Statistics of low flow: Theoretical derivation of the distribution of minimum streamflow series

SummaryA theoretical framework for minimum streamflow is developed. It grasps over the whole spectrum of the problem from the tail-behaviour of the parent distribution via the asymptotic theory of extreme value distributions to the derived distribution approach allowing a physical interpretation of its parameters. It is shown how the power parameter and also the moments of the Weibull distribution depend on the statistical properties of the parent and especially its coefficient of variation. For the application of the threshold method – Pit Under Threshold (PUT) – a mean deficit function is defined. It is a useful tool to analyse the tail towards minima and directs to the formulation of a generalised Pareto distribution for minima. A derived distribution function approach based on a linear recession leads to the same principal distribution but with a physical interpretation of its parameters. It is further generalised to include also a non-linear recession. The theoretical findings are tested in an exploratory manner against streamflow data from 35 sites in the Moselle drainage basin, France. Parameters estimated from PUT data describing distribution type and form show very stable behaviour with an appropriate determination of the threshold. Parameters estimated from annual minima show larger uncertainty due to existence of high outliers. Location parameters, the mean minimum streamflow and the threshold value, vary significantly between series. Only for large downstream basins common values appear. For small and medium sized basins low flow statistic are controlled by very local factors. In lack of very detailed information the link to parent statistics can be used for regionalisation purposes. The developed distributions are parsimonious with only one or two parameters besides the threshold value. Anyhow the hypothesis that the regional sample follows these distributions cannot be rejected. The distributions can be kept simple and give good results due to the good foundation in theory and process understanding.

► We develop a joint theoretical framework for streamflow minima.
► It is based on extreme value theory jointly with a derived distribution approach.
► We link the Weibull distribution power parameter with the properties of the parent.
► The derived distribution approach uses linear recession and non-linear recession.
► We tested theoretical findings on 35 runoff series from the Moselle River, France.