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

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.