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

W. Keller, W. Freeden: Mathematical Aspects of Geodetic Modelling 157
De-noising and smoothing makes use of the fact that the
noise of a signal is mainly concentrated on the smallest
scales of a wavelet representation of the corresponding
signal. Therefore, an energy reduction on the smallest scales
automatically reduces noise and smoothes the field. The
application of this idea is discussed in the publications
FREEDEN et al. (2003), FREEDEN and MAIER (2003), HESSE
and GUTTING (2003).
The majority of wavelet applications falls in the area of
recovery of the time-variable gravity field from the data
of the CHAMP and GRACE mission. Since the time
variability of the gravity field is not uniform but concentrated
on a small number of medium-size regions, wavelets
are the adequate tool for such kinds of investigations. A
wavelet analysis of CHAMP data was presented in the
contributions FREEDEN and MAIER (2003), FREEDEN and
MICHEL (2003), MAIER and MAYER (2003), MAYER and
MAIER (2003), FENGLER et al. (2004), FENGLER et al.
(2004a), FENGLER et al. (2004b), FREEDEN and MICHEL
(2004), SCHMIDT et al. (2005) and SCHMIDT et al. (2005a).
Similar studies for the GRACE mission are reported in
FENGLER et al. (2007), SCHMIDT et al. (2006), SCHMIDT et
al. (2007), MAYER-GÜRR et al. (2006) and MAYER-GÜRR
et al. (2007).
Besides the analysis of the gravitational field by wavelets
also the magnetic field is investigated with the same tools.
Results are published in MAYER and MAIER (2003), MAIER
and MAYER (2003) and finally, the CHAMP and GRACE
data are also used for the study of the ionosphere SCHMIDT
et al. (2007a).
A more exotic application of spherical wavelets is in the
field of inverse problems. Also here a couple of publications
have to be mentioned: MICHEL (2004), MAYER
(2004a), MICHEL (2005) and FENGLER et al. (2006a).
Overviews about the use of spherical wavelets in geosciences
are given in FREEDEN and MICHEL (2004c),
FREEDEN et al. (2003c) and FREEDEN and SCHREINER
(2005).
High-performance computing
The processing of data from the CHAMP and the GRACE
mission leads to linear systems of equations with a large
number of unknowns and an even much larger number of
observation data. A direct solution of theses systems of
equation requires very much computation time. The target
of parallelization is mainly the distribution of independent
parts of the computation on different CPUs of a parallelor
vector computer. Here the publications Austen and Keller
(2006), AUSTEN et al (2006), BAUR and KUSCHE (2006) and
BAUR et al. (2006) have to be mentioned.
Miscellaneous
Independent on the availability of new data type also older,
not satisfyingly solved problems have received attention.
Here a certainly incomplete list of those topics is to be
mentioned:
– A singularity-free alternative to Sanso's gravity space
approach is discussed in AUSTEN and KELLER (2007).
– Another topic is the closed solution of systems of polynomial
equations, which occur for example in the trilateration
problem. Here, new ideas based on Gröbner bases
and the Buchberger algorithm have been discussed in
AWANGE and GRAFAREND (2003, 2003a, 2003b, 2003c,
2003d, 2003e, 2003f), AWANGE et al. (2003, 2003a,
2003b, 2003c), AWANGE et al. (2004) and AWANGE et
al. (2005, 2005a, 2005b).
– For the processing of laser-scanning data new techniques,
close to spline approximation, have been proposed in
BORKOSKI and KELLER (2003) and BORKOWSKI and
KELLER (2005).
– A comprehensive textbook on map projections was published
GRAFAREND and KRUMM (2006) and the special
kind of harmonic maps was studied in GRAFAREND
(2005).
References
ABEYRATNE M.. K., FREEDEN W., MAYER C.: Multiscale Deformation
Analysis by Cauchy-Navier Wavelets. Journal of Applied
Mathematics, (12): 605-645, 2003.
AUSTEN G., BAUR O., KELLER W.: Use of High Performance
Computing in gravity field research. In Nagel WE, Jäger W
and Resch M (Eds.): High Performance Computing in
Science and Engineering’05, Springer-Verlag, Berlin
Heidelberg, 2006
AUSTEN G., KELLER W.: LSQR Tuning to Succeed in Computing
High Resolution Gravity Field Solutions. In: Alberigo P,
Erbacci G, Garofalo F (Eds.): Science and Supercomputing
in Europe, HPC-Europa Transnational Access Report 2005,
307-311, Monograf s.r.l., Bologna, Italy, 2006
AUSTEN G., KELLER W.: Numerical Implementation of the Gravity
Space Approach – Proof of Concept. In Rizos C, Tregoning
P (Eds.): Dynamic Planet – Monitoring and Understanding
a Dynamic Planet with Geodetic and Oceanographic Tools,
Conference of the IAG, Aug 22-26, 2005, Cairns, Australia,
IAG Symposia, Vol. 130, Springer-Verlag Berlin Heidelberg
New York, 2007
AWANGE J., GRAFAREND E.: Closed Form Solution of the Overdetermined
Nonlinear 7 Parameter Datum Transformation.
Allgemeine Vermessungsnachrichten 110 (2003) 130-149
AWANGE J., GRAFAREND E.: Explicit Solution of the Overdetermined
Three-Dimensional Resection Problem. Journal of
Geodesy 76 (2003) 605-616
AWANGE J., GRAFAREND E.: Groebner-Basis Solution of the
Three-Dimensional Resection Problem (P4P). Journal of
Geodesy 77 (2003) 327-337
AWANGE J., GRAFAREND E.: Multipolynomial Resultant Solution
of the 3d Resection Problem (P4P). Bollettino di Geodesia
e Scienze Affini 62 (2003) 79 – 102
AWANGE J., GRAFAREND E.: Nonlinear Analysis of the Three-
Dimensional Datum Transformation [Conformal Group
C7(3)]. Journal of Geodesy 77 (2003) 66-76
AWANGE J.: GRAFAREND E.: Polynomial Optimization of the 7-
Parameter Datum Transformation Problem when Only Three
Stations in Both Systems are Given. Zeitschrift für Vermessungswesen
128 (2003) 266-270