Age-ranked hydrological budgets and a travel time description of catchment hydrology

Age-ranked hydrological budgets and a travel time description of catchment hydrology

Abstract. The theory of travel time and residence time dis- tributions is reworked from the point of view of the hydro- logical storages and fluxes involved. The forward and back- ward travel time distribution functions are defined in terms of conditional probabilities. Previous approaches that used fixed travel time distributions are not consistent with our new derivation. We explain Niemi’s formula and show how it can be interpreted as an expression of the Bayes theorem. Some connections between this theory and population theory are identified by introducing an expression which connects life expectancy with travel times. The theory can be applied to conservative solutes, including a method of estimating the storage selection functions. An example, based on the Nash hydrograph, illustrates some key aspects of the theory. Gen- eralization to an arbitrary number of reservoirs is presented.

1 Introduction
Hydrological travel times have been studied extensively for many years. Some researchers (Rodriguez-Iturbe and Valdes, 1979; Rinaldo and Rodriguez-Iturbe, 1996), as reviewed by Rigon et al. (2016), looked at the construction of the hydro- logic response using geographical information. Others (e.g., Uhlenbrook and Leibundgut, 2002; Birkel et al., 2014) used travel times to understand catchment processes in relation to tracer experiments, while new experimental techniques were being developed (e.g., Berman et al., 2009; Birkel et al., 2011).
Based on these premises, Fenicia et al. (2008), Clark et al. (2011), McMillan et al. (2012), and Hrachowitz et al. (2013) aimed to describe both the spatial organization of the catch-
ment and the set of interactions between processes with an assembly of coupled storages (reservoirs) in the number and the organization necessary to give proper hydrological results without adding unwanted parametric complexity (e.g., Kle- meš, 1986; Kirchner, 2006). Despite the simplification ef- forts, the process of adding physical rigor to their models led to quite complex systems. Travel time analysis became a tool to disentangle flux complexities (e.g., Tetzlaff et al., 2008), opening the way for explicit unification of geomorphic theo- ries and storage-based modeling (Rigon et al., 2016).
A unique framework for understanding all catchment pro- cesses was made possible by the recent work of Rinaldo and others (Rinaldo et al., 2011; Botter et al., 2011). This new branch of research is the focus of the present work. In par- ticular, Botter et al. (2010, 2011) introduced a newly formu- lated StorAge Selection (SAS) function related to the prob- ability density function (pdf) of the water age or backward travel time distribution. With the aid of an a priori assigned SAS, they were able to write a “master equation” for the travel time probability distribution and solve it, thus system- atically connecting the solution of the catchment water bud- get to travel time aspects of the hydrological flows.