Wavelets in Geodesy and Geodynamics / Edition 1

by Wolfgang Keller

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
Publication date:	04/26/2004
Pages:	289
Product dimensions:	6.69(w) x 9.45(h) x (d)
Age Range:	18 Years

Table of Contents

	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