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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
High--accuracy modelling of the Earth's gravitational
potential from GOCE satellite gradiometry mission
M. S. Petrovskaya, A. N. Vershkov
Pulkovo Observatory, St. Petersburg, Russia
The Earth's gravitational potential is presented in form of a spherical harmon­
ic series with the coefficients C n;m which are the fundamental dynamical constants
for the Earth. The set of C n;m for degree 0 Ÿ n Ÿ N and order jmj Ÿ n consti­
tutes a geopotential model. A global geopotential model is a valuable source of
information about the nature and composition of our planet. It is strongly needed
for solving a large area of problems in satellite dynamics, geodynamics, oceanog­
raphy, navigation, for creating a universal height reference system, etc. The basic
data for constructing the geopotential models are provided by the measurements
of the terrestrial gravity anomalies. However these data cover only one third of
the Earth's surface. The application of perturbations of satellite orbits reached
its intrinsic limits: only the potential harmonics up to degree and order 70 can be
recovered by this approach while the contemporary demands are at least for 360.
The reason for it is the presence of damping factor q n = ( R
r ) n at the potential
harmonic of degree n where R and r are the Earth's mean radius and the satellite
geocentric distance, respectively.
A great progress in modelling the Earth's potential V is expected only from
satellite gradiometry missions -- measuring the gravity gradient components. They
represent six second order potential derivatives in the local reference frame cen­
tered at the satellite. In the expansions of these derivatives the n­th harmonic
has an additional factor n 2 which partly compensates the attenuation effect of
the factor q n in the series for V . The first dedicated satellite gradiometry mission
GOCE (Gravity field and Ocean Circulation Explorer) will be in the context of
the European Space Research Agency (ESA) program. It will allow to construct
a geopotential model whose accuracy and resolution is one order higher than for
the present models.
The problem of recovering the geopotential coefficients from satellite gradiom­
etry data can be easily solved if there are linear relations between the spectral
coefficients of the observables and the coefficients C n;m of V . However such rela­
tions have not been derived yet because the geopotential derivatives with respect
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to the north--oriented coordinates centered at the satellite contain singularities at
the poles. For this reason, instead of the simple spherical functions, the pure--spin
tensor harmonics are proposed for application. Due to very complicated struc­
ture of these basic functions, such analytical approach is extremely difficult for
practical constructing the geopotential models and only the numerical approach
is developed by the international scientific consortium. However this approach
will be also very problematic for the implementation since it assumes the simul­
taneous least squares processing of a huge amount of data (about 100 millions of
observations) and a great number (90 000) of unknown parameters.
In the present paper the authors managed to solve the problem of the geopo­
tential modelling from GOCE satellite mission by an analytical approach, on the
basis of the conventional spherical functions. First a procedure is elaborated for
removing the singularities in the expressions of the geopotential derivatives. Then
simple basic analytical relations are derived for the first time between the spectral
coefficients of six GOCE observables and the unknown geopotential coefficients
C n;m . The derived relations can be used for solving two (inverse) problems. So
long as GOCE mission is still in a preparation stage (until 2004), the spectra of the
geopotential derivatives are generated at the height h = 250 km of GOCE satel­
lite from the most advanced EGM96 geopotential model constructed by NASA
on the basis of a great number of the satellite and terrestrial data. Then from
the simulated spectra of GOCE observables the corresponding `output' geopoten­
tial model is recovered from the same basic relations. The `input' (EGM96) and
`output' (recovered) geopotential models fully coincide. Thus the derived basic
relations allow to perform the whole simulation experiment on the geopotential
modelling from GOCE mission with the estimation of different kinds of errors.
A number of numerical experiments are carried out on the basis of EGM96
geopotential model. At first the basic quantities, degree variances, are evaluated
for GOCE observables. They represent the mean square of the n­th degree har­
monic for each geopotential derivative. Very interesting specific regular behavior
of the spectra of six second derivatives is revealed.
The elaborated procedure of modelling the gravitational potential for the
Earth can be applied for constructing the expansions of the third and higher order
geopotential derivatives referred to the reference frame centered at the satellite,
which will allow to discover new properties of the Earth's gravitational field. The
present analytical results can be readily utilized for modelling the gravitational
fields of other planets and the Moon.
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