Introduction
The description of 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
. 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 . Among these methods, a
powerful tool is the use of travel-time distributions (TTDs), where storage
and transport in the catchment are modeled from a Lagrangian perspective
. This means that the catchment itself, or
meaningful parts of it, is treated as a control volume (CV). The spatially
complex array of different flow paths inside this CV is ignored and only
inlet and outlet fluxes are used for the analysis
. This observation-based description
of catchment dynamics makes TTDs a very robust tool. Although the application
of TTDs goes back many decades , recent
developments have strongly improved their theoretical foundations, turning
them into a versatile and coherent tool to characterize catchment dynamics
. Owing to this
progress, and have opined that TTDs
should be used routinely for hydrological model calibration, a notion that
has been picked up with tremendous speed
. Parallel to that,
has recently pointed out that the key to subsurface
characterization is to use all available information. From this
information-centered perspective, using TTDs has several advantages. First,
the travel-time behavior is controlled by different factors than the
hydrograph response. Whereas the latter relates rainfall–runoff events, the
former relates rainfall–runoff water .
Second, spatially distributed tracer experiments may dramatically increase
the information content available for catchment characterization
.
These advantages have led to a steady increase in both applied and
theoretical studies using TTDs for the description of catchment dynamics.
Applied studies here means that data from real-world sites are used
.
Compared to theoretical studies, the data do not suffer from model errors or
other conceptual limitations, but are often limited in amount (typically
limited to a few years only, although 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 find long-term trends,
establish connections between travel-time behavior and specific catchment
properties or investigate the impact of certain hydraulic regimes that only
occur rarely (e.g., extreme drought or storm events). The second category are
theoretical studies that either use a very simplified computational model to
focus on specific questions or employ more realistic hydrological
models that provide a large data set typically not available in real-world
sites . Such theoretical studies
allow a more thorough and detailed analysis of the involved processes and
their interdependence may suffer from an oversimplified model setup for
influx and outflux generation.
Our study falls into the latter category since we use a hydrological model,
i.e., the mesoscale Hydrological Model (mHM) , to generate the fluxes 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 fluxes (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 go beyond
prior studies to a spatially distributed travel-time analysis.
As a case study, we use a ca. 1000 km2 catchment in central Germany
for which detailed morphological and climatological data are available to
parameterize mHM. In addition, the chosen catchment is located in the Hainich
Critical Zone Exploratory, a comprehensive monitoring network used within
Collaborative Research Center AquaDiva . AquaDiva seeks to
elucidate the critical role of water fluxes connecting surface conditions
with biogeochemical functions in the subsurface. One of the goals of this
project is to understand how far signals 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 appropriate
analytical tool to investigate such questions.
By virtue of using the modeled data from mHM, we are able to address several
questions that have not been investigated before. First, how are spatially
distributed quantities, in particular land cover, precipitation and soil
type, impacting travel-time behavior in the soil? Unlike earlier model-based
studies, mHM is a spatially distributed hydrological model. We can therefore
improve the current knowledge by investigating the travel-time behavior for
every mHM grid cell and relate it to its geophysical and climatic properties.
Next, how do different hydrological regimes (wet vs. dry) impact travel-time
behavior in the soil? Here, we investigate the impact of changing external
conditions (meteorological factors) using the long time series of modeled
fluxes and states. Finally, what is the inter-connection between travel-time
behavior and specific conceptualization of different hydrological processes,
and how may these connections be used for further improvement of model
parameterization? Investigating the impact of model-specific
conceptualizations on the predicted travel-time behavior can provide a better
understanding of how actual measurement may be connected to certain model
parameters. For the quantitative analysis, we focus on soil moisture only;
i.e., we exclude groundwater. This is necessary due to the implementation of
groundwater in mHM as a linear reservoir. Although variations, i.e., fluxes,
of the groundwater level can be represented well , the
total storage remains uncertain. This is a common feature of hydrological
models and mHM is no exception. Furthermore, we consider this
restriction to be acceptable within the scope of our study, i.e., elucidation
of the spatio-temporal dynamics of TTDs. Groundwater by definition is far
less impacted by the spatial distribution of precipitation or land cover. In
addition, recently showed that TTDs show little
temporal variability compared to soil moisture.
To present our results on the above questions, the rest of the paper is
organized as follows: in Sect. we describe the numerical
and analytical tools used in this study. This comprises the framework of
travel-time distributions as applied in this study as well as the relevant
features of mHM. In Sect. , we present the results of our
study and demonstrate how they relate to the questions raised above. Finally
in Sect. , we summarize our main findings in light of these
questions and draw some conclusions.
Methods
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. To that end, we use the nomenclature as given by
and the theoretical framework by .
Then, we give a short overview of the numerical model (mHM) which was used
for the calculation of the states and fluxes. 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 CV. The definition of such a control
volume for a real-world situation is often arbitrary to some extent (see,
e.g., the schematic in Fig. ). 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 Sect. below).
Consequently, the boundaries of our CV were given by the grid cells of the
model.
Given that such a CV can be defined, it is clear that the dynamics of a water
parcel are determined by the influxes and outfluxes that are changing the
water content or storage. The time evolution of this storage S inside such
a CV is then given by the following balance equation:
ddtS(t)=Qin(t)-Qout(t)=J(t)-(ET(t)+Q(t)).
Equation () is a simple initial-value problem with the
influx Qin(t) given by the effective precipitation J(t),
whereas the outflux Qout(t) is given by
evapotranspiration ET(t) and runoff per grid cell Q(t).
To denote the different times involved in the dynamic of a water parcel, we
followed the notation of . Chronological time was
accordingly denoted with t, whereas the water parcel entered the CV at
tin and left at tex. At any given time t′ between
these two points, a water parcel can therefore be characterized by two
different properties: its age TA as well as its (remaining) life
expectancy TE (see Fig. ).
In their paper, emphasize the two interpretations that
originate from these two points of view. Age 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 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 are relevant, e.g., for tracer tests, since the concentration of an
ideal tracer at the outlet is given by the TTD of its associated water
parcel.
Water movement inside a hillslope (physical schematic on the left
and conceptual schematic on the right).
Schematic of different times associated with the travel-time
dynamics of a water parcel. Age TA is the time elapsed at t′
since the water parcel entered the CV at tin, whereas life
expectancy TE is the time remaining at t′ until the water
parcel leaves at tex.
To derive the TTDs associated with the forward and backward formulations,
presented a derivation using only the states and fluxes
inside the CV as well as what they call an age function (for more information
on their derivation, we also refer to , 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 outflow 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 different age functions to any
measurements and therefore determine the most adequate description. In the
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:
p→QTE,tin=Q(t)θtinS(t)exp-∫TEQ(t′)+ET(t′)S(t′)dt′
with TE=t-tin, t>tin, i.e., the time
from the moment the water parcel entered the reservoir until now. The
function θ in Eq. () is called the partition function
and can be derived using the following formula:
θtin=∫tin∞Q(t)S(t)exp-∫TEQ(t′)+ET(t′)S(t′)dt′dt.
This partition function describes the portion of the water parcel, entering
the CV at tin, that leaves eventually as discharge (as opposed to
leaving as evapotranspiration). It is consequently a dimensionless number
between 0 and 1.
For the backward formulation, we can state the following:
p←QTA,tex=J(t)S(t)exp-∫TAQ(t′)+ET(t′)S(t′)dt′,
with TA=tex-t, t<tex, 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 for the water leaving as evapotranspiration can be
determined in an analogous way and is not repeated here.
Numerical model
We used the spatially distributed, grid-based mesoscale Hydrological Model
(mHM; ) to generate the states and fluxes
needed for the TTD 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, soil moisture dynamics,
evapotranspiration, surface flow, interflows, recharge and baseflow. The
total runoff generated at each grid cell is routed to the neighboring
downstream cell following the river network using the Muskingum–Cunge
routing algorithm. Interested readers may refer to 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, the USA and Europe
.
Schematic of the mesoscale hydrological model used in the study,
depicting the different scales as well as the states and fluxes represented
in a single cell.
An important and unique feature of mHM is its multiscale parameter
regionalization (MPR), which explicitly accounts
for sub-grid variability of basin physical characteristics such as terrain,
soil, vegetation, and geological properties .
The model considers different levels of spatial resolution to better account
for spatial heterogeneity of inputs, forcings and the modeled hydrological
processes (see the schematic in Fig. ). The smallest scale
(called l0 within the mHM nomenclature) represents morphological factors
like elevation, soil type and land cover. On the other hand, meteorological
inputs can be represented on a larger scale (called l2 within mHM). The
modeling of the hydrology is done on a third scale (called l1 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 coefficients (or global parameters γ). In a
second step, the internal parameters are upscaled to the resolution of the
hydrological processes, i.e., l1, using parameter specific upscaling
operators. Thus, MPR takes sub-grid variabilities into account indirectly.
The global parameters 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
and .
Relevant to this study are near-surface and root-zone hydrological processes,
which are computed using different conceptualizations. In the top layer
(x3 in Fig. ), water content is estimated using the
infiltration excess approach similar to the HBV model ,
but enhanced to account for multiple sub-layers. Within these sub-layers, the
water is either percolating into deeper layers or evapotranspirating to the
atmosphere. Therefore, the root zone is characterized by effective parameters
for porosity, saturated hydraulic conductivity and the permanent wilting
point, which are estimated based on the pedotransfer functions of
. These effective parameters are estimated due to
transfer functions from the global parameters, which are determined during
the calibration process. Evapotranspiration is estimated based on potential
evapotranspiration, root water uptake and water availability in
layer x3. In the second layer (x5 in Fig. ), two
different types of interflow take place. Slow interflow q3 is
implemented using a power-law model, whereas fast interflow q2 is
triggered when a threshold value γTV is reached, i.e.,
q3=γsx5(1+α),q2=γfx5-γTV,ifx5>γTV0,otherwise.
In the third level (x5 in Fig. ) baseflow q4 is
generated using a simple reservoir model, i.e.,
q4=γbx6.
These runoff generation processes are represented at every grid cell of mHM.
The sum of direct runoff q1 (not used for the analysis), interflows and
baseflow constitutes the grid-specific total runoff which is then routed
through a river network. Interested readers may refer to
or http://www.ufz.de/mhm (user manual) for
further details on mHM.
As motivated in Sect. 1, we followed the concept of, e.g.,
and and divided the subsurface into two distinct zones:
the soil zone (called the root zone by , and shallow
storage by ) and the saturated zone (called
groundwater region by and deep storage by
). All analysis in our study was performed with
respect to the former. This was seen as necessary due to the large
uncertainties associated with storage estimation of the deeper regions.
Whereas mHM has been demonstrated to provide good estimates for soil moisture
, storage estimates for groundwater (x6 in
Fig. ) are far less reliable. Focusing on the soil zone only was
seen as justified since the focus of our study was the investigation of
spatially distributed factors like precipitation, land cover and soil type,
which have comparably little impact on groundwater dynamics. Furthermore,
recently demonstrated that travel-time behavior in the
deeper zone has comparably little temporal variability, too.
For computation of the TTDs according to Eqs. ()
through (), we used the combined estimates of layer x3
and x5 of mHM for the storage S (see Figs.
and ). For ET, we used the evapotranspiration fluxes from the
sub-layers of x3 and for Q we used q2, q3 and C.
Conceptually, the interflow is generated in the unsaturated zone
(reservoirs x3 and x5) within mHM. Thus, using the interflow as the
outflow from the unsaturated zone for deriving the travel times is a valid
assumption. Our delineation of shallow and deeper storage was therefore more
similar to than to .
Study area and model setup
In this study, we used a mesoscale catchment in central Germany with a
drainage area of approximately 1000 km2 to the gauging station at
Nägelstedt (see Fig. ). The catchment comprises the headwaters
of the Unstrut River basin and was selected in this study for its relevance
to Collaborative Research Center AquaDiva . The terrain
elevation within the catchment ranges between 170 and 520 m, with the
higher regions in the west and south being the forested hill chain of the
Hainich (see Fig. ). The forested area covers approximately
17 % of the catchment, while 78 % of the area is covered by
crop/grassland. The remaining 5 % is urban built-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.
States and fluxes as computed by mHM (see Fig. ) for the
derivation of TTDs using Eqs. ()–() (see
Fig. ).
Left panel: catchment (highlighted) used in the study shown within
the larger confines of the Unstrut catchment (area enclosed by continuous
line). The larger rivers of the catchment are shown in blue. The color bar
shows the elevation (in meters) of the study catchment. Right panel: Unstrut
catchment within the larger confines of Germany. The axis descriptions denote
the latitude and longitude values.
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 fields of
precipitation, air temperature and potential evapotranspiration. The point
data sets for the precipitation and air temperature at several rain gauges
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
. The potential evapotranspiration was estimated using
the method. Other data sets required to set up the model include
a digital elevation model (DEM) and derived terrain properties like slope,
aspect, flow direction and catchment boundary; soil and geological maps
provided by the Federal Institute for Geosciences and Natural Resources (BGR)
and metadata such as sand and clay contents, bulk density, horizon depths and
dominant hydrogeological classes; CORINE land-cover information for the
years 1990, 2000 and 2005 available from the European Environment
Agency (EEA); and runoff data for the catchment outlet provided by the
European Water Agency (EWA) and the Global Runoff Data Centre (GRDC).
The model simulations were performed for the period 1950–2005. The first
5 years of the data were used to warm up the model to acquire plausible
initial conditions. We therefore discarded the first 5 years of simulations
and the further analyses were performed using model outputs for the
period 1955–2005. The model showed quite good performance, with
NSE > 0.8 for the daily discharge simulations at the Nägelstedt
station. Other statistics such as 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 Fig. ). This comparison also showed a good
agreement between measurements and model prediction (see Fig. ).
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 exemplifying our general research procedure by virtue of a
single yet representative example.
Using the time series of soil moisture, evapotranspiration, interflow and
recharge, as computed by mHM, we computed the travel-time distributions for
every grid cell of the catchment (see Eq. and Fig. a).
One of the problems when computing forward TTDs by virtue of Eq. ()
is that all the water entering the CV at time tin must leave by
the end of the available time series. This means that a certain amount of
water at the end of these time series could not be used for the analysis. To
determine this period, we computed θin with respect to
discharge as well as to evapotranspiration. Adding up both values for a given
tin should add up to 1; i.e., all water that entered at
tin should leave within the available time frame. A value smaller
than 1 therefore indicates that some amount of the water is still inside the
CV with possible error-inducing effects on the calculation of the TTDs.
Analyzing this behavior, we concluded that close to 2 years at the end of the
available time series had to be excluded for the calculations of the TTDs
(data not shown). The shape of the resulting time-dependent distributions
varied strongly, depending particularly on rainfall events that triggered the
mobilization of older water stored within the soil. Another factor, although
not apparent from Fig. , was the water content, i.e., the state of
the soil itself. As has been demonstrated by , soil response
to rain events is strongly different between wet or dry conditions.
Comparison between monthly measured and modeled
evapotranspiration (ET) at eddy-covariance station Mehrsted (see
Fig. ).
Forward TTD of soil moisture with respect to mean travel time (in
months) for a single cell in the Nägelstedt catchment. (a) shows
the time-dependent TTD derived using Eq. () for a given
tin. (b) shows the stationary TTD derived using
Eq. ().
To disentangle these event-driven as well as state-dependent effects from
other factors that influence the water movement in the soil, we averaged
these time-dependent distributions. As a result, we got the stationary TTDs
for every cell:
p→QTE=∫Q(t)θtinS(t)exp-∫TEQ(t′)+ET(t′)S(t′)dt′dtin,
with TE=t-tin, t>tin.
In all investigated cases, these stationary TTDs could be well approximated
by an exponential-esque behavior (see Fig. b). Such behavior is
often assumed to be valid for TTDs in general such that they are consequently
modeled using exponential or gamma distributions .
Recent works, however, have questioned this generalization by emphasizing the
time-dependent nature of TTDs . The examples
given in Fig. exemplify these concerns by illustrating their
respective origins. Consequently, we acknowledged the inherent differences
between these two TTDs. Furthermore, the study area falls within a humid
region, with soils being generally wet and rainfall being evenly distributed
throughout the year. Under these conditions the assumption of (quasi)
stationary TTDs is reasonable . These
stationary TTDs provided the basis for all following analysis since they
allow the description of the average hydrological response of the catchment.
In addition, we also focused on travel-time behavior under specific
hydrological regimes, i.e., wet and dry conditions, providing a more detailed
understanding of the catchment.
Mean life expectancy (in months) of soil moisture derived by
Eq. () for the Nägelstedt catchment (see also
Fig. for comparison) once for all mHM cells (left panel) and for
all non-urban cells.
Kernel density estimate of the mean life expectancy (in months) of
soil moisture for several grid sizes in the Nägelstedt
catchment.
For our statistical analysis, we used these stationary TTDs, which, due to
their exponential-esque behavior, can be characterized by their expected
value τ. We call this value mean life expectancy (or mean age in the
case of backward TTDs) hereafter. Estimating this value for every mHM cell
provided therefore a single measure for the travel-time behavior in the soil
without the otherwise dominating impact of single precipitation events (see
Fig. ). One feature that became immediately apparent was the long
travel times in urban areas (see Fig. a). This can be explained by
the fact that these areas are largely sealed, resulting in low infiltration
rates and consequently low turnover rates inside the soil. To disentangle
this sealing effect from the soil behavior, we discarded cells inside urban
regions from our analysis (see Fig. b). This allowed us to
investigate the interplay between soil properties and travel-time behavior
apart from such artificial influences.
Impact of modeling resolution
Due to its multiscale parameterization, mHM is able to model catchment
dynamics at different spatial resolutions with the same set of calibration
parameters (see, e.g., , or ).
Within the context of TTDs, this feature may be used to investigate the
potential influence of age-dependent outflow generation. The mathematical
theory for including such age dependency has been developed independently by
different groups and recently been unified using the umbrella term of StorAge
Selection (SAS) functions . 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 a different water age should be
interpreted as the result of complex mixing processes taking place in the
subsurface of the catchment . To
determine the appropriate SAS function for a given catchment, predictions
using different functions would have to be compared with measurements.
Alternatively, the form of the SAS function can be determined by using a
physically based catchment model . 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 different scales
using the uniform SAS function. We motivated this choice by the principle of
least information (or principle of maximum entropy) stating that, among
different 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, thereby motivating the use of the
uniform SAS function.
To estimate the possible influence 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
multiscale nature of mHM, where sub-grid 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 sub-grid influence. Results from our simulations showed no
discernible differences in the statistical distribution of mean life
expectancy (see Fig. ). Using a higher resolution had a positive
effect on the 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 had no major impact on mean life
expectancy. We are aware that this assessment only covers one possible source
of age-dependent outflow behavior and that other unresolved heterogeneity (at
even smaller scales or due to other subsurface properties not accounted for
in mHM) would influence the outflow generation as well. We therefore regard
our conclusions as tentative and open to revision once actual measurements
become available.
Scatter plot of mean life expectancy (in months) of soil moisture
vs. monthly values (in millimeters) for precipitation (a) and
effective precipitation (b).
However, our investigation gave us the ability to find a good trade-off
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 . Contrary to these studies where
field measurements were used, we used results from computational simulations
only. This gave us a much larger data set, both in time and space, from which
we could infer the relative impact of different factors, in particular
meteorological (precipitation), land surface (land cover, leaf-area index)
and subsurface (soil) properties. Notably, our approach differs from
such that our analysis is based on model-derived
gridded simulations of TTDs as compared to the observation-based basin-wise
quantification of TTDs.
In the first 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 , we used the coefficient of
determination R2 to quantify the strength of the statistical
relationship. This quantity equals 1 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 statistical 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 (see Fig. ). We therefore expected this quantity
to exert strong controls on the steady-state TTDs as well. In our model, two
different quantities can be distinguished: first, the precipitation itself as
well as, second, the effective precipitation. The latter value is here
defined as the water flux that is actually entering the soil, i.e., corrected
by surface runoff (through sealing), canopy interception and snowmelt. While
the precipitation can be measured with high accuracy, it is the effective
precipitation that directly impacts soil-moisture dynamics.
The scatter plots of both data sets vs. the mean life expectancy show a
significant negative correlation between them (see Fig. ). 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 “flushing 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
Fig. ), i.e., longer life expectancy correlated with lower
heights of the terrain, and with a linear coefficient of determination of
R2=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 coefficient of determination of R2=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
effects to these covariates only.
Scatter plot of mean life expectancy (in months) of soil moisture
vs. elevation (in m).
Evapotranspiration
Evapotranspiration is directly influencing the form of a TTD (see, e.g.,
Eq. ). Consequently, we anticipated a strong correlation between
mean evapotranspiration rates and mean life expectancy.
With respect to evapotranspiration, two different definitions 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 . Using theoretical models,
good estimates can therefore be provided for PET at a given site
. On the other hand, 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 . As a
consequence, PET can often be estimated with higher accuracy than AET.
Scatter plot of mean life expectancy (in months) of soil moisture
vs. monthly evapotranspiration values (in millimeters). Displayed are both
potential evapotranspiration (a) and actual
evapotranspiration (b).
Scatter plots of both PET and AET show a positive correlation between
evapotranspiration and mean life expectancy in general (see
Fig. ). This correlation is more pronounced for AET, with a
coefficient of determination of R2=0.496 vs. only R2=0.259 for PET.
In contrast to precipitation, which is an inflow mechanism, ET is an outflow
mechanism. It does not push, but rather pulls the water out of the CV, which
explains the difference in behavior between 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 outflow mechanisms (the other
being discharge). The relative stronger impact of AET compared to PET was
also anticipated. AET is directly used in Eq. () for the
calculation of TTDs, 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 to therefore 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
fluxes through artificial and natural sealing. In mHM, three different
land-cover types are distinguished: forest, crop/grassland and urban area. As
explained above, we excluded mHM cells inside urban areas from our analysis
in order to better focus on the soil properties themselves. To further
elucidate possible influence of the remaining land-cover types, we separated
the catchment into forest and crop/grassland and calculated the mean travel
times separately.
Land cover in the Nägelstedt catchment (blue: forest;
green: urban; red: crop/grassland). (a) shows the spatial
distribution of land cover in the highest resolution l0 and
(b) shows the kernel density estimates of the mean life expectancy
(in months) of soil moisture for the land-cover types.
Leaf-area index (LAI) in the Nägelstedt catchment.
(a) shows the spatial distribution and (b) shows the
scatter plot of mean life expectancy (in months) of soil moisture vs.
LAI.
Estimating the probability density function (PDF) of the mean life expectancy
for both land-cover types separately revealed strong differences between them
both in shape of the respective PDF and the range of values (see
Fig. ). As shown above, results for the combined data set showed
a distinct bimodal behavior (see Fig. ). In contrast to that, the
PDFs for both land-cover types were almost unimodal. The most dominant peaks
of every single 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 difference may be partially due to a correlation with
precipitation values that have already been shown to exert a strong influence on
travel-time behavior. Forest in the study catchment (as well as in Germany in
general) is found disproportionately in hilly and mountainous regions. These
regions in turn show stronger precipitation values. The tendency depicted in
Fig. may therefore be caused by this covariate. However, this
correlation between forested and high-precipitation area would not explain
the distinct differences between both land-cover types. Another factor,
overlapping with the former, may be due to the differences 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 Fig. .
In addition to this classification scheme, mHM uses the leaf-area index (LAI)
to describe land-cover properties. The LAI describes the ratio of the cell
that is effectively covered by plant canopy. Due to the already established
influence on evapotranspiration (see above), it stands to reason that an
influence on the mean life expectancy exists as well. Comparing LAI class and
land cover reveals a strong overlap between both (see Figs. a
and a). Roughly, forest corresponds to LAI classes 1–4, urban
area corresponds to LAI class 5 and grassland corresponds to LAI classes
6–10.
Using the same approach as above, i.e., investigating the mean life
expectancy for every LAI class independently, 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
classification schemes. In addition, we saw little diversity for LAI classes
within the same land-cover class (data not shown).
Soil classes in the Nägelstedt catchment. (a) shows the
spatial distribution and (b) shows the kernel density estimate of
the mean life expectancy (in months) of soil moisture for selected soil
classes. The blue curve represents soil class 9 (36 % sand and 10 %
clay), the yellow curve represents soil class 38 (12 % sand and 15 %
clay), the orange curve represents soil class 40 (10 % sand and 19 %
clay), the red curve represents soil class 42 (7 % sand and 39 %
clay) and the brown curve represents soil class 51 (19 % sand and
70 % clay).
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 the case of coniferous forest) or vary strongly (e.g., in the case
of deciduous forest). To make values from different LAI classes comparable,
we averaged the respective values year-wise. A scatter plot of leaf-area
index vs. mean life expectancy does not show strong correlation between the
two, with similar ranges of values being found for almost all LAI values (see
Fig. 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 top soil layer, LAI values are only used
for interception and consequently do not directly influence travel-time
behavior. As a result, any possible relationship between LAI and TTDs 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 German soil database
Bodenübersichtskarte 1:1 000 000 (BÜK 1000) .
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
significant differences between them (see Fig. ). 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 to 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.
Scatter plot of mean life expectancy (in months) of soil moisture
vs. saturated soil moisture (in millimeters).
In addition, the soil class is a symbolic variable; i.e., its values only
indicate a certain type of soil but do 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
filled 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 coefficient of determination R2=0.675 (see Fig. ).
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 filled with water to have a
significant effect on the resulting travel-time behavior. On the other hand,
the soil types show a strong overlap with other factors like precipitation
levels and land-cover types that have already been discussed above.
Mean age of soil moisture in the Nägelstedt catchment.
(a) shows spatial distribution and (b) shows the scatter
plot of mean age vs. mean life expectancy (see Eq. ) in
months.
Statistical analysis of mean age
As described above, the difference between the forward and backward
formulations of travel time has long been acknowledged and
many studies have investigated their relationship
. Both
these formulations are linked by virtue of the so-called Niemi relation
J(tin)θ(tin)p→(t-tin|tin)=Q(t)p←(t-tin|t),
which can be derived by considering a water parcel entering the CV at
tin and leaving at t. Consequently, mean life expectancy and
age only coincide in the case of steady-state conditions. As a result, we
also investigated the behavior of mean age to elucidate connections and
differences between forward and backward formulations for our catchment.
Visually comparing mean age (see Fig. ) and mean life expectancy
(see Fig. b) in the Nägelstedt catchment showed strong
qualitative and quantitative similarities. Accordingly, we also got a very
strong statistical relationship between these two quantities, with a
coefficient of determination of R2=0.956. Overall, the relationship was
very linear, with mean age values falling short of mean life expectancy for
both small and large values.
Due to the mathematical and physical similarities, such a strong connection
was anticipated. To further investigate possible origins of their respective
differences, 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 influx, outflux and
state, respectively. Results showed overall the same trend for mean age and
life expectancy with respect to these predictors (see Table ).
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.,
, who emphasized the role of the outfluxes for the time
evolution of both age and life expectancy. In our analysis, we saw that proxy
variables for influx and state show the strongest correlations with mean
travel-time behavior. On the other hand, PET, which is a good proxy for one
of the two outfluxes, showed only moderately strong correlations with said
behavior. In the 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 outflux, i.e., discharge, we
excluded this quantity from our analysis.
R2 values for several predictors of mean travel
time.
Pre
PET
SSM
Mean life expectancy
0.860
0.260
0.675
Mean age
0.728
0.143
0.711
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 significant amount of the variability
contained in the data set. 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
first single-variable model consequently used precipitation as the variable
with the highest single R2 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.
R2 values for several regression models of increasing
complexity.
Pre
Pre + SSM
Pre + SSM + PET
Mean life
expectancy
0.860
0.911
0.913
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 ). This was expected since both variables alone
provided already high R2 values. In addition, precipitation and saturated
soil moisture only showed moderate correlation (R2=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 orographic effect; 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 ). 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), thereby adding only little new
information compared to the other two variables. Such low impact of outgoing
fluxes compared to precipitation has already been reported before, for the
case of synthetic toy models . Moreover, our results agree
with the findings of , 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 influx (i.e.,
precipitation) as well as the state variable (i.e., saturated soil moisture)
on the travel-time behavior in the soil. To further elucidate their impact,
we investigated the soil travel-time behavior independently for different
hydrological regimes during the considered period of time, i.e., from 1955 to
2005. To that end, we partitioned the available time series into regimes
based on soil moisture (state variable) and precipitation events (influx).
In the first 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 the 85th percentile of the time series
were labeled as wet years. In contrast, years with an average soil moisture
content below the 15th percentile of the time series were labels as dry
years. This year-wise partitioning was seen as necessary due to the strong
annual fluctuations of this variable. Finally, we performed the same analysis
as describe above for both – now smaller – data sets.
Mean life expectancy of soil moisture in the Nägelstedt
catchment in dry years. (a) shows the spatial distribution and
(b) shows the kernel density of mean life expectancy (in
months).
Mean life expectancy of soil moisture in the Nägelstedt
catchment in wet years. (a) shows the spatial distribution and
(b) shows the kernel density of mean life expectancy (in
months).
Mean life expectancy of soil moisture in the Nägelstedt
catchment caused by rainy months. (a) shows the spatial distribution
and (b) shows the kernel density of mean life expectancy (in
months).
Using results from dry years only (see Fig. ) showed a similar
qualitative travel-time behavior but strong quantitative contrast compared to
the mean travel-time behavior discussed above (see Fig. ).
Compared to this general case, mean life expectancy was much larger in dry
years. In addition, dry years exhibit a wider range of values, with the
largest one (over 50 months) being almost 4 times as large as the smallest
one (approximately 12 months).
Wet years, on the other hand, exhibit a very small range of values, with the
smallest value (approximately 5 months) being roughly half the largest value
(approximately 11 months) (see Fig. ). 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 discrepancy demonstrates again the
strong impact of the state variable (soil moisture) on travel-time behavior.
Another difference 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 strongly correlates with the different soil
types and therefore reflects the strong impact on this property on the
overall soil-moisture dynamics. The disappearance of this bimodal behavior is
therefore reflective of how the soil becomes more homogeneous when filled up
with water.
R2 values for several predictors of mean travel time (as caused
by wet and dry years).
Pre
PET
SSM
Wet years
0.374
0.084
0.388
Dry years
0.781
0.223
0.834
In addition, our results showed different statistical dependencies of
travel-time behavior with respect to precipitation, PET and SSM (see
Table ). Dry years showed very similar correlation values compared
to the general case (see Table ). On the other hand, correlation
values for wet years were remarkably smaller.
In the second case, we investigated travel-time behavior depending on extreme
values of the influx, i.e., for months having large 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 data set.
Results showed strong differences in mean life expectancy during rainy months
compared to the scenarios discussed above (compare Fig. with
Figs. and ). 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 flushing of the soil. We also saw a similarly
small variance and a nearly unimodal distribution of mean travel-time values.
R2 values for several predictors for mean travel time (as
caused by rainy months).
Pre
PET
SSM
Mean life expectancy
0.736
0.221
0.857
In addition to that, we saw differences in the statistical correlation of
mean life expectancy of precipitation, potential evapotranspiration and
saturated soil moisture (see Table ). Compared to the standard
travel-time behavior, precipitation was slightly correlated with mean life
expectancy. This was caused by lower overall variation in precipitation
values due to constraining our analysis to large values, thereby excluding
low and medium range rain events. In contrast to that, R2 values for PET
and SSM increased.
Relevance of TTDs for hydrological inference
The above results demonstrated the impact of certain soil properties, as
implemented in mHM, on mean travel times using the R2 metric as a measure.
In addition to that statistical analysis, their relationship can further be
elucidated by analyzing Eqs. () or (). Assuming for
example a simple linear relationship for both Q and ET with respect to S,
we get for Eq. () the following:
p→Q(TE,tin)=αQθ(tin)exp-αETTEexp-αQTE.
Equation () shows how under such simplified assumptions, the TTD of
such a CV would follow an exponential distribution with its mean travel time
being related to the recession constants αQ and
αET. As shown above, such an exponential behavior is visible
in the mean behavior (see Fig. right), whereas non-stationary
TTDs show this exponential behavior to be dominated by the event-based nature
of the governing fluxes (see Fig. left).
In addition to these differences, we also saw different mean travel-time
behavior for different regimes (see above). These differences can be
explained by the actual implementation of Q and ET in mHM, which is
generally nonlinear (see Sect. ). To assess the different roles
of each soil process in discharge generation, we calculated the relative
contribution of each outflow mechanism for each regime. The data in
Table show how much of the water that entered the soil during a
given time and left eventually as discharge was leaving as baseflow
QB, slow interflow QIs or fast interflow
QIf. On average, baseflow contributed the most to
discharge, with fast interflow having the smallest share. This overall
distribution became more strongly pronounced during dry years, with baseflow
taking the largest share of outflow generation and fast interflow becoming
negligible. For wet years this trend is reversed, with water entering the
soil during rainy months having an almost equal distribution. These different
weighs show the relative impact and therefore the relative information
content that travel-time distributions could contain; i.e., travel times in
dry years are mostly the results of the successive processes contributing
eventually to baseflow (see Fig. ), whereas travel times during
storm events contain information on all discharge processes equally.
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 different
regimes. Results confirmed the relationship described above, with mean travel
times during dry years showing the strongest correlation (see
Table ).
Such a high interdependency between certain model parameters and data from
different flow regimes is not unique for TTDs. Using discharge alone would
reveal similar overall tendencies; e.g., discharge data from droughts are
more informative for calibrating baseflow recession constants. What is new,
however, is the additional information content, which is not contained in
discharge data alone. Not only can this improve calibration efforts, it
allows the inference of additional system states. This is particularly
relevant with respect to, but not confined to, the total amount of stored
groundwater. Discharge data are not sensitive to, and therefore not
informative for, groundwater levels, but are to its variations, i.e., fluxes.
TTDs 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 one to
provide reasonable estimates of this quantity. Similarly, the estimation of
water in the root and vadose zone can be improved.
Relative contribution of the different fluxes to runoff
generation.
All years
Dry years
Wet years
QIf
0.150
0.061
0.173
QIs
0.298
0.239
0.317
QB
0.552
0.700
0.512
R2 values for recharge vs. mean travel times for different
regimes.
All years
Dry years
Wet years
R2
0.6059
0.6954
0.3619
In addition, 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 different timescales for different hydrological
regimes, varying by almost an order of magnitude. Despite this heterogeneity,
all travel times in our study remained within 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 and model parameters, it can be said that the
strong interlink between the travel-time behavior and outflow 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, have made the case for the usefulness of
TTDs for the parametrization of such models. The above presentations provide
empirical support for this notion.
Conclusions
In this study, we investigated the spatially
distributed soil-moisture dynamics in the Nägelstedt catchment using
travel-time distributions. The states and fluxes, needed for the derivation
of the travel times, were numerically computed using the mesoscale
Hydrological Model (mHM), which was calibrated against 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 1 order of magnitude), we did not see any significant difference in
their distribution. This indicates a general soundness of the
parameterization scheme of mHM used for the calculation of the states and
fluxes on the different 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
different water uptake mechanisms of trees vs. crop/grass.
We also investigated the travel-time behavior for different hydrological
regimes, i.e., for dry and wet conditions (using soil moisture and
precipitation as indicators). Our analysis revealed significantly different
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 baseflow on the generated outflow, whereas
the large variance shows the variety of soil responses under dry conditions.
Such conditions are therefore suited to inferring soil properties relating to
baseflow generation. In addition, due to the large variance of soil
responses, such conditions would allow the inference of 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 be 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 influence of the slow and fast interflows on the
total discharge behavior. As a result, TTDs derived under such conditions may
be suited to inferring the parameters relating to these hydrological
processes.
In the case of rainy months, which overlap with wet conditions to a
significant degree, we saw a similar distribution of mean travel times, but
with even shorter mean values. This indicates a stronger impact of fast
interflow on the total discharge behavior. Such information can therefore be
valuable for improving the parametrization of the fast interflow related
processes.
It is important to emphasize that our results have been derived with respect
to a single hydrological model, i.e., mHM, only. As a result, we also need to
critically assess the limitations of this approach and its impact on the
reliability of our conclusions. First, mHM treats the hydrological storage in
every compartment as fully mixed. In the absence of additional information,
we consequently assumed a uniform sampling scheme for the discharge
generation from every mHM cell. This may have introduced errors in the age
distribution of fluxes and therefore the travel-time behavior as discussed in
Sect. . Due to the well-established ability of mHM to take
sub-grid heterogeneity into account, we have confidence in the physical
plausibility of the spatially explicit soil moisture states and fluxes. In
the absence of, say, solute data, we have, however, to consider these
assumptions as tentative and open to revision. The other limitation of our
approach stems from the computational nature of our study, introducing a
number of uncertainties. Like any hydrological model, mHM may suffer from
three different sources of uncertainty: input uncertainty, structural
uncertainty and parametric uncertainty. We would therefore like to assess
their nature and potential impact on our results and conclusions. First,
input uncertainty refers to the uncertainties inherent in the forcing of the
model, i.e., precipitation. Our results have shown the strong impact of
precipitation on travel-time behavior. It would therefore stand to reason
that a strong impact of any uncertainty from precipitation propagates to the
resulting travel-time behavior. However, we investigated mean behavior only,
where time series from many months were averaged. We therefore consider
possible contributions to our results to be minor. Next, structural
uncertainty depends on the conceptual implementation of subsurface processes
within mHM and our choices of different mHM compartments for our analysis. In
Sect. , we discussed this issue by providing the rationales for,
e.g., including the interflow components in our analysis. Finally, parameter
analysis is probably the largest total source of uncertainty and several
studies have recently investigated its impact on mHM output generation
. The studies show that, while the
fluxes are typically well represented in mHM , the overall
soil moisture storage showed less accuracy, in particular during droughts
. For droughts, our results showed in general long
travel times and pronounced soil-specific behavior with comparably lesser
impact of precipitation. While we do not expect a major impact on the
qualitative nature of these results, we should consider the quantitative
aspect, i.e., the specific values for mean travel times, to be inconclusive.
In general, we consider the uncertainty stemming from the storage estimate to
be the most relevant due to having both comparably lower accuracy and the
strong impact on overall travel-time behavior demonstrated above. This is
exacerbated since the water content relevant for outflow generation may not
be the same as the one relevant for travel-time behavior. Immobile water due
to, e.g., dead-end pores, affects the latter but not the former. It is,
however, this connection between the total water content and the resulting
travel-time behavior that makes the use of TTDs an important tool for a
better calibration of hydrological models.
As an outlook, we can say that, 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., , 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 the comparison of 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 . 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.