Catchment travel time distributions (TTDs) are an efficient concept for summarizing the time-varying 3D transport of water and solutes towards an outlet in a single function of a water age and for estimating catchment storage by leveraging information contained in tracer data (e.g., deuterium

Sustainable water resource management is based upon a sound understanding of how much water is stored in catchments and how it is released to the streams. Isotopic tracers such as deuterium (

The determination of travel time distributions (TTDs) is the method that relies the most on isotopic tracers

A water molecule that reached an outlet has only one travel time, which is defined as the duration between entry and exit. The use of different methods of travel time analysis for stable isotopes of O and H and for

Besides methodological problems, the reasons for the travel time differences (hence the apparent storage or mixing) are still not well understood because little is known about the difference in the information content of

In this study, we go beyond the previous work and assess the differences between streamflow TTDs and the associated catchment storage (considering their uncertainties) when those are inferred from stable isotopes or from

Are the travel times and storage inferred from a common transport model for

Are the travel time information contents of

This study is carried out in the Weierbach catchment, which has been the focus of an increasing number of investigations in the last few years about streamflow generation

The Weierbach catchment is a forested headwater catchment of 42 ha located in northwestern Luxembourg (Fig.

Map of the Weierbach catchment and its location in Luxembourg. The weir is located at coordinates (5

Based on previous modeling

In this study, we use precipitation (

Data used in this study –

The 1088 stream grab samples analyzed for

Distribution of stream samples (

The 24 stream samples analyzed for

Since the stream grab samples were collected over a short time interval (seconds to minutes) using a weir, the associated concentrations,

Mathematically, the streamflow TTD is related to the stream tracer concentrations

Most of the previous travel time studies using tritium assumed steady-state flow conditions and an analytical shape for the streamflow TTD and fitted the parameters of the analytical function using the framework described in Sect.

Essentially, the master equation is a water balance equation in which storage and fluxes are labeled with age categories. The master equation is thus a partial differential equation. It expresses the fact that the amount of water in storage, with a given residence time, changes with calendar time. This change is due to new water introduced by precipitation

We assumed that

Model parameters.

Numerically solving the master equation requires an estimation of catchment mobile storage

The parameters of the SAS functions and the other model parameters were calibrated using a Monte Carlo technique. In total, 12 parameters were calibrated (Table

Unlike our previous modeling work in this catchment

The first step in the Monte Carlo procedure consisted of randomly sampling parameters from the uniform prior distributions with the ranges defined in Table

A total of 148 parameter sets were behavioral for deuterium simulations, with

The behavioral posterior parameter distributions constrained by deuterium or tritium or by both generally had similar ranges to their prior distributions, except, notably, for

Parameter ranges and information measures before and after calibration to isotopic data.

Distributions of SAS function mean (

Essentially, the results (Table

Simulations of stream

Simulations in deuterium.

Simulations of stream

Simulations of stream concentrations in tritium compared to observations of and variability in precipitation.

For each behavioral parameter set, we calculated

Flow-weighted (2015–2017) cumulative stream TTDs for the behavioral parameter sets constrained by

Statistics of

The mean and standard deviations are calculated from all retained behavioral solutions for a given criterion.

We defined the right-hand tail of the streamflow SAS function

Cumulative right-hand tail

Storage estimate

To quantitatively study the implications of different

Our work shows that streamflow TTDs and the related catchment mobile storage

These results emerged for a number of reasons. First, we treated

It seems likely that, the higher storage, the higher travel times, and the larger uncertainties for tritium are related to the lack of high-resolution data. Tritium simulations included many small peaks corresponding to flashy streamflow responses associated with young water (Fig.

The travel time and storage measures estimated from a joint use of

Sampling deuterium and tritium jointly provided substantial additional information, besides the similar travel time and storage measures derived using each tracer alone. Combining both tracers yielded a nonnegligible information gain of

With deuterium alone, we found 4.08 bits of information with 1385 samples. With tritium, we found 4.47 bits of information with only 24 samples. Thus, tritium was overall more informative than deuterium about travel times, even with a lower number of samples. This is because tritium considerably informed us about the travel times in ET. Tritium constrained the posterior of

Overall, stable and radioactive isotopes of H had different information contents on travel times. The positive

The storage value derived from unsteady travel times constrained by tracer data (Table

The visually satisfactory tracer simulations enhance our confidence that the model accurately simulates travel times in the Weierbach catchment. Still, the performance in

The isotopic simulations were better for decreasing

The tendency of the model to yield higher average tritium values than the observations in streamflow over 2015–2017 (Fig.

Finally, parameter distributions (Figs.

The highest flows that were not sampled for tritium (Fig.

We found much lower deviations for the travel time and storage measures constrained by deuterium and tritium together (Tables

Stable isotopes of O and H and tritium are indispensable tracers for inferring the streamflow TTD and deriving storage estimates in catchments. Our study addressed an emerging concern about the possible limitations of stable isotopes for inferring the whole streamflow TTD compared to tritium. We went beyond the previous data and methodological limitations, and we did not find that stable isotopes are blind to old water fractions, as suggested by earlier travel time studies. We found statistically significant differences between some travel time measures derived from each tracer, but these differences were considerably smaller than in previous studies. The differences we found can most likely be attributed to a higher number of stable isotope samples compared to tritium due to different analysis techniques. Based on the results in our experimental catchment in Luxembourg, we conclude that the perception that stable isotopes systematically truncate the tails of TTDs may not be valid. Instead, our results highlight that stable isotopes and tritium have different information contents on travel times, but they can still result in similar TTDs. In fact, inferring the streamflow TTD from a joint use of both tracers better exploits their information, which results in lower uncertainties and higher information gains. Although

In this section, we provide further details on the equations used in the model. The composite streamflow SAS function

Actual evapotranspiration

In the model, this equation is the only explicit partitioning condition of the tracer influx

The obtained differences in travel time and storage measures (Tables

Results from the Wilcoxon rank sum test comparing the travel time and storage measures between

All tests were made at the 5 % significance level.

The results show significant differences (at the 5 % level) between all measures except two. According to the statistical test, the youngest fractions of water (younger than

The codes implementing the composite SAS-based model for deuterium and tritium simulations can be found at the LIST GitLab (

The tritium input data used in this study can be obtained from the WISER database portal of the International Atomic Energy Agency. The rest of the data used in this study are the property of the Luxembourg Institute of Science and Technology (LIST) and can be obtained upon request to the corresponding author, after approval by LIST. Most of the data used in this study were uploaded to a Zenodo repository (

The supplement related to this article is available online at:

LP and JK designed the project and obtained the funding for this study. NBR and JK provided the experimental design for the study. NBR carried out the field and lab work. NBR and JK performed the modeling part. NBR, LP, EZ, and JK jointly structured the paper, and NBR wrote the paper, with contributions from JK, EZ, and LP.

The authors declare that they have no conflict of interest.

We thank Uwe Morgenstern from GNS Science and Axel Schmidt from the Bundesanstalt für Gewässerkunde (BfG) for providing access to the 2017 precipitation tritium data. We thank Laurent Gourdol for his help with the preparation of the tritium input data and for the useful discussions about estimating TTDs with tritium measurements. We thank Uwe Ehret for providing the MATLAB scripts to compute the information theory measures (

This research has been supported by the Fonds National de la Recherche Luxembourg (grant no. FNR/CORE/C14/SR/8353440/STORE-AGE) and the Fonds National de la Recherche Luxembourg, Deutsche Forschungsgeimeinschaft, and Fonds zur Förderung der wissenschaftlichen Forschung (grant no. FNR/INTER/DFG/14/02/CAOS‐2). The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Mariano Moreno de las Heras and reviewed by Francesc Gallart and two anonymous referees.