The theory of travel time and residence time distributions is reworked from the point of view of the hydrological storages and fluxes involved. The forward and backward 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. Generalization to an arbitrary number of reservoirs is presented.

Hydrological travel times have been studied extensively for many years. Some
researchers

Based on these premises,

A unique framework for understanding all catchment processes was made
possible by the recent work of Rinaldo and others

These approaches opened the possibility of exploring the nature of
storage–discharge relationships, which are usually parameterized within
rainfall–runoff models, and can provide fundamental insight into the
catchment functions invoked previously

There remains a need for theoretical developments that are clearly explained
and developed using a consistent set of notations. Questions arise, like does
the theory contain hidden parts that are not consistent or explained well?
How does it relate to the instantaneous unit hydrograph theory? How can it be
used? What generates time-varying backward probabilities? Does the theory
fully account for those phenomena which are involved in mobilizing old water

Questions also remain about how to apply the theory of age-ranked
distributions in terms of the model form and parameter estimation.

Our work includes a short review of existing concepts that were collected from many (mostly theoretical) papers, which used different conventions and approaches. In the following sections, the theory to date is synthesized into a framework using consistent notation. Besides presenting the concept in a new and organized way, our paper contains some non-trivial answers to the above questions. Clarifications and extensions will be presented and summarized in an integrated manner. These conceptual developments are followed by improved methods for selecting the appropriate form of SAS and estimating its parameters. Guidance for hierarchical approaches to parameter estimation is given, based on available data. Finally, the proposed framework and methods are illustrated using data from an experimental watershed.

Residence time, travel time and life expectancy of water particles and
associated constituents flowing through watersheds are three related
quantities whose meaning is well defined by the following equation:

A single control volume is considered in which the fluxes are the total precipitation, evapotranspiration and discharge.

Consider, for example, a control volume such as the one shown in
Fig.

Being interested in knowing the age of water, we need to consider a more general set of equations.

Assume that the water storage

Analogously,

Actual evapotranspiration, AE

“Backward” and “forward” are well-known concepts in the study of travel
time distributions. They were first introduced by

Based on the previous definitions, it is easy to define the pdfs of the
residence time, travel time, and evapotranspiration time. In particular, the
pdf of “residence time”, conditional on the actual time

It is evident that this probability is time-variant, since the integral and
the integrand in Eq. (

The pdf of “travel time” is

It is also possible to define the mean age of water for any of the two
outlets, which is given by

After the above definitions, the age-ranked equation
(Eq.

Therefore Eq. (

The conditional probability

Representation of the evolution of the backward pdf for three
injection times (

Representation of the backward cumulative distribution function for
three injection times (

Figure

Representation of the evolution of the backward pdf vs. the actual
time

Consider again the age-ranked Eq. (

Forward residence time probability distribution: in red the relative storage, in green the forward residence time distribution and in blue the relative discharge function.

It can be observed instead that

In order to normalize

Although

Representation of the forward probability of the outputs: in red the
relative storage,

The first plot in Fig.

Figure

Variation of the partitioning coefficient in time, for a single injection time in January: after a timescale of 5 months its oscillation became irrelevant and its value tended to its final value of 0.78.

As a result of definitions made in Sects.

Defining the total volume of water injected in the system
in [0,

Evolution of the partitioning coefficient in 1 year of hourly simulation: the highest values are achieved in January, with the lowest in June. However, the figure does not represent a simple oscillation. The March coefficient is lower than April. October and November present almost the same value.

Because the future is unknown, as remarked in
Sect.

As a byproduct, the SAS and the forward functions are shown to be related.
For discharge at any time

The forward probabilities can be related to the life expectancy, i.e., the expected time the water molecules remain in the storage.

In the control volume, we can conceptually denote the subsets of the storage
which contains the water molecules expected to exit at time

However,

This integral is not effectively known at time

The formalism developed in Sects. 2 to 6 applies in principle to any
conservative substance, indicated by a superscript

The bulk substance budget can instead be written as

The above is essentially the same as Eq. (12) in

The braces were added to emphasize that

In fact, in Eq. (

From a practical point of view there could be some obstacles in the correct
determination of the SAS, because the distribution of the input of the
substance can be unknown. In this case Eq. (

For the sake of simplicity we neglected evapotranspiration. However, now that
the concepts are established, we can observe that incorporating AE

Finally, in order to clarify this theory, an example of

With the scope to further clarify the formalism, we assume in this section
that the forward pdfs introduced in the previous sections are known. We use
the concept of linear reservoir, which has a long history in surface
hydrology, e.g.,

First consider only one outflow: the bulk equation for the water budget of a
single linear reservoir is

Because discharges are just linearly proportional to the storage, it is easy
to show that

Non-trivial

Even if semi-analytical results are not feasible using nonlinear reservoirs, suitably tuning the parameters of each age-ranked equation cannot change the form of the SAS, as is also suggested by arguments below.

Other aspects come into play when there are multiple outputs. Expanding the
previous linear case to include evapotranspiration, the bulk equation becomes

We reviewed existing concepts that were collected from many different papers, and presented them in a new systematic way. We established a consistent framework that offers a unified view of the travel time theories across surface water and groundwater.

It contains several clarifications and extensions.

Clarifications include the following.

The concepts of forward and backward conditional probabilities and a small but important change in notation.

Their one-to-one relation with the water budget (and the age-ranked functions) from which the probabilities were derived (after the choice of SASs).

The proper way to choose backward probabilities. Specifically, it was shown that the usual way to assign time-invariant backward probabilities is inappropriate. We also show how to do it correctly, and introduced a minimal time variability.

The fact that time-invariant forward probabilities usually imply time-varying backward probabilities, i.e., travel time distributions.

The rewriting of the master equation by Botter, Bertuzzo, and Rinaldo as an ordinary differential equation (instead of a partial differential equation).

The role and nature of the partitioning coefficient between discharge and evapotranspiration (which is unknown at any time except asymptotically).

The significance of the SAS functions with examples.

The relationship of the present theory with the well-known theory of the instantaneous unit hydrograph.

We added information and clarified some links of the present theory with

Extensions include

new relations among the probabilities (including the relation between expectancy of life and forward residence time probabilities);

an analysis of the partitioning coefficients (which are shown to vary seasonally);

an explicit formulation of the equations for solutes which would permit direct determination of the SAS on the basis of experimental data;

tests of the effects of various hypotheses, e.g., assuming a linear model of forward probability and a gamma model for the backward probabilities;

an extension of Niemi's relation (and a new normalization);

the presentation of Niemi's relation as a special case of the Bayes theorem; and

a system of equations from which to obtain the SAS experimentally.

Finally, as a proof of concept, this paper includes examples derived from a real case (Posina River basin) and comes with open-source code that implements the theory, available to any researcher (see Appendix E).

For interested researchers to replicate or extend our results, our codes are
made available at

Without the need to be comprehensive, since some reviews of the topic were
recently made available

We also do not mention travel time theories which emanate from the
instantaneous unit hydrograph, since they were extensively discussed in

One of the older papers on this subject is

All the researchers above worked at a finer scale than ours, describing
fields of properties dependent on location, time, and age, while we work at a
scale integrated over a whole control volume (a catchment or a hydrologic
response unit), where any reference to space disappears. Let us call their
approach “local” and our approach “spatially integrated”. Their local
approach directly used concentrations, while our spatially integrated one
placed the emphasis on residence (and travel) time probabilities. Both
concentration and probability vary between zero and one, but the first is
mass (volume) normalized over the total mass (volume) of all substances
present in a given location, the second is the mass (volume) of a substance
injected at a certain time over the mass (volume) of the same substance
coming from all the injection times. We have shown in Sect. 9 how the two
approaches match at the spatially integrated scale following the work of

Another relevant difference between the local and spatially integrated theories is the different parameterizations of the fluxes. In our treatment we distinguish the sources (precipitation, recharge, etc.) and the outputs (discharges and evapotranspiration). Local theories usually implement an advection–dispersion term and include a sink–source term, which is important only when solutes are involved. We also introduced a sink–source term, but when appropriate, in Sect. 9.

An explicit integration of the local theory to obtain the spatially
integrated one was recently presented in

A different but interleaved group of papers,
e.g.,

In the literature we cited in the main text, it seems usually recognized that a single reservoir is not able to reproduce proper discharge and tracer behavior, and a few “embedded” reservoirs are therefore used in models. For instance, concerns regarding the discrepancies between the velocity of the solute transport and celerity of the pressure signals that travel across the control volumes must be addressed with an appropriate choice of embedded “groundwater” reservoirs.

The theory developed in the main text can be easily extended to these cases.
As an illustrative example we take a simple model from

Their system is composed of three reservoirs (e.g., Fig. 2 in

The lower reservoir obeys the following budget equation:

Finally, the storage equation for the saturated storage is

Following the same arguments as for the other two reservoirs, the age-ranked
version of the budget becomes

The overall system is the sum of the three reservoirs where

It can be observed that the backward probability, as defined in
Eq. (

However, a better choice for the backward probability should be a little more
complex. For instance,

The authors acknowledge Trento University project CLIMAWARE
(