Wavelets in Geodesy and Geodynamics

Wolfgang Keller3-11-017546-0

In seismic analysis and other areas of the geosciences, wavelet analysis is superior to the signal processing governed by the theory of Fourier transformation because it is more valuable in understanding deviation from time and space statistical properties. Developed from a graduate course on wavelets held at the Department of Geomatics Engineering at the U. of Calgary (Canada), this textbook mathematically develops wavelet theory as applied in digital signal processing. Definitions and results are given in a mathematically rigorous form, while proofs are kept comparatively short. The reader is assumed to have a graduate-level mathematics education.

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Table of contents :
Preface……Page 6
Contents……Page 8
Notation……Page 10
1.1 Fourier analysis……Page 12
1.2 Linear filters……Page 25
2 Wavelets……Page 34
2.1 Motivation……Page 35
2.2 Continuous wavelet transformation……Page 41
2.3 Discrete wavelet transformation……Page 51
2.4 Multi-resolution analysis……Page 54
2.5 Mallat algorithm……Page 66
2.6 Wavelet packages……Page 74
2.7 Biorthogonal wavelets……Page 79
2.8 Compactly supported orthogonal wavelets……Page 93
2.9 Wavelet bases on an interval……Page 109
2.10 Two-dimensional wavelets……Page 113
2.11 Wavelets on a sphere……Page 121
3.1 Pattern recognition……Page 142
3.2 Data compression and denoising……Page 167
3.3 Sub-band coding, filtering and prediction……Page 192
3.4 Operator approximation……Page 207
3.5 Gravity field modelling……Page 223
3.1 Pattern recognition……Page 141
3.3 Sub-band coding, filtering and prediction……Page 191
A.1 Definition of Hilbert spaces……Page 228
A.2 Complete orthonormal systems in Hilbert spaces……Page 233
A.3 Linear functionals – dual space……Page 236
A.4 Examples of Hilbert spaces……Page 237
A.5 Linear operators – Galerkin method……Page 245
A.6 Hilbert space valued random variables……Page 247
B Distributions……Page 249
Exercises……Page 256
Bibliography……Page 280
Index……Page 288