Robust evidence for random fractal scaling of groundwater levels in unconfined aquifers

Abstract

This study introduces new approaches to improve the statistical robustness of techniques for quantifying the fractal scaling of groundwater levels, and uses these techniques to investigate scaling of groundwater levels from a consolidated permeable carbonate aquifer. Six groundwater level time series and an associated river stage time series from the unconfined Chalk aquifer (a dual-porosity, fractured limestone aquifer) in the Pang–Lambourn catchment, UK, have been analysed. Surrogate data of time series with known scaling properties have been used to estimate the probability distribution of the spectral and geometric scaling exponents determined by detrended fluctuation analysis (DFA) and power spectral density (PSD) respectively; robust regression techniques have been used to improve estimates of the scaling exponents; and robust non-parametric techniques have been used to correlate scaling exponents with features of the boreholes and catchments. Strong statistical support has been found for temporal scaling of groundwater levels over a wide range of time scales, however, bootstrap estimates of the scaling exponents indicate a much larger range of exponents than found by previous studies, suggesting that the uncertainty in existing estimates of scaling exponents may be too small. There is robust evidence that geometrical scaling properties at each borehole can be related to the depth of the observation boreholes and distance of those boreholes from the river in the catchment, but no such correlations were found for the spectral scaling exponents. The results build on the body of evidence that groundwater levels, as with many hydrogeological phenomena, may be well modelled with mathematical concepts from statistical mechanics that do not attempt to capture every detail of these highly heterogeneous and complex systems.