Tracking residence times in hydrological systems: forward and backward formulations

In recent years, several papers contributed to the development and clarification of key theoretical issues underlying the formulation of transport by travel time distributions in catchments. Such a formulation provides a robust description of the temporal evolution of ages contained in the catchment storage and sampled by the output fluxes. In particular, special attention has been devoted to general time-variant dynamics that are likely to occur in complex systems like catchments. In this context, important theoretical and practical implications arise from a proper distinction between backward and forward age distributions, which are based on the definition of diverse reference variables. The 'age' of a water particle represents the time elapsed since a previous injection, and as such, it is intrinsically a backward time concept. A forward approach, instead, requires the introduction of the particle's 'life expectancy', which quantifies the time a water particle will spend within the system before being sampled by one of the outflows (e.g. stream discharge or evapotranspiration). The sum of age and life expectancy is the particle's travel time. Despite forward and backward approaches being different, and that they only coincide in the special case of stationary systems, a proper distinction of these formulations has been sometimes overlooked in the literature. In this contribution, we review recent backward formulations using a unified notation and discuss a novel forward formulation. This paper illustrates how age and life-expectancy distributions naturally evolve in response to unsteady hydrologic fluxes and presents numerical applications relevant to catchment-scale solute circulation. In both forward and backward formulations, the mixing of ages, which is modelled through age-selection functions, plays a central role in describing the fate of solutes introduced in the system and measured at the system outlets. Copyright (C) 2015 John Wiley & Sons, Ltd.