Wavelets in Geodesy and Geodynamics

ISBN-10: 3110175460

ISBN-13: 9783110175462

Pub. Date: 04/28/2004

Publisher: De Gruyter, Walter, Inc.

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

Overview

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.

Product Details

ISBN-13:
9783110175462
Publisher:
De Gruyter, Walter, Inc.
Publication date:
04/28/2004
Pages:
279
Product dimensions:
6.90(w) x 9.68(h) x 0.81(d)
Age Range:
18 Years

Related Subjects

 Preface v Notation ix 1 Fourier analysis and filtering 1 1.1 Fourier analysis 1 1.2 Linear filters 14 2 Wavelets 24 2.1 Motivation 24 2.2 Continuous wavelet transformation 30 2.2.1 Concept 30 2.2.2 Time-frequency resolution 36 2.2.3 Approximation properties 37 2.3 Discrete wavelet transformation 40 2.3.1 Frames 40 2.4 Multi-resolution analysis 43 2.5 Mallat algorithm 55 2.6 Wavelet packages 63 2.7 Biorthogonal wavelets 68 2.8 Compactly supported orthogonal wavelets 82 2.8.1 Daubechies wavelets 83 2.8.2 Solution of scaling equations 96 2.9 Wavelet bases on an interval 98 2.10 Two-dimensional wavelets 102 2.10.1 Continuous two-dimensional wavelets 102 2.10.2 Discrete two-dimensional wavelets 104 2.11 Wavelets on a sphere 110 2.11.1 Harmonic wavelets 110 2.11.2 Triangulation based wavelets 123 3 Applications 131 3.1 Pattern recognition 131 3.1.1 Polar motion 131 3.1.2 Atmospheric turbulence 135 3.1.3 Fault scarps from seafloor bathymetry 139 3.1.4 Seismic reflection horizons 141 3.1.5 GPS cycle-slip detection 147 3.1.6 Edge detection in images 153 3.2 Data compression and denoising 156 3.2.1 Wavelet filters and estimation 156 3.2.2 Deconvolution thresholding 164 3.2.3 Image compression 173 3.3 Sub-band coding, filtering and prediction 181 3.3.1 QMF filter design and wavelets 181 3.3.2 Prediction of stationary signals with superimposed non-stationary noise 187 3.4 Operator approximation 196 3.4.1 Wavelet compression of operator equations 196 3.4.2 Multi-grid solvers for wavelet discretized operators 204 3.5 Gravity field modelling 212 A Hilbert spaces 217 A.1 Definition of Hilbert spaces 217 A.2 Complete orthonormal systems in Hilbert spaces 222 A.3 Linear functionals--dual space 225 A.4 Examples of Hilbert spaces 226 A.5 Linear operators--Galerkin method 234 A.6 Hilbert space valued random variables 236 B Distributions 238 Exercises 245 Bibliography 269 Index 277

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