David Labat

RAINFALL–RUNOFF RELATIONS FOR KARSTIC SPRINGS. PART II: CONTINUOUS WAVELET AND DISCRETE ORTHOGONAL MULTIRESOLUTION ANALYSES

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 Pyrenees Mountains (Ariege, 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.

Rainfall-runoff relations for karstic springs. Part I : convolution and spectral analyses

Karstic basins contain large reserves of subsurface water. In this paper, three karstic systems located in the Pyrenees Mountains (Ariege, France) are studied. Long records of rainfall and discharge rates for these karstic springs are available, sampled at different rates: daily, hourly and half-hourly. This study aims at illustrating and assessing the capabilities and limitations of linear black-box methods for analysing rainfall–runoff type relationships and reconstructing runoffs from rainfall rate data using such systems. In this study, precipitation and discharge rates are considered as two autocorrelated and cross-correlated stochastic processes. A linear and stationary rainfall–runoff model is adopted, which is used for identification and simulation purposes. Different versions are analysed, including a model based on a convolution integral between the precipitation rate P(τ) and a transfer function h(t−τ) which can be thought of as the unit impulse response of the system. It is shown that this linear stochastic model (i.e. the statistical version), although accurate in some respects, does not represent the hydraulic behaviour of the system very well during low flow episodes and floods. It is also shown that the use of Fourier analysis, alone, does not lead to a satisfactory reconstitution of observed runoff sequences. For these reasons, the use of non-linear random process input–output models based on Volterra integral series is proposed and discussed.

Rainfall–runoff relations for karstic springs: multifractal analyses

Karstic watersheds appear as highly as non-linear and non-stationary systems. The behaviour of karstic springs has been previously studied using non-linear simulation methods (Volterra expansion) and non-stationary analyses methods based on wavelet transforms. The main issue of karstic spring behaviour consists of the presence and the identification of characteristic time-scales. In order to highlight more precisely the scale-properties of the rainfall–runoff relations for karstic springs, the multifractal analysis is introduced. These methods are applied daily and half-hourly rainfall rates and runoffs measured on a three French karstic springs located in the Pyrenees Mountains (Ariege, France): Aliou, Baget and Fontestorbes. They are characterised by a variable development of the drainage systems. We have at our disposal long and uninterrupted series of data over period of several years, which constitute a high quality bank data. Multifractal analyses of both daily and half-hourly rainfall rates and runoffs give evident a scale-dependant behaviour. Effectively, it highlights the presence of different multifractal processes at each sampling rate. Using a universal class of multifractal models based on cascade multiplicative processes, the identified multifractal sub-processes are characterised by the classical parameters α and C1. All these results should lead to several improvements in karstic springflow simulation models.

Analyse en ondelettes en hydrologie karstique. 2e partie : analyse en ondelettes croisée pluie-débit

Abstract A method of cross-analysis of rainfall-runoff time series based on wavelet transforms (continuous Morlet wavelet transform and orthogonal multiresolution analysis), is presented and applied to rainfall rates and spring outflow rates measured on karstic Pyrenean watersheds. Results are compared to classical cross-correlation and Fourier cross-spectral analyses. The multiscale wavelet method appears as powerful in the study of the joint temporal variability and non-stationarity of the rainfall-runoff relationship, and it yields more precise results than Fourier cross-spectral analysis.

Adaptability of Mediterranean agricultural systems to climate change. The example of the Sierra Mágina olive growing region (Andalusia, Spain)

This research focuses on the adaptability of olive-growing systems to climate change in the Sierra Magina region of Andalusia. The authors combined a retrospective and prospective analysis, an interdisciplinary approach, collaboration among climatologists, geographers, and sociologists, and the participation of local farmers and stakeholders, all contributing their own knowledge. This paper assesses the adaptability of olive-growing systems to climate irregularities over the past 50 yr. First, a climatic study shows that rainfall decreased by 18% during the period 1955-2009. Water resource availability has declined 2 or 3 times more than rainfall, in part because of the expansion of irrigation, which ameliorated the effects of droughts and increased profitability. Second, relationships between rainfall and both irrigated and rainfed olive yields are assessed. These show that the cumulative rainfall of the 2 yr preceding the crop explains 41% of the variability of irrigated olive tree yields and 46% of rainfed yields; this result was unexpected for irrigated yields. Third, this study examines the perceptions of climate variability of 15 farmers, the views of 16 local stakeholders [developers, olive oil professionals, local authorities, a conservationist, and a representative of a local nongovernmental organization (NGO)]. The perceptions of the farmers are interpreted with respect to their socioeconomic status. All farmer and stakeholder interviewees know to a certain extent the climatic influence on olive yields, and most of them acknowledge the recent climatic changes. These findings will be valuable for future assessments of the adaptability of the agricultural and social systems to climate change.