In this study, we used the tran-SAS model for describing the catchment-scale water mixing and solute transport based on SAS functions. The catchment was conceptualized as a single storage S(t) (mm), whose water-age balance can be expressed as follows : 1S(t)=S0+V(t)2∂ST(T,t)∂t+∂ST(T,t)∂T=P(t)-Q(t)⋅ΩQ(ST,t)-ET(t)⋅ΩET(ST,t),3with an initial condition of ST(T,t=0)=ST0(T)4and a boundary condition of ST(0,t)=0. Here, S0 (mm) is the initial storage; V(t) (mm) represents the storage variations; P(t) (mm d-1), Q(t) (mm d-1), and ET(t) (mm d-1) are precipitation, discharge, and evapotranspiration, respectively; ST(T,t) (mm) is the age-ranked storage; ST0(T) (mm) is the initial age-ranked storage; and ΩQ(ST,t) (–) and ΩET(ST,t) (–) are the cumulative SAS functions for Q and ET, respectively.

By definition, the TTD of streamflow pQ(T,t) (d-1) is calculated as follows : 5pQ(T,t)=∂ΩQ(ST,t)∂ST⋅∂ST∂T.

The isotopic signature in streamflow CQ(t) (‰) can be obtained as follows : 6CQ(t)=∫0+∞CS(T,t)⋅pQ(T,t)⋅dT, where CS(T,t) (‰) is the isotopic signature of a water parcel in storage. Equations () and () also apply for ET.

In this study, we tested three SAS parameterizations: the power law time-invariant (PLTI; Eq. , ), power law time-variant (PLTV; Eq. , ), and beta time-invariant (BETATI; Eq. , ) distributions. Here, they are expressed as probability density functions in terms of the normalized age-ranked storage PS(T,t) (–), also known as fractional SAS functions (fSAS): 7ω(PS(T,t),t)=k⋅(PS(T,t))k-1,8ω(PS(T,t),t)=k(t)⋅(PS(T,t))k(t)-1,9ω(PS(T,t),t)=(PS(T,t))α-1⋅(1-PS(T,t))β-1B(α,β).

The parameters k, α, and β determine the catchment's water-age preference for outflows, while B(α,β) is the two-parameter beta function. If k<1, or if α<1 and β>1, the system tends to discharge young water. If k>1, or if α>1 and β<1, the catchment preferably releases old water. The case of k=1 or α=β=1 describes no selection preference (i.e. complete water mixing). PLTV is characterized by k(t) varying linearly over time between two extremes, k1 and k2, as a function of the catchment wetness wi (–), i.e. wi(t)=(S(t)-Smin)/(Smax-Smin), where Smin and Smax are the respective minimum and maximum storage values over the entire period.

We tested the model with two spatial representation and two temporal interpolation methods for δ18OP to explore the impact of input tracer data on model performance, results, and uncertainty. To evaluate the effect of spatial representation, we firstly used single-point δ18OP measurements, which we refer to in the following as raw δ18OP. These measurements, obtained from , were taken at the catchment outlet. The selection of δ18OP at the outlet assumes a precipitation collector close to the stream gauge at the outlet, which is a common occurrence in many catchments for logistical reasons. Indeed, the outlet, where in-stream δ18O is sampled, serves as the location where all precipitation inputs across the catchment are integrated. For convenience, precipitation monitoring is also often conducted at or near the gauging station at the outlet. Secondly, we used spatially interpolated δ18OP with kriging based on multiple locations. The spatial interpolation was conducted in using raw δ18OP from 24 precipitation collectors spread over the larger Bode region and using altitude as external drift. In a further step, the kriged δ18OP data were weighted with spatially distributed monthly precipitation to obtain representative estimates for the study catchment. In our study, the kriged δ18OP resulted in slightly more negative values than the raw δ18OP from the catchment outlet (Fig. ) because of the inclusion of more depleted δ18OP values from locations with higher altitudes during the kriging process. By considering these two options for the spatial representation of δ18OP, we intend to assess the influence of spatial variability and uncertainty in the simulated outputs between two opposing cases i.e. raw isotopes representing the simplest approach and kriged isotopes derived from a more sophisticated method. While there are other possibilities for the spatial representation of δ18OP, our choice allows us to effectively address our research question regarding the effects on SAS models of tracer data in precipitation collected at a single location within the catchment or spatially interpolated from multiple sites.

SAS model results are sensitive to the choice of the temporal resolution of input tracer data, and a finer resolution is generally recommended to achieve a satisfactory level of detail . Additionally, a forward Euler scheme was employed to solve Eq. (), whose precision increases with high-frequency time steps. For these reasons, we reconstructed daily δ18OP estimates from monthly values with two interpolation schemes. First, we used a step function in which the values between two consecutive samples assumed the value of the last sample. Second, we used a sine interpolation due to the fact that δ18OP samples typically exhibit pronounced seasonal variations with more depleted values in winter than in summer (Fig. ). The sine-wave function has been used in several studies to describe temporal variation in δ18OP . The seasonal pattern of δ18OP values over a period of 1 year can be described as follows : 10δ18OP(t)=aP⋅cos⁡(2⋅π⋅f⋅t)+bP⋅sin⁡(2⋅π⋅f⋅t)+kP, where a and b are regression coefficients (–), t is the time (decimal years), f is the frequency (yr-1), and k is the vertical offset of the isotope signal (‰). The coefficients a and b were estimated by fitting Eq. () to monthly δ18OP values using the iteratively re-weighted least-squares (IRLS) estimation . In our study, we reproduced the daily frequency isotopic data through the estimated regression coefficients of Eq. (). Figure  displays the daily kriged and raw δ18OP values simulated via step function and sine interpolation; by employing step function and sine interpolation as techniques to reconstruct tracer data in precipitation, we aim to analyze the effects on SAS-based results from two relatively simple, rather opposing approaches: one focusing on individual measurements and the other on seasonality.

In this study, different scenarios were used to quantify uncertainty in the modeled results. We tested 12 setups composed of three SAS functions (PLTI, PLTV, and BETATI), two temporal interpolations (step and sine function), and two spatial representations (raw and kriged values) of δ18OP (Table ). For each setup, we performed a Monte Carlo experiment by running the model with 10 000 parameter sets generated by Latin hypercube sampling LHS;. Model parameters and their search ranges are shown in Table . A 5-year warm-up period (i.e. repetition of the input data) from February 2008 to January 2013 was performed to reduce the impact of model initialization. The period from February 2013 to May 2015 was used to infer behavioral parameters (i.e. parameter sets giving acceptable predictions) and, subsequently, to interpret model results. The initial concentration of δ18O in storage was set to 9.2 ‰, coinciding with the mean δ18OQ over the study period.

The informal likelihood of the Sequential Uncertainty Fitting SUFI-2; procedure was applied to account for uncertainty in the parameter sets and resulting modeled estimates. In SUFI-2, the uncertainty is represented by a uniform distribution, which is gradually reduced until a specific criterion is reached. In our study, we calibrated the values of model parameters until the predicted output matched the measured δ18OQ to a satisfactory level, defined by an objective function. As the objective function, we employed the Kling–Gupta efficiency KGE;, and once the criterion of KGE ≥ 0.5 was satisfied, we defined a set of behavioral solutions for each model setup. However, as the aim of this study is to investigate the impact of various sources of uncertainty on simulated outputs, rather than to determine the best model setup, we decided to set a fixed sample size and narrow down those solutions generated by SUFI-2 in the previous step. Setting a fixed sample size ensures the comparability of results across the setups, as different sample sizes could influence the uncertainty analysis (i.e. the greater the number of behavioral solutions, the wider the uncertainty band). By fixing the sample size, we can still meet the requirement of a minimum acceptable KGE value (i.e. KGE ≥ 0.5). In this study, we determined the final behavioral solutions by using a fixed sample size that corresponds to the best 5 % parameter sets and modeled results in terms of the KGE.

To assess the range of possible behavioral solutions and understand the level of uncertainty associated with it, we calculated the 95 % confidence interval (CI) derived by computing the 2.5 % and 97.5 % percentile values of the cumulative distribution in the parameters and time series of output variables . These percentile values represent the lower and upper bounds of the CI, respectively. In our experimental setup, the main output variables were the in-stream δ18O signature and backward median transit time (TT50, in days, i.e. the time it takes for half of the water particles to leave the system as streamflow at the catchment outlet). Time series of TT50 were extracted directly from daily TTDs (Eq. ) and used as a metric for the streamflow age. This was done because TTDs are typically skewed with long tails ; hence, the median is often a more suitable metric than, for example, the MTT, as it is less impacted by the poor identifiability of the older water components .

Modeled δ18O values in streamflow (δ18OQ) represented by the 95 % confidence interval (CI) in the ensemble solution are displayed in Fig. . The results reveal how the predicted δ18OQ values enveloped the measured isotopic signature by reproducing its seasonal fluctuations, with depleted (i.e. more negative) values in winter and enriched (i.e. less negative) values in summer. Although the behavioral parameter sets were able to capture the seasonal isotopic trend, they poorly reproduced the exact values; therefore, the ensemble simulations are characterized by a non-negligible uncertainty.

Figure  shows the distinct effects of the interpolated input tracer data and model parameterization on the simulated δ18OQ values. The step function interpolation generated an erratic isotopic signature in streamflow with flashy fluctuations (Fig. a–f). On the other hand, sine interpolation of δ18OP values yielded a smooth response in the simulated δ18OQ values (Fig. g–l). No significant visual distinction was found between kriged (Fig. a–c) and raw (Fig. d–f) δ18OP samples when the step function interpolation was used, except for a slightly larger uncertainty observed with raw δ18OP samples. Furthermore, when employing the sine interpolation and raw δ18OP values (Fig. j–l), the simulations overestimated the in-stream measurements in comparison with kriged values (Fig. g–i). Finally, distinct SAS parameterizations did not produce remarkable differences in the simulated δ18OQ values, although PLTV generally yielded simulations that better enveloped the measured isotopic signature (Fig. b, e, h, k).

Despite the differences in the predicted δ18OQ values, all simulations can be considered satisfactory given the KGE values ranging between 0.55 and 0.72 across all tested setups (Fig. ). This aforementioned model performance can be classified as intermediate to good . When considering the best fit, the combination of step function interpolation and raw δ18OP values performed best. Additionally, PLTV generally yielded slightly better KGE values than PLTI or BETATI when grouping the setups with the same spatiotemporal interpolation of δ18OP. Differences in the mean KGEs were statistically insignificant (t test with p values > 0.05) only between setups a–g, c–i–k, and j–l (Table ), as the mean KGE values were nearly identical; this largely agrees with the visual analysis (Fig. ). Contrarily, the differences in the mean KGE values of the remaining setups were statistically significant (p values < 0.05), indicating that the a priori methodological choices (i.e. interpolation techniques of δ18OP values and/or SAS parameterization) strongly impact the overall results. Nonetheless, this does not mean that we can clearly identify the most suitable setup, but there is a need to carefully analyze the multiple potential choices with respect to SAS parameterization and tracer data interpolations as well as to evaluate the uncertainty range in modeled predictions.

Ranges of the behavioral SAS parameters for the tested setups are summarized in Table S1 in the Supplement. Parameters for the SAS functions of Q (i.e. kQ, kQ1, kQ2, α, and β) were different across the setups, although they were generally relatively narrow and well identified. However, the behavioral parameters were better constrained when using the step function interpolation, as their 95 % CI was, on average, 34 % narrower than that provided by the sine interpolation, across all the SAS parameterizations. The parameters kQ1 and α were also better identified than kQ2 and β, as their 95 % CI was, on average, 56 % narrower, across all tested setups. Conversely, there was no clear difference in the parameter ranges when using kriged or raw δ18OP values. The evapotranspiration parameter (i.e. kET) was poorly identified in all setups, as any value in the search range provided equally good results. The initial storage (i.e. S0) was only partially constrained, as any value between 335 and 2895 mm was considered acceptable.

Figure  illustrates the 95 % CI of the behavioral solutions for the predicted median transit time (TT50). The results show that the model simulated largely different ranges of TT50 based on the tested setups. When using PLTI and BETATI (Fig. a, c, d, f, g, i, j, l), the 95 % CI was relatively stable with smaller fluctuations throughout the simulation period compared with PLTV (Fig. b, e, h, k). However, minor differences emerged across the simulated TT50 as a result of the distinct interpolation techniques used for δ18OP. The 95 % CI of TT50 was, on average, 37 % larger, across all tested setups, when using raw δ18OP (Fig. d–f and j–l) rather than kriged δ18OP (Fig. a–c and g–i). This was especially visible when the step function was used (Fig. a–f). Moreover, the sine interpolation generated a 95 % CI of TT50 that was, on average, 62 % narrower across all tested setups (Fig. g–l) with respect to the step function (Fig. a–f). These differences were more evident for high-flow conditions where the 95 % CI of TT50 showed a significant reduction. In addition, the behavioral solutions obtained with the sine interpolation (Fig. g–l) were more skewed towards shorter mean TT50 values, across all tested setups, than those of the step function (Fig. a–f).

Behavioral solutions obtained with PLTV revealed a similar pattern regardless of the interpolation employed (Fig. b, e, h, k). Nonetheless, there was a noticeable difference in the 95 % CI of TT50 under distinct flow regimes. During low flows and dry periods (i.e. summer and autumn), the time series of predicted TT50 showed large uncertainties ranging at most between 259 and 1009 d across the different setups (Fig. e). Conversely, during high flows (i.e. winter and spring), the 95 % CI was much narrower and varied at least between 126 and 154 d (Fig. h). The large 95 % CI and the notable differences across the tested setups highlight the sensitivity and, in turn, the uncertainty in the predicted TT50 to the model parameterization, temporal interpolation of input data, hydrologic conditions, and nonspatially interpolated δ18OP.

In general, the variability in the predicted TT50 was controlled by the hydrological state of the system (Fig. ). High-discharge events reduced the TT50 values, whereas low-flow periods were associated with a longer estimated TT50. This is expected, as streamflow during high (low) flows is dominated by near-surface runoff (groundwater) with shallow (deep) flow paths leading to a shorter (longer) TT50. Such differences were particularly visible with PLTV (Fig. b, e, h, k), as the exponent kQ(t) shifts the water selection preference over time as a function of the wet/dry conditions. This resulted in the variability in TT50 being more pronounced than that of PLTI and BETATI, whose SAS parameters for Q are constant over time.

SAS functions provided valuable insights into the catchment-scale water release dynamics. Figure  presents the behavioral solutions releasing water of different ages and also shows that the catchment generally experienced a stronger affinity for releasing young water (i.e. kQ<1, or α<1 and β>1), rather than old water (i.e. kQ>1, or α>1 and β<1). These findings are in agreement with other studies in the upper Selke catchment . Nonetheless, there were differences in the water release scheme when comparing various combinations of SAS functions and spatiotemporal interpolation techniques of isotopes. The use of PLTV resulted in a substantial number of solutions, approximately 50 % of all behavioral solutions, suggesting a preference for both young and old water. On the other hand, only a few solutions showed affinity for old-water release, and this was more prominent when using the sine interpolation, raw δ18OP values, and PLTI across all tested setups.

In this study, we characterized the TTD uncertainty arising from some significant and critical aspects for the SAS modeling. These aspects are also the most directly linked to the data interpolation and SAS parameterization that we explored in this work. The uncertainty analysis was carried out across the 12 tested setups corresponding to different combinations of spatiotemporal data interpolation techniques and SAS parameterizations. Our results show that the uncertainty (i.e. 95 % CI) of the simulated TT50 (Fig. ) was firmly dependent on the choice of model setup, as the 95 % CI was primarily sensitive to the type of SAS function, temporal interpolation, and spatial representation of δ18OP.

Uncertainty in the simulated TT50 differed considerably between time-invariant (i.e. PLTI and BETATI; Fig. a, c, d, f, g, i, j, l) and time-variant (i.e. PLTV; Fig. b, e, h, k) SAS functions; thus, a large sensitivity is associated with the choice of the SAS parameterization. For example, PLTI and BETATI explicitly assume constant water selection preference over time, as these functions do not consider the temporal variability in the catchment wetness. As a consequence, the resulting TT50 had a moderately stable 95 % CI with smaller fluctuations compared with those of PLTV. On the other hand, including an explicit time dependence in the SAS function strongly affected the 95 % CI of TT50. Notably, PLTV produced a wider 95 % CI during low-flow conditions, which can hinder the TTDs ability to provide robust insights into flow and solute transport behaviors in the study area during low-flow conditions. This highlights the need to further constrain PLTV with additional data, which could involve obtaining tracer data at a finer resolution or additional information on the evapotranspiration and initial storage. In addition, the exceptionally old flow components associated with a very large 95 % CI of TT50 might be a distortion of the actual TT50 values, which can usually be more reliably estimated using radioactive tracers than with stable isotopes . Hence, PLTV-based TT50 greater than the observed period (828 d) should be interpreted carefully. It is important to note that we discussed the fractional SAS (fSAS) functions in this study, but another form of the SAS functions, such as the rank SAS (rSAS) functions, may have different uncertainty. This is mainly due to the difference in how the storage is considered: fSAS functions are expressed as function of the normalized age-ranked storage, which is equal to the cumulative residence time, whereas rSAS functions depend on the age-ranked storage, which is the volume of water in storage ranked from youngest to oldest .

Likewise, the high-frequency reconstruction of δ18OP from monthly samples via interpolation created further uncertainty. The sine interpolation effectively captured the dominant features of the observed δ18OP, such as seasonality. Consequently, sine interpolation successfully reproduced the seasonal trend in in-stream δ18O, although simulations overestimated the measurements (Fig. g–l). On the other hand, sine interpolation poorly reproduced rainfall isotopes during short-term flashy events and missed detailed characteristics of the tracer dataset by smoothing the variability in the observed δ18OP (Fig. ). As a result, high values of δ18OP are underestimated, whereas low values are overestimated. It is critical to recognize these limitations when interpreting modeling results, as uncertainty in the simulated δ18OP may conceal a more pronounced hydrological response of the system . Contrarily, step function interpolation preserved the maxima in the monthly observed δ18OP values by capturing their variation correctly (Fig. ). Simulations showed a better fit with measured in-stream δ18O (Fig. a–f) and higher model performance (Fig. ). However, combining step function with raw δ18OP resulted in larger uncertainty in the simulated TT50 (Fig. d–f). This reflects the need for a comprehensive exploration of the uncertainty range, rather than relying solely on the goodness of fit. Overall, the choice between step function and sine interpolation largely affected the reconstructed input data (Fig. ), leading to significant differences in the simulated TT50 and associated uncertainty. It is important to note that alternative methods, such as generalized additive models GAMs;, offer other options for interpolating tracer data. We conducted further tests with the SAS model using a GAM to reconstruct both kriged and raw δ18OP using smoothing functions; this provides a more sophisticated approach than the intuitive methods used in this study. However, the results, available in the Supplement, show that while a GAM provided more detailed reconstructed input tracer data (Fig. S1), it did not significantly alter the SAS-based results (Figs. S2, S3) or yield any new insights or conclusions about uncertainty with respect to using step function and sine interpolation. Therefore, we conclude that, while highly resolved input data may seem appealing, they do not necessarily lead to substantial benefits for the SAS-based output, supposedly due to the conceptual simplifications in the SAS model.

The spatial representation of δ18OP values had limited impact on the overall pattern of simulated TT50, as the time series were comparable with both kriged (Fig. a–c and g–i) and raw (Fig. d–f and j–l) δ18OP. Nonetheless, the spatial interpolation of δ18OP from different locations resulted in a reduction in the uncertainty in the TT50, which was particularly evident with the step function (Fig. a–f). This difference may be attributed to the fact that the upper Selke is a large (mesoscale) catchment with a substantial gradient in elevation, and, as a consequence, measurement for δ18OP from only one location may be generally overly simplistic. This finding highlights the importance of considering not only the model performance (Fig. ; raw values with a step function interpolation produced higher KGE values) but also the uncertainty range in predicted TT50.

Finally, we found that the uncertainty was larger under dry conditions when lower flow and longer TT50 were observed. This was especially visible when using the time-variant SAS function (Fig. b, e, h, k). It might be due to the fact that, under wet conditions, there is a high level of hydrologic connectivity within the catchment , which results in nearly all flow paths being active and contributing to the streamflow. This, ultimately, may make TT50 values easier to constrain. Conversely, under dry conditions, when usually only longer flow paths carrying older water are active , water partially flows through a drier soil zone where flow is more erratic (i.e. flow directions and patterns can vary widely) as the conductivity is controlled by soil moisture. As a result, wet areas can be patchy and water flows preferentially at certain locations only, as opposed to spatially uniform flow through the soil matrix; this might make it more challenging to constrain older water ages. Similarly, found higher uncertainty in the simulated SAS-based median water ages during drier periods, potentially due to higher uncertainty in the total storage. Moreover, non-SAS function studies have observed major uncertainties and deviations from observations in lumped modeled results during low-flow conditions . This was primarily due to the lack of spatial variability in the catchment characteristics in lumped models, which is a critical factor controlling low-flow regimes in rivers.

The dissimilarities in the simulated TT50 across the tested setups underline the importance of accounting for uncertainty in model-based TTDs. The uncertainty analysis with SUFI-2 performed in this study was essential to best describe the parameter identifiability and bounds of the behavioral solutions of each output variable. Furthermore, our results highlight the importance of gaining tracer datasets of good quality (i.e. tracer data with a finer temporal resolution), considering the spatial variability in the isotopic composition in precipitation, and, possibly, employing a model parameterization that best describes the catchment-specific storage and release dynamics. The last point can be defined according to a precise conceptual knowledge of the catchment’s functioning and information from previous studies in similar catchments.

Our results provide visually plausible seasonal fluctuations in the predicted δ18OQ samples (Fig. ) and satisfactory KGE values (Fig. ), despite the uncertainty arising from model inputs, structure, and parameters. The good match with observations provides confidence in the simulated TT50 for the upper Selke catchment. The magnitude of the uncertainty resulting from different setups cannot be generalized, but the overall approach for uncertainty assessment presented here could be extended to other areas and TTD studies. However, we recognize some limitations and indicate below possible reasons and, in turn, improvements that future work could achieve.

First, the limited length of the δ18O time series might not describe the system accurately; hence, implementing longer time series could improve the parameter identifiability and provide a more accurate estimation of the TTDs. Second, this study relied on stable water isotopes, which might underestimate the tails of the TTDs , although recent works have contested this . Possible advancements could be reached by using decaying tracers varying over a longer timescale than stable water isotopes e.g. tritium; and imparting more information on old water. Next, future work should retrieve more information on the evapotranspiration ET and initial storage S0, whose parameters were poorly identified. However, this issue is common in transport studies that rely on measurements of in-stream stable water isotopes . As a way forward, information on the ET isotopic compositions might help better constrain ET parameters and assess their affinity for young/old water. Regarding constraining the range of S0, further information can be gained from geophysical surveys in the study areas or groundwater modeling as well as by using decaying isotopes .

This study characterized the uncertainty in TTDs, which summarize the catchment's hydrologic transport behavior and, therefore, comprise decisive information for water managers. The value of TT50 has relevant implications for both water quantity and quality, as does its uncertainty. The larger the 95 % CI in the simulated TT50, the greater the difference in the TT50 values, which, ultimately, implies distinct hydrological processes, water availability, groundwater recharge, and solute export dynamics .

For example, knowing the TTD and its uncertainty may be crucial for characterizing the catchment's response to climatic change . Considering the increasing severity of droughts in the past decades , a catchment with a shorter TT50 and a dominant release of young water might be more affected by droughts than a catchment with a longer TT50, which means that its stream is fed by relatively old water sources. Therefore, a short TT50 reveals a low drought resilience of the catchment and limited water availability, which could limit streamflow generation processes and change the in-stream water quality status during drought periods . Likewise, TTD uncertainty may affect the understanding of the water infiltration rate, hydrological processes, and aquifer recharge, as a shorter TT50 suggests that water is quickly routed to the catchment outlet rather than infiltrating deeply into the groundwater. Finally, TTD uncertainty can have an impact on the quantification of the modern groundwater age, i.e. groundwater younger than 50 years . According to , the correct identification of modern groundwater abundance and distribution can help determine its renewal , groundwater wells and depths most likely to contain contaminants , and the part of the aquifer flushed more rapidly.

Uncertainty in TTDs also impacts on assessing the fate of dissolved solutes, such as nitrates , pesticides , and chlorides . These solutes constitute a crucial source of diffuse water pollution in agricultural areas , as they are spread on the soil in large quantities, especially during the growing season. The exposure time of solutes with the soil matrix has strong consequences for biogeochemical reactions, such as denitrification in the case of nitrates . A short TT50 suggests that water can be rapidly conveyed to the stream network , with limited time for denitrification. This explains the elevated in-stream concentration and short-term impact of nitrate export compared with that of a longer TT50, which is typically associated with old-water release and a low nitrate concentration . Similarly, pesticide transport is highly affected by the TTD uncertainty, as a long TT50 suggests little pesticide degradation due to decreased microbial activity along deeper flow paths . In other cases, a shorter TT50 may limit the time for degradation, causing a peak in the in-stream concentration . Overall, a longer TT50 can delay or buffer the catchment's reactive solute response at the outlet . This creates a long-term effect of hydrological legacies and a continuous problem with diffuse pollution of nitrates and pesticides , which can persist in the catchment for several years. Finally, TTD uncertainties also play an important role in chloride transport, although chlorides are commonly known to be conservative . A short TT50 may indicate rapid chloride mobilization, whereas a long TT50 implies chloride persistence in groundwater; therefore, chloride accumulates and is released at lower rates, with impacts on the ecosystem functions, vegetation uptake, and metabolism .

Understanding the uncertainty in TTDs is crucial for the aforementioned implications. While previous studies have used only a specific SAS function and/or specific tracer data interpolation technique in time and space, here we show that there could be a wide range of different results in terms of water ages, model performance, and parameter uncertainty. This is due to the specific choice regarding SAS parameterization and tracer data interpolation. With this, we want to convey that uncertainty is omnipresent in TTD-based models, and we need to recognize it, especially when dealing with sparse tracer data and multiple choices for model parameterization. Therefore, we want to encourage future studies to explore these uncertainties in other catchments and different geophysical settings, with the final aim to investigate whether these uncertainties may affect the conclusions of water quantity and quality studies for management purposes.