipsoid one needs information of the gravity field, thus making it possible to reduce the terrestrial observations to the reference surface. Neglect of gravity field data, such as deflections of the vertical and geoid heights, results in misclosure effects, which can be described using the object of anholonomity.     

The geodetic boundary value problem and the coordinate choice problem

Summary The geodetic boundary value problem (g.b.v.p.) is a free boundary value problem for the Laplace operator: however, under suitable change of coordinates, it can be transformed into a fixed boundary one. Thus a general coordinate choice problem arises: two particular cases are more closely analyzed, namely the gravity space approach and the intrinsic coordinates (Marussi) approach.     

On the solvability of the Stokes pseudo-boundary-value problem for geoid determination

A new form of boundary condition of the Stokes problem for geoid determination is derived. It has an unusual form, because it contains the unknown disturbing potential referred to both the Earth's surface and the geoid coupled by the topographical height. This is a consequence of the fact that the boundary condition utilizes the surface gravity data that has not been continued from the Earth's surface to the geoid. To emphasize the `two-boundary' character, this boundary-value problem is called the Stokes pseudo-boundary-value problem. The numerical analysis of this problem has revealed that the solution cannot be guaranteed for all wavelengths. We demonstrate that geoidal wavelengths shorter than some critical finite value must be excluded from the solution in order to ensure its existence and stability. This critical wavelength is, for instance, about 1 arcmin for the highest regions of the Earth's surface. Furthermore, we discuss various approaches frequently used in geodesy to convert the `two-boundary' condition to a `one-boundary' condition only, relating to the Earth's surface or the geoid. We show that, whereas the solution of the Stokes pseudo-boundary-value problem need not exist for geoidal wavelengths shorter than a critical wavelength of finite length, the solutions of approximately transformed boundary-value problems exist over a larger range of geoidal wavelengths. Hence, such regularizations change the nature of the original problem; namely, they define geoidal heights even for the wavelengths for which the original Stokes pseudo-boundary-value problem need not be solvable. Received 11 September 1995; Accepted 2 September 1996     

On the aliasing problem in the spherical harmonic analysis

To apply the least squares method for the interpolation of harmonic functions is a common practice in Geodesy. Since the method of least squares can be applied only to overdetermined problem, the interpolation problem which is always under-determined, is often reduced to an overdetermined form by truncating a series of spherical harmonics. When the data points are the knots of a regular grid it is easy to see that the estimated harmonic coefficients converge to the correct theoretical values, but when the observation density is not constant a significant bias is introduced. The result is obtained by assuming that the number of observations tends to infinity with points sampled from a given distribution. Under the same conditions it is shown that quadrature and “collocation-like” formulas displays a statistically consistent behaviour.     

Helmert's projection of a ground point onto the rotational reference ellipsoid in topocentric cartesian coordinates

Summary In order to derive the ellipsoidal height of a point P_t, on the physical surface of the earth, and the direction of ellipsoidal normal through P_t, we presente here an iterative procedure rapidly convergent to compute, in a topocentric Cartesian system, the coordinates of Helmert's projection of the ground point P_t onto the reference ellipsoid of revolution .We derive as well the cofactor matrix of total vector of the topocentric coordinates of the above ground point and of its Helmert's projection so that to compute the variance of ellipsoidal height.