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

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.