3.1. Transient Travel Time Distributions and Stream Water Concentration
 In section 2, we have derived a realistic parameterization of the discharge and evapotranspiration STOP functions for the Hupsel Brook catchment from a spatially distributed groundwater model. Because of the long computation times of the groundwater model and its many parameters, the groundwater model itself is not suited to explore parameter sensitivity on travel time distributions and stream water concentrations. Moreover, the groundwater model is constrained by its Darcian groundwater flow approximation and thus excludes preferential flow processes, which are likely to be very important in quantifying travel time distributions [Beven, 2010]. The STOP functions derived from the groundwater model can, however, be used as a first approximation that accounts for the mixing caused by topography and subsurface permeability in simple catchment-scale water balance approaches. Such water balance models can more accurately calculate fluxes (including preferential flow) and combined with STOP functions, these models can calculate time-varying flux travel time distributions and concentrations with higher temporal resolutions than the groundwater model and for many more parameter combinations. If tracer concentration measurements indicate that preferential flow is important and a larger preference for young water is evident from measurements, the parameters of the STOP functions can be adjusted to account for the mixing effect caused by preferential flow processes. This last step, however, is not performed in this study. Our approach is summarized in 3 steps.
 1. We used modeled hourly time series of storage, discharge and evapotranspiration derived with the catchment-scale water balance model described in the work of Van der Velde et al. . They defined 500 parameter sets that described observed discharges of the Hupsel Brook equally well. We used all 500 parameter combinations, which has the benefit that we directly obtain an indication of the travel time uncertainties originating from water balance uncertainties.
 2. For each time step and each parameter set, we numerically solved equations (3)–(9), as we were unaware of the analytical solutions derived by Botter et al.  at that time. Obviously, our numerical and Botter et al.  analytical approach should yield similar results, but this was not tested. The numerical approach yielded 500 time series of flux travel time distributions for discharge and evapotranspiration. The STOP functions and were derived from particle tracking simulations for the Hupsel brook catchment as explained in the previous paragraph.
 3. Based on the discharge flux travel time distributions of step 2, we calculated concentrations of a fictive solute that undergoes the same processes as nitrate, i.e., diffusion and degradation. This relative concentration, , which has a value between 0 and 1, is calculated by a convolution of temporally varying travel time distributions with a constant relation between travel time and concentration. This is the simplest way of relating travel time distributions to stream concentrations for solutes that do not enter a catchment by rainfall:
 The constant relation between travel time and relative concentration is given by , which can be interpreted as follows: a water parcel with zero concentration falls as rainfall on a catchment. Along its travel path inside the catchment, this water parcel is likely to receive some nitrate by diffusion from other parcels that have been in the catchment longer and already have higher nitrate concentrations. The catchment-scale average rate of receiving nitrate by diffusion as a function of the concentration difference between the water parcel and water parcels in its surroundings is described by rd [T−1]. If this water parcel continuous to travel inside the catchment, the chance that it encounters an anoxic zone where nitrate denitrifies into nitrogen gas increases. The catchment-scale average rate of this degradation as a function of travel time is described by rn [T−1]. A concentration of 1 is never reached and can be interpreted as the average concentration water parcels have on locations that need zero travel time for a new water parcel to reach. However, on these locations new water parcels need diffusion time to equilibrate with their surroundings, thus never reaching a concentration of 1 (i.e., water parcels fall on locations with potentially high nitrate concentrations, but due to overland flow and preferential flow paths do not spend enough time in contact with the soil to take up much nitrate). Of course this constant relation between travel time and concentration is a huge simplification of all processes and spatial and temporal dynamics herein. It should be regarded as a pragmatic approach that allows us to isolate the effects of temporal variability in travel time distributions on stream concentrations. This works well for short periods during which the total amount of nitrate in the catchment and its spatial distribution do not change significantly. More details on the underlying assumptions of a constant travel time concentration relation can be found in the work of Van der Velde et al. [2010a]. Example values for rd and rn for nitrate are adopted from the results of Van der Velde et al. [2010a]: rd = 0.2 d−1, rn = 0.0025 d−1.
 Figure 4b shows the calculated resident and flux travel time distributions at four moments during a storm event in March 2008 (Figure 4a). The contribution of this particular storm event to the resident (pS) and flux travel time distributions (pQ) is indicated in light blue (Figure 4b). From the distributions in Figure 4b, we conclude that this storm event mainly contributed to discharge (pQ) and only little to storage (pS): at the end of the storm event, the contribution of water from this storm event (in light blue) to the catchment storage (pS) is small. The catchment was already wet at the onset of the storm, which resulted in a discharge flux that was almost equal to the precipitation input. The discharge STOP function describes a preference for discharging the younger water from the storage, resulting in a relatively large contribution of event water to the discharge following the rainfall. This is typical for a lowland catchment under wet conditions with large contributions of overland flow [Van der Velde et al., 2010b].
Figure 4. (a) The discharge peak of the Hupsel Brook following a rainfall event in March 2008. The gray band indicates the results of 500 parameter sets all describing measured discharge equally well [Van der Velde et al., 2011]. (b) The calculated resident travel time distributions (pS), flux travel time distributions of discharge, pQ, and their cumulative distributions, pS and pQ for a single parameter set. The blue colors indicate the contribution of water from this storm event to the travel time distribution. (c) The observed nitrate concentration at the catchment outlet is shown together with the calculated relative concentration, . The gray band indicates the results of all 500 parameter combinations. Note that they axis of the cumulative resident time distribution (pS) equals g.
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 However, had the same event occurred following a relatively dry period with low groundwater tables, rainfall predominantly would have been stored in the catchment instead of producing a high discharge [Brauer et al., 2011]. In that case, rain would have contributed far more to the resident travel time distribution, as can be seen from previous rain events that indeed contributed substantially to the overall storage and remained to do so for a long period (spikes in pS shown in Figure 4b). The type of mixing thus controls that storage in this lowland catchment mainly consists of water from rainfall events that occurred after dry periods (large summer storms or the first rainfall events in autumn). Recognizing that antecedent conditions control the fate of rain water, Figure 4 nicely illustrates the versatility of our approach to derive time-varying travel time distributions, by combining rainfall, evapotranspiration, storage and discharge with a subsurface mixing scheme that is determined by landscape and vegetation properties.
 The calculated relative concentration and measured nitrate concentrations during the storm event of March 2008 are shown in Figure 4c and this period is extended in Figure 5. The dynamics in observed nitrate concentration is adequately reproduced by the calculated relative concentration. Although only the dynamics of modeled relative concentration and measured nitrate concentration can be compared, the potential of describing water quality dynamics with relatively simple travel time-based models is evident. Vice versa, Figure 5 also shows that event-based surface water quality records contain a wealth of information on flow routes and mixing processes inside a catchment and it seems feasible to infer the shape of the STOP functions directly from time series of tracers. Conservative tracers that enter a catchment by rainfall seem especially suited for such analysis [Kirchner et al., 2003; Lyon et al., 2008; McGuire et al., 2005]. Hydrological research has only scratched the surface of inferring subsurface information from water quality records with a high temporal resolution. It is clear that combining hydrological water balance models with travel time approaches such as the STOP functions can greatly improve our understanding of hydrological pathways within catchments and thus advance our predictive capabilities of solute spreading within catchments.
3.2. Relations Between STOP Functions, Landscape and Average Travel Time Distributions
 The effects of discharge and evapotranspiration STOP functions on the travel time distribution of discharge are explored by calculating long term (1983–2010) average travel time distributions for four different combinations of STOP functions.
 1. Mixing according to the Hupsel Brook catchment, the STOP functions derived from Figure 3.
 2. Evapotranspiration from a perfectly mixed reservoir, discharge STOP function from Figure 3.
 3. Discharge from a perfectly mixed reservoir, evapotranspiration STOP function from Figure 3.
 4. Discharge from a reservoir with mixing as could occur in a sloping catchment based on results from Figure 2 and the evapotranspiration STOP function from Figure 3.
 For each of these STOP function combinations we calculated flux travel time distributions for discharge following the procedure as described in section 3.1. From these time series of flux travel time distributions we calculated the temporally averaged flux travel time distribution over the entire period. Figure 6a shows that changing the subsurface mixing scheme for discharge from a preference for young water (mixing according to the Hupsel Brook catchment) to a perfectly mixed reservoir, or mixing as could occur in a sloping catchment, greatly decreases the contribution of water with travel times less than a 100 days to its discharge. Exactly the contributions of these relatively short travel times affect overall solute concentrations most, because during these short travel times the concentration differences between different types of water inside a catchment are largest. This partly explains the observed large dynamics in stream concentrations of lowland catchments [Tiemeyer et al., 2008; Rozemeijer et al., 2010] compared to surprisingly constant stream concentrations observed in many sloping catchments [e.g., Kirchner, 2003].
Figure 6. The effect of catchment subsurface mixing schemes for (a) discharge and evapotranspiration on average flux travel time distributions for discharge and (b) cumulative travel time distributions for discharge. The legend shows four numbers for each mixing scheme: the first 2 numbers correspond to the a and b of the discharge STOP-function, the second set of 2 numbers corresponds to the a and b values of the evapotranspiration STOP-function. The gray bands indicate the uncertainty stemming from the 500 different parameter sets describing the hydrology (see also Figure 4).
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 From Figure 6 we see that changing the subsurface mixing scheme of evapotranspiration strongly affects the travel time distribution of discharge as well. As grass and maize are the dominant types of vegetation in the Hupsel Brook catchment, it is reasonable to assume that the relatively short roots predominantly extract the younger water from the overall storage. Deep roots of trees on shallow soils, on the other hand, may extract a much more mixed subsample of storage. Figure 6b shows that the perfectly mixed assumption for evapotranspiration would decrease the mean flux travel time of stream water in the Hupsel Brook discharge from 2.6 to 1.1 year. This result demonstrates that a correct subsurface mixing scheme (STOP function) for evapotranspiration is just as important as the subsurface mixing scheme of discharge, even if we are only interested in the travel times of discharge. Also, it shows the large effect vegetation can have on discharge travel times and thus on solute concentrations in streams, even when the amount of evapotranspiration stays the same.
 Furthermore, Figure 6b shows that the overall shape of the discharge travel time distribution is strongly related to the mixing scheme for discharge. A preference for discharging old water yields a much narrower cumulative distribution than a preference for young water. The mean of the travel time distribution, however, is hardly affected by the type of subsurface mixing for discharge: the effect of a discharge volume increase of young water on the mean travel time is countered by increasing travel times of the oldest water (i.e., the oldest water becomes older). This result implies that the mean travel time of a catchment contains very little information on the type of subsurface mixing.
 Although assuming a constant STOP function is a much weaker assumption than assuming a constant travel time distribution or a constant Age function to characterize subsurface flow in a catchment, still, it is likely that in many catchments the STOP function will change with storage or subsurface flow patterns (as shown for the hillslopes in Figure 2). Therefore, more research is needed to understand the behavior of STOP functions with changing flow patterns and to understand the relation between STOP functions and landscape and vegetation properties. If it proves feasible to parameterize the STOP function based on landscape characteristics, the STOP function may become an indispensable tool for solute transport modeling in data-poor regions, because of its small number of parameters and its flexible and intuitive use in combination with a water balance approach.