Wavelets in Geodesy and Geodynamics

Wavelets in Geodesy and Geodynamics

by Wolfgang Keller
     
 

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

Table of Contents

Prefacev
Notationix
1Fourier analysis and filtering1
1.1Fourier analysis1
1.2Linear filters14
2Wavelets24
2.1Motivation24
2.2Continuous wavelet transformation30
2.2.1Concept30
2.2.2Time-frequency resolution36
2.2.3Approximation properties37
2.3Discrete wavelet transformation40
2.3.1Frames40
2.4Multi-resolution analysis43
2.5Mallat algorithm55
2.6Wavelet packages63
2.7Biorthogonal wavelets68
2.8Compactly supported orthogonal wavelets82
2.8.1Daubechies wavelets83
2.8.2Solution of scaling equations96
2.9Wavelet bases on an interval98
2.10Two-dimensional wavelets102
2.10.1Continuous two-dimensional wavelets102
2.10.2Discrete two-dimensional wavelets104
2.11Wavelets on a sphere110
2.11.1Harmonic wavelets110
2.11.2Triangulation based wavelets123
3Applications131
3.1Pattern recognition131
3.1.1Polar motion131
3.1.2Atmospheric turbulence135
3.1.3Fault scarps from seafloor bathymetry139
3.1.4Seismic reflection horizons141
3.1.5GPS cycle-slip detection147
3.1.6Edge detection in images153
3.2Data compression and denoising156
3.2.1Wavelet filters and estimation156
3.2.2Deconvolution thresholding164
3.2.3Image compression173
3.3Sub-band coding, filtering and prediction181
3.3.1QMF filter design and wavelets181
3.3.2Prediction of stationary signals with superimposed non-stationary noise187
3.4Operator approximation196
3.4.1Wavelet compression of operator equations196
3.4.2Multi-grid solvers for wavelet discretized operators204
3.5Gravity field modelling212
AHilbert spaces217
A.1Definition of Hilbert spaces217
A.2Complete orthonormal systems in Hilbert spaces222
A.3Linear functionals--dual space225
A.4Examples of Hilbert spaces226
A.5Linear operators--Galerkin method234
A.6Hilbert space valued random variables236
BDistributions238
Exercises245
Bibliography269
Index277

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