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1.
The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on
reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential
coefficients
n,m
. In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal
harmonic series is developed for the geopotential and then its coefficients
n,m
e
are transformed to the unknown
n,m
. The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral
formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied
numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From
two sets of
n,m
(n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation
are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution
appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics
of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential
model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068)
by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to
apply the LS technique for evaluating geopotential coefficients.
Received: 12 December 2000 / Accepted: 21 June 2001 相似文献
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Celestial Mechanics and Dynamical Astronomy - An expansion of the disturbing functionR is suggested for the special case when the orbits of a group of minor planets have approximately equal major... 相似文献
4.
This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84(3):165–178, 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first- and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first- and second-order derivatives in terms of the geopotential coefficients. These equations are converted into recurrent relations from which the coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission. 相似文献
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A set of spherical harmonics is the most widely used representation of the Earth’s gravity potential. This series converges outside and on the surface of a reference sphere enveloping the Earth. However, the Earth’s surface is better approximated by the reference ellipsoid—a compressed ellipsoid of revolution that covers the entire Earth. The gravity potential can be expanded in a series over ellipsoidal harmonics on the surface of the reference ellipsoid and on the surface of other external confocal ellipsoids of revolution. In contrast to spherical harmonics, depending on the associated Legendre functions of the first kind, ellipsoidal harmonics depend also on the associated Legendre functions of the second kind. The latter contain the very slowly converging hypergeometric Gauss series. The number of series increases with increasing the order of their derivatives. In this work, we derived new series for the gravitational potential of the Earth and its derivatives over ellipsoidal harmonics. Starting from the first order derivative, all the series corresponding to higher order derivatives depend on the same two hypergeometric Gauss series. The latter converges considerably faster than that for the hypergeometric series previously used when computing the gravity potential and its derivatives. 相似文献
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M. S. Petrovskaya 《Journal of Geodesy》1979,53(1):37-51
Summary The possibility of improving the convergence of Molodensky’s series is considered. Then new formulas are derived for the solution
of the geodetic boundary value problem. One of them presents an expression for the boundary condition which involves a linear
combination of Stokes’ constants and surface gravity anomalies. This differs from the usually used relation by the appearance
of additional terms dependent on second order terns with respect to the elevations of the earth’s surface. The formulas are
derived for Stokes’ constants and the anomalous potential in terms of surface anomalies. As compared to the Taylor’s series
of Molodensky, they are presented in the form of surface harmonic series. Due regard is made to the earth’s oblateness, in
addition to the terrain topography. 相似文献
9.
M. S. Petrovskaya 《Journal of Geodesy》1979,53(3):259-271
Summary The geopotential on and outside the earth is represented as a series in surface harmonics. The principal terms in it correspond
to the solid harmonics of the external potential expansion with the coefficients being Stokes’ constantsC
nm
andS
nm
. The additional terms which occur near the earth’s surface due to its non-sphericity and topography are expressed in terms
of Stokes’ constants too. This allows performing downward continuation of the potential derived from satellite observations.
In the boundary condition which correlates Stokes’ constants and the surface gravity anomalies there occur additional terms
due to the earth’s non-sphericity and topography. They are expressed in terms of Stokes’ constants as well. This improved
boundary condition can be used for upward and downward continuations of the gravity field. Simple expressions are found representingC
nm
andS
nm
as explicit functions of the surface anomalies and its derivatives. The formula for the disturbing potential on the surface
is derived in terms of the surface anomalies. All the formulas do not involve the earth’s surface in clinations. 相似文献
10.
Margarita Petrovskaya 《Journal of Geodesy》1988,62(2):161-170
A non-conventional treatment of Stokes’integral enables significant simplification of formulas for both the regional and global
contributions of the gravity field to the geoidal height. 相似文献