Spatially Distributed Characterization of Soil-Moisture Dynamics Using Travel-Time Distributions

Travel-time distributions are a comprehensive tool for the characterization of hydrological system dynamics. Unlike stream ow hydrographs, they describe the movement and storage of water inside and through the hydrological system. Until recently, studies using such travel-time distributions have generally either been applied to simple (arti cial toy) models or to real-world catchments using available time series, e.g., stable isotopes. Whereas the former are limited in their realism, 5 the latter are limited in their use of available data sets. In our study, we employ a middle ground by using the mesoscale Hydrological Model (mHM) and apply it to a catchment in Central Germany. Being able to draw on multiple large data sets for calibration and veri cation, we generate a large array of spatially distributed states and uxes. These hydrological outputs are then used to compute the travel-time distributions for every grid cell in the modeling domain. A statistical analysis shows the general soundness of the upscaling scheme employed in mHM and reveal precipitation, saturated soil moisture and po10 tential evapotranspiration as important predictors for explaining the spatial heterogeneity of mean travel times. In addition, we demonstrate and discuss the high information content of mean travel times for characterization of internal hydrological processes.

1. relating to comments of both reviewers, it remains unclear *how* pQ is constructed in your study.More specifically: which pQ does the reader actually get to see?Is it the age distribution of an incoming signal J that leaves a grid cell through *all* exit routes (i.e.Q and ET)? or is it only the age distribution of water leaving the unsaturated zone (Q and ET?) as I seem to understand from your reply No.8 to reviewer 2? In case groundwater storage was considered, how was the hydrologically passive storage accounted for (see e.g. the well known Fogure 1 in Zuber, 1986, Journal of Hydrology 86) -this is highly relevant for the age distributions as it exacerbates the dichotomy between celerity and velocity (McDonnell and Beven, 2014)!In addition, as you showed, mHM uses several soil layers.Is pQ a direct reflection of the total outflow from *all* soil layers?A further point is that only 5 years of warm up period were used.As the subsequently used "mean" age can be considerably sensitive to long tails, what is the potential of your estimates being biased towards too young water?In other words, even with 50 years modelling, is it conceivable that we miss some minor volumes of very old water here?These points need to be made much clearer -in the methods but also in results and discussion sections (particularily in the figures!).This first comment contains several distinct sub-comments.To address them appropriately, we will answer them individually.
(a) How is pQ determined.We are sorry for this misunderstanding.Due to the Editor's as well as the reviewers' comments, we are now aware that this crucial aspect was not properly explained in the earlier version of the mansucript.In the revised manuscript, we address these concerns by elaborating on the question what storages and fluxes were used for the analysis.It is now made clear in the Introduction, the Methods section as well as throughout the Results section (plus figure captions) that all our analysis was confined to the soil layer, only.In particular, we present the rationals for this decision both in the Introduction and further elaborate on its ramifications in the Methods section.(b) What about the soil layers.
The Editor is correct in his interpretation: we estimated the age distribution (TDDs) of water leaving the unsaturated zone (Q and ET).The groundwater part is not accounted in the analysis (see our elaborations above).(c) Warm-up period too short.
The problem of a warm-up period that might be too short is only relevant for backward TTDs.Most of the analysis in our study was performed with respect to forward TTDs.
In this case a similar problem exists with respect to a period of time at the end of the estimation period that may have to be discarded.We determined the length of this period by using the partitioning function θ as a measure.This quantity describes the ratio of a water parcel that is entering the CV at a given time and eventually leaves as discharge.Adding this value with the ratio of the water parcel that leaves as ET should always add up to one.We could therefore add up these quantities to determine how much of the time series we could use.Our investigation showed that only close to the last two years needed to be discarded from the analysis.In the revised version of the manuscript, we acknowledge this notion (see beginning of the Results and Discussion section in the revised manuscript).
2.Not everybody may be familiar with the details of mHM and of how the parameters are obtained in that model.It may therefore be important to clarify much earlier in the manuscript, how the soil moisture storage capacity is determined.Attentive readers will otherwise wonder if some of the storage is unaccounted for if the value is a "normal" calibration parameter (i.e.potential "passive" storage" that within you period of application does not become hydrologically active but may provide a mixing volume; see above and McDonnell and Beven, 2014).
With respect to the way the soil moisture storage is determined, we now address this issue early on in the manuscript (see the Introduction in the revised manuscript) and later we elaborate on it in the Methods section.We also expanded our explanation which parts of mHM are parameters determined during calibration and which are determined as model outputs (see the Methods section in the revised manuscript).In addition, we refer to Zacharias 2007, where important aspects of the calibration process are described in more detail.Finally, we discuss the difference between the water content relevant for outflow generation and for travel-time behavior in the Conclusion section of the revised mansucript.
3. I am missing a clear research question and/or research hypothesis this manuscript is looking at.Please add that at the end of the introduction.
We aggree that the manuscript in its previous form did not state concisly the main reserach questions and novel contributions of the study.From our current perspective, we would summarize the main questions as follows: (i) How are spatially distributed quantities, in particular land-cover, precipitation and soil type, impact travel-time behavior, (ii) how do different hydrological regimes impact travel-time behavior and (iii) what is the inter-connection between travel-time behavior and specific conceptualization of different hydrological processes and how may these connections be used for a better model calibration.These points are now better stated in the manuscript (see Introduction in the revised manuscript).
4. Note that figure captions should be stand-alone.In other words, the reader should be able to understand the figure by reading the captions alone.At the present the captions a very vague and uninformative.I would encourage you to be more specific and detailed.Only one example: what are the soil classes 9, 38, etc. in figure 14. what type of soils are we talking about?
We heeded the advise of the Editor and strongly revised the figures caption in the manuscript to make them more self explanatory (see figure captions in the revised manuscript).
In general, I would be glad if you could invest some more time in developing a more detailed and clear description of your methods and a stronger discussion of the limitations of the chosen approach (e.g.what about parameter uncertainty?uncertainty in age distributions?etc. and the effect on the interpretation) We thank the Editor for the for these suggestions.In the revised manuscript, we have now added more material to address these two main points.First, with regard to the methods, we have significantely expanded our description on how the states and fluxes were generated in mHM and how the TTDs were computed using these states and fluxes.This point was already addressed above.Second, we also added material to the Conclusions where we now ciritically assess the limitations of our approach.In particular, we discuss the different sources of uncertainty in traveltime estimation.These include uncertainties arising due to input data, simplified model structure and parametric uncertainty.With respect to input data, we refer to prior studies on the validety of the mHM input data.With respect to mHM model structure, we discuss the potential impact of our definition of soil moisture.However, to fully assess this aspect, a model-to-model comparison would be necessary.We are interested to pursue such a comparison with different candidate models being considered.The last point is parameteric uncertainty.Here we refer to the literature that exists on this topic with respect to mHM and discuss the main results from these studies and how they may relate to the estimation of TTDs.This last point is particularly interesting and a full study may be appropriate.These points are all discussed in the Conclusions section of the revised version of the manuscript.

Introduction
The description of storage and transport of both water and dissolved contaminants in catchments is a challenging subject due to the high heterogeneity of the subsurface properties that govern their fate (Dagan, 1989).This heterogeneity, combined with a limited knowledge about the subsurface, results in high degrees of uncertainty.As a result, stochastic methods are often applied, where the relevant processes are modeled as being random (Dagan, 1986;Rubin, 2003).Amongst these methods, a powerful tool is the use of travel-time distributions (TTDs ✿✿✿✿✿ TTD's), where storage and transport inside the catchment are modeled from a Lagrangian perspective (Rinaldo and Marani, 1987;Rinaldo et al., 1989).This means that the catchment itself or meaningful parts of it is treated as a control volume (CV).The spatially complex array of di erent ow paths inside such a CV is consequently ignored and only inlet and outlet uxes are used for the analysis (Botter et al., 2010;Rinaldo et al., 2011;Botter , 2012).This observation-based description of catchment dynamics makes TTDs ✿✿✿✿✿✿ TTD's a very robust tool.Although the application of TTDs ✿✿✿✿✿ TTD's ✿ goes back many decades (Danckwerts, 1953;Niemi, 1977), recent developments have strongly improved their theoretical foundations turning them into a versatile and coherent tool to characterize catchment dynamics (Bertuzzo et al., 2013;Benettin et al., 2015a;Rinaldo et al., 2015;Porporato and Calabrese, 2015).Owing to this progress, McMillan et al. (2012) and McDonnell and Beven (2014) have opined that TTDs ✿✿✿✿✿ TTD's should be used routinely for hydrological model calibration, a notion that has been picked up with tremendous speed (Windhorst et al., 2014;Vereecken et al., 2015;McGuire and McDonnell, 2015).Independently but somewhat parallel to that, Kitanidis (2015) has recently pointed out, that the key to subsurface characterization is to use all available information.From this information-centered perspective, using TTDs ✿✿✿✿✿ TTD's ✿ have several advantages.First, the travel-time behavior is controlled by di erent factors than the hydrograph response.Whereas the latter is relating rainfall-runo events the former is relating rainfall-runo water (McDonnell and Beven, 2014;Birkel and Soulsby, 2015).Second, spatially distributed tracer experiments may dramatically increase the information content available for catchment characterization (Birkel and Soulsby, 2015).
Here the advantage is that the data used for the analysis do not su er from model errors or other conceptual limitations.
However, such data are generally limited in amount (e.g., tracer or isotope time series limited to a few years only, although Hrachowitz et al. (2009) used time series of up to 17 years) and variety (only a limited number of data types are available).
As a result, such studies might fail to nd long-term trends, establish connections between travel-time behavior and speci c catchment properties or to investigate the impact of certain hydraulic regimes that are only rarely occurring (e.g., drought or extremely rainy months).In the second category, we nd theoretical studies, that either use a very simpli ed computational model to focus on speci c questions (Rinaldo et al., 2006;Du y, 2010;Botter et al., 2010;van der Velde et al., 2012;Benettin et al., 2015a;Porporato and Calabrese, 2015) or employ more realistic hydrological models that provide a large data set typically not available in real-world sites Sayama and McDonnell (2009);Fenicia et al. (2010);McMillan et al. (2012).Such theoretical studies allow a more thorough and detailed analysis of the involved processes and their interdependence but may su er from an oversimpli ed model setup for in-and out ux generation.
Our study falls into the latter category such that we use a hydrological model, i.e., the mesoscale Hydrological Model (mHM) (Samaniego et al., 2010a;Kumar et al., 2013a), to generate the uxes and states for the analysis.Using detailed data of precipitation, land cover, morphology and soil type as inputs, mHM is able to provide continuous simulations of spatially distributed uxes (e.g., groundwater recharge or evapotranspiration) and states (e.g., soil moisture) as outputs.By employing mHM, which is a spatially-distributed hydrological model, we are, however, able to extend these prior studies to a spatially-distributed travel-time analysis.This makes it possible to address several types of investigation.First, it allows for a comprehensive description of the ow and transport dynamics taking place in the catchment.The spatial distribution of such dynamics can then be related to e.g., land cover and physical properties of the soil as well as to driving forces like precipitation to determine dominant predictive factors.In addition, it allows to investigate how certain parametrizations of the mHM model are related to the travel-time behavior of the catchment.This opens the way for a more robust model calibration of hydrological models using additional datasets (McDonnell and Beven, 2014;Birkel and Soulsby, 2015;Kitanidis, 2015).
As a case study, we use a ca.1000 km 2 catchment in Central Germany for which detailed morphological and climatological data are available to parametrize mHM.In addition, the chosen catchment is the location of the Hainich Critical Zone Exploratory, a comprehensive monitoring network used within the Collaborative Research Center AquaDiva (Küsel et al., 2016).
AquaDiva seeks to elucidate the critical role of water uxes connecting surface conditions with biogeochemical functions in the subsurface.One of the goals of this project is to understand how far signal of surface properties, like land cover or land management, can be traced into the subsurface water and solute dynamics.Spatially explicit travel-time distributions are the perfect analytical tool to investigate such questions.To present our results on such questions, the rest of the paper is organized as follows: In Section 2 we describe the numerical and analytical tools used in this study.Thus comprises the framework of travel-time distributions, as applied in this study, as well as the relevant features of mHM.In Section 3, we present the results of our study and demonstrate how they relate to the questions raised above.Finally, in Section 4 we summarize our main ndings in light these questions and draw some conclusions.

By
In the following, we provide a short overview of the analytical and numerical tools and methods used in this study.We start by introducing the concept of travel-time distributions.In the following, we use the nomenclature as given by Benettin et al. (2015a) and the theoretical framework by Botter et al. (2010).In addition to that, we give a short overview of the numerical model (mHM) which was used for the calculation of the states and uxes.Finally, we introduce the catchment used in our study.

Travel-time distributions for a single control volume
Travel-time distributions are a stochastic description of the dynamic of a water parcel moving through a given control volume (CV).The de nition of such a control volume for real-world situation is often arbitrary to some extent (see e.g., the schematic in Figure 1).Within the context of this study, we used a spatially distributed model where the catchment is partitioned in regular grid cells (for more details see Section 2.2 below).Consequently, the boundaries of our CV were given by the grid cells of the model.Given that such a CV can be reasonably de ned, it is clear that the dynamics of a water parcel is determined by the inand out-uxes, that are changing the water content inside it.The time evolution of the water content S inside such a CV is then given by the following balance equation Equation ( 1) is a simple initial-value problem with the in-ux Q in (t) given by the e ective precipitation J(t) whereas the out-ux Q out (t) is given by evapotranspiration ET (t) and runo per grid cell Q(t).
To denote the di erent times involved in the dynamic of a water parcel, we followed the notation of Benettin et al. (2015a).
Chronological time was accordingly denoted with t, whereas the water parcel entered the CV at t in and left at t ex .At any given time t ′ in between these two points, any water parcel can therefore be characterized by two di erent properties; its age T A as well as its (remaining) life expectancy T E (see Figure 2).In their paper, Benettin et al. (2015a) emphasize the two interpretations that originate from these two points of view.Age on their derivation we also refer to Botter et al. (2010) and the references therein).In the following, we assume a uniform age function only.This means that the age distribution of the water leaving the CV is the same as the age distribution of the water inside the CV, i.e., no age preference of the out ow generating processes (discharge and ET) exists.This decision became necessary, since we could not yet draw on any data for the age distribution of water at the outlet of the catchment.As a result, we were not able to compare the predictions of di erent age functions to any measurements and therefore determine the most adequate description.In absence of such data, the most appropriate choice is the one involving the least amount of information, which is given by the assumption of uniform sampling.Using this assumption, we can state the following for the forward formulation with T E = t − t in , t > t in , i.e., the time from the moment the water parcel entered the reservoir until now.The function θ in Equation ( 2) is called the partition function (Botter et al., 2010(Botter et al., , 2011) ) and can be derived using the following formula This partition function describes the portion of the water parcel, entering the CV at t in , that is contributing eventually to discharge as opposed to evapotranspiration.It is consequently a dimensionless number between 0 and 1.
For the backward formulation, we can state the following with T A = t ex − t, t < t ex , i.e., the time from now until the moment the water parcel leaves the reservoir.
Both these formulations determine the travel time of the water leaving as discharge.The TTDs ✿✿✿✿✿✿ TTD's for the water leaving as evapotranspiration can be determined in an analogous way.

Numerical model
We used a spatially distributed, grid-based mesoscale Hydrological Model (mHM; Samaniego et al. (2010a); Kumar et al.
(2013a)) to generate the states and uxes needed for the TDD analysis described above.The model uses the grid cell as a primary hydrological unit and models the following dominant hydrological process: interception, snow accumulation and melting, root zone soil moisture dynamics, evapotranspiration, surface ow, inter ows, recharge and base ow.The total runo generated at each grid cell is routed to the neighboring downstream cell following the river network using the Muskingum-Cunge routing algorithm.Interested reader may refer to (Samaniego et al., 2010a) for further details on the model components.
The model code is open source and can be downloaded from www.ufz.de/mhm.The model has been successfully applied to a number of river basins across Germany, USA and Europe (Samaniego et al., 2010a, b;Kumar et al., 2010Kumar et al., , 2013a, b;, b;?;?;Thober et al., 2015; R An important and unique feature of mHM is its Multiscale Parameter Regionalization (MPR), that explicitly accounts for subgrid variability of basin physical characteristics such as terrain, soil, vegetation, and geological properties (Samaniego et al., 2010a;Kumar et al., 2013a).The model considers di erent levels of spatial resolution to better account for spatial heterogeneity of inputs, forcings and the modeled hydrological processes (see schematic in Figure 3).The smallest scale (called l 0 within the mHM nomenclature) is representing morphological factors, like elevation, soil type, land cover etc.On the other hand, meteorological inputs can be represented on a larger scale (called l 2 within mHM).The modeling of the hydrology is done on a third scale (called l 1 within mHM) that can vary depending, e.g., on catchment size or computational resources.Based on the MPR technique, morphological inputs are linked to internal model parameters (e.g., through the use of pedo-transfer functions) and a set of regional coe cients (or global parameters, γ).In a second step, the internal parameters are upscaled to the resolution of the hydrological processes, i.e., l 1 , using parameter speci c upscaling operators.Thus, MPR takes indirectly subgrid variabilities into account.The global parameters (γ) are space and time invariant and are inferred via a calibration procedure.mHM has 66 global parameters, which is a reasonable number for an optimization problem and is therefore able to avoid overparameterization.Further details on MPR can be found in Samaniego et al. (2010a); Kumar et al. (2013a).
Relevant to this study is near-surface and root-zone hydrological process, which are computed using di erent conceptualizations.In the upmost layer (x 3 in Figure 3) soil moisture ✿✿✿✿✿ water ✿✿✿✿✿✿✿ content is estimated using the in ltration excess ap- proach similar to the HBV model (Bergström, 1995), but enhanced to account for multiple soil layers.Within this layer ✿✿✿✿✿ these ✿✿✿✿✿ layers, the water is either percolating into deeper layers or evapotranspirates to the atmosphere.tion is estimated based on potential evapotranspiration, root water uptake and water availability in layer x 3 .In the second layer (x 5 in Figure 3) two di erent types of inter ow take place.Slow inter ow q 3 is implemented using a power-law model, whereas fast inter ow q 2 is triggered when a threshold value 10 In the third level (x 5 in Figure 3) base ow q 4 is generated using a simple reservoir model, i.e., However, due to the strong uncertainty typically associated with the estimation of the groundwater storage, we excluded this layer from our study.In the following all analysis in therefore con ned to the travel-time behavior of the soil only.
These runo generation processes are represented at every grid cell of mHM.The sum of direct runo q 1 (not used for the analyis), inter ows and base ow constitutes the grid speci c total runo which is then routed through a river network.

Study area and model set-up
In this study, we used a mesoscale catchment in Central Germany with a drainage area of approximately 1000 km 2 to the gauging station at Nägelstedt (see Figure 5).The catchment is the headwaters of the Unstrut river basin, and was selected in this study for its relevance to the Collaborative Research Center AquaDiva (Küsel et al., 2016).The terrain elevation within the catchment ranges between 170 m and 520 m with the higher regions in the west and south being the forested hill chain of the Hainich (see Figure 5).The forested area covers approximately 17% of the catchment, while 78% of the area is covered by crop/grassland.The remaining 5% is urban/build up area.The area is characterized by continental climatic conditions with a mean annual precipitation of approximately 660 mm and a mean temperature of approximately 8 • C. We established mHM over the study catchment and performed numerical simulations on several resolutions ranging from 200 m to 2 km.The model was forced using daily gridded elds of precipitation, air temperature and potential evapotranspi-5 ration.The point datasets for the precipitation and air temperature at several raingauges and weather stations located in and around the catchment were acquired from the German Meteorological Service (DWD).These point stations were then interpolated on regular grids using an external drift kriging interpolation procedure wherein the terrain elevation was used as an external drift (?) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ (Samaniego et al., 2013) .The potential evapotranspiration was estimated using the Hargreaves and Samani The model simulations were performed for the period 1950 to 2005.The rst ve years of the data were used to warm-up the model to acquire plausible initial conditions.We therefore discarded the rst ve years of simulations and the further analyses were performed using model outputs for the period 1955 − 2005.The model showed quite good performance with N SE > 0.8 for the daily discharge simulations at the Nägelstedt station.Other statistics like bias and correlations were also within a satisfactory range.To further validate our model prediction, we used measurements from a single Eddy Covariance measurement station inside the study area (see Figure 5).This comparison also showed a good agreement between both measurements and model prediction (see Figure 6).

Results and discussion
In this section, we present and discuss the results which have been derived using the methods described above.We will begin in the following by demonstrating and exemplify our general research procedure by virtue of a singly yet representative example.
The starting point for the derivation of soil travel times were the states and uxes as computed through mHM.Using the time series of soil moisture, evapotranspiration, inter ow and recharge, we used Equation ( 2 rainfall events that triggered the mobilization of older water stored within the soil.Another factor, although not apparent from Figure 7, was the water content, i.e., the state of the soil itself.As been demonstrated by Niemi (1977), soil response to rain events is strongly di erent between wet or dry conditions.
To disentangle this event-driven as well as state-dependent e ects from other factors that in uence the water movement in the soil, we averaged these time dependent distributions.As a result, we got the stationary TTDs ✿✿✿✿✿ TTD's for every cell with In all investigated cases, these stationary TTDs ✿✿✿✿✿ TTD's could be well approximated by an exponential-like behavior (see being generally wet and rainfall being evenly distributed throughout the year.Under these conditions the assumption of (quasi) stationary TDDs is reasonable (Tetzla et al., 2007;Hrachowitz et al., 2009).These stationary TTDs ✿✿✿✿✿ TTD's provided the basis for all following analysis, due to allowing the description of the average hydrological response of the catchment.In addition, we also focused on travel-time behavior under speci c hydrological regimes, i.e., wet and dry conditions, providing a more detailed understanding of the catchment.
For our statistical analysis, we used these stationary TTDs ✿✿✿✿✿ TTD's, which, due to their exponential-like behavior, can be characterized by its expected value τ .We will call this value mean life expectancy (or mean age respectively) in the following.
Estimating this value for every mHM cell provided a single measure for the travel-time behavior in the soil without the otherwise dominating impact of single precipitation events (see Figure 8).One feature that became immediately apparent were the long travel times in urban areas (see Figure 8 a).This can be explained by the fact that these areas are largely sealed, resulting in low in ltration rates and consequently low turnover rates inside the soil.To disentangle this sealing e ect from the soil behavior, we discarded cells inside urban regions from our analysis (see Figure 8 b).This allowed us to investigate the interplay between soil properties and travel-time behavior apart from such arti cial in uences.

Impact of modeling resolution
Due to its multiscale parameterization, mHM is able to model catchment dynamics at di erent spatial resolution with the same set of calibration parameters (see e.g., Samaniego et al. (2010a) or Kumar et al. (2013a)).Within the context of TTDs ✿✿✿✿✿ TTD's, this feature may be used to investigate the potential in uence of age-dependent out ow generation.The mathematical theory for including such age dependency has been developed independently by di erent groups and recently been uni ed using the umbrella term of StorAge Selection (SAS) functions (Rinaldo et al., 2015;Harman, 2015).These functions fully describe the sampling behavior of the catchment with respect to the age distribution of the stored water when discharge is generated.Discharge from a catchment may e.g., be primarily composed of younger or older water or it may show no preference to age whatsoever.SAS functions are therefore a concise mathematical representation of this behavior.
On a physical basis, such preference for di erent water age should be interpreted as the result of complex mixing processes taking place in the subsurface of the catchment (Botter, 2012;Benettin et al., 2013;van der Velde et al., 2012).In order to determine the appropriate SAS function for a given catchment, predictions using di erent functions could be compared with measurements.Alternatively, the form of the SAS function can be determined by using a physically based catchment model (Cornaton and Perrochet, 2006a, b).As already mentioned above, we could not directly infer, which form of a SAS function would be the most appropriate choice for our catchment.Instead we calculated the mean life expectancy for our catchment on di erent scales using the uniform SAS function.We motivated this choice by the principle of least information (or principle of maximum entropy) stating that amongst di erent alternatives, the one with the least amount of information should be chosen.Without any additional constraints, a uniform distribution is usually associated with maximum ignorance, therefore, motivating the use of the uniform SAS function.To estimate the possible in uence of this decision, we reasoned that a scale-dependent bias in the estimation of travel-time behavior would indicate the existence and possible strength of such an error.This is due to the multi-scale nature of mHM, where subgrid heterogeneity is taken into account by virtue of the Multiscale Parameter Regionalization.Using a smaller grid size would make this heterogeneity explicit and therefore reveal any possible unaccounted subgrid in uence.Results from our simulations showed no discernible di erences in the statistical distribution of mean life expectancy (see Figure 9).Using a smaller resolution had positive e ect on statistical estimation procedure due to the increase in data points.In addition, we saw more extreme values due to small scale features that were smeared out on coarser resolutions.Other than these two changes, we noted only minor changes in the statistics of mean life expectancy.We therefore concluded that, within the limits of the spatial scales tested here, mixing processes inside our catchment have no major impact on mean life expectancy.
We are aware, that this assessment is only covering one possible source of age-dependent out ow behavior and that other unresolved heterogeneity (at even smaller scales or due to other subsurface properties not accounted for in mHM) would in uence the out ow generation as well.We therefore regard our conclusions as tentative and open to revision once actual measurements become available.
However, our investigation gave us the ability to nd a good trade-o between computational costs and data amount for the following statistical analyses.We therefore used a data set from simulations using a grid size of 500 m.

Statistical analysis of mean life expectancy
The mean life expectancy τ of a water parcel inside a catchment is the result of a complex interplay of morphological and climatological factors.Several recent studies have therefore tried to determine their relative importance under varying conditions (McGuire et al., 2005;Cardenas, 2007;Broxton et al., 2009;Tetzla et al., 2009Tetzla et al., , 2011)).Contrary to these studies where eld measurements were used, we used results from computational simulations only.This gave us a much larger dataset, both in time and space, from which we could infer the relative impact of di erent factors, in particular meteorological (precipitation), land surface (land cover, leaf-are index) and subsurface (soil) properties.Notably, our approach di ers from Hrachowitz et al.
(2009) such that our analysis is based on model-derived gridded simulations of TDDs as compared to the observation-based basin-wise quanti cation of TDDs.
In the rst step, we determined for every cell the statistical relationship between the mean life expectancy τ and a number of potential predictors like average precipitation, soil depth, soil type or leaf-area index (LAI).Similar to Hrachowitz et al.
(2009), we used the coe cient of determination R 2 to quantify the strength of the statistical relationship.This quantity equals to one minus the ratio of the remaining variance vs. the total variance of the data themselves.It is therefore a measure of the variance explained by the model (which was always assumed to be linear in our study).

Precipitation
The analysis above showed the strong impact of precipitation on the event-based TTDs ✿✿✿✿✿ TTD's ✿ (see Figure 7).It is therefore to be expected to exert strong control on the steady-state TTDs ✿✿✿✿✿✿ TTD's as well.In our model two di erent quantities can be distinguished: rst, the precipitation itself as well as, second, the e ective precipitation.The latter value is here de ned as the water ux that is actually entering the soil, i.e., corrected by surface runo (through sealing), canopy interception and snowmelt.While the precipitation can be measured with high accuracy, it is the e ective precipitation that directly impacts the soil-moisture dynamics.
The scatterplot of both data sets against the mean life expectancy show a signi cant negative correlation between them (see Figure 10).This negative relationship can be explained such that precipitation events apply pressure to the water already stored in the soil.Instead of immediately traveling through the soil, the water from these events rather pushes older water out.Strong precipitation events therefore lead to a ' ushing out' of the soil and cause a shorter life expectancy.

Terrain elevation
In our next analysis, we used the physical elevation as a variable for our regression model.The height can simply be derived from the digital elevation model (DEM), which, in mHM, is represented using data obtained from the Shuttle Radar Topography Mission.
Using a scatter plot for visualizing the statistical relationship between mean life expectancy and the DEM showed a negative correlation (see Figure 11 a), i.e., longer life expectancy correlated with lower heights of the terrain, and with a linear coe cient of determination of R 2 = 0.668.Since no direct causal connection can be drawn between physical elevation and travel-time behavior, such a high value is indicative of underlying mechanisms.One of these is the aforementioned precipitation, since higher altitudes are correlated with stronger mean precipitation levels (linear coe cient of determination of R 2 = 0.812).Performing a multiple linear regression, including precipitation and saturated soil moisture (discussed below), showed strong correlation between these variables (data not shown).It therefore stands to reason to attribute potential causal e ects to these covariates, only.

Evapotranspiration
Evapotranspiration is directly in uencing the form of a TTD (see e.g., Equation 2).Consequently, we anticipated a strong correlation between mean evapotranspiration rates and mean life expectancy.
With respect to evapotranspiration, two di erent de nitions are typically distinguished: potential evapotranspiration (PET) and actual evapotranspiration (AET).As implied by its name, PET describes the maximum possible rate of evapotranspiration at a given site.This value is dependent on quantities like solar radiation and temperature that can generally be measured with good accuracy (Samani, 2000).Using theoretical models, good estimates can therefore be provided for PET at a given site (Almorox et al., 2015).On the other side, AET is a real quantity that can be measured.In principle, in situ measurements can therefore provide good estimates (e.g., the eddy-covariance method).In practice, however, exact measurements are hampered by a series of factors (Wang and Dickinson, 2012).As a consequence, PET can often be estimated with higher accuracy than AET.
Scatterplots of both PET and AET show a positive correlation between evapotranspiration and mean life expectancy in general (see Figure 12).This correlation is more pronounced for AET with a coe cient of determination of R 2 = 0.496 vs.only R 2 = 0.259 for PET.
Contrary to precipitation, which is an in ow mechanism, ET is an out ow mechanism.It is not pushing but rather sucking the water out of the CV, which explains the di erence in behavior of precipitation and ET.The lower relative strength of the correlation (compared to precipitation) can be explained such that ET is only one of the two out ow mechanisms (the other being discharge).The relative stronger impact of AET compared to PET was also anticipated.AET is directly used in Equation ( 2) for the calculation of TTDs ✿✿✿✿✿ TTD's, whereas PET is only coupled by virtue of an additional function.
As explained above, for real-world situations, better estimates can often be provided for PET.The higher explanatory power of AET has therefore to be balanced with its often less accurate estimate.Depending on the accuracy of measurements of AET, PET estimates may be a better predictor of mean life expectancy.

Land cover properties
Land cover is an important interface controlling the strength of incoming uxes through arti cial and natural sealing.In mHM, three di erent land-cover types are distinguished: forest, crop/grassland and urban area.As explained above, we excluded mHM cells inside urban area from our analysis in order to better focus on the soil properties themselves.To further elucidate possible in uence of the remaining land cover types, we separated the catchment into forest and crop/grassland and calculated the mean travel times separately.Estimating the PDF of the mean life expectancy for both land cover types separately, revealed strong di erences between them both in shape of the respective PDF and the range of values (see Figure 13).As shown above, results for the combined data set showed a distinct bimodal behavior (see Figure 9).In contrast to that, the PDF for both land cover types were almost unimodal.The most dominant peaks of every singly PDF coincided with the two peaks of the combined PDF.The behavior of the latter can therefore -to some degree -be considered to be a superposition of the former.
The relationship between these two land cover types was such that forest resulted in much shorter mean travel times compared to crop/grassland.This pronounced di erence may be partially due to a correlation with precipitation patters that have already been shown to exert a strong in uence on travel-time behavior.Forest in the study catchment (as well as in Germany in general) is disproportionately found in hilly and mountainous regions.These regions in turn show stronger precipitation values.The tendency depicted in Figure 13 may therefore be also caused by this covariate.However, this correlation between forested and high-precipitation area would not explain the distinct di erences between both land-cover types.Another factor, overlapping with the former, may be due to the di erences in water uptake.Trees are rooted into deeper soil layers compared to crop and grass and are therefore able to access a larger part of the subsurface water body.This larger access combined with the higher precipitation values as well as other factors would explain the almost non-overlapping travel-time behavior demonstrated in Figure 13.
In addition to this classi cation scheme, mHM uses the leaf area index (LAI) to describe land cover properties.The LAI describes the ratio of the cell that is e ectively covered by plant canopy.Due to the already established in uence on evapotranspiration (see above), it stands to reason to expect an in uence on the mean life expectancy as well.Comparing LAI class and land cover reveals a strong overlap between both (see Figure 13  Using the same approach as above, i.e., investigating the mean life expectancy for every LAI class independently, consequently revealed the same overall tendency for LAI classes compared to land cover types (data not shown).This was anticipated due to the aforementioned overlap between the two classi cation schemes.In addition, we saw little diversity for LAI classes within the same land cover class (data not shown).
However, this tendency was not present when using the actual leaf-area values associated with every LAI class.These values could be constant over the year e.g., in case of coniferous forest or vary strongly e.g., in case of deciduous forest.To make values from di erent LAI classes comparable, we averaged the respective values year-wise.A scatter plot of leaf area index vs.mean life expectancy does not show any strong correlation between the two with similar ranges of values being found for almost all LAI values (see Figure 14 (b)).This discrepancy can be explained by the implementation of the LAI in mHM.In contrast to the land cover type, that is used for the determination of ET processes in the upmost soil layer, LAI values are only used for interception and do consequently not directly in uence travel-time behavior.As a result, any possible relationship between LAI and TTDs ✿✿✿✿✿ TTD's ✿ is therefore biased and conclusions from our results must take into account this limitation critically.

Soil properties
An important input parameter in mHM is the soil type inside every cell.This property is implemented in mHM using the German soil data base Bodenübersichtskarte 1:1.000.000(BÜK 1000) (Federal Institute for Geosciences and Natural Resources (BGR),

1998).
Due to this relevance in the model, we anticipated a strong impact of the soil type in a cell on the resulting mean life expectancy.Estimating the PDF of mean travel times for every soil type individually, did indeed show signi cant di erences between them (see Figure 15).Soil classes found in the geographically lower regions of the catchment generally show longer mean travel times with a unimodal distribution shape, whereas soil types in the geographically higher regions correspond with generally shorter mean travel times with the shape of the distributions being less regular.This qualitative analysis reveals some overlap with the land cover distributions as well as mean precipitation rates.It is consequently not possible to directly infer causal correlation from statistical correlation.
In addition, the soil class is a symbolic variable, i.e., its values only indicate a certain type of soil but does not directly relate to any numerical quantity associated with this soil type.Consequently, we could not infer any quantitative connection between soil types and resulting travel-time behavior.To address this problem, we used the saturated soil moisture of the soil.This quantity is the amount of pore space per cell that can be potentially lled with water (porosity times the depth of root-zone soil layer).Its value is determined in mHM through pedo-transfer functions using the soil textural information on percentage of sand, clay and bulk density.Comparing these values in every single cell with the mean life expectancy shows a very strong statistical relationship with a coe cient of determination R 2 = 0.675.
The high correlation values of the saturated soil moisture can be explained by a mixture of causal and statistical factors.
On one hand, it is reasonable to expect the total amount of storage to be lled with water to have a signi cant e ect on the resulting travel-time bahvior.On the other hand, the soil tapes show a strong overlap with other factors like precipitation levels and land cove types that have already been discussed above.

Statistical analysis of mean age
As described above, the di erence between the forward and backward formulations of travel time has long been acknowledged (Niemi, 1977) and many studies have investigated their relationship (Cornaton and Perrochet, 2006a;Botter, 2012;Benettin et al., 2013;Harman, 2015;Benettin et al., 2015a).Both these formulations are linked by virtue of the so called Niemi relation age to elucidate connections and di erences between forward and backward formulations for our catchment.Visually comparing mean age (see Figure 17) and mean life expectancy (see Figure 8  Due to the mathematical and physical similarities, such a strong connection was anticipated.To further investigate possible origins of their respective di erences, we performed the same statistical analysis for mean age. To that end, we considered proxy variables that have already been shown to have a considerable impact on travel-time behavior.As demonstrated by the analysis above, these were precipitation (Pre), potential evapotranspiration (PET) and saturated soil moisture (SSM) as proxies for in ux, out ux and state respectively.Results showed overall the same trend for mean age and life expectancy with respect to these predictors (see Table 1).Precipitation was the most dominant factor for both quantities with the saturated soil moisture being a close second.This is in contrast to e.g., Benettin et al. (2015a), who emphasized the role of the out uxes for the time evolution of both age and life expectancy.In our analysis, we saw that proxy variables for in ux and state showed strongest correlations with mean travel-time behavior.On the other hand, PET, which is a good proxy for one of the two out uxes, showed only moderately strong correlations with said behavior.In case of mean age, this relationship was even weaker compared to the other two (precipitation and saturated soil moisture).Since we could not provide a proxy variable for the other out ux, i.e., discharge, we excluded this quantity from our analysis.

Joint impact of multiple variables on mean travel times
In the analysis above, the statistical relationship between mean travel-time behavior and a number of variables was presented and discussed.This was done for every variable individually to elucidate its possible impact on mean travel times.In addition to this simple analysis, we also investigated the joint impact of several variables.Such results can be of relevance for prediction, i.e., using a set of variables to predict travel times in a given CV.
To that end, we used the variables that had been shown to have the highest impact individually, i.e., precipitation, saturated soil moisture and potential evapotranspiration, and performed a multiple linear regression.Simple linear regression had already demonstrated that both precipitation and saturated soil moisture could explain a signi cant amount of the variability contained in the dataset.Combining these factors could therefore improve the predictability even further.We therefore applied Forward Stepwise Selection to generate a series of models with increasing complexity.The st single-variable model consequently used only precipitation as the variable with the highest single R 2 value.Next, the double-variable model used both precipitation and saturated soil moisture and the most complex three-variable model used precipitation, saturated soil moisture and potential evapotranspiration jointly.
Results for the default case, showed that, compared to using only one variable (precipitation), using two variables for the regression (precipitation and saturated soil moisture) improved the predictability of mean travel times (see Table 2).This was expected since both variables alone provided already high R 2 values.In addition, precipitation and saturated soil moisture did only show moderate correlation (R 2 = 0.451), so adding the latter variable added new information to the prediction model.
The correlation that existed between precipitation and saturated soil moisture is explained by the an orographic e ect, i.e., hilly regions in the catchment, with typically lower values of saturated soil moisture, also show higher precipitation values.In contrast, using three variables (precipitation, saturated soil moisture and potential evapotranspiration) resulted in almost negligible improvement (see Table 2).This is due to the already lower impact of PET compared to precipitation and saturated soil moisture.In addition, PET showed comparably stronger correlation with both precipitation and saturated soil moisture (data not shown), therefore adding only little new information compared to the other two variables.Such low impact of outgoing uxes compared to precipitation has already been reported before, for the case of synthetic toy models (Daly and Porporato, 2006).Moreover our results agree with the ndings of Hrachowitz et al. (2009), who also reported similarly strong explanatory power of climatic variables like precipitation as well as soil and land surface properties.

Impact of hydrological regime on travel-time behavior
The analysis above revealed the strong impact of the in ux (i.e., precipitation) as well as the state variable (i.e., saturated soil moisture) on the travel-time behavior.To further elucidate their impact, we investigated travel-time behavior independently for di erent hydrological regimes during the considered period of time, i.e., from 1955 -2005.To that end we partitioned the available time series into regimes based on soil moisture (state variable) and precipitation events (in ux).
In the rst case, we averaged the time series of mean saturated soil moisture in the whole Nägelstedt catchment for every year, i.e., 50 years in total.Next, we divided the resulting time series such that years with an average soil moisture content above 85th percentile of the time series were labeled as wet years.In contrast, years with an average soil moisture content below 15th percentile of the time series were labels as dry years.This annual partitioning was seen necessary due to the strong annual uctuations of this variable.Finally, we performed the same analysis as describe above for both -now smaller -datasets.
Using results from dry years only (see Figure 18), showed a similar qualitative travel-time behavior but strong quantitative contrast compared to the mean travel-time behavior discussed above (see Figure 8).Compared to the general case, mean life expectancy was much larger in dry years.In addition, dry years exhibit a wider range of possible values with the largest one (over 50 months) being almost 4 times a large as the smallest one (approximately 12 months).
Wet years on the other hand, exhibit a very small range with the smallest value (approximately 5 months) being roughly only half as large as the largest value (approximately 11 months) (see Figure 19).Compared to the general case, where the largest value (approximately 20 months) were roughly 3 times as large as the smallest value (approximately 6 months), these two scenarios fall on either side of this spectrum.This stark discrepancy demonstrates again the strong impact of the state variable (soil moisture) on travel-time behavior.Another di erence between the mean travel-time behavior in wet years and the general case is the unimodal distribution of the former.The analysis above revealed how the bimodal behavior is mostly due to the di erent soil types and therefore re ects the strong impact on this property on the overall soil-moisture dynamics.
The disappearance of this bimodal behavior is therefore re ective of how the soil becomes 'forced into line' when being lled Table 3. R 2 values for several predictors of mean travel time (as caused by wet and dry years).
In addition, results showed di erent statistical dependency of travel-time behavior with respect to precipitation, PET and SSM (see Table 3).Dry years showed very similar correlation values compared to the general case (see Table 1 a).On the other side, correlation values for wet years were remarkably smaller.
In the second case, we also investigated travel-time behavior depending in ux, i.e., in case of months having above average precipitation values (rainy months).To that end, we constrained our analysis to forward travel-time distributions which were triggered by heavy rain events.This means that, in analogy to the analysis above we only used months with precipitation values above the 97th percentile and performed again the same analysis for the reduced dataset.Results showed strong di erences in mean life expectancy during rainy months compared to the scenarios discussed above (compare Figure 20 with Figures 8 and 18).Compared to wet years, we saw even lower mean life expectancy.This can be explained by the strong impact of the rain on soil moisture leading to a ushing of the soil.We also saw a similarly small variance and a nearly unimodal distribution of mean travel-time values.In addition to that, we saw di erences for the statistical correlation of mean life expectancy with precipitation, potential evapotranspiration and saturated soil moisture (see Table 4).Compared to the standard travel-time behavior, precipitation was slightly less explanatory with mean life expectancy.This was caused by lower overall variation in precipitation values, due to constraining our analysis to large values therefore excluding low and medium range rain events.In contrast to that, R 2 values for PET and SSM increased.

TTD's for hydrological inference
The above results demonstrated the impact of certain soil properties, as implemented in mHM, on mean travel times using the R 2 metric as a measure.In addition to that statistical analysis, their relationship can further be elucidated by analyzing Equation ( 2) or (3).Assuming for example a very simple linear relationship for both Q and ET with respect to S we get for Equation (2) the following Equation ( 9) shows how under such simpli ed assumptions, the TTD of such a CV would follow an exponential distribution with its mean travel time being related to the recession constants α Q and α ET .As shown above, such an exponential behavior is visible in the mean behavior (see Figure 7  In addition to these di erences, we also saw di erent mean travel-time behavior for di erent regimes (see above).These di erences can be explained by the actual implementation of Q and ET in mHM, which is generally non linear (see Section 2.2).In order to assess the di erent roles of each soil process on discharge generation, we calculated the relative contribution of each out ow mechanism for each regime.The data in Table 5 show how much of the water that entered the soil during a given time and left eventually as discharge was leaving as base ow QB, slow inter ow QI s or fast inter ow QI f .On R 2 0.6059 0.6954 0.3619 Table 6.R 2 values for recharge vs. mean travel times for di erent regimes.
average, base ow contributed the most to discharge with fast inter ow having the smallest share.This overall distribution became stronger pronounced during dry years with base ow taking the largest share of out ow generation and fast inter ow becoming negligible.For wet years this trend is reversed, with water entering the soil during rainy months having an almost equal distribution.These di erent weighs show the relative impact and therefore the relative information content that traveltime distributions could contain, i.e., travel times in dry years are mostly the results of the successive processes leading eventually to base ow (see Figure 3), whereas travel times during storm events contain information on all discharge processes combined.
To further elucidate the relationship between the resulting mean travel times and certain model parameters, we performed a regression analysis comparing the recession constant for recharge with the mean travel times for di erent regimes.Results con rmed the relationship described above with mean travel times during dry years showing the strongest correlation (see Table 6).
Such a high interdependency between certain model parameters and data from di erent ow regimes is not unique for TTDs ✿✿✿✿✿ TTD's.Using discharge alone would reveal similar overall tendencies, e.g., discharge data from droughts is more informative for calibrating base ow recession constants.What is, however, new is the additional information content, which is not contained in discharge data alone.Not only can this improve calibration e orts, it allows the inference of additional system states.This is in particular relevant, but not con ned to, the total amount of stored groundwater.Discharge data are not sensitive to, and therefore not informative for, groundwater levels, but only to its relative changes.TTDs ✿✿✿✿✿ TTD's on the other hand, strongly depend on the total amount of water stored in every CV.Using both data types for inference would therefore allow to provide reasonable estimates of this quantity.Similarly, the estimation of water in the root and vadose zone can be improved.
In addition Birkel and Soulsby (2015) highlight the temporal aspects of travel times on model calibration.They point out, how the sampling frequency of the time series should match the expected travel times of the underlying process.Our results above revealed di erent time scales for di erent hydrological regimes, that di ered by almost an order of magnitude.Despite this heterogeneity, all travel times in our study remained in the range of months.Under such circumstances, a high resolution measurement campaign with daily or even hourly intervals would not be necessary.
Although the above explanations provide only a limited perspective on the relationship between TTDs The states and uxes, needed for the derivation of the travel times, were numerically computed using the mesoscale Hydrological Model (mHM), which was calibrated against 55 years of discharge data as well as using detailed data on soil properties, land cover and precipitation.We performed a statistical analysis of mean travel times to describe the soil response decoupled from the event-driven impact of precipitation.
Comparing the derived mean travel times for several modeling scales (spanning over one order of magnitude), we did not see any signi cant di erence in their distribution.This indicates the a ✿ general soundness of the parametrization scheme of mHM used for the calculation of the states and uxes on the di erent modeling scales.Our analysis shows that precipitation, saturated soil moisture and potential evapotranspiration are strong statistical predictors of mean travel time behavior.We also note that on average shorter mean travel times correspond to forested area and larger ones to crop/grassland, an observation that we linked to both correlations between forested and high-precipitation areas as well as the di erent water uptake mechanisms of trees vs. crop/grass.
We also investigated the travel-time behavior for di erent hydrological regimes, i.e., for dry and wet conditions (using soil moisture and precipitation as indicators).Our analysis revealed signi cantly di erent travel-time behavior for each of these regimes.Despite the strong heterogeneity of soil properties as well as (to a lesser extent) precipitation values, we could discriminate these regimes also in the resulting distribution of mean travel times.
Under dry conditions, we saw mean travel times having a pronounced bimodal distribution with long mean travel times and large variance.Such long travel times reveal the strong impact of base ow on the generated out ow, whereas the large variance shows the variety of soil responses under dry conditions.Such conditions are therefore suited for inferring soil properties relating to base ow generation.In addition, due to the large variance of soil responses, such conditions would allow to infer the spatial origin of solutes found in discharge streams.Such inferences are however, hampered by the long travel times involved.Not only are long time series needed, measurements must also being performed during such dry conditions.
Under wet conditions, we saw mean travel times having a unimodal distribution with shorter mean travel times and a smaller variance.This shorter travel times are caused by a larger in uence of the slow and fast inter ows on the total discharge behavior.As a result, TTDs ✿✿✿✿✿ TTD's derived under such conditions may be suited for inferring the parameters relating to these hydrological processes.
In case of rainy months, which overlap with wet conditions to a signi cant degree, we saw a similar distribution of travel times, but with even shorter mean values.This indicates a stronger impact of fast inter ow on the total discharge behavior.
Such information can therefore be valuable for improving the parametrization of the fast inter ow related processes.✿✿✿✿✿✿ having ✿ established a comprehensive description for the storage and release of water in the investigated catchment, the natural next step is the integration of reactive solute transport.As demonstrated by e.g., Botter et al. (2010), the concept of travel-time distributions can directly be adapted to account for the transport of both conservative and reactive solutes.This extension would facilitate to compare our predictions with the wealth of data that has been and continues to be collected within the AquaDiva center at the Hainich Critical Zone Exploratory (Küsel et al., 2016).
Thereby, we will be able to test our predictions by virtue of a large data set as well as initiate the collection of additional new data.
The shape of the SAS function is determining the preference of the uxes, e.g., discharge, for several ages of the water stored in the CV.In the backward formulation a at function would correspond to no preference with respect to age, a monotonously decreasing function would correspond to a preference for younger water and a monotonously increasing

Figure 1 .
Figure 1.Water movement inside a hill slope (physical schematic on the left and conceptual schematic on the right).
Figure 2. Interpretation ✿✿✿✿✿✿✿ Schematic ✿ of di erent times associated with ✿✿ the ✿✿✿✿✿✿✿✿✿ travel-time dynamics of a water parcel.✿✿✿ Age ✿✿✿ TA ✿✿ is ✿✿ the ✿✿✿✿ time ✿✿✿✿✿✿ elapsed is a backwards concept referring to the time passed since the beginning.The associated travel time distribution is therefore called the backward TTD.The concept of backward TTDs ✿✿✿✿✿ TTD's is of particular interest for the characterization of e.g., a water sample, since its composition is determined by the age of the water in the CV.Life expectancy, on the other hand, is a forward concept since it is referring to the time still left until exit from the CV.The associated travel time distribution is therefore called the forward TTD.Such forward TTDs ✿✿✿✿✿ TTD's ✿ are relevant e.g., for tracer test, since the concentration of an ideal tracer at the outlet is given by the TTD of its associated water parcel.In order to derive the TTDs ✿✿✿✿✿✿ TTD's associated with the forward and backward formulation, Botter et al. (2011) presented a derivation using only the states and uxes inside the CV as well as what they call an age function (for more information

Figure 3 .
Figure 3. Schematic of the mesocale hydrological model used in the study, depicting the di erent scales as well as the states and uxes represented in a single cell.

Figure 5 .
Figure 5. Left: Catchment (highlighted) used in the study shown within the larger con nes of the Unstrut catchment (area enclosed by continuous line).The larger rivers of the catchment are shown in blue.The colorbar shows the elevation (in m) of the study catchment.Right: Unstrut catchment within the larger con nes of Germany.The axis descriptions denote the latitude and longitude values.
Figure 7. Forward TTD ✿ of ✿✿✿ soil ✿✿✿✿✿✿✿ moisture ✿ with respect to mean travel time (in months) for a single cell within ✿ in ✿ the Nägelstedt catchment.
can be said that the strong interlink between the travel-time behavior and out ow generation indicates the high information content of the former with respect to the latter.As a result, travel-time distributions should be regarded as highly informative for the calibration of hydrological models.As mentioned in the Introduction,McDonnell and Beven (2014) TTD's for the parametrization of such models.The above presentations provide empirical support for this notion.Nägelstedt catchment by virtue of travel-time distributions.

Table 1 .
R 2 values for several predictors of mean travel time

Table 2 .
R 2 values of for several regression models of increasing complexity.

Table 4 .
R 2 values for several predictors for mean travel time (as caused by rainy months).

Table 5 .
Relative contribution of the di erent uxes to runo generation.