Introduction of Wavelet Analyses to Rainfall/Runoffs Relationship for a Karstic Basin: The Case of Licq‐Atherey Karstic System (France)

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Abstract
Karstic systems are highly heterogeneous geological formations characterized by a multiscale temporal and spatial hydrologic behavior with more or less localized temporal and spatial structures. Classical correlation and spectral analyses cannot take into account these properties. Therefore, it is proposed to introduce a new kind of transformation: the wavelet transform. Here we focus particularly on the use of wavelets to study temporal behavior of local precipitation and watershed runoffs from a part of the karstic system. In the first part of the paper, a brief mathematical overview of the continuous Morlet wavelet transform and of the multiresolution analysis is presented. An analogy with spectral analyses allows the introduction of concepts such as wavelet spectrum and cross‐spectrum. In the second part, classical methods (spectral and correlation analyses) and wavelet transforms are applied and compared for daily rainfall rates and runoffs measured on a French karstic watershed (Pyrenees) over a period of 30 years.Different characteristic time scales of the rainfall and runoff processes are determined. These time scales are typically on the order of a few days for floods, but they also include significant half‐year and one‐year components and multi‐annual components. The multiresolution cross‐analysis also provides a new interpretation of the impulse response of the system. To conclude, wavelet transforms provide a valuable amount of information, which may be now taken into account in both temporal and spatially distributed karst modeling of precipitation and runoff.

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