Elsevier

Journal of Hydrology

Volume 238, Issues 3–4, 5 December 2000, Pages 149-178
Journal of Hydrology

Rainfall–runoff relations for karstic springs. Part II: continuous wavelet and discrete orthogonal multiresolution analyses

https://doi.org/10.1016/S0022-1694(00)00322-XGet rights and content

Abstract

Karstic watersheds appear as highly non-linear and non-stationary systems. The main focus of this paper is a heuristic study of this non-stationarity using a time-scale localisation method called the wavelet transform.

First, a mathematical overview of these analysis methods is given. The wavelet transform methods used here can be divided into two main parts: the continuous Morlet wavelet transform and the multiresolution orthogonal analysis. A statistical interpretation of the wavelet coefficients is also presented, introducing wavelet spectrum analyses (univariate and cross-wavelet analyses).

These wavelet methods are applied to rainfall rates and runoffs measured at different sampling rates, from daily to half-hourly sampling rate. The karstic springs under study are located in the Pyrénées Mountains (Ariège, France) and in the Causses of Larzac (Aveyron, France).

They are first applied to a pumping and a naturally intermittent runoff process, allowing the separation of different sub-processes. Wavelet analyses of rainfall rates and runoffs and wavelet rainfall–runoff cross-analyses also give meaningful information on the temporal variability of the rainfall–runoff relationship. In particular, this kind of analysis provides a simple interpretation of the distribution of energy between the different scales. Finally, it is demonstrated that wavelet transforms make possible a physical explanation of the temporal structure of the basin response to rainfall allowing discrimination between a rapid response and recharge due to the karst drainage system and a slower one corresponding to infiltration response.

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 ({a0−j,kτ0a0−j}, (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:Cx(j,k)=∫−∞+∞x(t)ψ∗j,k

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.

References (46)

  • I. Daubechies

    Ten lectures on Wavelets

    (1992)
  • A. Davis et al.

    Multifractal characterizations of non-stationarity and intermittency in geophysical fields, observed, retrieved or simulated

    J. Geophys. Res.

    (1994)
  • D.L. Donoho et al.

    Wavelet shrinkage: asymptotia (with discussion)

    J. R. Stat. Soc. B

    (1995)
  • M. Farge et al.

    Transformée en ondelettes pour détecter et analyser les structures cohérentes dans les écoulements turbulents bidimensionnels

    C. R. Acad. Sci. Paris (II)

    (1998)
  • M. Farge

    Wavelet transforms and their applications to turbulence

    Ann. Rev. Fluid Dyn.

    (1992)
  • E. Foufoula-Georgiou et al.

    Wavelets in Geophysics

    (1995)
  • D. Gabor

    Theory of communication

    J. Inst. Elect. Eng. (London)

    (1946)
  • Gonzales, S.S., Gonzales, P.N., Garcia, G.S., 1996. Uvi-Wave wavelet toolbox for use with Matlab. Department of...
  • A. Grosmann et al.

    Decomposition of Hardy functions into square integrable wavelets of constant shape

    SIAM J. Math. Anal.

    (1984)
  • H. Higuchi et al.

    On the structure of a two-dimensional wake behind a pair of flat plates

    Phys. Fluids

    (1994)
  • I.M. Johnstone et al.

    Wavelet threshold estimators for data with correlated noise

    J. R. Stat. Soc. B

    (1997)
  • G.G. Katul et al.

    Intermittency, local isotropy, and non-gaussian statistics in atmospheric surface layer turbulence

    Phys. Fluids

    (1994)
  • G.G. Katul et al.

    Analysis of land surface heat fluxes using the orthonormal wavelet approach

    Water Resour. Res.

    (1995)

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