Geodetic Applications of Wavelets: Proposals and Simple Numerical Experiments

  • Laszlo Battha
  • Battista Benciolini
  • Paolo Zatelli
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 114)

Abstract

The aim of this paper is to treat at a review level some topics of the theory of wavelets that are relevant for some geodetic applications, to further specify some proposals already formulated by one of the authors (1994) and to enforce the proposals by mean of some numerical experiments. The papers about the geodetic applications of wavelets are quite numerous nowadays (see e.g. Barthelmes at al. 1994 and Ballani 1994), therefore it is worthwhile to better specify our present field of interest. Several authors developed efficient numerical algorithms based on the wavelet representation for the computation of linear operators in L 2 (Beylkin et al. 1991, Beylkin 1992, Alpert 1993). The application of these techniques for the computation of linear operators in physical geodesy is investigated in this paper. We only consider operators in planar approximation in order to rely on well established mathematical tools.

Keywords

Convolution 

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References

  1. 1.
    Alpert B.K., A Class of Bases in L 2 for the sparse representation of Integral Operators, SIAM J. Math. Anal.24-1 (1993), 246–262.CrossRefGoogle Scholar
  2. 2.
    Ballani L., Solving the Inverse Gravimetric Problem: on the Benefit of Wavelets, presented at the III Hotine-Marussi Symposium (1994).Google Scholar
  3. 3.
    Barthelmes L., L. Ballani, R. Klees, On the Application of Wavelets in Geodesy, presented at the III Hotine-Marussi Symposium (1994).Google Scholar
  4. 4.
    Beylkin G., On the Representation of Operators in Bases of Compactly Supported Wavelets, SIAM J. Numer. Anal.6-6 (1992), 1716–1740.CrossRefGoogle Scholar
  5. 5.
    Beylkin G., Coifman R., Rochlin V., Fast Wavelet Transform and Numerical Algorithms I, Communications on Pure and Applied Mathematics XLIV - 2 (1991), 141–183.CrossRefGoogle Scholar
  6. 6.
    Benciolini B., A Note on Some Possible Geodetic Applications of Wavelets, LAG Section IV Bulletin Vol.2 N. l (1994), 17–22.Google Scholar
  7. 7.
    Chui C.K., An Introduction to Wavelets, Academic Press, Inc., 1992, (ISBN 0-12-174584-8).Google Scholar
  8. 8.
    Cohen A., Daubechies I., Nonseparable Bidimensional Wavelets Bases, Revista Metemática Iberoamericana 9-1 (1993a), 51–137.CrossRefGoogle Scholar
  9. 9.
    Daubechtes I., Orthonormal bases of Compactly Supported Wavelets, Communications on Pure and Applied Mathematics XLI-7 (1988), 909–996.CrossRefGoogle Scholar
  10. 10.
    Daubechies Ingrid, Ten Lectures on Wavelets, SIAM - Philadelphia, 1992, (ISBN 0-89871-274-2).Google Scholar
  11. 11.
    Daubechies I., Lagarias J.C., Two-Scale Difference Equation. I Existence and Global Regularity of Solutions, SIAM J. on Mathematical Analysis 22-5 (1991), 1388–1410.CrossRefGoogle Scholar
  12. 12.
    Jawert Björn and Sweldens Wim, An Overview of Wavelet Based Multiresolution Analises, SIAM Review 36-3 (1994), 377–412.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Laszlo Battha
    • 1
  • Battista Benciolini
    • 2
  • Paolo Zatelli
    • 2
  1. 1.Geodetic and Geophysical Research Inst. of the Hungarian Academy of ScienceSopronHungary
  2. 2.Dipartimento di Ingegneria Civile e AmbientaleUniversità di TrentoTrentoItaly

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