This paper presents the “

The field of hydrologic transport focuses on how water flows
through a watershed and mobilizes solutes towards the catchment outlets. The
proper representation of transport processes is important for a number of
purposes such as understanding streamflow generation processes

Water trajectories within a catchment are usually considered from the time
water enters as precipitation to the time it leaves as discharge or
evapotranspiration. As watersheds are heterogeneous and subject to
time-variant atmospheric forcing, water flow paths have marked spatiotemporal
variability. For this reason, a formulation of transport by travel time
distributions

While early catchment-scale approaches

The new theoretical formulation has improved capabilities, including being
less biased to spatial aggregation

The specific objectives of this paper are to (i) provide a numerical model that solves the age master equation with any form of the SAS functions in a computationally efficient way, (ii) show the potential of the model for simulating catchment-scale solute transport, and (iii) assess the numerical accuracy of the model for different aggregation time steps.

The model implemented in

The general theoretical framework relies on the works by

The system state variable is the age distribution of the water storage.
Indeed, at any time

The key element of the formulation is the SAS function, which formalizes the
functional relationship between the age distribution of the system storage
and that of the outflows. Different forms have been proposed to express the
SAS function directly as a function of age or as a derived distribution of
the storage age distribution, (e.g., absolute, fractional or ranked SAS
functions; see

Conceptual illustration of the main variables of the theoretical
formulation. Precipitation volumes are represented through colored circles,
with darker colors indicating the older precipitations with respect to
current time

The age ME

The solution of Eq. (

The same reasoning applies to the age distributions and concentration of the evapotranspiration flux.

As explained in Sect.

In case all the outflows remove the stored ages proportionally to their
abundance, the outflow age distributions become a perfect sample (or
random sample, RS) of the storage age distribution. The SAS
functions in this case assume the linear form

Equation (

We discretize time and age using the same time steps

Illustration of the conventions used to discretize the time domain.
Time steps have a fixed length

To solve Eq. (

The model solves Eq. (

compute

compute

To compute the model output, further operations are required. In particular,

update

compute

compute

Starting from these basic routines, many additional operations can be implemented to, for example, characterize the nonconservative behavior of solutes or to compute some age distribution statistics.

A first issue that the model needs to take into account is that age
distributions are defined over an age domain

A second, connected problem regards the initial conditions of the system,
i.e., the unknown storage age distribution and solute concentration to be used
at the beginning of the calculations. In the absence of information, the
initial storage can be considered as one single old pool; hence the initial
number of age classes

The computational time of a simulation can be reduced by not accounting for zero-precipitation inputs as they have no influence in the balance but increase the number of operations required at each time step. In such a case, however, the position of an element in the vector does not correspond with its age anymore and age has to be counted separately. To keep the model intuitive, we decided to not remove zero-precipitation inputs.

Application of the approach requires knowledge of the input or output water fluxes to or from the catchment, the input solute concentration, and the initial conditions for the water storage magnitude and concentration. Then, an SAS function must be specified for each outflow. The code comes with example virtual data that can be used to evaluate the model capabilities. Hydrologic data for 4 years were obtained from recorded precipitation and streamflow at the Mebre-Aval catchment near Lausanne (CH). Evapotranspiration was obtained from regional daily estimates around the Lausanne area and modified to match the long-term mass balance. On average, yearly precipitation is 1100 mm, discharge is 580 mm (53 % of precipitation) and evapotranspiration is 520 mm. The storage variations, computed by solving the hydrologic balance, were normalized to the interval [0,1] to serve as a nondimensional metric of catchment wetness (variable wi). Overall, the data are not meant to be representative of a particular location, but they constitute a realistic set of hydrologic variables to test the model.

The code was run on the example data using the four illustrative shapes for the
discharge SAS function listed in Table

Description of the discharge SAS functions used in the application.
All the functions were tested with the same initial total storage

Two different examples of solute transport were simulated in the test. In the
first case, solute input concentration was generated by adding noise to a
sinusoidal wave with an annual cycle. This example can be representative of
atmospheric tracers with a yearly period (like stable water isotopes). In the
second case, the initial storage was set to a concentration of
100 mg L

Example of results that can be obtained from the model.

Each discharge SAS function simulates different transport mechanisms and
provides rather different outputs, both in terms of water age and streamflow
concentration. In the first solute transport example (Fig.

Overall, these quick examples were used to illustrate the model capabilities
and to show that results may change significantly depending on the choice of
the parameters. A sensitivity analysis is generally advised to identify the
parameters that have the highest impact on model results. For example,
previous catchment studies

Numerical errors on the storage age distribution

Here we evaluate the numerical accuracy of the model in computing the
solution of the age ME (i.e., the rank storage

Solute concentration (

For the RS comparison, the analytic solution was obtained by implementing
Eq. (

Results show that the numerical implementation of the ME is satisfactory for
the RS solution in terms of both accuracy and stability. However, solutions
other than the RS case may be more challenging owing to the nonuniform age
selection played by the outflows. For this reason, we tested power-law SAS
functions (Eq.

The standard deviations of the errors are shown in Fig.

Numerical errors on the storage age distribution

These examples suggest that the behavior of the system can be interpreted
using a (nonlinear) reservoir analogy. Each individual water parcel can be
seen as a depleting reservoir that decreases in time owing to the particular
outflow removal (Eq.

The model is based on a catchment-scale approach, so it only requires
catchment-scale fluxes like precipitation, discharge and evapotranspiration.
These fluxes can often be measured (or modeled in the case of ET) without the
need for a full hydrologic model. Moreover, the pure SAS function
approach implies that, differently from previous approaches

The use of an explicit numerical scheme has the potential of greatly reducing
the computational times. Short aggregation time steps are generally
recommended, especially when testing the affinity for younger storage volumes
(e.g., Eq. (

The model is based on a catchment-scale formulation of transport processes; thus it cannot provide spatial information unless the system is partitioned
into a series of spatial compartments

Although the numerical accuracy of the computations has to be evaluated for
each different application, Sect.

The codes implemented in the

The

The current model release, including example data and
documentation, is available at

The authors declare that they have no conflict of interest.

The authors thank Andrea Rinaldo and Gianluca Botter for the useful discussions that inspired this work and Damiano Pasetto for support in the numerical implementation of the model equations. Paolo Benettin thanks the ENAC school at EPFL for financial support. Edited by: Jeffrey Neal Reviewed by: three anonymous referees