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1.
Construction of Green's function to the external Dirichlet boundary-value problem for the Laplace equation on an ellipsoid of revolution 总被引：1，自引：0，他引：1
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 相似文献
2.
S. Ritter 《Journal of Geodesy》1998,72(2):101-106
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 相似文献
3.
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 相似文献
4.
Least-squares by observation equations is applied to the solution of geodetic boundary value problems (g.b.v.p.). The procedure is explained solving the vectorial Stokes problem in spherical and constant radius approximation. The results
are Stokes and Vening-Meinesz integrals and, in addition, the respective a posteriori variance-covariances.
Employing the same procedure the overdeterminedg.b.v.p. has been solved for observable functions potential, scalar gravity, astronomical latitude and longitude, gravity gradients
Гxz, Гyz, and Гzz and three-dimensional geocentric positions. The solutions of a large variety of uniquely and overdeterminedg.b.v.p.'s can be obtained from it by specializing weights. Interesting is that the anomalous potential can be determined—up to a
constant—from astronomical latitude and longitude in combination with either {Гxz,Гyz} or horizontal coordinate corrections Δx and Δy, or both. Dual to the formulation in terms of observation equations the overdeterminedg.b.v.p.'s can as well be solved by condition equations.
Constant radius approximation can be overcome in an iterative approach. For the Stokes problem this results in the solution
of the “simple” Molodenskii problem. Finally defining an error covariance model with a Krarup-type kernel first results were
obtained for a posteriori variance-covariance and reliability analysis. 相似文献
5.
A new local existence and uniqueness theorem is obtained for the scalar geodetic boundary-value problem in spherical coordinates.
The regularities H
α and H
1+α are assumed for the boundary data g (gravity) and v (gravitational potential) respectively.
Received: 27 July 1998 / Accepted: 19 April 1999 相似文献
6.
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. 相似文献
7.
The Somigliana–Pizzetti gravity field (the International gravity formula), namely the gravity field of the level ellipsoid
(the International Reference Ellipsoid), is derived to the sub-nanoGal accuracy level in order to fulfil the demands of modern
gravimetry (absolute gravimeters, super conducting gravimeters, atomic gravimeters). Equations (53), (54) and (59) summarise
Somigliana–Pizzetti gravity Γ(φ,u) as a function of Jacobi spheroidal latitude φ and height u to the order ?(10−10 Gal), and Γ(B,H) as a function of Gauss (surface normal) ellipsoidal latitude B and height H to the order ?(10−10 Gal) as determined by GPS (`global problem solver'). Within the test area of the state of Baden-Württemberg, Somigliana–Pizzetti
gravity disturbances of an average of 25.452 mGal were produced. Computer programs for an operational application of the new
international gravity formula with (L,B,H) or (λ,φ,u) coordinate inputs to a sub-nanoGal level of accuracy are available on the Internet.
Received: 23 June 2000 / Accepted: 2 January 2001 相似文献
8.
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 相似文献
9.
E. W. Grafarend 《Journal of Geodesy》2001,75(7-8):363-390
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 相似文献
10.
11.
Y. M. Wang 《Journal of Geodesy》1999,73(1):29-34
The formulas of the ellipsoidal corrections to the gravity anomalies computed using the inverse Stokes integral are derived.
The corrections are given in the integral formulas and expanded in the spherical harmonics series. If a coefficient model
such as the OSU91A is given, the corrections can be easily computed.
Received: 19 August 1996 / Accepted: 28 September 1998 相似文献
12.
When standard boundary element methods (BEM) are used in order to solve the linearized vector Molodensky problem we are confronted with
two problems: (1) the absence of O(|x|−2) terms in the decay condition is not taken into account, since the single-layer ansatz, which is commonly used as representation
of the disturbing potential, is of the order O(|x|−1) as x→∞. This implies that the standard theory of Galerkin BEM is not applicable since the injectivity of the integral operator
fails; (2) the N×N stiffness matrix is dense, with N typically of the order 105. Without fast algorithms, which provide suitable approximations to the stiffness matrix by a sparse one with O(N(logN)
s
), s≥0, non-zero elements, high-resolution global gravity field recovery is not feasible. Solutions to both problems are proposed.
(1) A proper variational formulation taking the decay condition into account is based on some closed subspace of co-dimension
3 of the space of square integrable functions on the boundary surface. Instead of imposing the constraints directly on the
boundary element trial space, they are incorporated into a variational formulation by penalization with a Lagrange multiplier.
The conforming discretization yields an augmented linear system of equations of dimension N+3×N+3. The penalty term guarantees the well-posedness of the problem, and gives precise information about the incompatibility
of the data. (2) Since the upper left submatrix of dimension N×N of the augmented system is the stiffness matrix of the standard BEM, the approach allows all techniques to be used to generate
sparse approximations to the stiffness matrix, such as wavelets, fast multipole methods, panel clustering etc., without any
modification. A combination of panel clustering and fast multipole method is used in order to solve the augmented linear system
of equations in O(N) operations. The method is based on an approximation of the kernel function of the integral operator by a degenerate kernel
in the far field, which is provided by a multipole expansion of the kernel function. Numerical experiments show that the fast
algorithm is superior to the standard BEM algorithm in terms of CPU time by about three orders of magnitude for N=65 538 unknowns. Similar holds for the storage requirements. About 30 iterations are necessary in order to solve the linear
system of equations using the generalized minimum residual method (GMRES). The number of iterations is almost independent
of the number of unknowns, which indicates good conditioning of the system matrix.
Received: 16 October 1999 / Accepted: 28 February 2001 相似文献
13.
确定似大地水准面的Hotine-Helmert边值解算模型 总被引：1，自引：1，他引：0
空间大地测量技术的发展使大地高的观测成为可能,从而为第二大地边值问题的研究带来了新的机遇,本文对基于Helmert第二压缩法的第二边值问题(简称为Hotine-Helmert边值问题)展开研究。首先介绍了地形直接、间接影响的定义与算法,然后推导了Hotine-Helmert边值问题的解算模型。Hotine-Helmert边值理论无须计算地形压缩对重力的次要间接影响,因而较Stokes-Helmert边值理论更简单。此外,文中引入了一种低阶修正的Hotine截断核函数,该核函数较传统的截断核函数能有效地改善似大地水准面的解算精度。为了验证本文构建的Hotine-Helmert边值解算模型的有效性和实用性,本文将EIGEN-6C4模型的前360阶作为参考模型,利用Hotine-Helmert边值解算模型构建了我国中部地区6°×4°范围、1.5′×1.5′分辨率的重力似大地水准面,其精度达到±4.8 cm。 相似文献
14.
L. de Witte 《Journal of Geodesy》1967,41(1):41-53
When the values of gravity anomalies are given at the geoid, Ag can be calculated at altitude by application of Poisson’s
integral theorem. The process requires integration of Δg multiplied by the Poisson kernel function over the entire globe.
It is common practice to add to the kernel function terms that will ensure removal of any zeroth and first order components
of Δg that may be present. The effects of trancating the integration at the boundary of a spherical cap of earth central half
angle ψo have been analyzed using an adaptation of Molodenskii’s procedure. The extension process without removal terms retains the
correct effects of inaccuracies in the constant term of the gravity reference model used in the definition of Δg. Furthermore,
the effects of ignoring remote zones or unmapped areas in the integration process are very much smaller for the extension
without removal terms than for the commonly used formula with removal terms. For these reasons the Poisson vertical extension
process without removal terms is to be preferred over the extension with the zeroth order term removal. Truncation of this
process at the point recommended for the Stokes integration, namely, the first zero crossing of the Stokes kernel function,
leaves negligible truncation errors. 相似文献
15.
XU Xinyu LI Jiancheng ZOU Xiancai CHU Yonghai 《地球空间信息科学学报》2007,10(3):168-172
The principle and method for solving three types of satellite gravity gradient boundary value problems by least-squares are discussed in detail. Also, kernel function expressions of the least-squares solution of three geodetic boundary value problems with the observations {Γ zz },{Γ xz , Γ yz} and {Γ xx -Γ yy ,2 Γxy}are presented. From the results of recovering gravity field using simulated gravity gradient tensor data, we can draw a conclusion that satellite gravity gradient integral formulas derived from least-squares are valid and rigorous for recovering the gravity field. 相似文献
16.
L. E. Sjöberg 《Journal of Geodesy》2003,77(3-4):139-147
Assuming that the gravity anomaly and disturbing potential are given on a reference ellipsoid, the result of Sjöberg (1988, Bull Geod 62:93–101) is applied to derive the potential coefficients on the bounding sphere of the ellipsoid to order e
2 (i.e. the square of the eccentricity of the ellipsoid). By adding the potential coefficients and continuing the potential downward to the reference ellipsoid, the spherical Stokes formula and its ellipsoidal correction are obtained. The correction is presented in terms of an integral over the unit sphere with the spherical approximation of geoidal height as the argument and only three well-known kernel functions, namely those of Stokes, Vening-Meinesz and the inverse Stokes, lending the correction to practical computations. Finally, the ellipsoidal correction is presented also in terms of spherical harmonic functions. The frequently applied and sometimes questioned approximation of the constant m, a convenient abbreviation in normal gravity field representations, by e
2/2, as introduced by Moritz, is also discussed. It is concluded that this approximation does not significantly affect the ellipsoidal corrections to potential coefficients and Stokes formula. However, whether this standard approach to correct the gravity anomaly agrees with the pure ellipsoidal solution to Stokes formula is still an open question. 相似文献
17.
Mohammad Asadullah Khan 《Journal of Geodesy》1973,47(3):227-235
An intrresting variation on the familiar method of determining the earth's equatorial radius ae, from a knowledge of the earth's equatorial gravity is suggested. The value of equatorial radius thus found is 6378,142±5
meters. The associated parameters are GM=3.986005±.000004 × 1020 cm3 sec-−2 which excludes the relative mass of atmosphere ≅10−6 ξ GM, the equatorial gravity γe 978,030.9 milligals (constrained in this solution by the Potsdam Correction of 13.67 milligals as the Potsdam Correction
is more directly, orless indirectly, measurable than the equatorial gravity) and an ellipsoidal flattening of f=1/298.255. 相似文献
18.
P. Holota 《Journal of Geodesy》1997,71(10):640-651
In this paper the linear gravimetric boundary-value problem is discussed in the sense of the so-called weak solution. For
this purpose a Sobolev weight space was constructed for an unbounded domain representing the exterior of the Earth and quantitative
estimates were deduced for the trace theorem and equivalent norms. In the generalized formulation of the problem a special
decomposition of the Laplace operator was used to express the oblique derivative in the boundary condition which has to be
met by the solution. The relation to the classical formulation was also shown. The main result concerns the coerciveness (ellipticity)
of a bilinear form associated with the problem under consideration. The Lax-Milgram theorem was used to decide about the existence,
uniqueness and stability of the weak solution of the problem. Finally, a clear geometrical interpretation was found for a
constant in the coerciveness inequality, and the convergence of approximation solutions constructed by means of the Galerkin
method was proved.
Received: 21 June 1996 / Accepted: 14 April 1997 相似文献
19.
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. 相似文献
20.
World Geodetic Datum 2000 总被引：7，自引：1，他引：6
Based on the current best estimates of fundamental geodetic parameters {W
0,GM,J
2,Ω} the form parameters of a Somigliana-Pizzetti level ellipsoid, namely the semi-major axis a and semi-minor axis b (or equivalently the linear eccentricity ) are computed and proposed as a new World Geodetic Datum 2000. There are six parameters namely the four fundamental geodetic
parameters {W
0,GM,J
2,Ω} and the two form parameters {a,b} or {a,ɛ}, which determine the ellipsoidal reference gravity field of Somigliana-Pizzetti type constraint to two nonlinear condition
equations. Their iterative solution leads to best estimates a=(6 378 136.572±0.053)m, b=(6 356 751.920 ± 0.052)m, ɛ=(521 853.580±0.013)m for the tide-free geoide of reference and a=(6 378 136.602±0.053)m, b=(6 356 751.860±0.052)m, ɛ=(521 854.674 ± 0.015)m for the zero-frequency tide geoid of reference. The best estimates of the
form parameters of a Somigliana-Pizzetti level ellipsoid, {a,b}, differ significantly by −0.39 m, −0.454 m, respectively, from the data of the Geodetic Reference System 1980.
Received: 1 February 1999 / Accepted: 31 August 1999 相似文献