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1.
Martinec and Grafarend (1997) have shown how the construction of Green's function in the Stokes boundary-value problem with gravity data distributed on an ellipsoid of revolution is approached in the O(e
0
2
)-approximation. They have also expressed the ellipsoidal Stokes function describing the effect of ellipticity of the boundary as a finite sum of elementary functions. We present an effective method of avoiding the singularity of spherical and the ellipsoidal Stokes functions, and also an analytical expression for the ellipsoidal Stokes integral around the computational point suitable for numerical solution. We give the numerical results of solving the ellipsoidal Stokes boundary-value problem and their difference with respect to the spherical Stoke boundary-value problem. 相似文献
2.
Bernhard Heck 《Studia Geophysica et Geodaetica》2011,55(3):441-454
For more than 150 years gravity anomalies have been used for the determination of geoidal heights, height anomalies and the
external gravity field. Due to the fact that precise ellipsoidal heights could not be observed directly, traditionally a free
geodetic boundary-value problem (GBVP) had to be formulated which after linearisation is related to gravity anomalies. Since
nowadays the three-dimensional positions of gravity points can be determined by global navigation satellite systems very precisely,
the modern formulation of the GBVP can be based on gravity disturbances which are related to a fixed GBVP using the known
topographical surface of the Earth as boundary surface. The paper discusses various approaches into the solution of the fixed
GBVP which after linearization corresponds to an oblique-derivative boundary-value problem for the Laplace equation. Among
the analytical solution approaches a Brovar-type solution is worked out in detail, showing many similarities with respect
to the classical solution of the scalar free GBVP. 相似文献
3.
We would like to solve the Stokes boundary-value problem taking into consideration the ellipsoidal corrections in the boundary condition in ellipsoidal coordinates The original problem, i.e., the ellipsoidal Stokes boundary-value problem has been solved by Martinec and Grafarend (1997) We use the same philosophy expressed by Martinec (1998) to solve the spherical Stokes boundary-value problem with ellipsoidal corrections in the boundary condition We wish to show the magnitude of the integration kernel describing the effect of the ellipsoidal corrections in the boundary condition in a cap around the computational point. 相似文献
4.
We have constructed Green's function to Stokes's boundary-value problem with the gravity data distributed over an ellipsoid of revolution. We show that the problem has a unique solution provided that the first eccentricity e0
of the ellipsoid of revolution is less than 0·65041. The ellipsoidal Stokes function describing the effect of ellipticity of the boundary is expressed in the
E-approximation as a finite sum of elementary functions which describe analytically the behaviour of the ellipsoidal Stokes function at the singular point = 0. We prove that the degree of singularity of the ellipsoidal Stokes function in the vicinity of its singular point is the same as that of the spherical Stokes function. 相似文献
5.
A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:
- - Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.
- - Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.
- - Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.
- - Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.
- - Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.
- - The removed effects at the application points are restored on the surface of reference ellipsoid.
- - Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.
- - Computation of the geoid of Iran has successfully tested this new methodology.
Keywords: Geoid computations; Ellipsoidal approximation; Ellipsoidal boundary value problem; Ellipsoidal Bruns’ formula; Satellite altimetry; Astronomical observations 相似文献
6.
In the first attempt to solve the Stokes boundary-value problem in ellipsoidal coordinates numerically (Ardestani and Martinec, 2000), we focused on the near-zone contribution since the effect of the ellipsoidal Stokes function in the far-zone contribution is not considered. We present a method for solving the ellipsoidal Stokes integral in far-zone contribution. The numerical results of computing the magnitude of this term for an area in north of Canada are presented. 相似文献
7.
Petr Holota 《Studia Geophysica et Geodaetica》2011,55(3):397-413
In the introductory part of the paper the importance of the topic for gravity field studies is outlined. Some concepts and tools often used for the representation of the solution of the respective boundary-value problems are mentioned. Subsequently a weak formulation of Neumann??s problem is considered with emphasis on a particular choice of function basis generated by the reproducing kernel of the respective Hilbert space of functions. The paper then focuses on the construction of the reproducing kernel for the solution domain given by the exterior of an oblate ellipsoid of revolution. First its exact structure is derived by means of the apparatus of ellipsoidal harmonics. In this case the structure of the kernel, similarly as of the entries of Galerkin??s matrix, becomes rather complex. Therefore, an approximation of ellipsoidal harmonics (limit layer approach), based on an approximation version of Legendre??s ordinary differential equation, resulting from the method of separation of variables in solving Laplace??s equation, is used. The kernel thus obtained shows some similar features, which the reproducing kernel has in the spherical case, i.e. for the solution domain represented by the exterior of a sphere. A numerical implementation of the exact structure of the reproducing kernel is mentioned as a driving impulse of running investigations. 相似文献
8.
Zdeněk Martinec 《Studia Geophysica et Geodaetica》1990,34(4):313-326
Summary A new method for computing the potential coefficients of the Earth's external gravity field is presented. The gravimetric
boundary-value problem with a free boundary is reduced to the problem with a fixed known telluroid. The main idea of the derivation
consists in a continuation of the quantities from the physical surface to the telluroid by means of Taylor's series expansion
in such a way that the terms whose magnitudes are comparable with the accuracy of today's gravity measurements are retained.
Thus not only linear, but also non-linear terms are taken into account. Explicitly, the terms up to the order of the third
power of the Earth's flattening are retained. The non-linear boundary-value problem on the telluroid is solved by an iteration
procedure with successive approximations. In each iteration step the solution of the non-linear problem is estimated by the
solutions of two linear problems utilizing the fact that the non-linear boundary condition may be split into two parts; the
linear spherical approximation of the gravity anomaly whose magnitude is significantly greater than the others and the non-linear
ellipsoidal corrections. Finally, in order to solve the problem in terms of spherical harmonics, the transform method composed
of the fast Fourier transform and Gauss Legendre quadrature is theoretically outlined. Immediate data processing of gravity
data measured on the physical Earth's surface without any continuation of gravity measurements to a reference level surface
belongs to the main advantage of the presented method. This implies that no preliminary data handling is needed and that the
error data propagation is, consequently, maximally suppressed. 相似文献
9.
Mehdi Najafi-Alamdary Alireza A. Ardalan Seyed-Rohallah Emadi 《Studia Geophysica et Geodaetica》2012,56(1):153-170
An ellipsoidal Neumann type geodetic boundary-value problem (GBVP) for the computation of disturbing potential on the surface
of the Earth based on the surface gravity disturbance as the boundary data is formulated. The solution methodology of the
GBVP can be algorithmically summarized as follows: (i) using global navigation satellite systems (GNSS) coordinates of the
gravity stations, the surface gravity disturbances are generated as the boundary data. (ii) Applying the deflection correction
to the gravity disturbances to arrive at the derivative of the surface disturbing potential along the ellipsoidal normal.
(iii) Removing the low frequencies part of the gravity field using harmonic expansion to degree and order 110. (iv) Using
the short wavelength part of the corrected gravity disturbances derived in the previous section as the boundary data within
the constructed GBVP to derive the short wavelength disturbing potential over the Earth surface. (v) The computed shortwave
length signals of disturbing potentials are converted to disturbing potential values by restoring the removed effects. 相似文献
10.
Robert Tenzer Ismael Foroughi Lars E. Sjöberg Mohammad Bagherbandi Christian Hirt Martin Pitoňák 《Surveys in Geophysics》2018,39(3):313-335
In planetary sciences, the geodetic (geometric) heights defined with respect to the reference surface (the sphere or the ellipsoid) or with respect to the center of the planet/moon are typically used for mapping topographic surface, compilation of global topographic models, detailed mapping of potential landing sites, and other space science and engineering purposes. Nevertheless, certain applications, such as studies of gravity-driven mass movements, require the physical heights to be defined with respect to the equipotential surface. Taking the analogy with terrestrial height systems, the realization of height systems for telluric planets and moons could be done by means of defining the orthometric and geoidal heights. In this case, however, the definition of the orthometric heights in principle differs. Whereas the terrestrial geoid is described as an equipotential surface that best approximates the mean sea level, such a definition for planets/moons is irrelevant in the absence of (liquid) global oceans. A more natural choice for planets and moons is to adopt the geoidal equipotential surface that closely approximates the geometric reference surface (the sphere or the ellipsoid). In this study, we address these aspects by proposing a more accurate approach for defining the orthometric heights for telluric planets and moons from available topographic and gravity models, while adopting the average crustal density in the absence of reliable crustal density models. In particular, we discuss a proper treatment of topographic masses in the context of gravimetric geoid determination. In numerical studies, we investigate differences between the geodetic and orthometric heights, represented by the geoidal heights, on Mercury, Venus, Mars, and Moon. Our results reveal that these differences are significant. The geoidal heights on Mercury vary from ? 132 to 166 m. On Venus, the geoidal heights are between ? 51 and 137 m with maxima on this planet at Atla Regio and Beta Regio. The largest geoid undulations between ? 747 and 1685 m were found on Mars, with the extreme positive geoidal heights under Olympus Mons in Tharsis region. Large variations in the geoidal geometry are also confirmed on the Moon, with the geoidal heights ranging from ? 298 to 461 m. For comparison, the terrestrial geoid undulations are mostly within ± 100 m. We also demonstrate that a commonly used method for computing the geoidal heights that disregards the differences between the gravity field outside and inside topographic masses yields relatively large errors. According to our estimates, these errors are ? 0.3/+ 3.4 m for Mercury, 0.0/+ 13.3 m for Venus, ? 1.4/+ 125.6 m for Mars, and ? 5.6/+ 45.2 m for the Moon. 相似文献
11.
The undulation of the geoid, the gravity anomaly and the deflection of the vertical are the three basic observations describing the shape and the gravity field of the earth. The Stokes’ formula that computes the undulation of the geoid using the gravity anomaly on the geoid under spherical approximate conditions was first put forward by Stokes[1]. According to Stokes’ theory, The Vening-Meinesz formula that computes the meridian and the prime vertical components of the deflection of the ve… 相似文献
12.
Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global
geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area
mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method
for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple
ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM
quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of
gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal
vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal
area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated
in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid
heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based
geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available
as area mean values. 相似文献
13.
14.
在涉及表面垂直位移的地球物理正演和反演问题的研究中,表面真垂直位移等于表面视垂直位移和大地水准面高变化的和.本文从广义Bruns公式、广义stokes公式和广义Ve-ning-Meinesz公式出发,导出了用表面垂直位移和重力变化确定大地水准面形变的公式.讨论了表面垂直位移和重力变化对大地水准面高变化的影响.在此基础上,给出了表面荷载源、几种不同充填介质的膨胀源和位错源所引起的大地水准面高变化对视垂直位移影响的数值结果,分析了局部地球物理事件引起的大地水准面高变化的特点.最后给出了使用辽南地区的实际观测资料确定的局部大地水准面高变化以及对视垂直位移影响的计算结果. 相似文献
15.
Parameters of the gravity field harmonics outside the geoid are sought in solving the Stokes boundary-value problem while
harmonics outside the Earth in solving the Molodensky boundary-value problem. The gravitational field generated by the atmosphere
is subtracted from the Earth’s gravity field in solving either the Stokes or Molodensky problem. The computation of the atmospheric
effect on the ground gravity anomaly is of a particular interest in this study. In this paper in particular the effect of
atmospheric masses is discussed for the Stokes problem. In this case the effect comprises two components, specifically the
direct and secondary indirect atmospheric effects. The numerical investigation is conducted at the territory of Canada. Numerical
results reveal that the complete effect of atmosphere on the ground gravity anomaly varies between 1.75 and 1.81 mGal. The
error propagation indicates that precise determination of the atmospheric effect on the gravity anomaly depends mainly on
the accuracy of the atmospheric mass density distribution model used for the computation. 相似文献
16.
Compilation of the bathymetrically and topographically corrected gravity disturbance, the so called BT disturbance, for the
purpose of gravity interpretation/inversion, is investigated from the numerical point of view, with special emphasis on regions
of negative heights. In regions of negative ellipsoidal (geodetic) heights, such as the Dead Sea region onshore or offshore
areas of negative geoidal heights, two issues complicate the compilation and subsequently the inversion of the BT disturbance.
The first is associated with the evaluation of normal gravity below the surface of the reference ellipsoid (RE). The latter
is tied to the legitimacy of the harmonic continuation of the BT disturbance in these regions. These two issues are proposed
to be resolved by the so called reference quasi-ellipsoid (RQE) approach. New bathymetric and topographic corrections are
derived based on the RQE and the inverse problem is formulated based on the RQE. The RQE approach enables the computation
of normal gravity by means of the international gravity formula, and makes the harmonic continuation in the regions of negative
heights of gravity stations legitimate. The gravimetric inversion is then transformed from the RQE approach back to the RE
approach, following the now legitimate harmonic upward continuation of the gravity data to stations on or above the RE. Stripping,
the removal of an effect of a known density contrast, is considered in the context of the RQE approach. A numerical case study
is presented for the RQE approach in a region of NW Canada. 相似文献
17.
Explicit formula for the geoid-quasigeoid separation 总被引：1，自引：0，他引：1
The explicit formula for the geoid-to-quasigeoid correction is derived in this paper. On comparing the geoidal height and
height anomaly, this correction is found to be a function of the mean value of gravity disturbance along the plumbline within
the topography. To evaluate the mean gravity disturbance, the gravity field of the Earth is decomposed into components generated
by masses within the geoid, topography and atmosphere. Newton’s integration is then used for the computation of topography-and
atmosphere-generated components of the mean gravity, while the combined solution for the downward continuation of gravity
anomalies and Stokes’ boundary-value problem is utilized in computing the component of mean gravity disturbance generated
by mass irregularities within the geoid. On application of this explicit formulism a theoretical accuracy of a few millimetres
can be achieved in evaluation of the geoid-to-quasigeoid correction. However, the real accuracy could be lower due to deficiencies
within the numerical methods and to errors within the input data (digital terrain and density models and gravity observations). 相似文献
18.
First, we present three different definitions of the vertical which relate to (i) astronomical longitude and astronomical
latitude as spherical coordinates in gravity space, (ii) Gauss surface normal coordinates (also called geodetic coordinates)
of type ellipsoidal longitude and ellipsoidal latitude and (iii) Jacobi ellipsoidal coordinates of type spheroidal longitude
and spheroidal latitude in geometry space. Up to terms of second order those vertical deflections agree to each other. Vertical
deflections and gravity disturbances relate to a reference gravity potential. In order to refer the horizontal and vertical
components of the disturbing gravity field to a reference gravity field, which is physically meaningful, we have chosen the
Somigliana-Pizzetti gravity potential as well as its gradient. Second, we give a new closed-form representation of Somigliana-Pizzetti
gravity, accurate to the sub Nano Gal level. Third, we represent the gravitational disturbing potential in terms of Jacobi
ellipsoidal harmonics. As soon as we take reference to a normal potential of Somigliana-Pizzetti type, the ellipsoidal harmonics
of degree/order (0,0), (1,0), (1, − 1), (1,1) and (2,0) are eliminated from the gravitational disturbing potential. Fourth,
we compute in all detail the gradient of the gravitational disturbing potential, in particular in orthonormal ellipsoidal
vector harmonics. Proper weighting functions for orthonormality on the International Reference Ellipsoid are constructed and
tabulated. In this way, we finally arrive at an ellipsoidal harmonic representation of vertical deflections and gravity disturbances.
Fifth, for an ellipsoidal harmonic Gravity Earth Model (SEGEN: http://www.uni-stuttgart.de/gi/research/paper/coefficients/coefficients.zip)
up to degree/order 360/360 we compute the global maps of ellipsoidal vertical deflections and ellipsoidal gravity disturbances
which transfer a great amount of geophysical information in a properly chosen equiareal ellipsoidal map projection. 相似文献
19.
A spherical approximation makes the basis for a majority of formulas in physical geodesy. However, the present-day accuracy
in determining the disturbing potential requires an ellipsoidal approximation. The paper deals with constructing Green’s function
for an ellipsoidal Earth by an ellipsoidal harmonic expansion and using it for determining the disturbing potential. From
the result obtained the part that corresponds to the spherical approximation has been extracted. Green’s function is known
to depend just on the geometry of the surface where boundary values are given. Thus, it can be calculated irrespective of
the gravity data completeness. No changes of gravity data have an effect on Green’s function and they can be easily taken
into account if the function has already been constructed. Such a method, therefore, can be useful in determining the disturbing
potential of an ellipsoidal Earth. 相似文献
20.
NOC model of the earth's main magnetic field 总被引：1，自引：0，他引：1
徐文耀 《中国科学D辑(英文版)》2003,46(9):882-894
The international geomagnetic reference field (IGRF) is a standard model for describing the spatial structure and temporal variation of the earth抯 main magnetic field[1—3]. The first IGRF model, designated IGRF 1965, was adopted by IAGA in 1968[4]. In l… 相似文献