Rainfall–runoff relations for karstic springs. Part II: continuous wavelet and discrete orthogonal multiresolution analyses
Section snippets
Introduction and objectives
Rainfall rates and runoffs are hydrological signals both characterised by high time-variability, as shown in Part I of this paper (Labat et al., 2000). Spectral and correlation analyses are often applied in karstic hydrology (e.g. Mangin, 1984) with some success. Nevertheless, these methods cannot fully take into account this temporal variability, which results from a spatial and dynamical heterogeneity. Operating globally on the signal, these analyses do not adequately convey the occurrence of
Continuous wavelet transforms
The basic objective of the wavelet transform is to achieve a complete time-scale representation of localised and transient phenomena occurring at different time scales. Time-scale discrimination is achieved in a more satisfactory way than with time–frequency decompositions such as windowed Fourier or Gabor methods (Gabor, 1946).
The first step consists of changing the projection basis. In continuous time, but on a finite interval [0,T], the classical orthonormal complex Fourier basis is composed
Discrete time wavelet transform and orthogonal multiresolution analysis
For practical applications, the hydrologist does not have at his or her disposal a continuous-time signal process but rather a discrete-time signal, which will be denoted by x(i). In order to apply the techniques introduced below, the time-scale domain needs to be discretised and one can choose the general form (k,j)∈Z2) where a0 (a0>1) and τ0 are constants. The discretised version of Eq. (1) defines the discrete wavelet transform coefficients, as follows:
Wavelet spectrum and cross-spectrum analysis
In this section, the statistics of the wavelet coefficients are studied in more detail in order to put in evidence some relations between different scales of two related signals. The characteristics of the wavelet coefficient statistics are of interest since they provide useful information on the time-scale energy distribution.
The concept of wavelet variance (wavelet spectrum) and covariance (wavelet cross-spectrum), first defined by Brunet and Collineau (1995), will be compared with the
Application of wavelets to intermittent karstic springflows (Larzac)
Both continuous and discrete orthogonal wavelet transforms (Morlet wavelet analysis and multiresolution analysis) are used to analyse the flow processes measured at the outlet of two karstic springs (Esperelle and Cernon) located in the Causse of Larzac (France). An extensive study of this karstic region, in terms of geochemistry, statistical hydrology, and tracers, has been recently conducted by the Bureau de Recherches Géologiques et Minières (BRGM): the reader is referred to this study (
Application of wavelets to karstic watersheds (Pyrénées)
In this section, the different wavelet techniques presented in the previous sections are applied to rainfall and springflow rates in three karstic basins. Two free Matlab-softwares are used to obtain the results of the practical study. The first one is provided by the University of Vigo (Uvi-Wave 3.0) and is available at the URL: http://ftp.tsc.uvigo.es with a download user manual (Gonzales et al., 1996); the second one was kindly provided by Torrence and Compo (1998) at the URL:
Study area and data description
Wavelet transforms are applied to three karstic springs located in the French Pyrénées mountains (Ariège): the Aliou, Baget, and Fontestorbes springs (see Labat et al., 2000, Part I). The three watersheds have a common median altitude of 1000 m, which put them in the category of mid-altitude basins. Aliou and Baget are two small basins of 13 km2 area whereas Fontestorbes is a more extended basin of 80 km2 area. Nevertheless, the specific runoffs of Aliou and Baget (36 l/s/km2) are higher than the
Wavelet analysis of daily and weekly data
The univariate and joint wavelet analysis methods described earlier are applied in this section to daily and weekly (obtained by aggregation of daily data) rainfall rates and runoffs. Compared to the Fourier and correlation analyses, wavelet transforms lead to more precise results especially in the temporal variability of the processes, which need to be taken into account in modelling such processes. Note that in all figures, the lighter the grey scales, the higher the value of the wavelet
Discussion and conclusions
Rainfall rates and karstic spring runoffs are both characterised by highly non-stationary and scale-dependent behaviour. The runoffs are of course influenced by the time structure of rainfall rates, but also, by the high non-linearity of the karstic system. It appears that the temporal structures of rainfall–runoff records cannot be taken into account adequately via classical spectral and correlation analyses, as demonstrated in Labat et al., Part I, Sections 4 and 5.
Wavelet analyses with both
Acknowledgements
This work is based on a research project on “Karstic Systems” funded in part (1996/97) by the French government national research programme in hydrology, “Programme National de Recherche en Hydrologie” of the CNRS — Institut National des Sciences de l'Univers. We acknowledge the material support of the authorities of the Parc Naturel Régional des Grands Causses, France. We also gratefully acknowledge the helpful review comments and suggestions by K. O'Connor and by an anonymous reviewer.
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