Transit time distributions (TTDs) integrate information
on timing, amount, storage, mixing and flow paths of water and thus
characterize hydrologic and hydrochemical catchment response unlike any
other descriptor. Here, we simulate the shape of TTDs in an idealized
low-order catchment and investigate whether it changes systematically with
certain catchment and climate properties. To this end, we used a physically
based, spatially explicit 3-D model, injected tracer with a precipitation
event and recorded the resulting forward TTDs at the outlet of a small
(

Transit time distributions (TTDs) characterize hydrologic catchment behavior unlike any other function or descriptor. They integrate information on timing, amount, storage, mixing and flow paths of water and can be modified to predict reactive solute transport (van der Velde et al., 2010; Harman et al., 2011; Musolff et al., 2017; Lutz et al., 2017). If observed in a time series, TTDs bridge the gap between hydrologic response (celerity) and hydrologic transport (velocity) in catchments by linking them via the change in water storage and the varying contributions of old (pre-event) and young (event) water to streamflow (Heidbüchel et al., 2012). TTDs are time and space-variant and hence no TTD of any individual precipitation event completely resembles another one. Therefore, in order to effectively utilize TTDs for the prediction of, for example, the effects of pollution events or water availability, it is necessary to find ways to understand and systematically describe the shape and scale of TTDs so that they are applicable in different locations and at different times. In this paper we look for first-order principles that describe how the shape and scale of TTDs change, both spatially and temporally. This way we hope to improve our understanding of the dominant factors affecting hydrologic transport and response behavior at the catchment scale.

Since the concept of TTDs was introduced, many studies have reported on their potential shapes and sought ways to describe them with different mathematical models like, for example, the piston-flow and exponential models (Begemann and Libby, 1957; Eriksson, 1958; Nauman, 1969), the advection–dispersion model (Nir, 1964; Małoszewski and Zuber, 1982), and the two parallel linear reservoirs model (Małoszewski et al., 1983; Stockinger et al., 2014). Dinçer et al. (1970) were the first to combine TTDs for individual precipitation events via the now commonly used convolution integral. Amin and Campana (1996) introduced the gamma distribution to transit time modeling.

Early studies reported that the outflow from entire catchments is
characterized best with the exponential model (Rodhe et al., 1996; McGuire
et al., 2005). However, neither the advection–dispersion nor the exponential
model is able to capture the observed heavier tails of solute signals in
streamflow (Kirchner et al., 2000). Instead, the more heavy-tailed TTDs
created by advection and dispersion of spatially distributed rainfall inputs
traveling toward the stream can be modeled with TTDs resembling gamma
distributions (Kirchner et al., 2001). Likewise, tracer time series from
many catchments exhibit fractal

From the application of conceptual and physically based models we know that individual TTDs are highly irregular and that they can rapidly change in time for successive precipitation events (van der Velde et al., 2010; Rinaldo et al., 2011; Heidbüchel et al., 2012; Harman and Kim, 2014). If the early part of TTDs (mainly controlled by unsaturated transport in the soil layer) resembles a power law while the subsoil is responsible for the exponential tailing, the combination of those two parts can result in TTD shapes that are similar to gamma distributions (Fiori et al., 2009). In the field of groundwater hydrology there have been intense discussions on the tailing of breakthrough curves (e.g., on the issue of whether they follow a power law or not) (Haggerty et al., 2000; Becker and Shapiro, 2003; Zhang et al., 2007; Pedretti et al., 2013; Fiori and Becker, 2015; Pedretti and Bianchi, 2018). If disregarded, heavy tails can constitute a significant problem when using TTDs to predict solute transport because the legacy of contamination can be underestimated (not so much from a total mass balance perspective but when providing risk assessments for highly toxic pollutants reaching further into the future). Furthermore, a truncation of heavy TTD tails should be avoided, especially when computing mean transit times (mTTs) since they are highly sensitive to the shape of the chosen transfer function (Seeger and Weiler, 2014). Other complicating matters are special cases of bimodal TTDs that can be caused by varying contributions from fast and slow storages (McMillan et al., 2012) or from urban and rural areas (Soulsby et al., 2015). Apart from individual catchment and event properties, mixing assumptions also affect TTD modeling since certain TTD shapes are inherently linked to specific mixing assumptions (e.g., a well-mixed system is best represented by an exponential distribution, partial mixing can be approximated with gamma distributions and no mixing with the piston-flow model) (van der Velde et al., 2015).

A number of studies reported on the best-fit shape of gamma distributions
generally ranging from

Conceptual and physically based models have also been used to investigate
the (temporally variable) shapes of TTDs. Haitjema (1995) found that the TTD
of groundwater can resemble an exponential distribution while Kollet and
Maxwell (2008) and Cardenas and Jiang (2010) derived a power-law form and
fractal behavior adding macrodispersion and systematic heterogeneity to the
domain in the form of depth-decreasing poromechanical properties. Increasing
the vertical gradient of conductivity decay in the soil decreased the shape
parameter

Replacing transit time with flow-weighted time or cumulative outflow (Niemi, 1977; Nyström, 1985) erases a substantial amount of the TTD shape variation associated with the external variability. However, since a change in the inflow often causes both fluctuations along and also a rearrangement between the flow pathways (i.e., internal variability), flow-weighted time approaches are not able to completely remove the influence of changes in the inflow rate. Still, Ali et al. (2014), providing a comprehensive assessment of different transit-time-based catchment transport models (where they compare several time-invariant to time-variable methods), conclude that applying a flow-weighted time approach can indeed yield adequate results for predicting catchment-scale transport.

To summarize, soil hydraulic conductivity, antecedent moisture conditions
(storage state), soil thickness, and precipitation amount and intensity are
amongst the most frequently cited factors that influence the shape of TTDs.
Obviously, there is not one single property that controls the TTD shape.
Instead, the interplay of several catchment, climate and event
characteristics results in the unique shape of every single TTD. One
approach to deal with this problem of multicausality is the use of
dimensionless numbers. Heidbüchel et al. (2013) introduced the flow path
number

Since McGuire and McDonnell (2006) stated a lack of theoretical work on the actual shapes of TTDs, quite a diverse range of research has been conducted to approach this problem from different angles and has yielded fragments of important knowledge. However, what is still missing is a coherent framework that enables us to structure our understanding of the nature of TTDs so that it eventually becomes applicable to real world hydrologic problems. Already in 2010, McDonnell et al. (2010) asked how the shape of TTDs could be generalized and how it would vary with ambient conditions, from time to time and from place to place. This study sets out to provide such a coherent framework which – although not exhaustive (or entirely correct for that matter) – will provide us with testable hypotheses on how the shape and scale of TTDs change spatially and temporally. As Hrachowitz et al. (2016) put it, “an explicit formulation of transport processes, based on the concept of transit times has the potential to improve the understanding of the integrated system dynamics […] and to provide a stronger link between […] hydrological and water quality models”.

In this study we will make use of a physically based, spatially explicit, 3-D model to systematically simulate how different catchment properties and climate characteristics and also their interplay control the shape of forward TTDs. We test which TTD shapes are most appropriate for capturing hydrologic and hydrochemical catchment response at different locations and for specific points in time. Furthermore we will try to interpret the results in the most general way possible, so that the theory can be extended to other potential controls of the TTD shape in the future. Our modeling does not explicitly include preferential flow within the soil and bedrock (like, for example, macropores or fractures), and therefore our TTDs mostly represent systems where water is transported via subsurface matrix flow coupled with overland flow. Still, the exclusion of these components can be considered legitimate and the results meaningful because of the important role that macrodispersion plays in shaping TTDs (Fiori et al., 2009). Hence, we consider our results the basis for further investigations approaching ever more realistic representations of the many hydrological processes taking place at the catchment scale.

We used HydroGeoSphere (HGS), a 3-D numerical model describing fully coupled surface–subsurface, variably saturated flow and advective–dispersive solute transport (Therrien et al., 2010). Groundwater flow in the 3-D subsurface is simulated with Richards' equation and Darcy's law, and surface runoff is simulated in the 2-D surface domain with Manning's equation and the diffusive-wave approximation of the Saint-Venant equations. The classical advection–dispersion equation for solute transport is solved in all domains. The surface and subsurface domains are numerically coupled using a dual node approach, allowing for the interaction of water and solutes between the surface and subsurface. The general functionality of HGS and its adequacy for solving analytical benchmark tests has been proven in several model intercomparison studies (Maxwell et al., 2014; Kollet et al., 2017), and its solute transport routines have been verified against laboratory (Chapman et al., 2012) and field measurements (Sudicky et al., 2010; Liggett et al., 2015; Gilfedder et al., 2019). Since our modeling approach entails subsurface flow only in porous media (no explicit fractures or macropores are included), the resulting TTDs have to be considered a special subset of distributions lacking some of the dynamics we can expect in real-world catchments while still providing a sound basis for further investigations (like, for example, adding more complex interaction dynamics along the flow pathways).

A small zero-order catchment was set up, 100 m long, 75 m wide
(

3-D model domain and shape of the virtual catchment from the top

Both the bottom and the sides of the domain were impermeable boundaries. A
constant head boundary condition (equal to the surface elevation) was
assigned to the lower front edge of the subsurface domain (nodes in the blue
square in Fig. 1), allowing outflow from both the bedrock and the soil. A
critical depth boundary was assigned to the lower edge of the surface domain
(on top of the constant head boundary) to allow for overland flow out of the
catchment. The surface of the catchment received spatially uniform
precipitation. We used a recorded time series of precipitation from the
northeast of Germany (maritime temperate climate: Cfb in the Köppen
climate classification) amounting to 690 mm a

The model runs were initialized with three different antecedent soil
moisture conditions

To investigate how different catchment and climate properties influence the
shape of forward TTDs we systematically varied four characteristic
properties from high to low values and looked at the resulting TTD shapes of
all the possible combinations (for a total number of 36 scenarios). The
properties we focused on were soil depth (

The four properties that were varied to explore their influence on
the shape and scale of TTDs: soil depth

We tested two soil depths

Metrics of the TTDs derived from the modeling of 36 scenarios with different combinations of catchment and climate properties. All times are given in days.

To complement the results obtained from the systematic variation of catchment and climate characteristics we tested the influence of seven other factors: (1) soil porosity, (2) bedrock hydraulic conductivity, (3) exponential decay in hydraulic conductivity with depth in the soil, (4) frequency of precipitation events, (5) soil water retention curve, (6) catchment shape and (7) effect of extreme precipitation after full saturation – conditions during which direct surface runoff may occur. These additional runs with altered soil properties and boundary and initial conditions were performed on the basis of some of the 36 initial runs (in the following sections we always indicate which runs form the basis of the specific scenarios; see also Table S2).

Notable catchment properties we did not test include topography, size, slope and curvature. Apart from investigating the effect of an exponential decay in soil hydraulic conductivity with depth we did not add heterogeneity to the subsurface hydraulic properties. Therefore we cannot make statements about how multiple soil layers or different spatial patterns of hydraulic conductivity would influence TTDs.

The influence of larger and smaller soil porosity was investigated with six
additional runs based on the three scenarios THDM, THIM and THWM. Three of
the additional runs had larger (0.54 m

Six runs were performed on the basis of the THDB scenario (which had a
bedrock hydraulic conductivity

Because all other model scenarios had a constant hydraulic conductivity
throughout the soil layer, we wanted to test whether the introduction of an
exponential decay in hydraulic conductivity with depth (from high
conductivity at the surface to low conductivity at the soil–bedrock
interface; see Bishop et al., 2004; Jiang et al., 2009) would have a large
influence on the TTD shapes. We based the conductivity decay test on four
scenarios (THDB, THWB, TLDB and TLWB), adding relationships of soil depth

Five time series with high precipitation event frequency and five time
series with low precipitation event frequency were created by means of the
rainfall generator used by Musolff et al. (2017) (Fig. 2b). It generates
Poisson effective rainfall (Cox and Isham, 1988) which is characterized by
exponentially distributed rainfall event amounts and interarrival times. The
mean interarrival time was set to 3 d and 15 d for the high-frequency scenarios (comparable to a humid precipitation distribution and
intensity pattern with lower intensities and more frequent events) and low-frequency scenarios (comparable to an arid precipitation distribution and
intensity pattern with higher intensities and less frequent events),
respectively. The total precipitation for all scenarios (both humid and arid
type) was 690 mm so that it matched our medium

All the base-case model scenarios were conducted with water retention curves
(WRCs) resembling silty soils:

In addition to the oval catchment we designed two more shapes to get an idea whether it would have a significant impact on the resulting TTDs (Fig. 1d). One of the catchments had the center of gravity located farther away from the outlet (top; 60 m) the other catchment had the center of gravity located closer to the outlet (bottom; 40 m). This increased the average flow path length from 61 to 70 m for top and decreased it to 55 m for bottom – while catchment length, area and slope stayed the same for all cases. The four additional runs we conducted were based on the scenarios THWM and THDM.

We tested these effects for two scenarios (THWB and TLWB) since both of
these scenarios were already close to creating overland flow. Full
saturation in this case means that the initial condition for these model
runs consisted of a fully saturated domain (both in the bedrock and in the
soil); i.e.,

We also tested to what extent the sequences of subsequent precipitation
events with different magnitude, intensity and interarrival time influence
TTD shapes. This was necessary to assure that our resulting TTD shapes were
not primarily a product of the point in time – within the sequence of
precipitation events – at which the tracer was applied to the catchment. To
this end 15 precipitation event time series were created by means of the
rainfall generator used by Musolff et al. (2017). The mean interarrival time
was set to 3 d (comparable to a precipitation distribution and
intensity pattern found in humid environments with low intensities and more
frequent events) and the total precipitation amount for all scenarios was
690 mm, matching our medium

The output data from HGS were mainly processed with Microsoft Excel. We
summed surface and subsurface flows, computed total tracer outflow from the
catchment, created the probability density and cumulative probability
density distribution for tracer outflow, calculated the metrics of the
forward TTDs, fitted theoretical distributions to our data and smoothed the
original TTDs for better visual comparability of the shapes. HGS keeps track
of the mass balance of inflow, outflow and storage and calculates the
discrepancy (mass balance error) between the three terms (Fig. S4). The mean absolute mass balance error for the 36 runs was
negligible (

The probability density distributions of transit time (the forward TTDs,
d

For each TTD we calculated seven metrics to characterize its shape: the
first quartile (

We fitted predefined mathematical probability density functions to the modeled data since condensing the main characteristics of an observed probability distribution into just one to three parameters of a mathematical function is appealing and eases the potential of transferability of the findings. Massoudieh et al. (2014) explored the use of freeform histograms as groundwater age distributions and concluded that mathematical distributions performed better in terms of their ability to capture the observed tracer data relative to their complexity. In order to determine which theoretical probability density function best captures the shape of our modeled TTDs, we chose two probability density functions that are commonly used to describe the transit of water through catchments (inverse Gaussian and gamma), as well as the less common log-normal distribution which also has just two adjustable parameters:

Inverse Gaussian distribution (a particular solution of the
advection–dispersion equation) with dispersion parameter

Gamma distribution with shape parameter

Log-normal distribution with standard deviation

Three-parameter beta distribution with shape parameters

Truncated log-normal distribution with the time of truncation

For visual examples of all five types of distributions please refer to Fig. S7.

The method of least squares was used to find the best fit between the modeled TTDs and the theoretical distribution functions (i.e., minimizing the sum of the squared residuals with the Solver function in Excel using one value for each of the 12 000 d of the modeled TTDs).

The fitting was performed on the cumulative probability distributions since their shape is not subject to the more extreme internal variability that probability distributions can experience.

Smoothing was only applied to enhance the visual comparability of the TTDs. All calculations were performed on the unsmoothed TTDs. For details on the smoothing method see Sect. S5 and Fig. S8.

The flow path number

The flow path number

Output from the model runs comprised subsurface discharge, overland discharge and tracer mass outflux in the discharge from which we derived TTDs (for an example see Fig. S9). Additionally, the model provided spatially and temporally resolved tracer concentrations throughout the entire domain. The differences emerging between the individual TTDs can be tracked by looking at the spatiotemporal evolution of the applied tracer impulse throughout the entire catchment. For a detailed example please refer to Sect. S6 and Fig. S10.

Changing the sequence of precipitation events affects the shape of TTDs to a
certain degree. In particular the timing and magnitude of the first
precipitation event determines how strong the early response turns out. This
can be observed in Fig. 5, where the different TTDs split up into different
branches according to the arrival and magnitude of the first event after
tracer application. However, following this initial split – with more and
more precipitation events taking place – all TTDs tend to converge towards
a single line. Examining the cumulative TTDs in Fig. 5 it is obvious that
the variability in the TTD shape introduced by different precipitation event
sequences is much smaller than the variability introduced by the other
catchment and climate properties. While the range of

15 TTDs resulting from 15 different precipitation time series with all other catchment and climate properties being equal. The first few events have the largest influence on the TTD shapes, while subsequent events gradually even out the differences. Inset shows cumulative distributions.

We found that

Influence of different properties on different parts of the TTD.
Shown is the average percentage decrease in transit time for each quartile
(

Dry

Results of the 36 model runs. TTDs are grouped by soil depth
(

High

Big

Decreasing

Influence of soil depth

The simulation results suggest that the TTDs can be visually divided into
four distinct parts (Fig. 7), where the shape of three parts is clearly
controlled by the catchment and climate properties and the fourth is a
transition zone. The shape of the initial part of the TTD (up to

Practical implications can be drawn from our results concerning, for example, pollution events. Some catchments are more vulnerable to pollution in the sense that they tend to store pollutants for a longer period of time and hence exhibit long legacy effects. In particular catchments with TTDs with heavy tails belong in that category (i.e., catchments with deeper soils and a moderate hydraulic conductivity difference between soil and bedrock). Also, certain moments in time are worse for pollution events to happen – a spill occurring during dry conditions will stay in the catchment longer than a spill during wet conditions because it is more likely to reach the bedrock and stay in contact with it before it is flushed out of the soils. Accordingly, locations and situations that lead to a longer storage of decaying pollutants will eventually result in the release of fewer solutes downstream.

We also plotted the probability density replacing the actual transit time
with the cumulative outflow to check whether this would eradicate the
differences between the different distributions (see Fig. S11). We made two interesting observations: (1) For the scenarios with
high

Shape parameters of the best-fit inverse Gaussian (

Shape parameters of the best-fit inverse Gaussian (

Then we included the two probability distributions with three parameters
(beta, truncated log-normal) into the analysis and investigated how they
compared to the two-parameter distributions. The performance of the beta was
quite similar to the one of the gamma in terms of representing TTD shapes
and the median transit times. However, it was able to capture the mTTs a lot
better than the gamma, even surpassing the performance of the inverse Gaussian on average (average deviation 4 %, 13 d, maximum deviation 38 d), especially in environments with low

Average and maximum deviations of mean and median transit times between the best-fit theoretical probability distributions and the modeled TTDs (given as the ratio of average deviation of the fitted distributions to the average modeled mean and median transit times as well as the average deviation in days). The sum of the squared residuals indicates the goodness of fit between the shape of theoretical probability distributions and modeled TTDs.

Gamma shape parameters (

Figure 9 gives an overview of the shape and scale of our modeled TTDs (using the best-fit gamma distribution parameters).

Figure 10 shows how the shape and scale of TTDs change with the individual
catchment and climate properties. For increasing

Change of gamma shape parameters (

Nonlinear regression analysis relating the shape and scale parameters of
the fitted log-normal and gamma distributions to any single soil,
precipitation or storage property (

Relationship between the dimensionless flow path number

The influence that soil porosity exerts on the shape of TTDs is quite
similar to the influence of

Overview of how certain catchment and climate characteristics
influence the shape of TTDs.

Variations in the saturated hydraulic conductivity of the bedrock

For catchments that already have highly conductive soils, adding a decay in

The shape of TTDs is not influenced significantly by precipitation frequency
since the mean values of all distribution metrics for the low-frequency
(arid type) and the high-frequency (humid type) scenarios are quite similar
to each other (Fig. 12d and Table S9). However, transit
times in the high-frequency (humid) environment are shorter (

The TTDs from the scenarios with sand-type WRCs have higher initial peaks
and lighter tails compared to the ones with silt-type WRCs (Fig. 13a and b).
Their transit times are consistently shorter over all distributions,
and the influence of other parameters (like

Overview of how certain catchment and climate characteristics
influence the shape of TTDs (continued). Panels

We observe unexpectedly little variation between the TTDs of the differently
shaped catchments (Fig. 13c). While

Starting runs with fully saturated soils increased the fractions of overland
flow for both the high and the low

The fact that TTDs under dry

Modeled TTDs for low

None of the theoretical distributions we tested captures the shape
of all of the observed TTDs adequately over the entire age range. On the one
hand, this is due to the misfit after the quite sudden break in slope at the
tail end of the distributions; on the other hand – and this is more
relevant from a mass balance perspective – it results from a
misrepresentation of the initial response. Looking at Fig. 7, 8, 12 and 13,
it becomes clear that all TTDs are humped distributions, with none of them
exhibiting an initial maximum (with a monotonically decreasing limb
afterwards) and none of them possessing a value of 0 after 1.5 min (the
first time step reported). Since all inverse Gaussian and log-normal
distributions start with a value of 0 and all gamma and beta distributions
are either monotonically decreasing or start with a value of 0 they cannot
be perfect representations of the modeled TTDs for porous media. Instead, a
set of probability distributions – with initial values larger than 0, a
rising limb to a maximum probability density and a falling limb with lighter
or heavier tails – would theoretically be the best option to represent
variable TTDs. We can confirm this expectation since the truncated
log-normal distributions we tested do indeed capture the modeled TTD shapes
best in most of our scenarios. Still they too are not able to reproduce the
break in the TTD tails we observed in the model output after which the tails
initially seem to follow a power law. This, however, does not constitute a
substantial problem with regard to the correct mass balance since these
heavier tails only comprise a very small fraction of the mass that was added
to the system as a tracer. Still, if the tailing of the TTDs is relevant to
a problem (e.g., when dealing with legacy contamination) one can add the
observed breaks in the tails to the distributions (for a description see
Sect. S7 and Fig. S7). As for the application of
three-parameter distributions, although the beta model performed better than
the two-parameter models overall (by a slim margin) we do not recommend
using it due to its additional fitting parameter (the upper limit

Further theoretical developments should include the use of TTDs for nonconservative solute transport. This could be achieved by considering the TTD a basic function to which different reaction terms can be added (like “cutting the tail” of solutes that decay after a certain time in the catchment or shifting, damping and extending the TTD for solutes that experience retardation). An example is provided for an exponential decay reaction in Sect. S8 and Fig. S12.

Predicted TTD shapes based on their relationship to the flow path
number

We can pretty accurately predict the general shape of a TTD within the
parameter range of our model scenarios using

A flow path number between

If the system receives more water than it can remove during

With increasing

If the system has the capacity to remove more water in the subsurface
than it receives during

When

The theoretical framework around the flow path number

This research can also contribute to the field of catchment evolution. One
could argue that in low-order catchments positive flow path numbers are not
sustainable over longer periods of time because that would mean that the
subsurface outflow capacity of the (zero-order) catchment is permanently
insufficient and the catchment is not capable of efficiently discharging all
of the incoming precipitation via the subsurface. Consequently, the
catchment storage would be filled up completely and overland flow would be
occurring on a regular basis. Since widespread overland flow is rarely
observed in most catchments it could be argued that most catchments have
already evolved towards negative flow path numbers (e.g., by increasing

For certain scenarios we still see differences in the probability
distributions if we replace transit time with cumulative outflow (see Fig. S11). This observation can be explained by the fact that
for the high

Our results can be considered valid for systems that do not experience a
large fraction of preferential flow in the soil and bedrock since we only
model flow taking place in the porous matrix of the subsurface domain. This
is the likely reason that we also encounter

Gamma distributions (solid lines) capture the middle part of the
modeled TTDs (dashed lines; thickness corresponds to

An interesting question that remains is whether backward TTDs can be linked to catchment and climate properties in a similar fashion to the one we used, since backward TTDs are comprised of many individual water inputs that entered the catchment over a very long period of time with potentially greatly varying initial conditions. That leads to the question of whether it is more important to know the conditions at the time of entry to the catchment or the conditions at the time of exit from the catchment (or both) in order to make predictions about TTD shapes and mTTs. Remondi et al. (2018) were among the first to tackle this problem by water flux tracking with a distributed model. They found that mainly soil saturation and groundwater storage affected backward TTDs.

In our simulations for a virtual low-order catchment we observed that the
shape of TTDs changes systematically with the four investigated catchment
and climate properties (

The shape of TTDs converges towards L-shaped distributions with high initial values if a catchment's capacity to store inflow decreases or if the actual inflow to a catchment does not equal its subsurface outflow capacity.

Heavier tails are produced when the system is in a more “relaxed” state,
where all potential flow paths (deep and shallow, slower and faster) are
equally used for transport. This is generally the case if

Gamma functions are able to capture the time variance of TTDs in an
appropriate way, especially for low

Ideally, this work will help to generate new or to expand existing hypotheses on hydrologic and hydrochemical catchment response that can be tested in future field experiments.

All data used in this study are presented either in the main paper or in the Supplement.

The supplement related to this article is available online at:

IH, PT and TF conceptualized the study. Formal analysis was carried out by IH. Funding acquisition was organized by JHF. The investigation was carried out by IH, AM, JY and JHF. JY edited the software. IH wrote the original draft of the paper, and further writing, reviewing and editing was performed by IH, AM, JHF, JY, PT and TF.

The authors declare that they have no conflict of interest.

We would like to thank Carlotta Scudeler for her guidance on hydrologic modeling and her contribution to a previous version of this paper. Thanks also to René Therrien for his help with the HGS modeling and to Ilja van Meerveld and Stefanie Lutz for excellent discussions of the manuscript. Finally, we would like to acknowledge the work of at least six anonymous reviewers that provided necessary criticism and valuable suggestions for improvement.

This research was supported by the Helmholtz Research Programme “Terrestrial Environment”, topic 3: “Sustainable Water Resources Management”, with the integrated project: “Water and Matter Flux Dynamics in Catchments”. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Nunzio Romano and reviewed by three anonymous referees.