the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A comparison of catchment travel times and storage deduced from deuterium and tritium tracers using StorAge Selection functions
Nicolas Björn Rodriguez
Laurent Pfister
Erwin Zehe
Julian Klaus
Catchment travel time distributions (TTDs) are an efficient concept for summarizing the timevarying 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 ^{2}H and tritium ^{3}H). It is argued that the preferential use of the stable isotopes of O and H as tracers, compared to tritium, has truncated our vision of streamflow TTDs, meaning that the long tails of the distribution associated with old water tend to be neglected. However, the reasons for the truncation of the TTD tails are still obscured by methodological and data limitations. In this study, we went beyond these limitations and evaluated the differences between streamflow TTDs calculated using only deuterium (^{2}H) or only tritium (^{3}H). We also compared mobile catchment storage (derived from the TTDs) associated with each tracer. For this, we additionally constrained a model that successfully simulated highfrequency stream deuterium measurements with 24 stream tritium measurements over the same period (2015–2017). We used data from the forested headwater Weierbach catchment (42 ha) in Luxembourg. Timevarying streamflow TTDs were estimated by consistently using both tracers within a framework based on StorAge Selection (SAS) functions. We found similar TTDs and similar mobile storage between the ^{2}H and ^{3}Hderived estimates, despite statistically significant differences for certain measures of TTDs and storage. The streamflow mean travel time was estimated at 2.90±0.54 years, using ^{2}H, and 3.12±0.59 years, using ^{3}H (mean ± 1 SD – standard deviation). Both tracers consistently suggested that less than 10 % of the stream water in the Weierbach catchment is older than 5 years. The travel time differences between the tracers were small compared to previous studies in other catchments, and contrary to prior expectations, we found that these differences were more pronounced for young water than for old water. The found differences could be explained by the calculation uncertainties and by a limited sampling frequency for tritium. We conclude that stable isotopes do not seem to systematically underestimate travel times or storage compared to tritium. Using both stable and radioactive isotopes of H as tracers reduced the travel time and storage calculation uncertainties. Tritium and stable isotopes both had the ability to reveal short travel times in streamflow. Using both tracers together better exploited the more specific information about longer travel times that ^{3}H inherently contains due to its radioactive decay. The two tracers thus had different information contents overall. Tritium was slightly more informative than stable isotopes for travel time analysis, despite a lower number of tracer samples. In the future, it would be useful to similarly test the consistency of travel time estimates and the potential differences in travel time information contents between those tracers in catchments with other characteristics, or with a considerable fraction of stream water older than 5 years, since this could emphasize the role of the radioactive decay of tritium in discriminating younger water from older water.
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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 (^{2}H), oxygen 18 (^{18}O), and tritium (^{3}H) have become the cornerstone of several approaches for tackling these two critical questions (Kendall and McDonnell, 1998). For instance, hydrograph separation, using the stable isotopes of O and H (Buttle, 1994; Klaus and McDonnell, 2013), has unfolded the difference between catchments hydraulic response (i.e., streamflow) and chemical response (e.g., solutes; Kirchner, 2003) related to the different concepts of water celerity and water velocity (McDonnell and Beven, 2014). Isotopic tracers have also been the backbone for unraveling water flow paths in soils (Sprenger et al., 2016) and distinguishing between soil water going back to the atmosphere and flowing to the streams (Brooks et al., 2010; McDonnell, 2014; McCutcheon et al., 2017; Berry et al., 2018; Dubbert et al., 2019).
The determination of travel time distributions (TTDs) is the method that relies the most on isotopic tracers (McGuire and McDonnell, 2006). TTDs provide a concise summary of water flow paths to an outlet by leveraging the information on storage and release contained in tracer input–output relationships. TTDs are essential for linking water quantity to water quality (Hrachowitz et al., 2016), for example, by allowing calculations of stream solute dynamics from a hydrological model (Rinaldo and Marani, 1987; Maher, 2011; Benettin et al., 2015a, 2017a). TTDs are commonly calculated from isotopic tracers in many subdisciplines of hydrology and, thus, have the potential to link the individual studies focused on the various compartments of the critical zone (e.g., groundwater and surface water; Sprenger et al., 2019). ^{3}H has been used as an environmental tracer since the late 1950s (Begemann and Libby, 1957; Eriksson, 1958; Dinçer et al., 1970; Hubert et al., 1969; Martinec, 1975), and it gained particular momentum in the 1980s with its use in diverse TTD models (Małoszewski and Zuber, 1982; Stewart et al., 2010). It is argued that ^{3}H contains more information on travel times than stable isotopes due to its radioactive decay (Stewart et al., 2012). For example, low tritium content generally indicates old water in which most of the ^{3}H from nuclear tests has decayed. Despite its potential, ^{3}H is used only rarely in travel time studies nowadays (Stewart et al., 2010), most likely because highprecision analyses are laborious (Morgenstern and Taylor, 2009) and rather expensive. In contrast, the use of stable isotopes in travel time studies has soared in the last 30 years (Kendall and McDonnell, 1998; McGuire and McDonnell, 2006; Fenicia et al., 2010; Heidbuechel et al., 2012; Klaus et al., 2015a; Benettin et al., 2015a; Pfister et al., 2017; Rodriguez et al., 2018). This is notably due to the fast and lowcost analyses provided by recent advances in laser spectroscopy (e.g., Lis et al., 2008; Gupta et al., 2009; Keim et al., 2014) and the associated technological progress in the sampling techniques of various water sources (Berman et al., 2009; Koehler and Wassenaar, 2011; Herbstritt et al., 2012; Munksgaard et al., 2011; Pangle et al., 2013; Herbstritt et al., 2019). According to Stewart et al. (2012); Stewart and Morgenstern (2016), the limited use of ^{3}H may have cause a biased or truncated vision of stream TTDs in which the long TTD tails remain mostly undetected by stable isotopes. Longer mean travel times (MTTs) were inferred from ^{3}H than from stable isotopes in several studies employing both tracers (Stewart et al., 2010). Longer MTTs may have profound consequences for catchment storage, which is usually estimated from TTDs as $S=Q\times \mathrm{MTT}$ (with Q as the flux through the catchment), assuming steadystate flow conditions (i.e., $S\left(t\right)=\stackrel{\mathrm{\u203e}}{S\left(t\right)}=S$, $Q\left(t\right)=\stackrel{\mathrm{\u203e}}{Q\left(t\right)}=Q$, $\mathrm{MTT}\left(t\right)=\stackrel{\mathrm{\u203e}}{\mathrm{MTT}\left(t\right)}=\mathrm{MTT}$; McGuire and McDonnell, 2006; Soulsby et al., 2009; Birkel et al., 2015; Pfister et al., 2017). Under this assumption, a truncated TTD would result in an underestimated MTT and, thus, an underestimated catchment storage. A different perspective on catchment storage and on its relation with travel times may, however, be adopted by calculating storage from unsteady TTDs.
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 ^{3}H (e.g., amplitudes of seasonal variations vs. radioactive decay) was first pointed out as a main reason for the discrepancies in MTT (Stewart et al., 2012). Further research is thus needed to develop mathematical frameworks that coherently incorporate stable isotopes of O and H and ^{3}H in travel time calculations. Moreover, several limiting assumptions were used in previous studies that employed ^{3}H to derive the MTT, which is in itself an insufficient statistic for describing various aspects (e.g., shape, modes, and percentiles) of the TTDs. For example, the steadystate flow assumption has been used in almost all ^{3}H travel time studies (McGuire and McDonnell, 2006; Stewart et al., 2010; Cartwright and Morgenstern, 2016; Duvert et al., 2016; Gallart et al., 2016). Yet, time variance is a fundamental characteristic of TTDs (Botter et al., 2011; Rinaldo et al., 2015), and it has been acknowledged in simulations of stream ^{3}H only very recently (Visser et al., 2019). Hydrological recharge models or tracerweighting functions have also been employed to account for the influence of the mixing of precipitation tracer values in the unsaturated zone and for the influence of the seasonal (hence, timevarying) losses to the atmosphere via ET(t) (e.g., Małoszewski and Zuber, 1982) on the catchment inputs in ^{3}H (Stewart et al., 2007). However, these methods do not explicitly represent the influence of the TTD of ET on the agelabeled water balance and, thus, represent indirect approximations. In contrast, explicit considerations of ET and of the influence of its TTD on the streamflow TTD are becoming common for stable isotopes (van der Velde et al., 2015; Visser et al., 2019). Finally, more guidance on the calibration of the TTD models against ^{3}H measurements is needed (see, for example, Gallart et al., 2016). The uncertainties of ^{3}Hinferred travel times, in particular, may have been overlooked, while these could explain the differences in the stable isotopeinferred travel time estimates.
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 ^{3}H compared to stable isotopes when determining TTDs. First, ^{3}H sampling in catchments typically differs from stable isotope sampling in terms of frequency and flow conditions. Stable isotope records in precipitation and in the streams have lately shown an increasing resolution, covering a wide range of flow conditions (McGuire et al., 2005; Benettin et al., 2015a; Birkel et al., 2015; Pfister et al., 2017; von Freyberg et al., 2017; Visser et al., 2019; Rodriguez and Klaus, 2019). Tritium records in precipitation and streams are, on the other hand, usually at a monthly resolution in many places around the globe (IAEA and WMO, 2019; IAEA, 2019; Halder et al., 2015). Only a handful of travel time studies employing ^{3}H report more than a dozen stream samples for a given site and for conditions other than baseflow (e.g., Małoszewski et al., 1983; Visser et al., 2019). This general focus on baseflow ^{3}H sampling introduces, by design, a bias towards older water. Second, the natural variability in ^{3}H compared to that of stable isotopes has rarely been documented. ^{3}H in precipitation has returned to prebomb levels, and like stable isotopes, it shows a clear yearly seasonality (e.g., Stamoulis et al., 2005; Bajjali, 2012). However, ambiguous travel time estimates may still be obtained with ^{3}H in the Northern Hemisphere because the current precipitation has similar ^{3}H concentrations to water recharged in the 1980s (Stewart et al., 2012). Higher sampling frequencies of precipitation ^{3}H are almost nonexistent. Rank and Papesch (2005) revealed a shortterm variability in precipitation ^{3}H, which is likely due to different air masses. This variability was also observed during complex meteorological conditions, such as hurricanes (Östlund, 2013). ^{3}H in streams also exhibits yearly seasonality (Różański et al., 2001; Rank et al., 2018), but shortterm dynamics are not understood well because highfrequency data sets are limited. Dinçer et al. (1970) showed that shortterm stream tritium variations can be caused by the melting of the snowpack from the current and the previous winters. In addition, the seasonally higher values of precipitation ^{3}H in spring could explain some of the ^{3}H peaks observed in the large rivers (Rank et al., 2018). More studies employing both ^{3}H and stable isotopes and comparing their travel time information content are therefore crucial for understanding travel times in catchments from a multitracer perspective.
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 ^{3}H measurements used in a coherent mathematical framework for both tracers. For this, we use highfrequency isotopic tracer data from an experimental headwater catchment in Luxembourg. Here, we focus on the stable isotope of H (deuterium ^{2}H) for which we have more precise measurements than for oxygen18 (^{18}O). A transport model based on TTDs was recently developed and successfully applied to simulate a 2year, highfrequency (subdaily) record of δ^{2}H in the stream (Rodriguez and Klaus, 2019). Here, we additionally constrain the same model within the same mathematical framework against 24 stream samples of ^{3}H collected during highly varying flow conditions over the same period as for ^{2}H. We do not assume steadystate flow conditions and we employ StorAge Selection functions (SAS) to account for the type of and the variability in the TTDs in Q and ET that affect the water age balance in the catchment. The tracer input–output relationships and the ^{3}H radioactive decay are accounted for in this method, which reduces ^{3}Hderived travel time ambiguities usually due to similar tritium activities between recent precipitation and the water recharged since the 1980s. We provide guidance on how to jointly calibrate the model to both tracers and on how to derive likely ranges of storage estimates and travel time measures other than the MTT. This work addresses the following related research questions:

Are the travel times and storage inferred from a common transport model for ^{2}H and ^{3}H in disagreement?

Are the travel time information contents of ^{2}H and ^{3}H similar?
2.1 Study site description
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 (Glaser et al., 2016, 2019; Scaini et al., 2017, 2018; Carrer et al., 2019; Rodriguez and Klaus, 2019), biogeochemistry (MoraguesQuiroga et al., 2017; Schwab et al., 2018), and pedology and geology (Juilleret et al., 2011).
The Weierbach catchment is a forested headwater catchment of 42 ha located in northwestern Luxembourg (Fig. 1). The vegetation consists mostly of deciduous hardwood trees (European beech and oak) and conifers (Picea abies and Pseudotsuga menziesii). Short vegetation covers a riparian area that is up to 3 m wide and surrounds most of the stream. The catchment morphology is a deep vshaped valley in a gently sloping plateau. The geology is essentially Devonian slate of the Ardennes massif, phyllite, and quartzite (Juilleret et al., 2011). Pleistocene periglacial slope deposits (PPSDs) cover the bedrock and are oriented parallel to the slope (Juilleret et al., 2011). The upper part of the PPSDs (∼0–50 cm) has higher drainable porosity than the lower part of the PPSDs (∼50–140 cm; MartínezCarreras et al., 2016). Fractured and weathered bedrock lies from ∼140 cm depth to ∼5 m depth on average. Below ∼5 m depth lies the fresh bedrock that can be considered impervious. The climate is temperate and semioceanic. The flow regime is governed by the interplay of seasonality between precipitation and evapotranspiration. Precipitation is fairly uniformly distributed over the year and averaged 953 mm yr^{−1} over 2006–2014 (Pfister et al., 2017). The runoff coefficient over the same period is 50 %. Streamflow (Q) is double peaked during wetter periods (MartínezCarreras et al., 2016) and single peaked during drier periods that normally occur in summer when evapotranspiration (ET) is high.
Based on previous modeling (e.g., Fenicia et al., 2014; Glaser et al., 2019) and experimental studies (e.g., MartínezCarreras et al., 2016; Juilleret et al., 2016; Scaini et al., 2017; Glaser et al., 2018), Rodriguez and Klaus (2019) proposed a perceptual model of streamflow generation in the Weierbach catchment. In this model, the first and flashy peaks of doublepeaked hydrographs are generated by precipitation falling directly into the stream, by saturation excess flow from the nearstream soils, and by infiltration excess overland flow in the riparian area. The second peaks are generated by delayed lateral subsurface flow. The lateral fluxes are assumed higher at the PPSD/bedrock interface due to the hydraulic conductivity contrasts (Glaser et al., 2016, 2019; Loritz et al., 2017). Lateral subsurface flows are, thus, accelerated when groundwater rises after a rapid vertical infiltration through the soils (Rodriguez and Klaus, 2019). The model based on travel times presented in this study was developed in a stepwise manner, based on this hypothesis of streamflow generation, and the consistency between simulated and observed δ^{2}H points toward a robust representation of the key processes. Water flow paths and streamflowgeneration processes in this catchment are, however, not completely resolved. Other studies carried out in the Colpach catchment (containing the Weierbach) suggested that the first peaks are caused by a lateral subsurface flow through a highly conductive soil layer, and that second peaks are caused by groundwater flow in the bedrock (Angermann et al., 2017; Loritz et al., 2017). This is contrary to the conclusions from other studies in the Weierbach catchment (Glaser et al., 2016, 2020), showing that the key processes are still under debate.
2.2 Hydrometric and tracer data
In this study, we use precipitation (J; in mm h^{−1}), ET (mm h^{−1}), Q (mm h^{−1}), and δ^{2}H (‰) and ^{3}H (tritium units – TUs) measurements in precipitation (C_{P,2} and C_{P,3}, respectively) and streamflow (C_{Q,2} and C_{Q,3}, respectively). Here, the subscript 2 indicates deuterium (^{2}H) and the subscript 3 indicates tritium (^{3}H). The analysis in this study focuses on the period October 2015–October 2017 (Fig. 2). Details on the hydrometric data collection (J, ET, and Q), and on the ^{2}H sample collection and analysis are given in Rodriguez and Klaus (2019).
The 1088 stream grab samples analyzed for ^{2}H were instantaneous samples collected manually or automatically with an autosampler (AS; Fig. 1), resulting in samples taken, on average, every 15 h over October 2015–October 2017. These stream samples represent most flow conditions in the catchment in terms of frequency of occurrence (Fig. 3). The 525 precipitation samples analyzed for ^{2}H were collected approximately every 2.5 mm rain increment (i.e., every 23 h on average) with a sequential rainfall sampler (SRS) and, in addition, as cumulative bulk samples on an approximately biweekly basis (but ranging from 1 to 4 weeks in some occasions). Both the sequential rainfall samples and the cumulative bulk samples represent a precipitationweighted average δ^{2}H over different time intervals (approximately daily intervals for sequential rainfall samples and approximately biweekly intervals for bulk samples). The samples were analyzed at the Luxembourg Institute of Science and Technology (LIST), using a Los Gatos Research (LGR) isotope water analyzer, yielding an analytical accuracy of 0.5 ‰ (equal to the LGR standard accuracy) for ^{2}H and a precisionmaintained <0.5 ‰ (quantified as 1 standard deviation of the measured samples and standards).
The 24 stream samples analyzed for ^{3}H were instantaneous grab samples selected from manual biweekly sampling campaigns to cover various flow ranges. The manual selection was not based on flows ranked by exceedance probabilities but rather on the streamflow time series itself. The selected samples represent various hydrological conditions (e.g., the beginning of a wet period after a long dry spell or small but flashy streamflow responses) based on data available for this catchment (see Sect. 2 and Rodriguez and Klaus, 2019). The 24 tritium samples cover a wide portion of the flow frequencies (see Fig. 3; all sampled flow conditions occurred more than 90 % of the time). This number of ^{3}H samples is one of the highest used in travel time studies (see Małoszewski and Zuber, 1993; Uhlenbrook et al., 2002; Stewart et al., 2007; Gallart et al., 2016; Gabrielli et al., 2018; Visser et al., 2019), and it is limited by the analytical costs. The samples were analyzed by the GNS Science water dating laboratory (Lower Hutt, New Zealand), which provides highprecision tritium measurements using electrolytic enrichment and liquid scintillation counting (Morgenstern and Taylor, 2009). The precision of the stream samples varies from roughly 0.07 TU to roughly 0.3 TU, but it is usually around 0.1 TU. ^{3}H in the precipitation was obtained for the Trier station (Global Network of Isotopes in Precipitation, GNIP, station; 60 km away from the Weierbach) until 2016 from the WISER database of the International Atomic Energy Agency (IAEA; IAEA and WMO, 2019; Stumpp et al., 2014). The 2017 values were obtained from the radiology group of the Bundesanstalt für Gewässerkunde (Schmidt et al., 2020). ^{3}H in precipitation before 1978 was calculated by regression with data from Vienna, Austria (Stewart et al., 2017). ^{3}H in precipitation obtained from the IAEA corresponds to monthly integrated sampling made with an evaporationfree rain totalizer, as described in the GNIP station operations manual.
Since the stream grab samples were collected over a short time interval (seconds to minutes) using a weir, the associated concentrations, C_{Q,2}(t) and C_{Q,3}(t), represent the instantaneous value of the deuterium and tritium concentrations in the stream at time t, equivalent to the concept of flux concentrations in Kreft and Zuber (1978) and Małoszewski and Zuber (1982). For both ^{2}H and ^{3}H, the time series of tracer in precipitation was interpolated between two consecutive samples (e.g., A and B) as being equal to the value of the next sample (i.e., B). This was necessary in order to obtain a continuous tracer input time series (required for Eq. (1) to work). For ^{2}H, the signal obtained from cumulative bulk samples was continuous by design and, thus, used as a baseline representing the steps of constant δ^{2}H over 2 weeks on average. Then, the discontinuous signal with higher frequency variations provided by the sequential rainfall samples was inserted into the continuous baseline for the periods when sequential rainfall samples were available (this higher frequency signal is not continuous because of the periods of the absence of samples when the SRS failed). Therefore, in this study, C_{P,2}(t) represents the instantaneous value of δ^{2}H in precipitation at time t, equal to the precipitationweighted average value over varying time intervals. Also, C_{P,3}(t) represents the instantaneous value of ^{3}H in precipitation at time t, equal to the precipitationweighted average value over monthly intervals. Assuming uniform precipitation over the catchment, C_{P,2}(t) and C_{P,3}(t) are also equivalent to flux concentrations (Małoszewski and Zuber, 1982). Since no measurements of J, Q, ET, and C_{P,2} are available before 2010, we periodically looped back their values from the period of October 2010–October 2015 for the time period before 2010 as a best estimate of their past values (Figs. S16 and S17). We aggregated the input data (J, ET, Q, C_{P,2}, and C_{P,3}) to a resolution Δt=4 h, which is small enough to capture the variability in the flows and tracers in the input and simulate the variability in the flows and tracers in the output.
2.3 Mathematical framework
Mathematically, the streamflow TTD is related to the stream tracer concentrations C_{Q}(t), according to the following Eq. (1):
where T is the travel time (the age of water at the outlet), t is time of observation, C_{Q}(t) is the stream tracer concentration, ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q}\end{array}}$ (probability distribution function – pdf) is the stream backward TTD (Benettin et al., 2015b), and ${C}_{\mathrm{P}}^{*}(T,t)$ is the tracer concentration of the water parcel reaching the outlet at time t with travel time T (this parcel was in the inflow at time t−T). The concentrations in this equation need to be flux concentrations, i.e., representative of the tracer mass fluxes in inflows and outflows (see Sect. 2.2). This equation is always true for the exact (usually unknown) TTD because it simply expresses the fact that the stream concentration is the volumeweighted arithmetic mean of the concentrations of the water parcels with different travel times at the outlet (the weighting of tracer concentrations by hydrological fluxes is thus implicit in ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q}\end{array}}(T,t)$). Thus, contrary to most of the past travel time studies using a steadystate version of Eq. (1), no weighting of the concentrations by fluxes is necessary because the timevarying TTD ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q}\end{array}}(T,t)$ already accounts for the timevarying fractions of precipitation not reaching the stream (due to either ET or storage; see Sect. 2.4) and for timevarying streamflow rates. ${C}_{\mathrm{P}}^{*}(T,t)$ depends on T and t as separate variables if the tracer concentration of a water parcel in the catchment changes between injection time t−T and observation time t. For solutes like silicon and sodium, the concentration can increase with travel time (Benettin et al., 2015a). For ^{3}H, radioactive decay with a constant α=0.0563 yr^{−1} implies ${C}_{\mathrm{P},\mathrm{3}}^{*}(T,t)={C}_{\mathrm{P},\mathrm{3}}(tT){e}^{\mathit{\alpha}T}$, where ${C}_{\mathrm{P},\mathrm{3}}(tT)$ is the concentration in precipitation measured at t−T. For ^{2}H, ${C}_{\mathrm{P},\mathrm{2}}^{*}(T,t)={C}_{\mathrm{P},\mathrm{2}}(tT)$. Thus, the streamflow TTD simultaneously verifies Eqs. (2) and (3) as follows:
Practically, when measurements of ^{2}H and ^{3}H are used to inversely deduce the TTD by using Eqs. (2) and (3), different TTDs may be found. These different TTDs may be called ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q,\mathrm{2}}\end{array}}$ and ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q,\mathrm{3}}\end{array}}$ for instance, referring to ^{2}H and ^{3}H, respectively. To avoid introducing more variables and to avoid confusion, we do not use the names ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q,\mathrm{2}}\end{array}}$ and ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q,\mathrm{3}}\end{array}}$, and we instead refer to the TTDs constrained by a given tracer using a common symbol ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q}\end{array}}$. We do this also to stress that the exact (true) TTD must simultaneously verify both Eqs. (2) and (3), and that two different TTDs ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q,\mathrm{2}}\end{array}}$ and ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q,\mathrm{3}}\end{array}}$ cannot physically exist. This is a fundamental difference from previous work that assumed two different TTDs, using for example Eq. (3) for ^{3}H and another method for ^{2}H (the sine wave approach; e.g., Małoszewski et al., 1983). The framework in this study also uses the fact that the same functional form of streamflow TTD needs to simultaneously explain both tracers to be valid, unlike previous work that used different TTD models for different tracers (Stewart and Thomas, 2008).
2.4 Transport model based on TTDs
Most of the previous travel time studies using tritium assumed steadystate flow conditions and an analytical shape for the streamflow TTD and fitted the parameters of the analytical function using the framework described in Sect. 2.3. In this study, the TTDs are unsteady (i.e., time varying or transient) and cannot be analytically described. Still, they can be calculated by numerically solving the master equation (Botter et al., 2011). This method has been applied in several recent studies (e.g., van der Velde et al., 2015; Harman, 2015; Benettin et al., 2017b) and is described in more detail by Benettin and Bertuzzo (2018). The numerical method used to solve this equation in this study is described by Rodriguez and Klaus (2019). The biggest difference, compared to many previous travel time studies, is that timevarying TTDs can be obtained from the master equation without using tracer information (i.e., Eqs. 2 and 3). In this case, tracer equations (Eqs. 2 and 3) simply become a constraint on the solutions found by solving the master equation.
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 J(t), water aging, and losses to catchment outflows ET(t) and Q(t). Solving the master equation requires knowledge (or an assumption about the shape) of the SAS functions Ω_{Q} and Ω_{ET} of outflows Q and ET, which conceptually represent how likely it is that water ages in storage (residence times) are to be present in the outflows at a given time. Solving the master equation yields the distribution of residence times in storage at every moment that can be represented in a cumulative form with ageranked storage S_{T}. It is defined as the amount of water in storage (e.g., 10 mm) younger than T (e.g., 1 year) at time t. T→S_{T} is just a mathematical change of a variable, and it has no meaning respective to the location or depth of a water parcel with a certain residence time in the catchment. By definition $\underset{T\to +\mathrm{\infty}}{lim}{S}_{T}=S\left(t\right)$, where S(t) is catchment storage. Ω_{Q} and Ω_{ET} are functions of S_{T} and cumulative distributions functions (cdf's) for numerical convenience. SAS functions are closely linked to TTDs, such that one can be found from the other using the following expression (here it has been used for Q, but it is also valid for other outflows):
The partial derivative with respect to travel time T ensures the transition from cdf to pdf. Assuming a parameterized form for Ω_{Q} and Ω_{ET}, and calibrating their parameters using the framework defined in Sect. 2.3, yields timevarying TTDs constrained by the tracers in the outflows. In this study, the parameters of Ω_{Q} are directly calibrated by using Eq. (1) for C_{Q}. Since no tracer data C_{ET} are available, the parameters of Ω_{ET} are indirectly deduced from Eq. (1) using the tracer measurements in streamflow only. This is made possible by the indirect influence of Ω_{ET} on the tracer partitioning between Q and ET and on the tracer mass balance (Appendix A2).
We assumed that Ω_{ET} is a function of only S_{T}, and it is gamma distributed with a mean parameter μ_{ET} (in millimeters) and a scale parameter θ_{ET} (in millimeters). Rodriguez and Klaus (2019) showed that, in the Weierbach catchment, a weighted sum of three components in the streamflow SAS function is more consistent with the superposition of streamflowgeneration processes (i.e., saturation excess flow, saturation overland flow, and lateral subsurface flow; see Sect. 2.1) than a single component. This means that Ω_{Q} is written as a weighted sum of three cdf's (see Appendix A1; Rodriguez and Klaus, 2019) as follows:
λ_{1}(t), λ_{2}(t), and λ_{3}(t) are timevarying weights summing to one. λ_{1}(t) is parameterized to sharply increase during flashy streamflow events, using parameters ${\mathit{\lambda}}_{\mathrm{1}}^{*}$, f_{0}, S_{th} (in millimeters), and ΔS_{th} (in millimeters; see Appendix A1). λ_{2}(t)=λ_{2} is calibrated, and λ_{3}(t) is just deduced by difference. Ω_{1} is a cumulative uniform distribution over S_{T} in [0, S_{u}] (with S_{u} a parameter in millimeters). Ω_{1} represents the young water contributions associated with short flow paths during flashy streamflow events. We chose rather low values of ${\mathit{\lambda}}_{\mathrm{1}}^{*}$ (see Table 1), such that λ_{1}(t) is generally the smallest weight (because ${\mathit{\lambda}}_{\mathrm{1}}\left(t\right)\le {\mathit{\lambda}}_{\mathrm{1}}^{*}$). The lower values of λ_{1}(t) compared to other weights are consistent with tracer data, suggesting limited contributions of event water to streamflow (MartínezCarreras et al., 2015; Wrede et al., 2015). Ω_{1} corresponds to the following processes in the nearstream area: saturation excess flow, saturation overland flow, and rain on the stream (Rodriguez and Klaus, 2019). Ω_{2} and Ω_{3} are gamma distributed, with mean parameters μ_{2} and μ_{3} (in millimeters) and scale parameters θ_{2} and θ_{3} (in millimeters), respectively. Ω_{2} and Ω_{3} represent older water that is always contributing to the stream. This older water consists of groundwater stored in the weathered bedrock that flows laterally in the subsurface. Note that we used the same functional form of Ω_{Q}(S_{T},t) for ^{2}H and ^{3}H to keep the functional form of the TTDs consistent between the tracers. Although composite SAS functions may considerably increase the complexity of the model compared to traditional SAS functions, they are necessary to account for different streamflowgeneration processes (Rodriguez and Klaus, 2019). These processes are potentially associated with contrasting flow path lengths and/or water velocities, hence the contrasting travel times. The accurate representation of these contrasting travel times is most likely vital for reliable simulations of stream chemistry (Rodriguez et al., 2020).
^{a} Details about the equations involving these parameters are given in Appendix A1 and in Rodriguez and Klaus (2019). ^{b} ${\mathit{\lambda}}_{\mathrm{1}}^{*}$ is, in fact, uniformly sampled between 0 and $\mathrm{1}{\mathit{\lambda}}_{\mathrm{2}}\le \mathrm{1}$ to ensure that $\sum _{n=\mathrm{1}}^{\mathrm{3}}{\mathit{\lambda}}_{k}\left(t\right)=\mathrm{1}$. This also ensures that values close to 0 are more often sampled than values close to 1 for ${\mathit{\lambda}}_{\mathrm{1}}^{*}$. ^{c} λ_{1}(t) varies, λ_{2} is constant, and λ_{3}(t) varies, and it is deduced using ${\mathit{\lambda}}_{\mathrm{3}}\left(t\right)=\mathrm{1}{\mathit{\lambda}}_{\mathrm{2}}{\mathit{\lambda}}_{\mathrm{1}}\left(t\right)$.
2.5 Model initialization and numerical details
Numerically solving the master equation requires an estimation of catchment mobile storage S(t). Here, S(t) represents the sum of dynamic (or active) storage and inactive (or passive) storage (Fenicia et al., 2010; Birkel et al., 2011; Soulsby et al., 2011; Hrachowitz et al., 2013). In this study, the model is initialized with storage $S(t=\mathrm{0})={S}_{\mathrm{ref}}=\mathrm{2000}$ mm. This initial value is chosen to be large enough to sustain Q and ET during drier periods and to store water that is sufficiently old to satisfy Eq. (1). S(t) is then simply deduced from the water balance as $S\left(t\right)={S}_{\mathrm{ref}}+\underset{x=\mathrm{0}}{\overset{t}{\int}}\left(J\right(x)Q(x)\mathrm{ET}(x\left)\right)\mathrm{d}x$. The initial residence time distribution in storage p_{S}(T,t) is exponential with a mean of 1.7 years, the mean residence time (MRT) by Pfister et al. (2017). Initial conditions need not be specified for the SAS functions, since these are directly calculated from the initial state variables (${S}_{T}(t=\mathrm{0})=S(t=\mathrm{0})\underset{x=\mathrm{0}}{\overset{T}{\int}}{\mathit{p}}_{\mathrm{S}}(x,t=\mathrm{0})\mathrm{d}x$), assuming a parametric form and a set of parameter values. The model is then run with the time steps Δt=4 h and age resolution ΔT=8 h. In this way, the computational cost is balanced with the resolution of the simulations in δ^{2}H. A 100year spinup is used to numerically allow the presence of water of up to 100 years old in storage and to avoid a numerical truncation of the TTDs. This spinup is also long enough to completely remove the impact of the initial conditions. This means that S_{ref} and the initial residence time distribution in storage do not influence the results over October 2015–October 2017. ET(t) is taken to be equal to potential evapotranspiration PET(t), except that it tends nonlinearly towards 0 (using a constant smoothing parameter n) when storage S(t) decreases below S_{root} (in millimeters) and where S_{root} is a parameter accounting for the water amount accessible by ET (Appendix A2).
2.6 Model calibration
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 1). The initial ranges were selected based on parameter feasible values (e.g., f_{0} between 0 and 1 by definition), on previous estimations (e.g., S_{th}), on hydrological data (e.g., S_{u} and ΔS_{th} deduced from average precipitation depths), and on initial tests done on the parameter ranges (e.g., μ and θ). These ranges allow a wide range of shapes of SAS functions, while minimizing numerical errors (occurring, for example, for S_{T}>S(t)).
Unlike our previous modeling work in this catchment (Rodriguez and Klaus, 2019), we fixed the initial storage in the model S_{ref} (to 2000 mm). We did this to reduce the degrees of freedom when sampling the parameter space in order to limit the impact of numerical errors on the calibration. These errors are due to the numerical truncation of Ω_{Q}(S_{T},t) when a considerable part (e.g., a few percent) of its tail extends above S(t). This occurs when parameters μ_{2}, μ_{3}, θ_{2}, and θ_{3} are too large compared to S_{ref} when the latter is also randomly sampled. Choosing a constant large value for S_{ref} thus guarantees the absence of truncation errors. S_{ref} has little influence on the storage deduced from travel times, since the ages sampled from storage by streamflow are governed only by μ_{2}, μ_{3}, θ_{2}, and θ_{3}. These parameters are independent of S_{ref} as long at it allows sufficiently old water to reside in storage, which is ensured by its large value and by the long spinup period we used (100 years).
The first step in the Monte Carlo procedure consisted of randomly sampling parameters from the uniform prior distributions with the ranges defined in Table 1. A total of 12 096 sets of the 12 calibrated parameters were sampled as a Latin hypercube (LHS; Helton and Davis, 2003). This sampling technique has the advantages of a stratified sampling technique and the simplicity and objectivity of a purely random sampling technique (Helton and Davis, 2003). It was chosen to make sure that the parameter samples are as evenly distributed as possible, despite their relatively small number with respect to the high number of dimensions (due to computational constraints enhanced by the required long spinup period). The model was then run over the 100year spinup followed by October 2015–October 2017, and its performance was evaluated over October 2015–October 2017. We evaluated model performance in a multiobjective manner, by using separate objective functions for ^{2}H and ^{3}H. For deuterium, we used the Nash–Sutcliffe efficiency (NSE) as follows:
where N_{2}=1016 is the number of deuterium observations in the stream. For tritium, we used the mean absolute error (MAE) as follows:
where N_{3}=24 is the number of tritium observations in the stream. We used the MAE for tritium because it is common to report errors in TU and because of the limited variance in stream ^{3}H (due to the limited number of samples and the low variability), making the NSE less appropriate (Gallart et al., 2016). The behavioral parameter sets that are used for uncertainty calculations and further analysis were selected based on threshold values L_{2} and L_{3} for the performance measures E_{2} and E_{3}, respectively (Beven and Binley, 1992). Parameter sets were considered behavioral for deuterium simulations, if ${E}_{\mathrm{2}}>{L}_{\mathrm{2}}=\mathrm{0}$, and behavioral for tritium simulations, if ${E}_{\mathrm{3}}<{L}_{\mathrm{3}}=\mathrm{0.5}$ TU. We subsequently refer to these parameter sets and corresponding simulations as constrained by deuterium, constrained by tritium, and constrained by both when both performance criteria were used. We chose these constraints to obtain reasonable model fits to the data, to obtain a comparable number of behavioral parameter sets for ^{2}H and ^{3}H, and to maximize the amount of information gained about the parameters when adding a constraint on the model performance for a tracer. This information gain was assessed with the Kullback–Leibler divergence D_{KL} between the parameter distributions inferred from various combinations of constraints L_{2} and L_{3} (Sect. 2.7).
2.7 Information contents of ^{2}H and ^{3}H
Loritz et al. (2018, 2019) recently used information theory to detect the hydrological similarity between hillslopes of the Colpach catchment and to compare topographic indexes in the Attert catchment in Luxembourg. Thiesen et al. (2019) used information theory to build an efficient predictor of rainfall–runoff events. In this study, we leverage information theory to evaluate our model parameter uncertainty (Beven and Binley, 1992) and to assess the added value of δ^{2}H and ^{3}H tracers for information gains on travel times. First, we calculated the expected information content of the prior and posterior parameter distributions, using the Shannon entropy ℋ as follows:
In this equation, the parameter X (e.g., μ_{2}) takes values (e.g., 125 mm) falling in intervals I_{k} (e.g., [100, 150] mm) that do not intersect each other and which union ${\cup}_{k=\mathrm{1}}^{{n}_{\mathrm{I}}}{I}_{k}$ equals I_{X}, the total interval of values on which X is defined (e.g., [50, 500] mm). The definitions of the n_{I} intervals I_{k} for each parameter depend on the binning of the parameter values (given in Table 2). The distribution f defines the probability of the parameter X to be in a certain state (i.e., to take a value falling in an interval I_{k}) when constrained by the criterion ${E}_{\mathrm{2}}>{L}_{\mathrm{2}}(i=\mathrm{2})$ or ${E}_{\mathrm{3}}<{L}_{\mathrm{3}}(i=\mathrm{3})$ (posterior distribution) or none of those (prior distribution). f can also be calculated for a combination of these criteria ($\mathcal{H}\left(X\right({}^{\mathrm{2}}\mathrm{H}\cap {}^{\mathrm{3}}\mathrm{H}))$). When using the logarithm of base 2, ℋ is expressed in bits of information contained in the distribution f. The uniform distribution over I_{X} has the maximum possible entropy. Lower values of ℋ thus indicate that the distribution is not flat; hence, it is less uncertain than the uniform prior distribution. In general, lower values of ℋ indicate lower parameter uncertainty. Lower values of ℋ for the posteriors also indicate that information on travel times was extracted from the tracer time series. We used the Kullback–Leibler divergence D_{KL} to precisely evaluate the information gain from prior to posterior distributions as follows:
where f is the posterior distribution constrained by E_{2}>L_{2} and/or E_{3}<L_{3}, and g is the prior distribution. D_{KL} is expressed in bits of information gained when the knowledge about a parameter distribution is updated by using tracer data. Summing the ${D}_{\mathrm{KL}}(X{}^{i}\mathrm{H},X)$ for all the parameters and for a given tracer (i=2 or i=3) yields the total amount of information learned on travel times from that tracer. We also used the Kullback–Leibler divergence D_{KL} to evaluate the gain of information when ^{3}H is used in addition to ^{2}H to constrain model predictions, or vice versa, in the following:
where f is the posterior distribution constrained by E_{2}>L_{2} and E_{3}<L_{3}, and g is the posterior distribution constrained only by ${E}_{\mathrm{2}}>{L}_{\mathrm{2}}(i=\mathrm{2})$ or only by ${E}_{\mathrm{3}}<{L}_{\mathrm{3}}(i=\mathrm{3})$. D_{KL} is expressed in bits of information gained when the knowledge about a parameter posterior distribution is updated by adding another tracer. Calculating D_{KL} also requires binning the parameter values to define the intervals I_{k} and calculate the distributions f and g. The binning for each parameter (Table 2) was chosen such that the resulting histograms visually reveal the underlying structure of the parameter values, while avoiding uneven features and irregularities (e.g., very spiky histograms).
3.1 Calibration results
A total of 148 parameter sets were behavioral for deuterium simulations, with E_{2} ranging from L_{2}=0 to 0.24. A total of 181 parameter sets were behavioral for tritium simulations, with E_{3} ranging from 0.24 to L_{3}=0.5 TU. Additionally, 16 parameter sets were behavioral for both tritium and deuterium simulations, with E_{2} ranging from L_{2}=0 to 0.19 and E_{3} ranging from 0.36 to L_{3}=0.5 TU. These solutions show that a reasonable agreement between the model fit to ^{2}H and the model fit to ^{3}H can be found.
The behavioral posterior parameter distributions constrained by deuterium or tritium or by both generally had similar ranges to their prior distributions, except, notably, for μ_{2}, θ_{2}, μ_{3}, and θ_{3} (Table 2). To assess the reduction in parameter uncertainty, we calculated and compared the entropy of the prior and of the posterior distributions (Table 2). A visual inspection of the posterior distributions was also made. Here, we show only the parameters μ_{2}, θ_{2}, μ_{3}, and θ_{3} (Fig. 4) that directly control the range of longer travel times in streamflow, since they act mostly on the righthand tail of the gamma components in Ω_{Q}. These parameters thus have a direct influence on the catchment storage inferred via ageranked storage S_{T}. The distributions of μ_{2}, θ_{2}, μ_{3}, and θ_{3} are clearly not uniform. The distributions of the other parameters are provided in the Supplement (Figs. S12 and S13). Most distributions are not uniform, indicating that the parameters are identifiable.
^{a} Binning is indicated as $[a:b:c]$, where a is the left edge of the first bin, b is the bin width, and c is the right edge of the last bin. For instance, $[\mathrm{7}:\mathrm{2}:\mathrm{11}]$ indicates data sorted with the two bins [7, 9] and [9, 11]. ^{b} ℋ and D_{KL} are expressed in bits.
Essentially, the results (Table 2 and Fig. 4) reveal that the parameter ranges decreased by adding information on ^{2}H or ^{3}H or both. This effect is particularly noticeable for f_{0} and ${\mathit{\lambda}}_{\mathrm{1}}^{*}$, which saw their upper boundary decrease, and for μ_{2} and μ_{3}, which saw their lower boundary increase considerably. These results also show that the posterior distributions depart from the uniform prior distributions when considering ^{2}H alone or ^{3}H alone (i.e., $\mathcal{H}\left(X{}^{i}\mathrm{H}\right)<\mathcal{H}\left(X\right)$ and ${D}_{\mathrm{KL}}(X{}^{i}\mathrm{H},X)>\mathrm{0}$ in Table 2). This effect is not very pronounced for most parameters, but it is clearly visible for ${\mathit{\lambda}}_{\mathrm{1}}^{*}$, μ_{2} and μ_{3} (e.g., uneven distributions of points in Fig. 4), and for μ_{ET}. The posterior distributions become considerably narrower when both tracers are considered, since $\mathcal{H}\left(X\right({}^{\mathrm{2}}\mathrm{H}\cap {}^{\mathrm{3}}\mathrm{H}))$ is much lower than ℋ(X), which is visually represented by the distribution of points tending to cluster towards a corner in Fig. 4. Generally, more was learned about the likely parameter values by adding a constraint on ^{2}H simulations after constraining ^{3}H simulations to the opposite (i.e., generally ${D}_{\mathrm{KL}}\left(X\right({}^{\mathrm{2}}\mathrm{H}\cap {}^{\mathrm{3}}\mathrm{H}),X{}^{\mathrm{3}}\mathrm{H})\ge {D}_{\mathrm{KL}}(X\left\right({}^{\mathrm{2}}\mathrm{H}\cap {}^{\mathrm{3}}\mathrm{H}),X{}^{\mathrm{2}}\mathrm{H})$). Noticeable exceptions to this are the parameters μ_{2}, θ_{2}, and θ_{3}, which are more related to the longer travel times in streamflow and to catchment storage than the other parameters.
Simulations of stream δ^{2}H captured both the slow and the fast dynamics of the observations when constrained by E_{2}>0 (blue bands and blue curve in Fig. 5a), although some variability is not fully reproduced. The Nash–Sutcliffe efficiency (E_{2}) is limited to 0.24 despite visually satisfying simulations (Sect. 4.4.2). Most flashy responses in δ^{2}H (associated with flashy streamflow responses) were reproduced to some extent by the behavioral simulations (the very thin peaks of the blue bands in Fig. 5a; more visible in Figs. S1–S9 in the Supplement). Nevertheless, about 3 % of δ^{2}H data points were visibly underestimated, pointing at a partial limitation of the composite SAS functions to simulate the variability in the streamflow TTD at these few instances (see Sect. 4.4.2). Behavioral simulations that were selected using the other performance criterion instead (E_{3}<0.5 TU; red bands in Fig. 5) did not match the δ^{2}H observations well. This shows that ^{3}H contains some information on travel times that is not in common with ^{2}H. Yet, these behavioral simulations are able to match all observed δ^{2}H flashy responses in amplitude, suggesting that, like δ^{2}H, ^{3}H contains information on young water contributions to streamflow (Sect. 4.3). Additionally, δ^{2}H simulations that were constrained by both criteria (green bands) have a smaller variability than those constrained only by E_{2}>0, suggesting that ^{3}H contains some information that is common with ^{2}H.
Simulations of stream ^{3}H generally matched the observations better in 2017 than before 2017 (red bands and red curve in Fig. 6). Some simulations (red bands) nevertheless matched the observations before 2017 relatively well. Similar to δ^{2}H simulations, both the slow and the fast tracer responses seemed necessary to reproduce the variability in ^{3}H observations (especially in 2017), although additional stream samples would be needed to confirm that the model is accurate between the current measurement points. The higher stream ^{3}H values in 2017 that were better reproduced by the model correspond to an extended dry period during which streamflow responses were mostly flashy and shortlasting hydrographs. The ^{3}H values in 2017 were closer to precipitation ^{3}H, mostly around 10 TU (see also Fig. S15). The stream reaction to those higher values suggests a considerable influence of recent rainfall events on the stream. Steadystate TTD models relying only on tritium decay would probably struggle to simulate these fast responses. This also suggests a stronger influence of old water in 2016 than in 2017 (see Sect. 4.4.2). Simulations constrained by deuterium (blue bands) tended to overestimate stream ^{3}H. Simulations constrained by both criteria (green bands) worked well in 2017, but they overestimated stream ^{3}H before 2017. Similar to δ^{2}H simulations, this suggests that ^{2}H and ^{3}H have common but also distinct information contents on travel times. The tendency of the model constrained by deuterium and/or by tritium to overestimate the tritium content in streamflow suggests a nonnegligible influence of the isotopic partitioning of inputs between Q and ET (Sect. 4.4.2; Appendix A2; Fig. S15).
3.2 Storage and travel time results
For each behavioral parameter set, we calculated ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$, the average streamflow TTD weighted by Q(t) (over 2015–2017) in cumulative form (Fig. 7). Visually, there are no striking differences between ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$ constrained by deuterium or by tritium, except a slightly wider spread for simulations constrained by tritium. The ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$ constrained by both tracers clearly differ. The associated curves (Fig. 7c) show a much narrower spread. The travel time uncertainties are thus visually much lower than when using each tracer individually, highlighting the benefit of using both tracers together. The ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$ constrained by both tracers also slightly shifted towards higher travel times. We calculated various statistics of the ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$ constrained by the different performance criteria to quantitatively compare the distributions (Table 3). This showed that the ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$ constrained only by tritium systematically corresponded to higher travel times (and lower young water fractions) than those constrained only by deuterium. A Wilcoxon rank sum test revealed that some statistically significant differences exist between the ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$ constrained by deuterium and the ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$ constrained by tritium (Appendix B). Even if these differences are statistically significant, they remain lower than in previous studies (Sect. 4.1). In addition, the youngest water fractions and the oldest water fractions of ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$ did not significantly differ according to the Wilcoxon rank sum test (Appendix B).
We defined the righthand tail of the streamflow SAS function Ω_{tail} as being the weighted sum of the two gamma components in Ω_{Q} as follows:
where ${\mathit{\lambda}}_{\mathrm{3}}^{*}=\mathrm{1}{\mathit{\lambda}}_{\mathrm{2}}{\mathit{\lambda}}_{\mathrm{1}}^{*}$. Ω_{tail} thus allows us to study the asymptotic behavior of the function Ω_{Q} in detail. In particular, this asymptotic behavior is time invariant when plotted against S_{T} because Ω_{2} and Ω_{3} are functions of S_{T} only. The behavioral parameter sets were thus directly used to calculate the curves (S_{T}; Ω_{tail}(S_{T})). These curves show similar differences for ^{2}H and ^{3}H to the curves (T; ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{P}}_{Q}\end{array}}\left(T\right)$; see Fig. 8) because a slightly wider spread is observed for Ω_{tail} constrained by tritium than that constrained by deuterium (Fig. 8b), and the Ω_{tail} constrained by both tracers tend to converge to a narrow envelope of curves slightly shifted towards higher storage values (Fig. 8c).
S_{95P} is calculated as the 95th percentile of Ω_{tail} (Eq. 11).
To quantitatively study the implications of different Ω_{tail} for storage estimations, we computed the statistics of a storage measure derived from these curves (Table 4). The 95th percentile of Ω_{tail}, called S_{95P} (black crosses in Fig. 8), allows us to estimate the total mobile storage S(t) from Ω_{tail}. On average, the Ω_{tail} constrained by tritium or by both tracers yielded significantly higher mobile storage S(t) and smaller spread in S(t) (Fig. 8; Tables 4 and B1). Yet, the mobile storage S(t) values estimated from the tracers are mutually consistent when considering the uncertainties.
4.1 Consistency between TTDs derived from stable and radioactive isotopes of H
Our work shows that streamflow TTDs and the related catchment mobile storage S(t) can be estimated in unsteady conditions by using ranked SAS functions (Ω(S_{T},t); Harman, 2015). Similar to Visser et al. (2019), we propose coherently using the measurements of stream ^{2}H and ^{3}H to calibrate the parameters of the SAS functions, which are here defined in the ageranked domain ${S}_{T}\in [\mathrm{0},+\mathrm{\infty}[$ instead of the cumulative residence time domain ${P}_{\mathrm{S}}\in [\mathrm{0},\mathrm{1}]$. The calibrated tail of the streamflow SAS function Ω_{Q} (here called Ω_{tail}) could thus be used to approximate the mobile storage S(t) instead of defining the value a priori. The SAS functions also allowed us to estimate the unsteady TTDs defined in the travel time domain T and their statistics (e.g., mean, median, and young water fraction). There were statistically significant differences between some TTD measures (e.g., mean and median) constrained by deuterium or by tritium (Wilcoxon rank sum test; Appendix B). Yet, the statistical significance may be questioned due to the contrasting number of ^{3}H samples (24) compared to δ^{2}H (>1000), which is not accounted for in the statistical test. The Wilcoxon rank sum test only compares an equivalent number of accepted simulations (148 for deuterium against 181 for tritium), regardless of data considerations. The TTDs obtained from each tracer were broadly consistent in shape, and the travel time differences were considerably smaller (i.e., <1 year) in the Weierbach catchment than in a previous comparison study in four catchments from Germany and New Zealand (up to 5 years; Stewart et al., 2010). This is particularly true for the MTT (only 8 % difference in this study), which was the only travel time measure compared in the previous study (up to 200 % difference in MTT for Stewart et al., 2010). In addition, our travel time differences were smaller for the 75th and 90th percentiles of the TTD than for the 10th and 25th percentiles. The 90th percentile difference was not statistically significant. This is not consistent with previous statements (Stewart and Morgenstern, 2016) that tritium would reveal the long tails of the TTD that remain undetected by stable isotopes. Finally, our travel time differences were smaller than the calculation uncertainties. The storage estimates derived from ^{2}H or ^{3}H were also statistically different, but the differences were also small compared to the calculation uncertainties.
These results emerged for a number of reasons. First, we treated ^{2}H and ^{3}H equally by calculating TTDs using a coherent mathematical framework for both tracers (i.e., the same method and same functional form of SAS function). However, sampling frequency and model efficiency criteria needed tracerspecific adaptations (see Sects. 4.4.2 and 4.4.3). Second, we did not derive the travel times solely based on the radioactive decay of tritium in order to avoid biases due to mixing at the outlet (Bethke and Johnson, 2008) and in order to avoid the travel time ambiguity caused by tritium from nuclear tests (Stewart et al., 2012). Moreover, we did not use multiple control volumes with different TTDs determined by tracer measurements in their input and output (Małoszewski et al., 1983; Uhlenbrook et al., 2002; Stewart et al., 2007; Stewart and Thomas, 2008). In this way, we avoided uncertainties related to difficulties in characterizing end members and gathering representative samples (Delsman et al., 2013). Third, we explicitly accounted for unsteady flow conditions, which has been done in only one previous study using tritium (Visser et al., 2019). This allowed us to estimate realistic average TTDs corresponding to the catchment inflows, outflows, and internal flows that are highly time variant. Fourth, our tritium stream sampling was not focused solely on baseflow and, hence, not biased towards old water. Fifth, we considered the entire TTDs by using various percentiles and statistics and not only the MTT, which is highly influenced by the improbable extreme values of T. This means that, even if there is water older than, for example, 1000 years in the streamflow, it can be neglected if it represents less than, for example, 0.000001 % of the volume. Finally, we explicitly accounted for parameter uncertainty. This is important because absolute values without an uncertainty estimate are difficult to interpret.
4.2 Does tritium help to reveal the presence of older water?
^{3}H systematically resulted in higher travel time and storage estimates (Tables 3 and 4). Isotopic effects on the transport of water molecules containing deuterium or tritium (i.e., on different isotopologues) seem insufficient to explain these travel time differences because the selfdiffusion of these isotopologues in water is nearly equal (Devell, 1962), and their advective velocities are the same. However, flow paths in the relatively small Weierbach catchment are probably too short to allow travel time differences due to isotopic effects on selfdiffusion coefficients.
It seems likely that, the higher storage, the higher travel times, and the larger uncertainties for tritium are related to the lack of highresolution data. Tritium simulations included many small peaks corresponding to flashy streamflow responses associated with young water (Fig. 6). Only few simulated flashy peaks could be confirmed by the presence of stream ^{3}H measurements, especially in 2016. More stream ^{3}H samples during flashy ^{3}H events may further validate these simulations of young water in streamflow and shift the TTDs constrained by tritium towards younger water. This is consistent with the fact that the larger travel time differences were found for the 10th and 25th percentiles of the TTDs. The limited tritium sampling resolution (biweekly) covered most of the flow probabilities (Fig. 3), but it may still be slightly biased towards hydrological recessions during which the youngest water fractions are absent by definition (Sect. 4.4.3). Tritium and stable isotopes of O and H synchronously sampled at a high resolution would pave the way for further research on stream water travel times from a multitracer perspective.
The travel time and storage measures estimated from a joint use of ^{2}H and ^{3}H are higher than those with individual tracers (Tables 3 and 4). These measures are not intermediate values (i.e., the average of the results from the individual tracers) because deuterium and tritium have information in common about longer travel times (i.e., the simulations constrained by both tracers are a specific selection among all accepted simulations; see Fig. 4). Tritium may, thus, have helped to reveal the presence of old water in streamflow. However, it did so only when combined with deuterium. It is commonly assumed that ^{3}H carries more information on old water because of radioactive decay that relates lower tritium activities to increasing travel times (Stewart et al., 2010). However, as shown by Stewart et al. (2012) and in Fig. 2, current tritium values of the water recharged in 1980–2000 are similar to the tritium values of the water recharged today. Thus, the younger water disrupts the relationship between travel times and tritium values. It seems that using the highfrequency δ^{2}H measurements reduced the ambiguity of tritiumderived travel times by helping to discriminate between young and old water contributions to streamflow. The travel times being below ∼5 years in the Weierbach catchment (Table 3) could be another reason for the limited information on ^{3}H in older water. ^{3}H decays by only about 25 % in 5 years, meaning that all the tritium activities of the water in the Weierbach catchment have varied by, at most, ∼2 TU since water entered the catchment. This is much lower than the 10 TU amplitude of tritium variations in precipitation. Thus, in catchments with relatively short residence times, radioactive decay may give information that is redundant with respect to the natural variability in the tracer signal of precipitation. In a few decades, water recharged in 1980–2000 may have completely left many catchments or may be a negligible part of storage, such that the log (^{3}H) of stored water may increase linearly with residence time (see the recent increasing trend in ${C}_{\mathrm{P},\mathrm{3}}^{*}$ in Fig. 2). Thus, in a few decades, tritium could be even more informative about old water contributions because there may be no travel time ambiguity anymore. Furthermore, the oscillations of tritium in precipitation over long timescales (>10 years) recently detected and related to cycles of solar magnetic activity (Palcsu et al., 2018) may give stream tritium concentrations even more agespecific meaning. Therefore, it is important to reiterate the call of Stewart et al. (2012) to start sampling tritium in streams now and for the next few decades to use in travel time analyses.
4.3 Travel time information contents of stable and radioactive isotopes
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 ∼10 % of the initial ℋ(X) for most parameters. In total, 12.7 bits of information on travel times were found by combining the two tracers. This is more than twice the amount found in each individual tracer (around 4 bits; see the paragraph below). This amount of information can be calculated for a given tracer by summing ${D}_{\mathrm{KL}}\left(X\right({}^{\mathrm{2}}\mathrm{H}\cap {}^{\mathrm{3}}\mathrm{H}),X)$ for all parameters (see Sect. 2.7; Table 2). Combining the tracers also resulted in lower uncertainties (lowest entropy $\mathcal{H}\left(X\right({}^{\mathrm{2}}\mathrm{H}\cap {}^{\mathrm{3}}\mathrm{H})$) in Table 2; narrower groups of curves in Figs. 7 and 8; lower standard deviations in Tables 3 and 4). This information gain on travel times was possible because composite SAS functions (Eq. 5) allowed us to constrain three nearly independent components (Ω_{1}, Ω_{2}, and Ω_{3}) of the same streamflow TTD with one tracer or the other. This reduced the potential tradeoffs between the shapes suggested by one tracer or the other. These three components are formally related only by the requirement for having ${\mathit{\lambda}}_{\mathrm{1}}\left(t\right)+{\mathit{\lambda}}_{\mathrm{2}}\left(t\right)+{\mathit{\lambda}}_{\mathrm{3}}\left(t\right)=\mathrm{1}$. Thus, all their other parameters are independent.
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 μ_{ET} even better than deuterium (Table 2). The particularly large information gains on μ_{ET} and θ_{ET} with tritium reveal a stronger influence of Ω_{ET} on the accuracy of stream tracer simulations than for deuterium via an indirect influence on isotopic partitioning (Appendix A2). This also highlights the importance of explicitly considering ET in streamflow travel time calculations (van der Velde et al., 2015; Visser et al., 2019; Buzacott et al., 2020). However, ^{2}H resulted in lower uncertainties for nearly all other parameters (e.g., lower Shannon entropy ℋ(X^{2}H); Table 2). This is most likely due to the much higher sampling frequency for deuterium that allows for constraining the simulations better than with biweekly tritium measurements (see the simulation envelopes Figs. 5 and 6). From our experience in the Weierbach catchment, we estimate that, for ^{2}H, a weekly sampling to cover the damped variations in δ^{2}H (i.e., about 100 samples over 2015–2017) complemented with an eventbased, highfrequency sampling (every 15 h) of the flashy responses (i.e., about 300 samples over 2015–2017) could have given us as much information as the complete time series. This suggests that a more strategic sampling of ^{2}H may outperform ^{3}H. The amount of information gained from the isotopic data necessarily grows with an increasing number of samples. Yet, we do not know whether it scales linearly or nonlinearly and whether it quickly reaches a plateau as the number of observation points grows (Fig. S14). In the future, it would be useful to further use information theory (e.g., entropy conditional on sample size) to know how this information scales, how many measurements are enough, and when to sample isotopes for maximum information gain on travel times. This would imply artificially resampling a higherfrequency isotopic time series using various strategies (e.g., Pool et al., 2017; Etter et al., 2018) and recalibrating the model many times, which would involve much subjectivity and come with an exorbitant computational price. Finally, the information contents on travel times that we have derived depend on our model structure (number of control volumes and SAS functional form). More work is needed in developing modelfree (e.g., databased) unsteady TTD estimation methods in order to reduce the dependence of the results on modeling assumptions.
Overall, stable and radioactive isotopes of H had different information contents on travel times. The positive D_{KL} values are not simply due to different performance measures for deuterium and for tritium (see Table S1 in the Supplement) but due to nonredundant information on travel times for each tracer. Performance measures E_{2} and E_{3} are not identical but are both based on minimizing a sum of residuals and, thus, do not considerably influence what can be learned from tracer data (see Table S2). Moreover, the parameters corresponding to the best simulations in ^{2}H did not correspond to those for ^{3}H and vice versa. Yet, stable and radioactive isotopes had some information in common with respect to young water. This is consistent with early tritium studies that tried to show its potential for detecting young water contributions to streamflow (Hubert et al., 1969; Crouzet et al., 1970; Dinçer et al., 1970; Klaus and McDonnell, 2013). This has been overlooked in recent travel time studies because of the sampling focused on periods outside events (Stewart et al., 2010). The theoretical span of 0–4 years pointed out in Stewart et al. (2010) should, however, not be taken as the only range of travel times in which ^{18}O, ^{2}H, and ^{3}H may have redundant information. As clearly written by Stewart et al. (2010), this limit corresponds to a steadystate exponential TTD only, while other TTD shapes (or unsteady TTDs) could yield much higher limits. More importantly, this limit can be lowered by the seasonality of the input function (see Stewart et al., 2010, p. 1647). Finally, stable and radioactive isotopes had some information in common with old water as well. This is clearly shown by the increased travel time and storage measures when both tracers are used, which also highlights that they can give similar results.
4.4 Limitations and way forward
4.4.1 Hydrometric versus tracerinferred storage
The storage value derived from unsteady travel times constrained by tracer data (Table 4; ∼1200–1700 mm) is noticeably larger than the maximum storage (≃250 mm) estimated from point measurements of porosity and water content (MartínezCarreras et al., 2016), from water balance analyses (Pfister et al., 2017), from water balance analyses combined with recession techniques (Carrer et al., 2019), and from a distributed hydrological model (≤700 mm, Glaser et al., 2016, 2020). Our storage value is more consistent with the ∼1600 mm derived from depth to bedrock and porosity data used for the Colpach catchment (containing the Weierbach) that was modeled with CATFLOW (Loritz et al., 2017). Large differences between hydrometrically derived and tracerderived storage estimates are not uncommon (Soulsby et al., 2009; Fenicia et al., 2010; Birkel et al., 2011) and, in fact, highlight the ability of tracers to reveal the existence of stored water that is not directly involved in streamflow generation (Dralle et al., 2018; Carrer et al., 2019). This hydraulically disconnected storage is nevertheless important for explaining the long residence times in catchments (Zuber, 1986). More research is needed to improve the conceptualization of storage and to unify storage terminology and the various estimates obtained from tracers or other techniques. The storage value we found is not in complete contradiction with the previous estimates if we consider their uncertainties. Hydrological measurements (J, Q, and especially ET) are highly uncertain (Waichler et al., 2005; Graham et al., 2010; Buttafuoco et al., 2010; McMillan et al., 2012; McMahon et al., 2013), and their errors are accumulated in longterm water balance calculations. An explicit consideration of those uncertainties in the future could reconcile the different storage estimates. Furthermore, it is worth remembering that simplifying storage from a complex spatially distributed quantity to a simple, compact 1D water column neglects the importance of subsurface heterogeneity, surface topography, and bedrock topography for the storage and release of water. As a result, upscaling local point measurements of storage capacity that are not representative of the whole subsurface is very likely to under or overestimate the true storage capacity of the whole catchment. This is even more true if the new techniques used to scan the subsurface over larger areas, such as electrical resistivity tomography (ERT), are themselves associated with uncertainties, thus requiring adaptations and sitespecific independent knowledge (Parsekian et al., 2015).
4.4.2 Model performance and uncertainty
The visually satisfactory tracer simulations enhance our confidence that the model accurately simulates travel times in the Weierbach catchment. Still, the performance in δ^{2}H or in ^{3}H could be improved in the future by testing other models of composite SAS functions. The best NSE for deuterium simulations (E_{2}) was 0.24, which is lower than several other studies using SAS functions (van der Velde et al., 2015; Harman, 2015; Benettin et al., 2017b). E_{2} is penalizing for the δ^{2}H time series in the Weierbach catchment because the observed stream δ^{2}H has many more points corresponding to damped seasonal fluctuations (Fig. 5a) compared to large, flashy fluctuations (Fig. 5b). E_{2} also overemphasizes the timing errors, even if the shape of the simulation is perfect (Klaus and Zehe, 2010; Seibert et al., 2016). In addition, E_{2} is not an absolute measure of model performance that allows comparisons between different studies (Seibert, 2001; Schaefli and Gupta, 2007; Criss and Winston, 2008). Future work needs to develop more appropriate objective functions for δ^{2}H, especially with respect to the information gained from model calibration. This implies either accounting for expert knowledge, intuition, and visual experience with simulations in a customized performance measure (Ehret and Zehe, 2011; Seibert et al., 2016), finding an adequate benchmark model for δ^{2}H (Schaefli and Gupta, 2007), or correctly defining the statistical properties of the model errors (Schoups and Vrugt, 2010). The best MAE for tritium simulations (called E_{3}) was 0.24. This is slightly higher than values of the root mean square error (RMSE; close to 0.10) reported in a number of studies using tritium (Stewart et al., 2007; Stewart and Thomas, 2008; Duvert et al., 2016). However these studies had only a few stream samples, while Gusyev et al. (2013) report, for instance, a RMSE of 1.62 TU for 15 stream samples. Stream δ^{2}H data seem to suggest a larger fraction of young water than the simulations (see the underestimation of flashy events in Fig. 5). Stream ^{3}H data seem to suggest larger fractions of old water than the simulations (see the overestimation of tritium activities over March–September 2016 in Fig. 6). A model passing through all observation points may, thus, show larger differences between the TTDs constrained by deuterium and the TTDs constrained by tritium. It is important to recall that there are fewer ^{3}H stream samples compared to ^{2}H; thus, a comparison of the TTDs from this hypothetical ideal model could be misleading. Furthermore, the different scaling for the units for δ^{2}H and ^{3}H may also mislead the visual comparisons and interpretations on young water contributions based on the different amplitude of flashy tracer responses. We believe that a higher resolution of stream ^{3}H would unambiguously show the potential of tritium for revealing young water in the stream, as shown in the early tritium studies (Hubert et al., 1969; Crouzet et al., 1970; Dinçer et al., 1970). Our choice of performance measures (E_{2}=NSE and E_{3}=MAE) and selection criteria (L_{2}=0 and L_{3}=0.5) resulted in slightly more TTDs constrained by tritium than TTDs constrained by deuterium (148 curves for E_{2}>0 against 181 curves for E_{3}<0.5). These numbers are highly sensitive to performance thresholds, and we chose to represent the closest match in the number of accepted solutions for each tracer, while considering only meaningful performance criteria variations (i.e., ≥0.1) and acceptable model performance. This guarantees a similar treatment of the two tracers (i.e., it avoids biases in travel times for a given tracer), while accepting only satisfying simulations for both tracers. Future work could assess the sensitivity of travel time differences between tracers for other performance measures and thresholds and for contrasting numbers of accepted solutions.
The isotopic simulations were better for decreasing δ^{2}H than for increasing δ^{2}H (better simulations of the flashy events in δ^{2}H pointing downwards; Fig. 5). This is probably because the increases in δ^{2}H generally correspond to drier periods, during which C_{Q,2} starts reacting more strongly to C_{P,2}, indicating that young water fractions (controlled by λ_{1}(t) in the model) are higher than expected. C_{P,2} can explain only about 30 % of the variations in C_{Q,2}, but this can increase to 44 % during drier periods (Figs. S10 and S11). The low explanatory power of C_{P,2} is linked to the larger influence of groundwater for streamflow responses in the Weierbach catchment (conceptualized with Ω_{2} and Ω_{3} and having larger weights in λ_{2} and λ_{3}). During drier periods, we expect an increase in the nonlinearity of the processes delivering young water to the stream. For example, the decreasing extent of the stream network and of saturated areas observed in the Weierbach catchment during drier conditions (Antonelli et al., 2020a, b) is likely caused by decreasing groundwater levels (Glaser et al., 2020), and it could reduce the amounts of young water reaching the stream (see van Meerveld et al., 2019). However, streamflow is lower during drier conditions; thus, the fractions of young water can still increase because of a less pronounced dilution of the young water in streamflow compared to wet periods. On the other hand, preferential flow observed in the soils of the Weierbach catchment and in the direct vicinity (Jackisch et al., 2017; Angermann et al., 2017; Scaini et al., 2017, 2018) may become more relevant during drier conditions and could increase the amount of young water contributing to streamflow, especially because precipitation intensities can be much higher in summer (due to thunderstorms) than in winter. The parameterization of streamflow SAS functions via λ_{1}(t) (Eq. A5) includes – to some extent – the effect of wet vs. dry conditions and the role of precipitation intensity, but it seems not fully able to capture how these factors influence young water fractions in the stream. Testing other parameterizations of λ_{1}(t), or including additional information such as soil moisture or groundwater levels in the current parameterization of λ_{1}(t), may improve the simulations. Finally, the uncertainty in precipitation δ^{2}H could be higher during drier periods because precipitation amounts can be too small (e.g., <1 mm) over several weeks or because the precipitation intensities can be too high (e.g., >5 mm h^{−1}) to be captured efficiently by the sequential rainfall sampler. This may lead to inaccuracies in the input data and, thus, to the inability of the model to simulate the corresponding flashy events in stream δ^{2}H. The representation of precipitation δ^{2}H should be improved in the future by using more recent sampling techniques (e.g., Michelsen et al., 2019).
The tendency of the model to yield higher average tritium values than the observations in streamflow over 2015–2017 (Fig. 6) and lower average tritium values than precipitation (see Fig. S15 where this is more visible) seems related to either not enough tritium residing in storage or to not enough tritium being removed by ET. The latter mechanism is only indirectly controlled by Ω_{ET}, which loosely acts on the isotopic partitioning between Q and ET (Appendix A2). Unfortunately, no tracer data in ET can be used to close the tracer mass balance and to draw firm conclusions on the correct mechanism. In any case, an accumulation of tritium in storage for decreasing the average stream tritium content is not a realistic behavior in the long term. The average stream ^{3}H is higher for the simulations constrained by E_{2}>L_{2} than E_{3}<L_{3}, probably because of the lower resolution of ^{3}H measurements. The simulations overestimated ^{3}H in the stream, particularly in 2015–2016 compared to 2017 (Fig. 6). In 2017, the simulations were better because the model used more young water (<7 d old; using Ω_{1}) to simulate the variability in and higher values of stream ^{3}H compared to 2016. The lower ^{3}H in 2015–2016 could be caused by an increased travel time in the older water components in 2015–2016 compared to 2017, due to changes in the importance of different subsurface flow paths in the Weierbach catchment caused by a wetter period. The old water components of Ω_{2} and Ω_{3} (Eq. 5) represent subsurface flow paths likely occurring in the lower soils, following bedrock topography (Glaser et al., 2016; Rodriguez and Klaus, 2019) and potentially in weathered bedrock fractures (Scaini et al., 2018), or in the bedrock (Angermann et al., 2017; Loritz et al., 2017). We used the functions of S_{T} only for Ω_{2} and Ω_{3}, meaning that the ranges of ages that they select do not change considerably with time (because the distribution of S_{T} is rather stable). Including an explicit dependence on time for Ω_{2} and Ω_{3} could help to better represent deeper flow paths in the catchment and improve ^{3}H simulations in 2015–2016. Eventually, the monthly resolution of ^{3}H in precipitation is coarser than the biweekly sampling in the stream, which can hinder accurate simulations. An increase in the sampling resolution of tritium in precipitation will be necessary in the future (Rank and Papesch, 2005).
Finally, parameter distributions (Figs. 4 and S12–S13) and information measures (Table 2) suggest that some parameters are not strongly constrained by tracer data (but they are not unidentifiable either). This may result from the larger number of parameters compared to the traditional SAS functions. Nevertheless, all these parameters are necessary for representing the array of nonlinear and timevarying processes leading to the selection of particular ages from storage (numerically represented by ∼10^{5} control volumes) to generate both outflows Q and ET. This is essential to avoid neglecting certain travel times that may become important for accurate water chemistry simulations (Rodriguez et al., 2020). Other methods for exploring parameters (using Markov chains), such as DREAM (Vrugt, 2016) or PEST (Doherty and Johnston, 2003), could yield narrower posterior distributions. Nevertheless, these more advanced algorithms would need to be adapted to allow parameter constraints, numerically diverging solutions (typically for randomly selected combinations of parameters values that are incompatible), and multiobjective calibration.
4.4.3 Data constraints
The highest flows that were not sampled for tritium (Fig. 3) represent about 50 % of the water that left the catchment via streamflow over 2015–2017. The high flows are mostly second delayed streamflow peaks in this catchment where doublepeaked hydrographs occur in wet conditions (Sect. 2.1). Previous studies in the Weierbach catchment, using various tracers, suggest that second peaks are more likely composed of older water than first peaks (Wrede et al., 2015; MartínezCarreras et al., 2015). Nevertheless, the high flows in the second peaks may be associated with shorter travel times than low flows. Loritz et al. (2017) described the subsurface of the Weierbach catchment as highly permeable and hypothesized that it is able to rapidly transmit large amounts of young water during high streamflow events. This may explain the higher tritiumderived travel times due to the limited ^{3}H sampling in this study (e.g., 25 % difference in median travel time). For deuterium, the highest flows are associated with 40 samples (about 4 % of the samples) which represent about 20 % of the water leaving via streamflow over 2015–2017 (Fig. 3). It is important to note that weighting the available stream samples by streamflow in the calibration (i.e., calibrating on tracer loads instead of concentrations) would not compensate for this relative absence of samples during highflow conditions. In addition, it would bias the calibrated TTDs towards highflow conditions, while our goal is to have TTDs which accurately represent the functioning of the catchment over all flow conditions (the whole 2015–2017 study period). An adaptive sampling frequency based on accumulated flows (e.g., one sample every dozen m^{3}) could improve the representativity of the samples with respect to the flow volumes. This would not improve the results because the TTDs already account for the flow volumes by definition and because the larger water mass not sampled for tritium is not leading to a strong bias towards young or old water when compared to deuterium. The latter is shown by the good agreement between the TTDs constrained by deuterium and the TTDs constrained by tritium. Flowproportional sampling would also lead to a much larger number of samples, rapidly exceeding the current field and laboratory capacities. This is why nearly continuous in situ measurements would be preferable (e.g., Pangle et al., 2013; von Freyberg et al., 2017). Nevertheless, in situ measurements are currently not available for tritium.
We found much lower deviations for the travel time and storage measures constrained by deuterium and tritium together (Tables 3 and 4). However, it has to be acknowledged that there are only few accepted solutions (16), while there about 10 times more when using ^{2}H alone or ^{3}H alone. We should expect a higher standard deviation due to a lower number of accepted solutions for calculating this statistic when using both tracers. On the contrary, the associated TTDs (Figs. 7c and 8c) fall close to each other, resulting in lower deviations that clearly point to lower uncertainties. A lower number of accepted solutions is inevitable in the end as it is an inherent consequence of using several performance measures independently as opposed to using a combined objective function (e.g., Hrachowitz et al., 2013; Rodriguez et al., 2018). Fewer accepted simulations are also an advantage for identifying behavioral parameter sets (Klaus and Zehe, 2010). Less strict threshold criteria for behavioral solutions could increase the number of accepted solutions, but they would accept less accurate simulations, which could lead to misleading conclusions. More stream ^{3}H measurements would, on the other hand, allow the use of more advanced objective functions, which could lead to more accepted solutions. The input data, measured over 2010–2017 and used to spin up the model from 1960 to 2010 (J, ET, Q, and C_{P,2}), could be unrepresentative of the real hydrometeorological and isotopic conditions of 1960–2015 due, for instance, to nonstationarity or climate change. These changing conditions could affect the modeled residence times in storage and, thus, the estimated streamflow travel times (Wilusz et al., 2017). Different methods to spin up the model could be tested in the future (Hrachowitz et al., 2011), especially for assessing the effect of changing hydrometeorological and isotopic conditions on the estimation of travel times. For this, isotope tracer records that span several decades, like the ones that can be reconstructed from pearl mussels shells (Pfister et al., 2018, 2019), represent a crucial asset. Eventually, the precipitation tritium samples were taken about 60 km away from the catchment and may introduce some uncertainty.
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 ^{3}H appeared to be slightly more informative than ^{2}H, even with fewer samples, a different sampling strategy of the stable isotopes could outperform tritium. Future work could, additionally, compare streamflow TTD and storage from the two tracers in larger catchments where older water is expected in order to give tritium more time to decay and to better leverage its ability to point out the presence of very old water. More work is also needed with respect to comparing the information of the tracers on travel times using databased approaches in order to avoid a dependence on model structure. We therefore recommend that (1) tritium keeps being sampled in as many places as possible, as emphasized by Stewart et al. (2012), but also that (2) tritium keeps being sampled at the highest frequency possible and also synchronously with stable isotopes, if possible. This is particularly important for the isotopic measurements in precipitation that drive all model simulations, regardless of functional forms of TTD and their parameter values. Overall, our work shows that more tracer data are naturally better for gathering more information about the catchments functions of storage and release.
A1 Parameterization of the SAS functions
In this section, we provide further details on the equations used in the model. The composite streamflow SAS function Ω_{Q} used in this study is as follows:
Ω_{1}(S_{T}) is a cumulative uniform distribution for S_{T} in [0, S_{u}], where S_{u} (in millimeters) is a calibrated parameter representing the amount of stored young water potentially contributing to flashy streamflow responses. Thus, in the following:
Ω_{2}(S_{T}) and Ω_{3}(S_{T}) are direct functions of S_{T} and are gamma distributed as follows:
where Γ is the gamma function, γ is the lower incomplete gamma function, μ_{2} and μ_{3} (in millimeters) are mean parameters (calibrated), and θ_{2} and θ_{3} (in millimeters) are scale parameters (calibrated).
λ_{1}(t), λ_{2}(t), and λ_{3}(t) sum to 1. These are simply timevarying weights giving each component (i.e., cdf Ω) a dynamic contribution to streamflow generation. In particular, λ_{1}(t) is made highly time variant to represent the flashy hydrographs that have an on–off type of response to precipitation. λ_{2}(t) is considered constant and is calibrated to keep the parameterization parsimonious. ${\mathit{\lambda}}_{\mathrm{3}}\left(t\right)=\mathrm{1}{\mathit{\lambda}}_{\mathrm{2}}{\mathit{\lambda}}_{\mathrm{1}}\left(t\right)$ is deduced by the difference for parsimony as well. Since Ω_{1}(S_{T}) represents young water contributions, and previous studies in the Weierbach catchment showed that event water contributions depend on the catchment wetness and on precipitation intensity (Wrede et al., 2015; MartínezCarreras et al., 2015), λ_{1}(t) was parameterized using storage S(t) and a proxy storage variations $\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}S\left(t\right)}$ (see Rodriguez and Klaus, 2019, for more details) as follows:
where ${\mathit{\lambda}}_{\mathrm{1}}^{*}\in [$0, 1] (no unit) is a calibrated parameter representing the maximum value of λ_{1}(t), and f(t)∈[0, 1] and g(t)∈[0, 1] are given by the following:
f_{0}∈[0, 1] (no unit) is a calibrated parameter guaranteeing a minimum for λ_{1}(t) during dry periods, S_{min}=min(S(t)), and S_{th} (in millimeters; calibrated parameter) is a storage threshold relative to the minimum storage S_{min} separating wet ($S\left(t\right)>{S}_{\mathrm{min}}+{S}_{\mathrm{th}}$) from dry periods ($S\left(t\right)<{S}_{\mathrm{min}}+{S}_{\mathrm{th}}$). m=1000 is a fixed parameter used to smooth the function f with respect to S(t). $\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}S\left(t\right)}$ is a proxy of storage variations calculated as a moving average of storage variations over a time window $\mathrm{\Delta}{t}^{*}=\mathrm{2}\mathrm{\Delta}t$ as follows:
with $\mathrm{\Delta}S\left(t\right)=\mathrm{\Delta}t\left(J\right(t)Q(t)\mathrm{ET}(t\left)\right)$. $\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}S\left(t\right)}$ essentially increases during precipitation events and decreases when Q(t) or ET(t) are high. ΔS_{th} is a threshold in $\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}S\left(t\right)}$ above which g(t) tends to 1, allowing λ_{1}(t) to increase and decrease sharply during flashy streamflow events.
A2 Actual evapotranspiration and tracer partitioning between Q and ET
Actual evapotranspiration ET(t) is calculated from potential evapotranspiration PET(t) using the following formula:
where ${S}_{\mathrm{root}}={S}_{\mathrm{ref}}\mathrm{150}$ is a fixed parameter (in millimeters) representing the storage threshold S(t)=S_{root} below which ET(t) starts decreasing from PET(t) towards 0. A similar strategy was employed, for instance, by Fenicia et al. (2016) and Pfister et al. (2017) in the Weierbach and neighboring Luxembourgish catchments. This decrease is smoothed by the fixed coefficient n=20. S_{root} accounts for the water available for evaporation and plant transpiration until the capillary forces offer too much resistance. This formula thus represents the decrease in water losses to the atmosphere under waterlimited conditions.
In the model, this equation is the only explicit partitioning condition of the tracer influx J×C_{P} between evaporative losses ET×C_{ET} and streamflow Q×C_{Q}. An implicit partitioning nevertheless exists for the following reason. The tracer mass balance equation is as follows:
where M(t) is the tracer mass in the catchment, and C_{P}(t) is the tracer concentration in precipitation at time t. J(t)C_{P}(t) is given by the input data, and Q(t)C_{Q}(t) and ET(t)C_{ET}(t) are partly determined by the SAS functions Ω_{Q} and Ω_{ET}. For Q(t)C_{Q}(t), Q(t) is the measured data, and C_{Q}(t) is directly related to Ω_{Q} through the related TTD ${\scriptscriptstyle \begin{array}{c}\leftarrow \\ {\mathit{p}}_{Q}\end{array}}$ (Eqs. 1 and 4). The parameters of Ω_{Q} are thus directly determined by the fit of the simulations to observed C_{Q}(t). Tracer data for C_{ET}(t) are not available. Thus, the parameters of Ω_{ET} cannot be directly determined from data the same way it can be done for Ω_{Q}. Still, the parameters of Ω_{ET} need to yield C_{ET} values which satisfy the tracer mass balance (Eq. A10) in the long term (when $\frac{\mathrm{d}M}{\mathrm{d}t}\left(t\right)$ becomes negligible). If the parameters of Ω_{ET} do not allow the closure of the tracer mass balance, the simulations in C_{Q}(t) will be affected and will not match the observations. Therefore, the fit between observed and simulated C_{Q}(t) can be used also to indirectly deduce the parameters of Ω_{ET}, using the implicit tracer partitioning Ω_{ET} exerts. This partitioning is only indirect (or implicit) because there is no onetoone relationship between T and ${C}_{P}^{*}(T$, t) (Eq. 1), meaning that ageselection patterns expressed by the SAS functions do not uniquely determine the average values of Q(t)C_{Q}(t) and ET(t)C_{ET}(t). Tritium was more informative on travel times than deuterium due to its stronger constraint on the parameter values of Ω_{ET}, μ_{ET} and θ_{ET}. Based on the reasoning above, this is simply due to the fact that the relationship between T and ${C}_{\mathrm{P}}^{*}(T$, t) is clearer for tritium due to its radioactive decay than for deuterium, for which there is essentially no relationship between travel time and tracer concentrations. In conclusion, information on the parameters of Ω_{ET} exists in the time series of C_{Q}(t) and can be extracted by calibrating the model based on SAS functions, particularly from using tritium.
The obtained differences in travel time and storage measures (Tables 3 and 4) were further compared to assess their statistical significance (Table B1). For this, we used a Wilcoxon rank sum test (also known as the Mann–Whitney U test) for each of the timeaveraged (flowweighted over 2015–2017) statistics (e.g., the 10th percentile) of the distributions calculated from ^{2}H (148 distributions) or ^{3}H (181 distributions) and shown in Figs. 7a–b and 8a–b. This tested the null hypothesis that the two underlying median TTDs or SAS functions obtained from each tracer are equal (i.e., the distribution obtained as the median of all the flowweighted, timeaveraged distributions over 2015–2017 corresponding to the behavioral parameter sets for a given tracer). We chose this test because it is nonparametric, and because it allows the taking into account of travel time and storage uncertainties by including all the behavioral distributions. 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 ∼2 months) and the oldest fractions of water (90th percentile; older than about 4 years) are most likely drawn from a common TTD, regardless of the tracer used. Despite significant differences in all other measures, this test suggests that the truncation of the long TTD tail when using only deuterium is not statistically plausible.
The codes implementing the composite SASbased model for deuterium and tritium simulations can be found at the LIST GitLab (https://git.list.lu/catchmentecohydro/composite_sas_model_2h_3h_weierbach, last access: 23 January 2021) (Rodriguez, 2021).
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 (https://doi.org/10.5281/zenodo.4061554) (Hissler et al., 2020). The rest of the data are the property of Luxembourg Institute of Science and Technology and can be obtained upon request to the corresponding author.
The supplement related to this article is available online at: https://doi.org/10.5194/hess254012021supplement.
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 (ℋ and D_{KL}). Nicolas Rodriguez, Julian Klaus, and Laurent Pfister (grant no. FNR/CORE/C14/SR/8353440/STOREAGE) acknowledge funding for this study from the Luxembourg National Research Fund (FNR). This study also contributes to and benefited from the “Catchments as Organized Systems” (CAOS FOR 1598; grant no. FNR/INTER/DFG/14/02/CAOS‐2) research unit funded by DFG, FNR, and FWF. We thank Jérôme Juilleret for his help in collecting tritium samples in the field. The authors acknowledge support from the state of BadenWürttemberg, Germany, through the bwHPC project. We thank Barbara Glaser for her help with the input data (PET).
This research has been supported by the Fonds National de la Recherche Luxembourg (grant no. FNR/CORE/C14/SR/8353440/STOREAGE) 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 openaccess
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.
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 Abstract
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Appendix A: Model equations
 Appendix B: Statistical significance of travel time and storage differences
 Code availability
 Data availability
 Author contributions
 Competing interests
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Conclusions
 Appendix A: Model equations
 Appendix B: Statistical significance of travel time and storage differences
 Code availability
 Data availability
 Author contributions
 Competing interests
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement