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 Pyrénées Mountains (Ariège, 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.