NATIONAL REPORT OF THE FEDERAL REPUBLIC OF ... - IAG Office

156
Introduction
Mathematical Aspects of Geodetic Modelling
Mathematical modelling always reacts to newly available
data-types or tries to solve problems in modelling, which
have not been treated satisfactory so far. From this background
the advances in mathematical modelling were
primarily triggered by the new data from the CHAMP and
the GRACE mission. These new data types generated a
need for
– more efficient methods for data screening and correction,
– higher resolution of the data analysis products, both in
space and in time.
Both requirements can be met by the use of wavelets.
Therefore, the majority of contributions deals in one or
another way with construction and use of wavelets.
Independent on the problem at hand, the enormous amount
of data requires the application of high-performance
computing. A fair amount of publications is devoted to the
parallelization of mathematical models of geodetic
problems.
Finally, various new ideas for old problems were presented
and numerically investigated.
Classical Wavelet Theory
The origin of wavelet analysis was on the real line and on
the plane. It is only a couple of years since the main ideas
of the classical wavelet theory were generalized to curved
manifolds, especially to the sphere. In geodetic context,
classical wavelet theory was mainly used for two purposes:
– outlier detection and elimination,
– data or/and operator compression.
The new gravity field missions provide the user with an
enormous amount of data. Therefore, numerically efficient
methods for outlier detection and elimination are needed.
Here the ability of wavelets for time and scale localization
is a useful tool. For the data of the CHAMP mission this
problem is treated in GÖTZELMANN et al. (2006). Similar
questions for the pre-processing of laser-scanning data are
studied in BORKOWSKI and Keller (2006).
Another application of wavelet theory is the compression
of data or the compression of operators transforming this
data. For different kind of geodetic operators this subject
is addressed in KELLER (2004) and in KUROISHI and KELLER
W. KELLER 1 , W. FREEDEN 2
(2004). An overview about classical wavelet analysis and
geodetic applications is given in KELLER (2004).
Spherical wavelets
Spherical wavelets are base functions, which express both:
The scale of a signal-pattern and the place of occurrence
of this pattern. This makes them particularly useful for
localized modelling of various fields on the sphere. Roughly
speaking, in the spherical wavelet modelling two different
cases can be distinguished:
– the construction of tailored wavelets,
– the use of wavelets for the space-time evaluation of
different fields of geodetic relevance.
Wavelets can be tailored for a big variety of applications.
In ABEYRATNE et al. (2003) and FREEDEN and MICHEL
(2005) the focus was put on deformation analysis. The paper
FREEDEN and MAYER (2003) addresses the construction of
smooth harmonic spherical wavelets. If vectorial and
tensorial quantities are to be analyzed on the sphere, the
corresponding wavelets are developed in FREEDEN and
MICHEL (2004), and FREEDEN and MAYER (2006). Wavelets
on more complicated surfaces than spheres are constructed
in MAYER (2004) and MAYER (2006). Spherical wavelets,
which are derived from Bernstein instead of Legendre
polynomials, are discussed in FENGLER et al. (2006). One
of the deficiencies of spherical wavelets is their lack of
orthogonality. But at least bi-orthogonality can be achieved,
as it is shown in FREEDEN and SCHREINER (2006).
Among the use of spherical wavelets for the study of
geodetic fields, four different targets can be distinguished:
– the deformation field of the Earth,
– the ocean circulation,
– and the time-variable gravity field of the Earth, and
– the issue of de-noising and smoothing of different types
of data.
Wavelets are used for the investigation of deformations in
the publications MICHEL (2003), MICHEL (2004), FREEDEN
and MICHEL (2005a) and KAMMAN and MICHEL (2006). The
steady-state ocean circulation was studied by means of
wavelets in the papers FREEDEN and MICHEL (2004a),
FREEDEN et al. (2005), FEHLINGER et al. (2007) and
FENGLER and FREEDEN (2005).
1 Wolfgang Keller: Geodetic Institute, Universität Stuttgart, Geschwister-Scholl-Str. 24/D, D - 70174 Stuttgart, Germany,
Tel. +49 - 711 - 6858 3459, Fax +49 - 711 - 6858 3285, e-mail wolfgang.keller@gis.uni-stuttgart.de
2 Willi Freeden: AG Geomathematik, Fachbereich Mathematik, Universität Kaiserslautern, Kurt-Schumacher-Str.26, D - 67653 Kaiserslautern,
Germany, Tel. +49 - 631 - 205-2852 / -3867, Fax +49 - 631 - 205-4736, e-mail freeden@mathematik.uni-kl.de