Recent advances in wavelet analyses: Part 1. A review of concepts

International audienceThis contribution provides a review of the most recent wavelet applications in the field of earth sciences and is devoted to introducing and illustrating new wavelet analysis methods in the field of hydrology. Wavelet analysis remains unknown in the field of hydrology even though it clearly overcomes the well-known limits of the classical Fourier analysis. New wavelet-based tools are proposed to hydrologists in order to make wavelet analysis more attractive. First, a multiresolution continuous wavelet analysis method is shown to significantly improve the determination of the temporal-scale structure of a given signal. Second, the concept of wavelet entropy in both continuous and multiresolution frameworks is introduced allowing for an estimation of the temporal evolution of a given hydrological or climatologic signal's complexity. New insights in the scale-dependence of the relationship are exposed by introducing wavelet cross-correlation and wavelet coherence. Continuous wavelet cross-correlation provides a time scale distribution of the correlation between two signals, whereas continuous wavelet coherence provides a qualitative estimator of the temporal evolution of the degree of linearity of the relationship between two signals on a given scale. These methods are applied to four large river runoffs and two global climatic indexes in a companion paper