Reference : Application of wavelet transforms to geosciences: Extraction of functional and freque... |

Dissertations and theses : Doctoral thesis | |||

Physical, chemical, mathematical & earth Sciences : Mathematics Physical, chemical, mathematical & earth Sciences : Multidisciplinary, general & others | |||

http://hdl.handle.net/2268/209501 | |||

Application of wavelet transforms to geosciences: Extraction of functional and frequential information | |

English | |

Deliège, Adrien [Université de Liège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes >] | |

18-May-2017 | |

Université de Liège, Liège, Belgique | |

Doctorat en Sciences | |

Nicolay, Samuel | |

Bastin, Françoise | |

Arneodo, Alain | |

Flandrin, Patrick | |

Fettweis, Xavier | |

Mabille, Georges | |

[en] Wavelets ; Multifractal analysis ; Wavelet leaders method ; Mars topography ; Hölder exponent ; Temperature time series ; Time-frequency analysis ; Border effects ; Oceanic Niño Index ; Forecast ; Mode extraction ; WIME | |

[en] It is now well-known that there exist functions that are continuous but nowhere differentiable.
Still, it appears that some of them are less “irregular” than others. The pointwise regularity of a function can be characterized by its Hölder exponent at each point. For the sake of practicability, it is more appropriate to determine the “size” of the sets of points sharing a same exponent, through their Hausdorff measure. By doing so, one gets the multifractal spectrum of a function, which characterizes in particular its monofractal or multifractal nature. The first part of this work is based on the so-called “wavelet leaders method” (WLM), recently developed in the context of multifractal analysis, and aims at its application to concrete situations in geosciences. First, we present the WLM and we insist particularly on the major differences between theory and practice in its use and in the interpretation of the results. Then, we show that the WLM turns out to be an efficient tool for the analysis of Mars topography from a unidimensional and bidimensional point of view; the first approach allowing to recover information consistent with previous works, the second being new and highlighting some areas of interest on Mars. Then, we study the regularity of temperature signals related to various climate stations spread across Europe. In a first phase, we show that the WLM allows to detect a strong correlation with pressure anomalies. Then we show that the Hölder exponents obtained are directly linked to the underlying climate and we establish criteria that compare them with their climate characteristics as defined by the Köppen-Geiger classification. On the other hand, the continuous version of the wavelet transform (CWT), developed in the context of time-frequency analysis, is also studied in this work. The objective here is the determination of dominant periods and the extraction of the associated oscillating components that constitute a given signal. The CWT allows, unlike the Fourier transform, to obtain a representation in time and in frequency of the considered signal, which thus opens new research perspectives. Moreover, with a Morlet-like wavelet, a simple reconstruction formula can be used to extract components. Therefore, the second part of the manuscript presents the CWT and focuses mainly on the border effects inherent to this technique. We illustrate the advantages of the zero-padding and introduce an iterative method allowing to alleviate significantly reconstruction errors at the borders of the signals. Then, we study in detail the El Niño Southern Oscillation (ENSO) signal related to temperature anomalies in the Pacific Ocean and responsible for extreme climate events called El Niño (EN) and La Niña (LN). Through the CWT, we distinguish its main periods and we extract its dominant components, which reflect well-known geophysical mechanisms. A meticulous study of these components allows us to elaborate a forecasting algorithm for EN and LN events with lead times larger than one year, which is a much better performance than current models. After, we generalize the method used to extract components by developing a procedure that detects ridges in the CWT. The algorithm, called WIME (Wavelet-Induced Mode Extraction), is illustrated on several highly non-stationary examples. Its ability to recover target components from a given signal is tested and compared with the Empirical Mode Decomposition. It appears that WIME has a better adaptability in various situations. Finally, we show that WIME can be used in real-life cases such as an electrocardiogram and the ENSO signal. | |

Fonds de la Recherche Scientifique (Communauté française de Belgique) - F.R.S.-FNRS | |

Researchers ; Professionals | |

http://hdl.handle.net/2268/209501 |

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