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1.
The Legendre functions of the second kind, renormalized by Jekeli, are considered in the external space on a set of ellipsoids of revolution which are confocal with respect to the normal ellipsoid. Among these ellipsoids a reference one is chosen which bounds the Earth. New expressions for the first and second order derivatives of the Legendre functions are derived. They depend on two very quickly convergent Gauss hypergeometric series which are obtained by transforming the slowly convergent initial hypergeometric series. The derived expressions are applied for constructing the ellipsoidal harmonic series for the Earth disturbing gravitational potential and its derivatives of the first and second orders. Since outside the chosen reference ellipsoid there are no Earth masses (as compared to the normal ellipsoid) then it is more appropriate for constructing the boundary-value equation and solving it on the basis of surface gravity data reduced to this ellipsoid. 相似文献
2.
Due to the complicated structure of their expressions, the ellipsoidal harmonic series for the derivatives of the Earth’s gravitational potential are commonly applied only on a reference ellipsoid. They depend on the first- and second-order derivatives of the associated Legendre functions of both kinds and contain a few singular terms. We construct ellipsoidal harmonic expansions in the exterior space for the first and second potential derivatives, which are similar to the series on the reference ellipsoid enveloping the Earth. We take a point P at an arbitrary altitude above the reference ellipsoid and construct the ellipsoid of revolution confocal to it, which passes through this point. The conventional complicated singular expressions for the first and second potential derivatives in the local north-oriented ellipsoidal reference frame, with the origin at the point P, are transformed into non-singular ellipsoidal harmonic series, which do not contain the first- and second-order derivatives of the associated Legendre functions. The resulting series have an accuracy of the squared eccentricity. These series can be applied for constructing a geopotential model, which is based, simultaneously, on the surface gravity data and the data of satellite missions, which provide measurements of the accelerations and/or the gravitational gradients. When the eccentricity of the considered external ellipsoid is equated to zero, the ellipsoid becomes an external sphere passing through the point P and the constructed ellipsoidal harmonic expansions are converted into non-singular spherical harmonic series for the first and second potential derivatives in the local north-oriented spherical reference frame. 相似文献
3.
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
4.
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
5.
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
6.
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
7.
The ellipsoidal Stokes boundary-value problem is used to compute the geoidal heights. The low degree part of the geoidal heights can be represented more accurately by Global Geopotential Models (GGM). So the disturbing potential is splitted into a low-degree reference potential and a higher-degree potential. To compute the low-degree part, the global geopotential model is used, and for the high-degree part, the solution of the ellipsoidal Stokes boundary-value problem in the form of the surface integral is used. We present an effective method to remove the singularity of the high-degree of the spherical and ellipsoidal Stokes functions around the computational point. Finally, the numerical results of solving the ellipsoidal Stokes boundary-value problem and the difference between the high-degree part of the solution of the ellipsoidal Stokes boundary-value problem and that of the spherical Stokes boundary-value problem is presented. 相似文献
8.
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
9.
Layer-Based Modelling of the Earth’s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation 总被引：1，自引：0，他引：1
Moritz Rexer Christian Hirt Sten Claessens Robert Tenzer 《Surveys in Geophysics》2016,37(6):1035-1074
Global forward modelling of the Earth’s gravitational potential, a classical problem in geophysics and geodesy, is relevant for a range of applications such as gravity interpretation, isostatic hypothesis testing or combined gravity field modelling with high and ultra-high resolution. This study presents spectral forward modelling with volumetric mass layers to degree 2190 for the first time based on two different levels of approximation. In spherical approximation, the mass layers are referred to a sphere, yielding the spherical topographic potential. In ellipsoidal approximation where an ellipsoid of revolution provides the reference, the ellipsoidal topographic potential (ETP) is obtained. For both types of approximation, we derive a mass layer concept and study it with layered data from the Earth2014 topography model at 5-arc-min resolution. We show that the layer concept can be applied with either actual layer density or density contrasts w.r.t. a reference density, without discernible differences in the computed gravity functionals. To avoid aliasing and truncation errors, we carefully account for increased sampling requirements due to the exponentiation of the boundary functions and consider all numerically relevant terms of the involved binominal series expansions. The main outcome of our work is a set of new spectral models of the Earth’s topographic potential relying on mass layer modelling in spherical and in ellipsoidal approximation. We compare both levels of approximations geometrically, spectrally and numerically and quantify the benefits over the frequently used rock-equivalent topography (RET) method. We show that by using the ETP it is possible to avoid any displacement of masses and quantify also the benefit of mapping-free modelling. The layer-based forward modelling is corroborated by GOCE satellite gradiometry, by in-situ gravity observations from recently released Antarctic gravity anomaly grids and degree correlations with spectral models of the Earth’s observed geopotential. As the main conclusion of this work, the mass layer approach allows more accurate modelling of the topographic potential because it avoids 10–20-mGal approximation errors associated with RET techniques. The spherical approximation is suited for a range of geophysical applications, while the ellipsoidal approximation is preferable for applications requiring high accuracy or high resolution. 相似文献
10.
L. B. PEDERSEN 《Geophysical Prospecting》1985,33(2):263-281
Wavenumber domain expressions for bodies with elliptical cross-section and of ellipsoidal shape have been developed both for homogeneous bodies and for certain bodies of density/magnetization varying linearly with depth or, more generally, according to a polynomial with depth. The simple expressions thus obtained lend themselves to an easy analysis, especially for long and short wavelengths. At the long-wavelength end of the spectra their decay is governed by an exponential with a decay “depth” equal to the depth to the center of mass. At the short-wavelength end this depth is replaced by the depth to the upper focus of the ellipsoid (or the elliptic cross-section). For vertically inhomogeneous ellipsoids the decay rate is also dependent on the product of the vertical gradient of density/magnetization and their semi-axes. 相似文献
11.
12.
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
13.
14.
Mariya I. Yurkina 《Studia Geophysica et Geodaetica》1994,38(4):325-332
Summary A relation is established between coefficients of an expansion of the gravitational potential into a series of Legendre's
function of the second kind and coefficients of an expansion of gravity anomalies on the surface of the reference ellipsoid
into a series of the same functions. This connection can be useful in geodetic computations which take into account the Earth's
flattening. 相似文献
15.
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
16.
Satellite gradiometry is an observation technique providing data that allow for evaluation of Stokes’ (geopotential) coefficients.
This technique is capable of determining higher degrees/orders of the geopotential coefficients than can be achieved by traditional
dynamic satellite geodesy. The satellite gradiometry data include topographic and atmospheric effects. By removing those effects,
the satellite data becomes smoother and harmonic outside sea level and therefore more suitable for downward continuation to
the Earth’s surface. For example, in this way one may determine a set of spherical harmonics of the gravity field that is
harmonic in the exterior to sea level.
This article deals with the above effects on the satellite gravity gradients in the local north-oriented frame. The conventional
expressions of the gradients in this frame have a rather complicated form, depending on the first-and second-order derivatives
of the associated Legendre functions, which contain singular factors when approaching the poles. On the contrary, we express
the harmonic series of atmospheric and topographic effects as non-singular expressions. The theory is applied to the regions
of Fennoscandia and Iran, where maps of such effects and their statistics are presented and discussed. 相似文献
17.
The paper presents two algorithms for the computation of intersection of geodesics and minimum distance from a point to a geodesic on the ellipsoid, respectively. They are based on the iterative use of direct and inverse problems of geodesy by means of their implementations with machine-precision accuracy in GeographicLib. The algorithms yield the same results as those obtained by Karney’s approach based on the use of auxiliary ellipsoidal gnomonic projections, with the advantage on our side that the algorithms are not limited to distances below 10000 km. This results in our algorithm being the only general solution for the problem of minimum distance from a point to a geodesic on the ellipsoid. 相似文献
18.
This paper presents an approach to determine the gradient of curvature of the normal plumblines at a point P above the ellipsoid and introduces a new geometrical object which is the isocurvature line. The assumed facts are the coordinates of the point P and the formula for the normal gravity potential U. For the determination of the gradient of the normal plumbline curvature k at the point P we define a small circle on the meridian plane of P whose center is at the point P. The circle has the radius of one meter and interior D. In this circle we construct a curvature replacement function to approximate the curvature function k. This replacement function is a quotient of polynomials hence it is easy to find its partial derivatives at the point P. For the construction of replacement function we make the assumption that in the interior of the circle D the first order partial derivatives of U behave linearly and the second order partial derivatives have constant values which equal their value at the point P. Then we set the gradient of the curvature function to be equal with the gradient of the aforementioned replacement function at P. An isocurvature line of the normal gravity field passing through a point P is a curve such that the value of the function of the plumblines’ curvature k is constant and equals k(P). We give a formula to find the direction of the isocurvature line on the meridian plane and we prove that there are infinitely many isocurvature lines passing through the point P and they all lie on a special surface, the isocurvature surface. 相似文献
19.
A simple expression is presented on the capability of storage-treatment systems to reduce non-point pollutant runoff load to natural waters. Their efficiency depends on the capacities of the facilities and probabilistic properties of runoff, such as interval, duration, volume, and concentration of runoff events. Assuming the compound Poisson process for runoff time series, the exact expressions of the ratio of treated load in terms of storage and treatment capacities are theoretically derived on the neighbourhoods of all boundaries of the domain on which the problem is defined. Then, an approximate expression over the whole domain is presented, of which the value and the first-order derivative coincide with those of the exact derived expressions near the boundaries. Accuracy is checked by Monte Carlo simulations. 相似文献
20.
Michele Caputo 《Pure and Applied Geophysics》1964,57(1):66-82
Summary Adopting thePizzetti-Somigliana method and using elliptic integrals we have obtained closed formulas for the space gravity field in which one of the equipotential surfaces is a triaxial ellipsoid. The same formulas are also obtained in first approximation of the equatorial flattening avoiding the use of the elliptic integrals. Using data from satellites and Earth gravity data the gravitational and geometric bulge of the Earth's equator are computed. On the basis of these results and on the basis of recent gravity data taken around the equator between the longitudes 50° to 100° E, 155° to 180° E, and 145° to 180° W, we question the advantage of using a triaxial gravity formula and a triaxial ellipsoid in geodesy. Closed formulas for the space field in which a biaxial ellipsoid is an equipotential surface are also derived in polar coordinates and its parameters are specialized to give the international gravity formula values on the international ellipsoid. The possibility to compute the Earth's dimensions from the present Earth gravity data is the discussed and the value ofMG=(3.98603×1020 cm3 sec–2) (M mass of the Earth,G gravitational constant) is computed. The agreement of this value with others computed from the mean distance Earth-Moon is discussed. The Legendre polinomials series expansion of the gravitational potential is also added. In this series the coefficients of the polinomials are closed formulas in terms of the flattening andMG.Publication Number 327, and Istituto di Geodesia e Geofisica of Università di Trieste. 相似文献