Chaos theory applied to the modelling of karst springs: first results from univariate time series

The hydrological dynamics of karstic systems are generally highly nonlinear and weakly predictable. This paper introduces, for the first time, a hydrological modelling approach based on chaos theory. Although this modelling approach may be extended to multi-variables, as a first step in exploring its applicability, the focus is on the simple case of single-variable modelling for karst springs, which in practice corresponds to basins where rainfall is ungauged or poorly constrained. Chaos modelling is applied to the discharge of two karstic springs, the Doubs and the Lez springs in France, selected because they represent very different geological settings, climatic conditions, anthropogenic forcings and discharge dynamics. A deterministic model of autonomous ordinary differential equations is obtained for each spring. The models have chaotic behavior in both cases. The forecasting skills of these chaotic models are assessed. Forecasting performance estimates suggest that, under real conditions, forecasting could be performed for time horizons of ~16 h for Doubs and ~19 h for Lez (±1,000 L/s, 95% of confidence). This analysis offers new evidence for chaos in hydrogeology: the dynamic of discharge of karst springs is both deterministic and highly sensitive to the initial conditions, and it can be approximated by low-dimensional models