Nonparametric lower bounds to mean transit times

Abstract. Mean transit time μ_T, also called mean residence time, has been used widely in hydrological studies as an indicator of catchment water storage characteristics. Typically μ_T is estimated by the nature of catchment transformation of a natural input tracer time series. For example, increased damping and delaying of ¹⁸O seasonal isotopic variation may be taken to indicate longer mean transit times. Part of a μ_T estimation process involves specification of a lumped parameter flow model which provides the basis for a parametric transit time distribution. However, μ_T estimation has been called into question because catchment flow systems have a degree of complexity which may not justify use of simple parametric distributions. Moving toward a related index, the question is raised here as to the extent to which an arbitrary transit time distribution might enable a model mean transit time to be minimized before the fit to catchment output tracer data becomes unacceptably poor. This minimized mean value μ^* represents a lower bound to μ_T, whatever the true transit time distribution might be. The lower bound is not necessarily an approximation to μ_T but might serve as an index for catchment comparisons or detect when μ_T is large. For a linear catchment system a simple nonparametric linear programming (LP) approach can be utilised to obtain μ^*, which is conditional on a user-specified acceptable level of data fit. The LP method presented is applicable to both steady state and time-varying catchment systems and has the advantage of not requiring specification of lumped parameter models or use of explicit transit time distributions.