4.1 Data and methods

Extensive information about the experiment including hydrological and tracer data can be found in Queloz, Bertuzzo, et al. (2015). In brief, the experiment was carried out at EPFL campus (CH) between 2013 and 2014 in a large (2.5 m-depth, 1.2 m²-diameter) vegetated lysimeter. The experiment was forced by controlled, non-stationary irrigation. Two small willow trees planted within the lysimeter triggered intense evapotranspiration fluxes that regularly exceeded 30 mm/day during warm summer days. Flow from the bottom drainage was discontinuous, with peaks following irrigation and no flow during dry days (Figure 2). A number of five fluorobenzoate (FBA) tracers were applied in Spring 2013 and showed different recoveries. We focus here on the tracer that had the most conservative behaviour (2,6-DFBA, 60% recovery). The tracer arrived at the bottom of the lysimeter over two distinct timescales that resulted in two different major peaks with duration of weeks to months (Figure 2). It is worth to recall here that the TTD is the normalised mass breakthrough curve (MBTC) and not the concentration breakthrough curve (CBTC). The MBTC is influenced by the variability of the hydrological fluxes and would resemble the drainage time series, with multiple individual peaks corresponding to high flow events. Yet, the CBTC better reveals the two longer-lasting peaks and we consider those as representative of contrasting transport processes occurring within the lysimeter. The major of the two peaks occurred through the summer (roughly 3 months after application) when flow was very discontinuous, while the other peak occurred at the beginning of the winter (roughly 6 months after application), when evapotranspiration became significantly lower. Although the second peak is smaller in magnitude compared with the first peak, it drained more tracer mass because flows were higher.

Tracer transport for this experiment was previously modelled by (Queloz, Carraro, et al., 2015), who used an approach based on SAS functions and conceptualised the lysimeter as two systems in series: the unsaturated soil (modelled through a power-function SAS) and the saturated gravel filter (modelled as a fully mixed compartment). In this study, we use one single control volume for the entire lysimeter system (unsaturated and saturated) and we test a new functional shape for the SAS function Ω _Q(P _S). We introduce the truncated normal SAS function (formula in S1.1), which has two parameters: m, corresponding to the mode of the distribution and s, which is proportional to the standard deviation of the distribution. Depending on the value of the parameters, the normal SAS function can be bell-shaped or not. Note that even a symmetric SAS function does not imply that the resulting TTDs are symmetric. As the parameters of the normal SAS function are strongly related to the function mode, it is an appealing candidate to explore multimodality. Thus, we generated a composite SAS function as the linear combination of two normal SAS functions (see section S1.1). The composite function allows for two distinct peaks and has a total of five calibration parameters (two for each normal distribution and one for the partitioning). For comparison, we also used a single normally distributed SAS function (two parameters) and a power-function (one parameter) for Ω _Q. The SAS function for the ET flux (Ω _ET) is assumed in all cases to be a uniform distribution over the younger storage fraction defined by the parameter u, as proposed by Harman (2015). The model further assumes that the tracer is conservative but only partially taken up by vegetation, through a parameter α ≤ 1 (see Section 3). This partial uptake of the FBA tracer by the willow trees in the lysimeter was revealed by Queloz, Bertuzzo, et al. (2015) who found small amounts of tracer in plant tissues, and whose breakthrough curves were not described well either by an absence of vegetation uptake, or by a complete vegetation uptake of the tracer. The simulation was run using tran-SAS (Benettin & Bertuzzo, 2018) and calibrated to measured tracer data through the DREAM _ZS algorithm (ter Braak & Vrugt, 2008; Vrugt et al., 2009). Calibration was based on the assumption that residuals were independent and normally distributed. Further details on the model and calibration setup can be found in Section S2. For each of the three tested SAS models, the best parameter combination (Table S1) was retained to obtain the instantaneous TTDs and compute the marginal TTD.

4.2 Results

The results of the tracer breakthrough simulations are shown in Figure 3 together with their corresponding SAS function Ω _Q. Breakthrough curves are discontinuous because of the frequent absence of flow during the dry summer months. The three curves all match the beginning of the early breakthrough in mid-May 2013. This 1-month delay since tracer application is obtained through SAS functions that, in all cases (see Figure 3 inset), are almost null up to P _S = 0.2, meaning that drainage does not comprise the 20% younger soil waters. The three simulations are also fairly similar at reproducing the first wide peak from June to October, with the power-function simulation less-variable and slightly higher than the tracer peak, but overall broadly comparable with measurements. The main difference occurs upon the second peak, which is entirely missed by the simpler models while it is captured by the double normal SAS function. The higher accuracy obviously comes at a cost (five SAS parameters instead of 1 or 2) but, besides a much better performance (see additional analyses in Section S2), the double normal model is more appropriate because it allows expressing the multimodality of the data. As a result, model errors are symmetric and much less correlated than in the other cases (Figure S2 and Table S2). The marginal TTDs resulting from the three models are illustrated in Figure 4. As expected, the marginal TTD resulting from the multimodal SAS function has two distinct peaks: the first at 90–100 days and the second at 200–250 days. By contrast, the marginal TTDs resulting from other SAS functions have a single and higher peak at 90–100 days and underestimate the older water contribution in the right tail of the distribution. These simulations ultimately tell us that in all cases, water found at the bottom drainage is representative of the older water stored in the lysimeter (Figure 3, inset). However, two major transport mechanisms emerge from the data and the simulations: some water is relatively fast and it is able to percolate during the dry summer season (c.f. first peak of the CBTC), likely due to the occasional activation of preferential pathways; the rest of the tracer mass moves with the matrix flow and takes more than twice as much to reach the lysimeter bottom (c.f. second peak of the CBTC).

5.1 Methods

The additional tracer concentrations were obtained from the following equation (similar to Equation 1):

(4)

where C _S(T, t) has a different expression depending on the considered solute (Section 3, and SI, section S3). These additional tracers have Damköhler numbers between roughly 0.5 and 6 for the considered catchment (ratio of MTT to kinetic constant, see Oldham, Farrow, & Peiffer, 2013; Maher & Chamberlain, 2014; Ameli et al., 2017), representing from reaction-limited to transport-limited tracer dynamics, respectively. The pesticide, with a low Damköhler number, will thus be highly sensitive to the accuracy of the model in rapidly transporting water to the outlet. Conversely, the weathering product, with a high Damköhler number, will be highly sensitive to the accuracy of the model in retaining water for a long time in storage until it nearly reaches chemical equilibrium.

5.2 Results

The unimodal gamma-distributed Ω _Q simulated the observed stream δ²H well, with a Nash-Sutcliffe Efficiency of NSE = 0.89 (Figure 5a). The model only failed to reproduce some of the rapid δ²H responses, especially at the beginning of wetter periods. The corresponding calibrated Ω _Q is similar to the old water component Ω₂ of the SAS function Ω^ref (Figure 5). As such, it clearly neglects the young water component Ω₁ ( S _T ≲ 200 mm), and it overestimates the old water fractions ( S _T ≳ 600 mm) compared with Ω₂. This is also visible in the corresponding marginal TTDs (average TTD over 100 years) (Figure 6), which have the largest differences over both younger ( T ≲ 0.2 year) and older ( T ≳ 1 year) water components. The observed differences between Ω^ref and Ω _Q did not have a large influence on the simulations of the stream δ²H. To assess the consequences of those differences for other tracers, we compared the simulations from those two models for the three additional solutes in the stream. The observed differences between the unimodal Ω _Q calibrated to δ²H and the bimodal Ω ^ref are well reflected in the simulations of these three tracers. Because of the mismatch in old water fractions ( T ≳ 1 year, Figure 6), Ω _Q constantly overestimates the concentration in weathering product (Figure 5b). The mismatch in intermediate ages (0.25 ≲ T ≲ 1 year) results in a slower middle-term response of Ω _Q to a change in the nitrate input (Figure 5c), with a 6 months delay to return to concentrations below the WHO guideline of 50 mg/L and about 2 years more to return to nitrate-free water. Finally, without a young water component ( T ≲ 0.2 years), the unimodal Ω _Q does not simulate the quick pesticide breakthrough (Figure 5d) nor does it simulate a long-term response because of the quick decay of the pesticide (τ_dec = 50 days). The differences in water age between the two models are quantified through some statistics of the marginal age distributions (Table 1). The unimodal Ω _Q results in a much larger mean travel time (MTT) (explaining the delay in middle-term response in nitrate), in a fraction of water younger than τ_dec 58 times lower, and in a higher fraction (13% more) of water older than τ _eq.

These results show that, despite the ability to simulate the observed δ²H, the calibrated unimodal Ω _Q fails to simulate other tracers. Overall, we can expect that a deviation of a model (here Ω _Q) from the ‘truth’ (usually unknown, but here Ω^ref) around a given age T will create a mismatch for all tracers whose stored concentrations vary with a time scale close to T. For instance, the unimodal approximation is reliable only for travel times close to 1 year (Figure 6, the curves cross around T = 1 year). δ²H in storage depends on the δ²H of precipitation, which essentially varies over a whole year. Thus, the unimodal approximation matches the time scale of δ²H dynamics, and the simulations of δ²H in the stream are accurate. For all the other solutes, the large differences between the unimodal approximation and the true age description exactly correspond to the time scales of the tracer dynamics in storage, and the simulations are wrong.

6.1 Relevance of water age multimodality for tracers

Our results confirm the ability of age-based lumped models to simulate tracer dynamics in discharge. Unimodal SAS functions only had one or two parameters but they provided satisfactory simulations with good performance measures. This is because they managed to capture the average tracer dynamics (e.g., as seen through a moving average) resulting from the range of travel times that characterises the majority of the discharge. For the lysimeter, 50% of the water left via the bottom drainage in less than 100 days, which is captured by the power function and the single normal function well enough (Figure 4). For the virtual catchment, 50% of the water left via discharge in less than 1 year, which is captured by the single gamma distribution well enough (Figure 6). Detailed characteristics of the tracer observations such as flashy responses associated with small discharge volumes (e.g., short-term dynamics within each peak in Figure 3, months 24–30, 12–16, and 52–56 in Figure 5), or secondary peaks (Oct–Jan in Figure 3) do not need to be reproduced to obtain satisfactory values of the standard objective functions used for calibration. Yet, these detailed characteristics represent transport processes acting at different time scales and they can be highly informative on the deficiencies of a too simple model. They could also be used as ‘soft data’ (Seibert & McDonnell, 2002) along with experimental knowledge to justify the selection of more complex age models accounting for more hydrological processes (as in Rodriguez & Klaus, 2019).

Composite SAS functions increased model complexity but in return gave more accurate process representation. For instance, the double normal model applied to the lysimeter experiment (Section 4) was able to capture the two main timescales at which the tracer was transported through the lysimeter, while the simpler models underestimated the impact of longer transit times (Figures 3 and 4). In some cases, the increase in performance could be insufficient to justify the use of additional parameters. For example, for the catchment-scale virtual experiment, the performance in δ²H of the single gamma model does not motivate the use of a more complex model. This highlights that a single tracer measured in inflows and outflows may not allow the identification of all peaks of the TTDs. Furthermore, the virtual catchment experiment showed that the simpler model, although efficient enough in reproducing δ²H, essentially failed to simulate the three other tracers. The unimodal approximation neglected the short term response of the catchment, it delayed the middle term response, and it overestimated the long-term response. This shows that simulations of solute chemistry require an accurate estimation of the TTDs that goes beyond conservative tracer input–output relationships, and that accounts for the potential multiple modes of transport associated with different age fractions (Benettin et al., 2017). On the other hand, this also means that stream chemistry data (Neal et al., 2013) and new available tracers (Abbott et al., 2016; Hissler, Stille, Guignard, Iffly, & Pfister, 2014; Pfister et al., 2017; Visser et al., 2019) can be used to validate TTDs deduced from a conservative tracer input–output relationship (Guillet et al., 2019). Overall, these results highlight the need for new methods that do not calibrate TTDs using tracer input–output relationships but instead measure the TTDs directly, such as (Kim et al., 2016; Kirchner, 2019).

Several characteristics shared by the two hydrological systems described in Sections 4 and 5 made the comparison between unimodal and multimodal models possible. Contrary to previous studies on the shape of TTDs and SAS functions, our work was not impacted by large unknowns such as total storage (Birkel, Soulsby, & Tetzlaff, 2011; Rodriguez, McGuire, & Klaus, 2018) and unknown past inputs and initial conditions (Hrachowitz, Soulsby, Tetzlaff, & Malcolm, 2011). In addition, there was no uncertainty related to hydrological and tracer data regarding the spatial homogeneity of inputs, the calculations of ET, the limited tracer sampling frequency (Hrachowitz et al., 2011; Kirchner, Feng, Neal, & Robson, 2004; Stockinger et al., 2016) and record length (McGuire & McDonnell, 2006). In addition, we avoided using several common modelling assumptions that may have a noticeable impact on the inferred age distributions. These assumptions include the no-flow boundaries defining the control volume to ensure the closure of the mass balance (related to the calculation of actual ET and/or deep water losses), and the ages of water used by ET. van der Velde et al. (2015) and Ali, Fiori, and Russo (2014) showed that the ET flux and its TTD influence the TTD of discharge. Further research is needed to improve our understanding of the role of the water ages abstracted by ET on the shapes of the discharge TTD (via the effect of Ω _ET on S _T or P _S hence p _Q).

6.2 Interpretations of the water age uni- or multi-modality

To understand the possible reasons for the multimodal character of TTDs not simply caused by time-varying inflows and outflows, it may useful to see the travel time T of an individual water parcel from a Lagrangian point of view (i.e., following the water parcel on its journey). The travel time can be expressed as T = L/v, where L is the total length of the flow path taken by that particle (from entry into the system to exit), and v is the harmonic mean of Lagrangian velocities of that parcel along the considered flow path of length L (see SI, section S4). The harmonic mean of velocities is used to obtain an average that appropriately summarises the entire history of Lagrangian velocities along the flow path in a single measure. We use this particular point of view as a simplification that helps us focusing on the role of flow path lengths and velocities in shaping the TTD, whereas potentially more complex Lagrangian representations of water transport are possible (Berkowitz, Cortis, Dentz, & Scher, 2006; Botter, Bertuzzo, Bellin, & Rinaldo, 2005; Cvetkovic, Carstens, Selroos, & Destouni, 2012; Davies, Beven, Rodhe, Nyberg, & Bishop, 2013; Sternagel, Loritz, Wilcke, & Zehe, 2019; Zehe & Jackisch, 2016). Using a similar approach, Kirchner, Feng, and Neal (2001) calculated the shape of TTDs for theoretical catchment geometries corresponding to various distributions of L [called width function w(x), with x the distance to the stream]. Seeing both L and v as random variables makes the TTD a ratio distribution (Rinaldo, Vodel, Rigon, & Rodriguez-Iturbe, 1995). Ratio distributions generally do not have simple analytical expressions (Curtiss, 1941), meaning that the shape of the TTDs can be complex, with potentially many peaks and often a heavy tail. In particular, if the distributions of L and v are multimodal for various physical reasons (see Section 2), it can be expected that the TTD is multimodal as well (see Figure 7, note, however, the implicit assumption of independence between L and v). This has been recently observed in a catchment in Switzerland where the extension and the contraction of the flow network for different wetness conditions creates complex patterns of flow paths leading to multimodal TTDs (assuming a constant, homogeneous velocity field) (van Meerveld, Kirchner, Vis, Assendelft, & Seibert, 2019). In reality, the shape of TTDs will depend on the relationship between L and v. In catchments, it can be expected that the location of a given input will determine L, but also v. For instance, in heterogeneous media, dispersion causes the distribution of v to broaden and flatten with increasing L (Le Borgne, de Dreuzy, Davy, & Bour, 2007). This can be interpreted as an increased probability with time that the advective velocity of a water particle has changed along its transport (e.g., as in Davies et al., 2013; Scaini, Amvrosiadi, Hissler, Pfister, & Beven, 2019; Zehe & Jackisch, 2016). Therefore, L and v are probably not independent, making the TTD, as the pdf of their ratio, difficult to estimate a priori (Curtiss, 1941). Consequently, even a complete knowledge of the distributions of L and v may not allow to determine the TTD as it was done in Figure 7 by assuming that L and v are independent.

At large scales (e.g., hillslope, catchment), L can be much larger than the scale of the heterogeneous properties of the porous media (i.e., different pore sizes, macropore network, rocks, roots, etc.). As a result, dispersion is strong and the differences in travel time T between the water particles coming from different parts of the catchment (with different L) can be attenuated. This may explain the general validity of unimodal TTDs in catchments (McGuire & McDonnell, 2006). For instance, Kirchner et al. (2001) found that low Péclet numbers (i.e., advection ≪ dispersion) are needed to explain the gamma shape of TTDs suggested by the spectral slopes of stream chemistry. At smaller scales, L can be small compared with the scale of heterogeneities (i.e., advection ≫ dispersion) such that the TTDs are more likely to be multimodal. For example, preferential flow in soils can cause a strongly bimodal distribution for v. In the lysimeter at EPFL (Section 4), flow paths are mostly vertical and the transport distance is probably narrowly distributed around L = 2.5 m (the depth of the soil column). The multimodal TTDs obtained in the lysimeter are thus due to contrasting velocities coming from preferential flow vs. matrix flow in the lysimeter (Figure 7a), which was clearly identified from comparisons of tracers measured in soil water vs. discharge (Benettin, Queloz, Bensimon, McDonnell, & Rinaldo, 2019). The importance of preferential flow in soils for catchment-scale TTDs may be difficult to evaluate. Glaser, Jackisch, Hopp, and Klaus (2019) showed in the Weierbach catchment (Luxembourg) that rainfall-runoff simulations were not improved by modelling vertical preferential flow at the catchment scale, using the soil preferential flow parameters deduced from plot-scale tracer breakthrough experiments. However, non-uniform lateral flow (similar to lateral preferential flow) appeared to be an important mechanism for discharge generation in this catchment, and it could be implemented in the model by using a highly conductive horizontal soil layer (Glaser et al., 2019). Different conclusions regarding preferential flow were found by van Schaik et al. (2014), who highlighted its non-negligible role for runoff simulations during extreme rainfall events in a semi-arid catchment in Spain. This suggests that preferential flow has an influence on a catchment TTD only if it provides sufficient volumes of water to the outlet.

Our results suggest that different hydrological processes generate multimodal TTDs only if these processes result in contrasting water ages and provide comparable volumes of water. For instance, if surface runoff reaches the stream in 1 hour, there may be no detectable peak in the TTD around T = 1 hour if 99% of discharge is generated by subsurface stormflow having water of at least 1 week of age. Similarly, if two recharge areas to a groundwater well provide similar water ages, the TTD of the well will be unimodal regardless of the sizes of each recharging area. Composite SAS functions provide an intuitive mathematical representation: multiple peaks are observed in the marginal TTDs if the different probability distribution functions summed together in the SAS function are far from each other in the age domain (T, P _S, or S _T), and if they are associated with weights λ (Equations S3 and S6) that are comparable (e.g., 25% and 75%).

6.3 Process knowledge and the shape of composite SAS functions

The characteristics (number of peaks, peak amplitudes, and age ranges) of multimodal age distributions can be deduced from experimental knowledge of hydrological processes and are useful to test hypotheses about these processes (Rodriguez & Klaus, 2019). In this case, a step-wise increase in the complexity of the age distributions may allow a better understanding of the complexity of the hydrological system enciphered in the tracer data. For instance, one may start with an age component for young water after detecting surface runoff in the catchment and mapping its extent from manual survey and/or aerial images. The characteristics of this age component may then be deduced from GIS analysis, using initially simple assumptions of flow paths and water velocities (c.f., van Meerveld et al., 2019). Spectral slopes of stream chemistry suggest that a gamma distribution with a shape parameter k < 1 may be an appropriate candidate for this young water component (Kirchner et al., 2000). However, a uniform distribution on a small age range may perform well for tracer simulations, and may avoid numerical issues due to the asymptote at age 0 of the gamma distribution with k < 1 (Rodriguez & Klaus, 2019). Older water components may then be deduced from estimates of soil water storage and age (based for instance on soil moisture measurements, hydraulic turnover time) and hydrogeological knowledge of the aquifer (no-flow boundaries, hydraulic conductivity, groundwater depth) for the groundwater component. The functional form of older components may be chosen among popular functions, such as the power function (Benettin, Soulsby, et al., 2017; Queloz, Carraro, et al., 2015) or the beta distribution (Danesh-Yazdi, Botter, & Foufoula-Georgiou, 2017; Kaandorp et al., 2018; van der Velde et al., 2015; Yang et al., 2018) to use P _S as a variable. The uniform distribution or the gamma distribution (Harman, 2015; Rodriguez & Klaus, 2019; Smith, Tetzlaff, & Soulsby, 2018) may be chosen to use S _T instead, which may reduce the uncertainties related to the usually unknown total storage.

On the other hand, composite SAS functions may more easily lead to over-parameterization and the related parameter uncertainties (hence age uncertainties) compared with simpler models. In some cases, it may not be possible to conclude that age multimodality exists in the investigated hydrological system (even if it is true) because the available data does not justify the use of more complex models (c.f., for the virtual catchment when considering only δ²H). This may happen in particular when the employed tracer data is at insufficient resolution or when its record length is too short to reveal the influence of contrasting water age contributions to the outlet. Furthermore, lumped models based on water ages often do not allow identifying hydrological processes on their own due to the ‘compression’ of information about the hydrological system. This compression is the consequence of the current inability of mathematical models to perfectly reflect all the perceptual information contained in process knowledge. A given composite TTD or SAS function can correspond to many configurations of processes (c.f., Figure 7a,c). This means that the details on the various hydrological processes aggregating to catchment response cannot be derived from water ages alone. Only precise knowledge of potential water flow paths or velocities (such as for the EPFL lysimeter or Biosphere 2 hillslope) may allow such inductive reasoning. Alternatively, such inferences may be possible by linking parts of the TTDs or SAS functions (e.g., components and age ranges) inferred from one tracer to different end-members and spatial contributions identified from other independent tracers. This could be done by sampling more than just catchment inputs (precipitation) and outputs (e.g., streamflow), and by using numerous tracers in End Member Mixing Analysis (Christophersen, Neal, Hooper, Vogt, & Andersen, 1990; Correa et al., 2017; Martinez-Carreras et al., 2015), or by using Bayesian Mixing (Evaristo et al., 2019; Parnell et al., 2013; Zhang, Evaristo, Li, Si, & McDonnell, 2017) when the number of tracers is limited.

6.4 Modelling complex hydrological systems with complex age distributions

The composite TTDs (Leray et al., 2016) and the composite SAS functions proposed in this study (see also Rodriguez & Klaus, 2019) have the ability to embed a larger variety of processes compared with traditional lumped models, but they are still analytical approximations of potentially more complex distributions. Shape-free distributions are more flexible and could better represent the range of transport processes in hydrological systems. Shape-free distributions can be determined from time series analyses (Turner & Macpherson, 1990), by deconvoluting the tracer output signal (Cirpka et al., 2007; Gooseff et al., 2011; Luo & Cirpka, 2008; McCallum, Engdahl, Ginn, & Cook, 2014), from Bayesian inference using several tracers (Massoudieh et al., 2012; Massoudieh, Visser, Sharifi, & Broers, 2014), or from multilinear regression analysis (Kirchner, 2019). However, these methods require a large number of tracers (Massoudieh et al., 2014), or tracer time series that are well conditioned for numerical reasons (Kirchner, 2019), and the resulting solutions can be highly non-unique (McCallum et al., 2014). Eventually, analytical solutions can perform better than shape-free solutions in some cases (Massoudieh et al., 2014). Composite TTDs and SAS functions could thus offer a good balance between model parsimony and sufficient representation of the relevant processes at the scale of interest, which is precisely the aim of lumped models.

Spatially explicit models allow a detailed representation of flow paths and transport media properties. These suggest that complex SAS functions and TTD shapes emerge from specific flow paths (Kaandorp et al., 2018; Pangle et al., 2017; Wilusz et al., 2020; Yang et al., 2018) or from various conductivity (or velocity) distributions in the system (Danesh-Yazdi et al., 2018; Davies et al., 2013). Distributed models also allow linking water sources to travel times, providing complementary spatial information that cannot be visualised in lumped models (Sayama & McDonnell, 2009; van Huijgevoort, Tetzlaff, Sutanudjaja, & Soulsby, 2016). However, these models usually require more detailed initial and boundary conditions (Delavau, Stadnyk, & Holmes, 2017; Jing et al., 2019), which get more uncertain at larger scales. In addition, these models are more computationally expensive than their lumped counterparts, and thus require efficient parallelised numerical solutions (Maxwell, Condon, Danesh-Yazdi, & Bearup, 2018). While we often do not have the means to validate the age results from these models, they are useful to explore the possible complexities of the age distributions and relate them to the dominant hydrological processes (Wilusz et al., 2020). This is especially true as these models can much better represent the small-scale mechanisms leading to the observed hydrological processes compared with lumped models, provided that sufficient experimental data is available to characterise well the associated small-scale heterogeneities. In particular, this means that contrary to lumped models, spatially explicit models allow identifying which hydrological processes may lead to complex TTDs and SAS functions.