Keller W. Wavelets in Geodesy and Geodynamics

Berlin: Walter de Gruyter, 2004. — 298 p.For many years, digital signal processing has been governed by the theory of Fourier transform and its numerical implementation. The main disadvantage of Fourier theory is the underlying assumption that the signals have time-wise or space-wise invariant statistical properties. In many applications the deviation from a stationary behavior is precisely the information to be extracted from the signals. Wavelets were developed to serve the purpose of analysing such instationary signals. The book gives an introduction to wavelet theory both in the continuous and the discrete case. After developing the theoretical fundament, typical examples of wavelet analysis in the Geosciences are presented. The book has developed from a graduate course held at The University of Calgary and is directed to graduate students who are interested in digital signal processing. The reader is assumed to have a mathematical background on the graduate level.Preface.
Notation.
Fourier analysis and filtering.
Fourier analysis.
Linear filters.
Wavelets.
Motivation.
Continuous wavelet transformation.
Discrete wavelet transformation.
Multi-resolution analysis.
Mallat algorithm.
Wavelet packages.
Biorthogonal wavelets.
Compactly supported orthogonal wavelets.
Wavelet bases on an interval.
Two-dimensional wavelets.
Wavelets on a sphere.
Applications.
Pattern recognition.
Data compression and denoising.
Sub-band coding, filtering and prediction.
Operator approximation.
Gravity field modelling.
Hilbert spaces.
Definition of Hilbert spaces.
Complete orthonormal systems in Hilbert spaces.
Linear functionals – dual space.
Examples of Hilbert spaces.
Linear operators – Galerkin method.
Hilbert space valued random variables.
Distributions.
Exercises.

Index.