## Wavelets in the GeosciencesThis book contains state-of-the-art continuous wavelet analysis of one and more dimensional (geophysical) signals. Special attention is given to the reconaissance of specific properties of a signal. It also contains an extension of standard wavelet approximation to the application of so-called second generation wavelets for efficient representation of signals at various scales even on the sphere and more complex geometries. Furthermore, the book discusses the application of harmonic (spherical) wavelets in potential field analysis with emphasis on the gravity field of the Earth. Many examples are given for practical application of these tools; to support the text exercises and demonstrations are available on the Web. |

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### Contents

Introduction to Continuous Wavelet Analysis | 1 |

Building Your Own Wavelets at Home | 72 |

Factoring Wavelet Transforms into Lifting Steps | 131 |

Efficiently Representing Functions on a Sphere | 158 |

Leastsquares Geopotential Approximation by Windowed Fourier Transform and Wavelet Transform | 189 |

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### Common terms and phrases

Aext applications approximation average average-interpolation B-spline basis functions Bio-Haar biorthogonal boundary BRDF coarser Computer Graphics constant construction convolution cubic B-spline defined detail coefficients dual lifting dual scaling functions dual wavelet equation Euclidean algorithm example factor fast wavelet transform filter h finite Fourier transform Freeden frequency Geodesy given Haar transform Haar wavelets half-plane inner product integral interpolating subdivision interval inverse transform Laurent polynomial Lazy wavelet Lemma lifting scheme lifting steps linear combination locations Mathematics modulus Morlet wavelet multiresolution analysis Note obtain operator original signal outer harmonics phase polyphase matrix potential predictor properties Quadratic real line reconstruction refinement relation representation reproducing kernel result scale continuous scale discrete second generation wavelets sequence Signal Process smooth space Speech Signal Process sphere subdivision scheme surface Sweldens synthesis Theorem tion translates and dilates upper half-plane values vanishing moments wavelet analysis wavelet coefficients wavelet transform windowed Fourier transform zero