Ellipsoidal Wavelet Representation of the Gravity Field
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Date
2008-01
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Ohio State University. Division of Geodetic Science
Abstract
The determination and the representation of the gravity field of the Earth are some of the most important
topics of physical geodesy. Traditionally in satellite gravity recovery problems the global gravity
field of the Earth is modeled as a series expansion in terms of spherical harmonics. Since the Earth’s
gravity field shows heterogeneous structures over the globe, a multi-resolution representation is an
appropriate candidate for an alternative spatial modeling. In the last years several approaches were
pursued to generate a multi-resolution representation of the geopotential by means of spherical base
functions.
Spherical harmonics are mostly used in global geodetic applications, because they are simple and the
surface of Earth is nearly a sphere. However, an ellipsoid of rotation, i.e., a spheroid, means a better
approximation of the Earth’s shape. Consequently, ellipsoidal harmonics are more appropriate than
spherical harmonics to model the gravity field of the Earth. However, the computation of the coefficients
of a series expansion for the geopotential in terms of both, spherical or ellipsoidal harmonics,
requires preferably homogeneous distributed global data sets.
Gravity field modeling in terms of spherical (radial) base functions has long been proposed as an
alternative to the classical spherical harmonic expansion and is nowadays successfully used in regional
or local applications. Applying scaling and wavelet functions as spherical base functions a
multi-resolution representation can be established. Scaling and wavelet functions are characterized
by the ability to localize both in the spatial and in the frequency domain. Thus, regional or even local
structures of the gravity field can be modeled by means of an appropriate wavelet expansion. To be
more specific, the application of the wavelet transform allows the decomposition of a given data set
into a certain number of frequency-dependent detail signals. As mentioned before the spheroid means
a better approximation of the Earth than a sphere. Consequently, we treat in this report the ellipsoidal
wavelet theory to model the Earth’s geopotential.
Modern satellite gravity missions such as the Gravity Recovery And Climate Experiment (GRACE)
allow the determination of spatio-temporal, i.e., four-dimensional gravity fields. This issue is of
great importance in the context of observing time-variable phenomena, especially for monitoring the
climate change. Global spatio-temporal gravity fields are usually computed for fixed time intervals
such as one month or ten days. In the last part of this report we outline regional spatio-temporal
ellipsoidal modeling. To be more specific, we represent the time-dependent part of our ellipsoidal
(spatial) wavelet model by series expansions in terms of one-dimensional B-spline functions. Thus,
our concept allows to establish a four-dimensional multi-resolution representation of the gravity field
by applying the tensor product technique
Description
The research which led to this report was initiated during a visit to the Ohio State
University (OSU) from February 2002 through January 2003, and partially supported by
grants from the National Geospatial-Intelligence Agency's (NGA's) University Research
Initiative (NURI), entitled 'Application of spherical wavelets to the solution of the
terrestrial gravity field model' (NMA201-00-1-2006, 2000-2005, PIs: C. Shum), and from
the National Science Foundation's Collaboration in Mathematics and Geosciences (CMG)
Program, entitled 'Multi-resolution inversion of tectonically driven spatio-temporal
gravity signals using wavelets and satellite data' (EAR-0327633, 2003-2007, PIs: C.K.
Shum).
The authors benefited very much from discussions with many researchers at the OSU and
thank C.K. Shum and the OSU for hospitality. Further thanks go to the German Geodetic
Research Institute (DGFI) and the University of Munich (LMU), at which parts of the
work were conducted. Finally we thank Erik W. Grafarend for many fruitful discussions,
which were the actual starting point for this project.