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
(∼6000 m2) catchment for different scenarios. We found
that the TTDs can be subdivided into four parts: (1) early part – controlled
by soil hydraulic conductivity and antecedent soil moisture content, (2) middle part – a transition zone with no clear pattern or control, (3) later
part – influenced by soil hydraulic conductivity and subsequent
precipitation amount, and (4) very late tail of the breakthrough curve –
governed by bedrock hydraulic conductivity. The modeled TTD shapes can be
predicted using a dimensionless number: higher initial peaks are observed if
the inflow of water to a catchment is not equal to its capacity to discharge
water via subsurface flow paths, and lower initial peaks are connected to
increasing available storage. In most cases the modeled TTDs were humped
with nonzero initial values and varying weights of the tails. Therefore,
none of the best-fit theoretical probability functions could describe the
entire TTD shape exactly. Still, we found that generally gamma and
log-normal distributions work better for scenarios of low and high soil
hydraulic conductivity, respectively.
Introduction
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.
Initial use of theoretical probability distributions
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 1/f scaling, which is consistent with gamma
TTDs with shape parameter α≈0.5 (Kirchner, 2016).
General observations on the shape of TTDs
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).
Controls on shape variations
A number of studies reported on the best-fit shape of gamma distributions
generally ranging from α0.01 to 0.90 (Hrachowitz et al., 2009;
Godsey et al., 2010; Berghuijs and Kirchner, 2017; Birkel et al., 2016),
which indicates L-shaped distributions with high initial values and heavier
tails. Several studies found that α values decrease with increasing
wetness conditions (e.g., Birkel et al., 2012; Tetzlaff et al., 2014),
causing higher initial values and heavier tails. However, the opposite was
observed in a boreal headwater catchment (Peralta-Tapia et al., 2016) where
α ranged between 0.43 and 0.76 for all years except the wettest year
(α=0.98). In the Scottish highlands α showed little
temporal variability (and therefore no link to precipitation intensity) but
was closely related to catchment landscape organization – especially soil
parameters and drainage density – where a high percentage of responsive
soils and a high drainage density resulted in small values of α
(Hrachowitz et al., 2010).
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 α (from 0.95 for homogeneous conditions down to a value of
0.5 for extreme gradients) in a study by Ameli et al. (2016). Somewhat
surprisingly, the level of “unstructured” heterogeneity within the soil
and the bedrock was found to only have a weak influence on the shape of TTDs
(Fiori and Russo, 2008) since the dispersion is predominantly ruled by the
distribution of flow path lengths within a catchment. Antecedent moisture
conditions and event characteristics influence catchment TTDs at short
timescales while land use affects both short and long timescales (Weiler et
al., 2003; Roa-García and Weiler, 2010); generally TTD shapes appear highly
sensitive to catchment wetness history and available storage, mixing
mechanisms and flow path connectivity (Hrachowitz et al., 2013). Kim et al. (2016) recorded actual TTDs in a sloping lysimeter and reported that their
shapes varied both with storage state and the history of inflows and
outflows. They argued that “the observed time variability […]
can be decomposed into two parts: [1] “internal” […] –
associated with changes in the arrangement of, and partitioning between,
flow pathways; and [2] “external” […] – driven by fluctuations
in the flow rate along all flow pathways”. From these partly contradictory
findings, it is clear that relating best-fit values for the shape parameter
α of the gamma distribution to catchment, climate or precipitation
event properties does not yield a consistent picture yet. Moreover, the
shape of TTDs is also dependent on the resolution of time series data
(sampling frequency). While α can decrease with longer sampling
intervals – since the nonlinearity of the flow system is overestimated when
sampling becomes more infrequent (Hrachowitz et al., 2011) – higher
α values can also result from lowering the sampling frequency in
both input (precipitation) and output (streamflow) (Timbe et al., 2015).
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.
TTD theory
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 F which combines several catchment, climate and event properties into
one index relating flows in and out of the catchment to the available
subsurface storage. It was originally designed to monitor the exceedance of
certain storage thresholds for the activation of different dominant flow
paths (groundwater flow, interflow, overland flow) at the catchment scale
but can also help to categorize and predict TTD shapes. Moreover, from
continuous time series of TTDs one can mathematically derive residence time
distributions (describing the age distribution of water stored in the
catchment), storage selection functions (describing the selection preference
of the catchment discharge for younger or older stored water) (Botter et
al., 2010, 2011; van der Velde et al., 2012; Benettin et al., 2015; Harman,
2015; Pangle et al., 2017; Danesh-Yazdi et al., 2018; Yang et al., 2018) and
master transit time distributions (MTTDs) (representing the flow-weighted
average of all TTDs of a catchment) (Heidbüchel et al., 2012; Sprenger
et al., 2016; Benettin et al., 2017), which can all take on different shapes
depending on climate and catchment properties, just like the individual
TTDs. Hence the results presented in this paper can also provide insights
into the use of these descriptors of catchment hydrologic processes.
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”.
Our approach
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.
Methods
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).
Model setup
A small zero-order catchment was set up, 100 m long, 75 m wide
(∼6000 m2), with an average slope of 20 % towards the
outlet and elliptical in shape (Fig. 1). The catchment converges slightly
towards the center, creating a gradient that concentrates flow. The bedrock
is 10 m thick and has a saturated hydraulic conductivity of KBr,x=KBr,y=10-5 m d-1 (horizontal) and KBr,z=10-6 m d-1 (vertical). The soil layer is isotropic, of uniform
thickness and has a higher hydraulic conductivity. All other parameters are
uniform across the entire model domain (based on values typically found in
many catchments in central Europe): porosity n=0.39 m3 m-3, van
Genuchten parameters alpha αvG=0.5 m-1, beta βvG=1.6, saturated water content θs=
0.39 m3 m-3, residual water content θr=
0.05 m3 m-3, pore-connectivity parameter lp= 0.5, and
longitudinal and transverse dispersivity αL=5 m and
αT=0.5 m, respectively. The magnitude for αL
was estimated with regard to the length of the model catchment (100 m) using
the relationship described in Gelhar et al. (1992) and Schulze-Makuch (2005). Newer research by Zech et al. (2015) shows that the longitudinal
dispersivity is probably up to an order of magnitude smaller than that reported
by Gelhar et al. (1992). Still, we do not suspect the value of the local
dispersivity to have a large impact on the TTDs (see also Fiori and Russo,
2008) since the local dispersivity is usually negligible compared to the
dispersion caused by the spatial distribution of rainfall (the “source zone
dispersion” in Fiori et al., 2009). Nevertheless, we tested whether changing
αL from 5 to 0.5 m would change our results significantly
(see Sect. S1, Fig. S1 and Table S1 in the Supplement). Both bedrock and soil
are exclusively porous media without any potential preferential flow paths
like macropores or rock fractures.
3-D model domain and shape of the virtual catchment from the top
(a), front (b) and side (c). The blue square
indicates the outflow boundary with constant head condition. The red layer
represents the soil, which has a much higher hydraulic conductivity than the
underlying bedrock (grey). The orange lines indicate the zone of convergence
(but no explicit channel). The two additional catchment shapes (top-heavy
and bottom-heavy) we tested in Sect. 2.2.1 are shown in the black box (d).
Boundary conditions
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-1 (Fig. 2a). The time
series was 1 year long and repeated 32 more times to cover the entire
modeling period, which lasted a total of ∼33 years (12 000 d). We made sure that the looping of the precipitation time series would
not cause any unwanted artifacts in the resulting TTDs (see Sect. S2 and Fig. S2). Neither evaporation nor transpiration was considered
during the simulations. This means that all precipitation we applied was
effective precipitation that would eventually discharge at the catchment
outlet. The addition of the process of evapotranspiration is planned in a
follow-up modeling study to investigate what influence it exerts on
catchment TTDs. The tracer was applied uniformly over the entire catchment
during a precipitation event that lasted 1 h and had an intensity of
0.1 mm h-1 and a tracer concentration of 1 kg m-3. This resulted
in a total applied tracer mass of 0.589 kg.
(a) 1-year time series of subsequent precipitation (looped 32
more times for the entire modeling period and rescaled for smaller or larger
amounts of subsequent precipitation Psub). Tracer application took place
during the first hour of the model runs. (b) Time series of subsequent
precipitation for a high-frequency scenario (humid) and a low-frequency
scenario (arid). The total precipitation amount is the same for both
scenarios.
Initial conditions
The model runs were initialized with three different antecedent soil
moisture conditions θant: a dry one (θant=22.0 %, corresponding to an average effective saturation of the soil
layer Seff≈50 %), an intermediate one (θant=28.8 %; Seff≈70 %) and a wet one (θant=35.6 %; Seff≈90 %). To obtain realistic
distributions of soil moisture, we first ran the model starting with full
saturation and without any precipitation input and let the soils drain until
the average effective saturation reached the states for our initial
conditions. We recorded these conditions and used them as initial conditions
of the virtual experiment runs. In general, the soil remained wetter close
to the outlet in the lower part and became drier in the upper part of the
catchment. Note that the process of evapotranspiration was excluded from the
modeling so that the lowest achievable saturation was essentially defined by
the field capacity. An average effective saturation Seff of
approximately 50 % was the lowest that could be achieved by draining the
soil layer since the lower part of the catchment stayed highly saturated due
to the constant head boundary condition being equal to the surface elevation
at the outlet. The upper parts of the catchment, however, were initiated
with much lower Seff values (≈30 % in the dry scenarios).
That means that although an Seff value of 50 % seems to be quite
high, it actually represents an overall dry state of the catchment soil.
Throughout the modeling runs the dry initial condition did not occur again
as that would have taken 13 years of drainage without any precipitation for
the scenarios with high soil hydraulic conductivity KS and almost 1500
years for the scenarios with low KS. The inclusion of evapotranspiration
would, however, speed up the drying process of the soil and hence make these
initial conditions more realistic.
Model scenarios
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 (Dsoil), saturated soil
hydraulic conductivity (KS), antecedent soil moisture content (θant) and subsequent precipitation amount (Psub, essentially a
measure of the amount of precipitation that falls after the delivery of the
traced event) (Fig. 3).
The four properties that were varied to explore their influence on
the shape and scale of TTDs: soil depth Dsoil, saturated soil hydraulic
conductivity KS, antecedent soil moisture θant and
subsequent precipitation amount Psub. The bedrock hydraulic conductivity
KBr was kept constant for all of these base-case scenarios.
We tested two soil depths Dsoil, namely depths of 0.5 and 1.0 m,
evenly distributed across the entire catchment. Similarly, we chose two
saturated soil hydraulic conductivities KS, a high one with
2.0 m d-1 (similar to fine sand) and a low one with 0.02 m d-1
(similar to silt). Three states of antecedent moisture content θant were selected to represent initial conditions – 50, 70 and
90 % of effective saturation. Finally the subsequent precipitation amount
Psub was varied in three steps from 345 to 690 and up to 1380 mm a-1.
The original precipitation time series (690 mm a-1, Fig. 2a) was
rescaled to obtain time series with smaller and larger amounts. With two
soil depths, two soil hydraulic conductivities, three antecedent moisture
conditions and three subsequent precipitation amounts this resulted in 36
model scenarios. Based on these 36 runs we evaluated the differences in the
shape of the TTDs. The abbreviated names of the 36 model runs consist of
four letters, each representing one of the properties that we varied: the
first one describes Dsoil (T = thick; F = flat), the second one
KS (H = high; L = low), the third one θant (W = wet;
I = intermediate; D = dry) and the fourth one Psub (S = small; M = medium; B = big). For example the name FHIB would indicate a run with
a flat (“F”) (shallow) soil, a high (“H”) KS, an intermediate (“I”) antecedent moisture content and a big (“B”) subsequent precipitation amount (see Table 1 for
an overview of the names of all 36 scenarios). We are well aware that
“thick” and “flat” are technically incorrect descriptions of soil depth.
However, in order to have unique identifiers (i.e., individual letters) for
all 10 property states we decided to use T and F for describing deep and
shallow soils, respectively.
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.
Soil porosity
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 m3 m-3) and three had smaller
soil porosity (0.24 m3 m-3) than the base-case scenarios
(0.39 m3 m-3).
Bedrock hydraulic conductivity
Six runs were performed on the basis of the THDB scenario (which had a
bedrock hydraulic conductivity KBr of 10-5 m d-1). In the
first run KBr was decreased to 10-7 m d-1, in the following
runs it was successively increased to 10-3, 10-2, 10-1,
100 and 2×100 m d-1, matching the KS of the soil
layer in the final run.
Decay in saturated hydraulic conductivity with depth
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 z
and saturated hydraulic conductivity KS with a shape parameter f=
0.29 m and saturated hydraulic conductivity at the surface KS0=
7 m d-1 (for the high conductivity scenarios) or KS0=0.07 m d-1 (for the low conductivity scenarios) (Fig. 4a):
KS(z)=KS0e-zf.
This preserved the mean KS values of 2×10-0 (high) and
2×10-2 m d-1 (low) (from the base-case scenarios).
(a) Exponential decay in saturated soil hydraulic conductivity with
depth for the high (blue) and the low (red) KS scenario. The x axis in
the inset has a log scale. The spatial mean KS is indicated by the
vertical black lines. (b) Water retention curves (solid) and relative
hydraulic conductivities (dotted) for sandy and silty soils. The permanent
wilting point (PWP) and the field capacity (FC) are marked as references
(dashed).
Precipitation frequency
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 Psub scenarios.
Water retention curve
All the base-case model scenarios were conducted with water retention curves
(WRCs) resembling silty soils:
θ=θr+θs-θr1+αvGψβvGν,
with van Genuchten parameters αvG (m-1) and βvG (dimensionless), saturated water content θs, residual
water content θr (both m3 m-3), pressure head ψ
(m) and ν=1-1/βvG (see Sect. 2.1 for van Genuchten
parameter values). However, we also wanted to investigate how a different
WRC in the soil layer (see Fig. 4b) would influence the shape of TTDs. We
chose to test a sand-type WRC since it can, in some aspects and to a certain
extent, also indicate how a system with the threshold-like initiation of
rapid preferential flow behaves. The sand-type WRC causes an increase in
hydraulic conductivity at relatively lower soil water contents
compared to the silt-type WRC. Hence, for the same precipitation event
lateral flow is initiated faster (at lower saturations) in sandy soils since
water reaches the soil–bedrock interface more quickly, where it is diverted
from vertical to lateral flow. The relative hydraulic conductivity kr
was derived with Eq. (3):
kr=Sefflp1-1-Seffν-1ν2,
with effective saturation Seff and pore-connectivity parameter lp
(both dimensionless). Other aspects of preferential flow – like bypass flow
through macropores in deeper soil layers – are, however, not captured by
sand-type WRCs. The van Genuchten parameters for the sand-type WRC were
defined as follows: αvG=14.5 m-1 and βvG=2.68. We based the additional eight runs on the scenarios THDB, THWB,
THDS, THWS, TLDB, TLWB, TLDS and TLWS.
Catchment shape
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.
Full saturation and extreme precipitation intensity
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., Seff was 100 % (θant=39 %).
Additionally, we increased the intensity of the input precipitation event
(delivering the tracer) from 0.1 mm h-1 (normal) to 10 mm h-1
(very large, +) and up to 100 mm h-1 (extreme, +++), in an
attempt to create infiltration excess overland flow and record its influence
on the shape of TTDs.
Influence of the sequence of precipitation events
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 Psub scenarios (Fig. S3).
The generated precipitation time series resembled our original time series
of precipitation, which also had an interarrival time close to 3 d.
All other parameters and properties of the 15 model runs were based on the
THDM scenario.
Processing of the output data
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 (6.8×10-2±7.2×10-2 %).
Creation of TTDs
The probability density distributions of transit time (the forward TTDs,
d-1) were created by normalizing the mass outflux Jout
(kg d-1) for each time step by the total inflow mass Min (kg):
TTD(t)=Joutnorm(t)=Jout(t)Min.
The cumulative TTDs (dimensionless) were created by multiplying the
normalized mass outflux (d-1) of each time step by the associated time
step length Δt (d) before cumulating it:
TTDcml(t)=∑t=0t(Joutnorm(t)⋅Δt).
Calculation of TTD metrics
For each TTD we calculated seven metrics to characterize its shape: the
first quartile (Q1), the median (Q2), the mean (mTT), the third
quartile (Q3), the standard deviation (σ), the skewness (ν) and the excess kurtosis (γ) (see Sect. S3 and Fig. S5 for details on the calculation and for visual comparison of the
metrics). Furthermore we determined the young water fraction Fyw as the
fraction of water leaving the catchment after 2.3 months (Jasechko et al.,
2016; Kirchner, 2016; Wilusz et al., 2017). For more details on how
Fyw changes with catchment and climate properties, see Sect. S4, Fig. S6
and Table S3.
Fitting
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 D (dimensionless)
and mean mTT (d):
InvGau(t)=4πDtmTT-0.51texp{-[1-tmTT2⋅mTT4Dt]}.
Gamma distribution with shape parameter α (dimensionless) and
scale parameter β (d) (with mean mTT =αβ):
Gamma(t)=tα-1e-t/ββαΓ(α).
Gamma distributions can take on very different shapes when α is
changed: α<1, highly skewed distributions with an initial
maximum and heavier (i.e., sub-exponential) tails; α=1,
exponential distribution; α>1, less skewed, “humped”
distributions with an initial value of 0, a mode and lighter tails. They can
be stretched or compressed with scale parameter β. Thus when using
gamma distributions for the determination of mTTs, it is necessary to choose
the correct shape parameter α to avoid problems of equifinality. The
same holds true for all multiple parameter distributions.
Log-normal distribution with standard deviation σ and mean
μ (both dimensionless) of the natural logarithm of the variable (with mean
mTT = exp(μ+σ2/2)):
LogN(t)=1tσ2πexp-lnt-μ22σ2.
We tested two more probability density functions both having three (instead
of just two) adjustable parameters:
Three-parameter beta distribution with shape parameters α and
β (dimensionless) and upper limit c (d) (with mean mTT =αc/(α+β)):
beta(t)=tα-1(c-t)β-1cα+β-1B(α,β).
The fourth parameter of the beta distribution could be the lower limit a. It
is not included in the above definition since in our case it is zero.
Truncated log-normal distribution with the time of truncation λ (d) as the third parameter:
Trunc(t)=1t+λσ2πexp-ln(t+λ)-μ22σ2/1-∫t=0λ1tσ2πexp-lnt-μ22σ2dt.
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
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.
Flow path number
The flow path number F is a dimensionless number proposed by Heidbüchel
et al. (2013) that relates catchment inflow to outflow (in the numerator)
while simultaneously assessing available storage space (in the denominator)
for each point in time and at the catchment scale. It was introduced to
define thresholds for the activation and deactivation of different flow
paths that transport water more slowly (e.g., groundwater flow), faster
(interflow) or very fast (macropore flow, overland flow). For this paper we
modified F slightly so that both numerator and denominator have the
dimensions (m3):
F(t)=Pdr(t)-KremDsoil(n-θant(t))Ain,
where soil depth Dsoil (m), catchment surface area Ain (m2),
porosity n (m3 m-3) and antecedent moisture content θant (m3 m-3) are paired with the driving precipitation
amount Pdr (m3), which is calculated as the average subsequent
precipitation amount Psub (m a-1) over the average event duration
tEv (d):
Pdr(t)=tEvPsub(t)Ain365.25.
The subsequent precipitation amount Psub (m a-1) is calculated for
every time step as the amount of precipitation falling within the year that
follows this time step using a moving window. Note that differing from
Heidbüchel et al. (2013) we used the event duration tEv instead of
the interevent duration tIe to compute Pdr since it better
represents the amount of precipitation falling during an average event
filling up the available storage. Furthermore, the subsurface discharge
capacity of the soil Krem (m3) consists of the effective saturated
soil hydraulic conductivity KS (m d-1), the sum of the average
interevent and event duration tIe+tEv (d), the porosity n
(m3 m-3), and the cross-sectional area of the soil layer at the
outlet of the catchment Aout (m2):
Krem=tIe+tEvKSnAout.
The cross-sectional area of the soil layer at the outlet of the catchment
Aout can be considered to represent the connection of the catchment to
either a river channel, riparian zone or the alluvial valley fill where
medium to rapid subsurface outflow from the catchment can occur. Note that
differing from Heidbüchel et al. (2013) we used the sum of the
interevent and event duration tIe+tEv instead of just the event
duration tEv to compute Krem since it better represents the amount
of water that can be removed from the catchment during an average
precipitation cycle.
The flow path number F varies in time mainly due to the changes in antecedent
moisture content θant, since variations in the amount of driving
precipitation Pdr are damped due to the moving window approach that is
used to compute it. That means F can vary quite rapidly (towards either more
positive or negative values) during the wet-up phase of a catchment and change
more slowly (towards 0) during the dry-down phase. A positive flow path
number F indicates that there is a surplus of water entering the catchment
that cannot be removed by subsurface transport at the same rate. Hence, the
storage fills up. Conversely, a negative F indicates that the drainage
capacity of the catchment exceeds the water inputs and the amount of stored
water decreases. Furthermore, values between 0 and 1 signal that the
available soil storage space is able to accommodate the net inflow of water,
while values larger than 1 mean that the catchment receives more water than
it can discharge or store in the subsurface. In turn, the larger the storage
capacity in the subsoil, the more F converges towards 0. There is only one
notable important exception to this last rule: in highly conductive soils
the increase in discharge capacity (caused by an increase in soil depth and
the consequential increase in the cross-sectional area of the soil layer at
the outlet Aout) can be larger than the increase in storage capacity
itself – leading to F becoming even more negative with increasing storage
capacity.
Results
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.
Influence of the sequence of precipitation events
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 Q1 observed for
the 15 scenarios with different event sequences is still 14 % of the
total range observed for the 36 base-case scenarios, this percentage
decreases down to 2 % for Q3. The other distribution metrics
describing the shape of the TTDs also vary a lot less between the scenarios
with different event sequences compared to the scenarios with different
catchment and climate properties (the range of all event sequences is only
1.1 % of the range of all base-case scenarios for the standard deviation,
1.6 % for the skewness and 1.0 % for the excess kurtosis). A table
with the distribution metrics for all 15 scenarios can be found in the
Supplement (Table S4). Therefore we can assume that the shape of TTDs is not
significantly influenced by the precipitation event sequence – at least in
environments with a naturally short interarrival time resembling humid
climate conditions and an event amount distribution that is exponential.
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.
Effects on TTD metrics
We found that θant affects the young parts of TTDs (the first
10 d) a lot more than the older parts (its influence is hardly
discernible after approximately 100 d; see Fig. 6a). By
contrast, KS affects the older parts more than the young parts. This
difference is due to the fact that θant constitutes one of the
initial conditions that also directly influences the current soil hydraulic
conductivity while the influence of different KS values gains more
importance later when the soil moisture conditions become more similar.
Dsoil and Psub influence all parts of the TTDs equally strongly and
hence have the smallest influence on the actual shape of the distributions.
As can be observed in (b), the influence of KS is a lot
stronger in scenarios with wet θant while the influence of
Psub decreases with increasing θant. Panel (c)
shows that both θant and Psub have a larger influence when
KS is high, but for Psub this increased influence is only seen for
the longer transit times. The influence of the initial condition θant is larger when KS is high because the relative differences in
flow through a dry soil and a wet soil are larger for soils with high
KS compared to soils with low KS. Panel (d) confirms the
impression that Dsoil only has a minor influence on the shape of TTDs –
all parts of the TTDs are equally affected and it does not make a
significant difference for the influence of the other factors whether the
soils are deeper or shallower. Finally in the lower right panel it is
demonstrated that Psub has opposite effects on the influence of
θant and KS: Larger Psub causes the influence of
KS to increase for the longer transit times while the influence of
θant decreases when Psub becomes larger. The fact that
different catchment and climate properties have varying degrees of control
on transit times depending on current conditions and the interplay of
dominant hydrologic processes has already been observed in the field
(Heidbüchel et al., 2013). Table 1 lists all metrics of the 36 TTDs
resulting from the base-case scenarios.
Influence of different properties on different parts of the TTD.
Shown is the average percentage decrease in transit time for each quartile
(Q1, Q2, Q3) and the mean (μ) of the TTD caused by a
decrease in Dsoil from 1 to 0.5 m (green), an increase in KS from
0.02 to 2 m d-1 (purple), an increase in θant from
50 % to 90 % effective saturation Seff (red) and an increase in
Psub from 0.3 to 1.4 m a-1 (blue). Panel (a) shows the average decrease in transit time for changing each of
the four properties, and panels (b–e) show the decrease in
transit time conditional on the variation of one of the four properties
(θant, KS, Dsoil and Psub). Two
examples are illustrated by the black circles: (1) The dashed blue line in
(c) shows that the increase in Psub has a larger
influence on the third quartile transit time (Q3) – a decrease of
∼75 % instead of just ∼50 % – for a
catchment with a high KS compared to a catchment with a low KS. (2) The thick red line in (e) shows that the increase in
θant from 50 % to 90 % Seff has a smaller influence
on the second quartile transit time (Q2) – a decrease of just
∼15 % instead of ∼35 % – for a
catchment with a big Psub compared to a catchment with a small
Psub.
Antecedent moisture content
Dry θant results in a lower probability for shorter transit
times while wet θant triggers faster responses and higher
initial peaks for TTDs (Fig. 7). When increasing θant by
14 % (from Seff 50 % to 90 %), on average Q1 decreases by 44 %, Q2 by 27 %, the mTT by 19 % and Q3 by
15 % (Fig. 6 center, Table 1). The median Fyw increases by 16 %.
Neither the standard deviation (and hence the width) nor the skewness nor
the kurtosis values of the TTDs are affected much by θant
though. Higher θant initially promotes faster lateral transport
(both on the surface and in the subsurface) while impeding percolation of
tracer towards the bedrock, and therefore more tracer is transported fast
towards the outlet and less tracer is entering the deeper soil layers and
the bedrock. Long-term trends or interannual shifts in Psub can cause
temporal changes in TTDs but substantial short-term variations are derived
mainly from differences in θant. Therefore variations in TTD
shape and scale can be high even in relatively small catchments. Generally,
the influence of θant is stronger for catchments with higher
KS and for climates with smaller Psub (Fig. 6).
Results of the 36 model runs. TTDs are grouped by soil depth
(a and b= deep (thick); c and d= shallow
(flat)) and soil hydraulic conductivity (a and c = high; b and d = low). Yellow colors indicate dry, green intermediate and
blue wet antecedent moisture conditions; thick lines indicate large,
mid-sized lines medium and thin lines small amounts of subsequent
precipitation amounts. Dashed black lines divide TTDs into four parts, each
part controlled by different properties. Note the log–log axes. Insets show
cumulative TTDs.
Saturated hydraulic conductivity
High KS values are associated with TTDs that have higher initial values
and lighter tails (Fig. 7). Also, a decrease in KS causes more
pronounced ups and downs in the TTD, with the effect of individual rainfall
events being better discernible even in the later parts of the TTD (Fig. 8b). Increasing KS by 2 orders of magnitude on average
shortens Q1 by 44 %, Q2 by 58 %, the mTT by 59 % and
Q3 by 62 % (Fig. 6 center, Table 1). The median Fyw increases by
13 %. The standard deviation increases with decreasing KS, while the
skewness and kurtosis both decrease significantly – TTDs become less skewed
and more platykurtic (flatter). The interplay between KS and θant is obvious in that the influence of θant decreases
over time while the influence of KS increases. Initially θant controls the soil hydraulic conductivity, the partitioning of the
tracer into surface and subsurface flow, and also the spreading within the
soil. Later on, as moisture conditions become more similar for scenarios
with identical Psub and Dsoil, KS gains in importance while
θant becomes less relevant. The influence of KS increases
for wet θant (especially for short transit times) and for big
Psub (especially for long transit times) since both maximize the
differences in hydraulic conductivity between catchments – the drier the
conditions, the more similar the unsaturated hydraulic conductivities in
general (Fig. 6).
Subsequent precipitation amount
Big Psub compresses the TTDs (Fig. 7). Doubling Psub
decreases Q1 by 63 % on average, Q2 by 61 %, the mTT by
57 % and Q3 by 58 % (Fig. 6 center, Table 1). The median
Fyw increases by 22 %. The standard deviation (and hence the width)
decreases by 42 %, while the skewness of the TTDs more than doubles.
Bigger Psub causes more leptokurtic (peaked) TTDs. Big amounts of
Psub increase the total flow through the catchment (both in the soil and
bedrock) and hence control how effectively tracer is flushed out of the
system. TTDs will have lighter tails and shorter mTTs mainly due to the fact
that a bigger Psub flushes the soils faster and only allows a smaller
fraction of the precipitation events to infiltrate into the bedrock. The
fraction of water entering the bedrock depends strongly on the contact time
of that water with the soil–bedrock interface. That means that in regions
with small Psub a larger fraction of precipitation has the chance to
infiltrate into the bedrock before it is flushed out of the soil layer by
subsequent precipitation. Therefore the tails of TTDs in more arid regions
tend to be heavier than the TTD tails in humid regions. The influence of
Psub is larger for dry θant and high KS (especially for
the longer transit times) (Fig. 6).
Soil depth
Decreasing Dsoil causes a larger fraction of tracer to arrive at the
outlet faster (Fig. 8a). Halving Dsoil shortens all the
quartiles and the mTT of the TTDs on average by approximately 40 % (Fig. 6a, Table 1), while the median Fyw increases by 10 %. The
standard deviation (the width of the TTD) is decreased by 19 % and the
skewness is increased by about 56 %. Shallower soils cause more
leptokurtic (peaked) TTDs, almost doubling the excess kurtosis. Shallower
soils saturate faster than deeper soils; they also redirect tracer more
quickly from vertical to lateral flow, and therefore the early response in
shallower soils is slightly stronger. According to our findings,
Dsoil has only a small amount of influence on TTD shape. In catchments with deeper
soils we should, however, expect longer transport times.
Influence of soil depth (a) and saturated soil hydraulic
conductivity (b) on the shape of TTDs. Lighter shades of one color
indicate shallower soils; dashed lines indicate higher hydraulic
conductivity. Insets show cumulative TTDs.
General observations on the shape of TTDs
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
∼10 d) depends strongly on θant and KS
(in accordance with Fiori et al., 2009) and less strongly on Dsoil. TTDs
in soils with wet θant or high KS exhibit higher initial
peaks with a larger probability for short transit times. Starting
after approximately 10 d a transition period follows where no individual
parameter dominates. During this period precipitation drives the emptying of
the uppermost soil layers with the presence of faster and/or larger flows
(in catchments with higher KS/bigger Psub) being gradually
compensated by higher remaining concentrations of tracer (in catchments with
lower KS/smaller Psub), so that the tracer mass outflux at the
catchment outlet converges towards a very similar value at around 120 days
before diverging again. After the transition period, the shape of the TTDs
is governed by Psub (i.e., essentially the climate) and KS, with
larger Psub and higher KS causing a more rapid decline of outflow
and hence a compression of the TTDs. Finally, the shape of the tails of the
TTDs is controlled by the hydraulic conductivity of the bedrock KBr (not
the soil KS) (see also Fiori et al., 2009). In many cases these tails
constitute straight lines in the log–log plots (which is necessary but
insufficient for identifying power-law functions). Furthermore, all modeled
TTDs share one common feature – for every subsequent precipitation event
there is a more or less discernible spike. Generally, larger subsequent
events cause higher spikes (i.e., a higher proportion of outflow during
those events) while the size of the spikes decreases at later times. And
although this multitude of local maxima in the probability density curve
does invoke a sense of irregularity, the general pattern of shapes of the
TTDs is not influenced by the individual subsequent events (Fig. 5 and Table S4), which is why we decided to smooth the TTDs for visual
comparison so that the underlying systematic changes in shapes are more
clearly visible (see Fig. S8).
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 KS, the differences between the distributions were reduced
considerably. For the cumulative probability distributions in particular there
were hardly any discernible differences left. The largest discrepancies
could still be found in the early part of the distributions where the
distributions with high θant continued to have larger outflow
probabilities. (2) For the scenarios with low KS, the individual
distributions did not collapse into a single cumulative probability
distribution. They rather split up into three distributions according to
their Psub values. That means that for the scenarios with larger
Psub a larger amount of cumulative outflow was necessary to flush out
the same amount of tracer compared to the scenarios with smaller
Psub.
Distribution fitting
Shape parameters of the best-fit inverse Gaussian (D), gamma (α) and
log-normal (σ) distributions as well as flow path numbers (F) for the
36 different scenarios are listed in Table 2. The parameters D, α and
σ range from 0.15 to 0.98, from 0.78 to 3.66 and from 0.51 to 1.15,
respectively. F ranges from -0.22 to 0.63. First we compared the
performances of only these three probability distributions with two
parameters. Out of the 36 model scenarios, the inverse Gaussian yielded the
best fit 5 times, the gamma 13 times and the log-normal 18 times. In
general, the log-normal works a little better for high KS, dry θant and small Psub and the gamma for low KS, wet θant and big Psub, while the inverse Gaussian is less ideal for
capturing the shape of the modeled TTDs (Tables 3 and S5).
Contrary to that, the inverse Gaussian represents the mean transit time
(mTT) better than the other two distributions. On average, the mTT of the
fitted gamma deviates from the observed mean by 24 % (88 d) with a
maximum deviation of 423 d for one scenario, underpredicting for dry and
overpredicting for wet θant, while the inverse Gaussian
performs much better in this regard with an average deviation from the mTT
of only 5 % (17 d) and a maximum deviation of 102 d. The gamma
especially underpredicts the mean when Psub is small. The correct
identification of the median transit time works much better for the gamma –
here the average deviation of the fitted median from the observed median is
only 4 % (12 d) with a maximum deviation of 59 d. The inverse Gaussian and log-normal yield average deviations from the median transit
time of 6 % and 5 % (15 and 13 d) with maximum deviations of 50 and 43 d, respectively.
Shape parameters of the best-fit inverse Gaussian (D), gamma
(α) and log-normal (σ) distributions and associated flow
path numbers (F) for the 36 different scenarios.
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 KS values. Finally, the
truncated log-normal distribution performed best in every regard, capturing
TTD shapes, mTTs and median transit times better than all other
distributions (mTT average deviation 3 %, 10 d, maximum deviation 91 d; median transit time average deviation 4 %, 11 d, maximum
deviation 36 d) (Table 3).
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 (α) and mean transit times (mTTs)
for individual scenarios with different combinations of catchment and
climate properties. The red boxes contain exemplary gamma distributions with
shape and scale corresponding to the red dot location. The dotted black line
marks the shape parameter value of 1 that corresponds to an exponential
distribution.
Figure 9 gives an overview of the shape and scale of our modeled TTDs (using
the best-fit gamma distribution parameters).
Predicting the shape of TTDs
Figure 10 shows how the shape and scale of TTDs change with the individual
catchment and climate properties. For increasing θant, TTDs
converge towards L-shaped distributions with shorter mTTs (in highly
conductive soils the shape is more affected than the scale, in soils with
low KS the scale is more affected than the shape). When KS is
increasing, mTT is decreasing (in the case that Psub is big, the shapes of
the TTDs also change towards having lighter tails). Quite similar patterns
can be observed for increasing Dsoil and decreasing Psub – with
mTTs becoming longer and TTD shapes increasing in tail weight when KS is
high and becoming more humped when KS is low.
Change of gamma shape parameters (α) and mean transit
times (mTTs) for four catchment and climate properties: yellow colors
indicate dry, green intermediate and blue wet θant; thick
marker lines indicate large, mid-sized lines medium and thin lines small
Psub; solid lines indicate low, dashed lines high KS; lighter shades
of a color indicate shallow, darker shades deep Dsoil. The dotted black
line marks the shape parameter value of 1 that corresponds to an exponential
distribution.
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 (Dsoil, KS, θant,
Psub) did not yield satisfying relations that could be used to predict
TTD shapes. Here, we would like to present the significant nonlinear
relationships we found between the shape parameters of the fitted TTDs and
the flow path number F (R2=0.90), mainly because we
can draw much more general conclusions on TTD shapes using a dimensionless
number (Fig. 11):
14shapeparameterαF=0.64F-0.20,ifKS<0.2md-1,15shapeparameterσF=0.12lnF+1.19,ifKS≥0.2md-1.
Generally, for similar catchments with low KS, gamma distributions are
more likely to fit the TTDs. The relatively higher proportion of surface
flow within and surface outflow from these catchments seems to favor flow
and transport dynamics that are best represented by the shapes of gamma
distributions because they are able to capture both rapid response (high
initial values) as well as the relatively slow outflow from the soils and
the bedrock (long tails). In contrast, similar catchments with high
KS and only small proportions of surface flow are more likely to behave
according to log-normal distributions with less rapid response from surface
flow (low initial values) and faster outflow from the more conductive soils
(higher and narrower modes at intermediate transit times). A notable
exception is scenarios where catchments with highly conductive soils still
experience larger proportions of surface outflow (> 25 %; F > 0.05) due to large amounts of Psub – these dynamics cannot
be predicted by the same relationship since they produce distributions with
larger contributions of advective transport and lighter tails and hence
smaller values of σ (indicated by the black circle in Fig. 11).
Relationship between the dimensionless flow path number F and the
shape parameters α (upper panel, scenarios with low KS) and
σ (lower panel, scenarios with high KS) of the gamma and the
log-normal distribution, respectively. The dotted trend lines are the
best-fit regressions for the relationship between the flow path number and
the shape parameters α (light blue) and σ (orange). The
points in the black circle are excluded from the regression analysis since
they are associated with scenarios of excessive surface outflow.
Effects of other factors on the shape of TTDsPorosity
The influence that soil porosity exerts on the shape of TTDs is quite
similar to the influence of Dsoil. Larger soil porosity causes a
dampening of the initial response and increasing transit times in all parts
of the TTD (just like deeper soils; see Fig. 12a and Table S6). Increasing porosity also causes larger standard deviations,
smaller skewness and smaller kurtosis (i.e., less peaked TTDs).
Overview of how certain catchment and climate characteristics
influence the shape of TTDs. (a) Porosity – solid lines indicate small,
dotted lines large porosity. (b) Hydraulic conductivity of the bedrock –
being equal or lower than the KS of the soil layer. (c) Decay in
saturated soil hydraulic conductivity with depth – darker shades of one
color represent scenarios with decay, lighter shades scenarios without
decay. (d) Precipitation frequency – orange TTDs are low-frequency (“arid
type”) scenarios, blue TTDs are high-frequency (“humid type”) scenarios.
The shaded areas between the lines illustrate the higher shape variability
for the low-frequency TTDs. Insets show cumulative TTDs.
Hydraulic conductivity of the bedrock
Variations in the saturated hydraulic conductivity of the bedrock KBr
affect the shape of TTDs both in the initial part of the distributions but
even more so in the tail (Fig. 12b and Table S7). If
KBr is increased so that it equals the KS of the soil layer, we
basically create one large continuum of homogeneous bedrock (or soil).
Hence, the resulting TTD does not contain any abrupt breaks in slope and
basically resembles outflow from a larger homogeneous reservoir. For lower
KBr breaks in the slope of the TTD tails start to appear indicating that
the soil layers have already been emptied while the bedrock still contains
water from the traced input precipitation event. For scenarios where
KBr is at least 3 orders of magnitude smaller than the soil KS, the
tails initially resemble power-law distributions with constants (a) around
0.2 and exponents (k) around 1.6 for longer periods of time:
TTD(t)=at-k.
An exponent k smaller than 2 indicates that a mean value of the power-law
distribution cannot be defined since it is basically infinite; however, in
our simulation results, the power-law tails eventually break down when the
bedrock domain is almost empty. Somewhat counterintuitively, the scenario
with the lowest KBr (“very low”) exhibits the shortest quartile and mean
transit times. This is clearly an effect of a smaller fraction of water
infiltrating into the bedrock and more water being transported laterally in
the relatively conductive soil layer. We observe the longest quartile
transit times in the scenario where KBr is 1 order of magnitude lower
than KS (“high”) and the longest mean transit time when it is 2 orders
of magnitude lower (“med. high”). This is due to the fact that for these
cases the higher KBr causing faster transport within the bedrock is
counterbalanced by the larger fraction of event water that enters into the
bedrock, where it is transported more slowly than in the soil. Therefore what
seems paradoxical in the first place – longer mTTs when KBr is higher
– can be explained by differences in the runoff partitioning between soil
and bedrock. This also explains the observation that the standard deviation
of the TTDs initially increases with increasing KBr while both skewness
and excess kurtosis decrease.
Decay in saturated hydraulic conductivity with depth
For catchments that already have highly conductive soils, adding a decay in
KS (with higher KS close to the surface and lower KS close to
the soil–bedrock interface) does not change the shape of TTDs to a great
extent – all shape metrics remain rather similar and transit times across
the entire TTD are moderately shortened (Fig. 12c and Table S8). We observe a larger impact if soil KS is low. In these
cases adding a decay reduces the standard deviation and increases the
skewness and the kurtosis of the resulting TTDs (i.e., they become narrower,
more skewed and more peaked). Additionally, the difference in transit times
increases towards the late part of the TTD with mTT and Q3 being
considerably shorter when there is a decay in KS. This difference
between the smaller effects of a KS decay in an already highly
conductive soil compared to the larger effects for a low conductivity soil
can be explained by the fact that the additional soil zones of higher
conductivity are more effectively used for scenarios of generally low
conductivity – in soils that are already quite conductive, a larger
fraction of the incoming event water will still infiltrate to deeper soil
layers before moving laterally, whereas in low conductivity soils the faster
lateral transport possible due to the KS decay will be triggered much
sooner and for a larger fraction of the incoming event water.
Precipitation frequency
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 (Q1=-17 %, Q2=-11 %, mTT =-9 %, Q3=-3 %).
Additionally, the higher the precipitation frequency, the smaller the
variation between individual TTDs. This is mainly due to two facts: when the
precipitation frequency is high (1) the interarrival times are shorter, which
will more often mobilize event water and avoid longer periods of relative
inactivity when the water “just sits” in the soil, and (2) the amounts of
precipitation events are on average smaller so that there is a smaller
chance of a very big event “flushing” the entire system, creating very
short transit times for a preceding event followed by a long period of no or
only small precipitation events. These transit time dynamics with regard to
different patterns of precipitation have already been observed in the field
(Heidbüchel et al., 2013).
Water retention curve
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 KS and θant) on
their shape is reduced. Sand-type TTDs are more skewed and more peaked than
silt-type TTDs (Table S10). Therefore they more closely
resemble TTDs that we would expect in environments where preferential flow
is present. Generally, the differences in TTDs between the different WRCs
are more pronounced in the scenarios with low KS because the wetting of
the upper soil layers and hence the increase in the hydraulic conductivity
takes relatively more time such that the differences between the two WRC
scenarios are amplified. In the scenarios with silt-type WRCs the saturation
process causes a slower increase in hydraulic conductivity since soil water
potential decreases more gently with increasing soil water content.
Overview of how certain catchment and climate characteristics
influence the shape of TTDs (continued). Panels (a–b): Water retention
curves (WRCs) – light blue and yellow lines indicate silt-type soil WRCs,
dark blue and orange lines indicate sand-type soil WRCs. (a) Scenarios with
high KS, (b) scenarios with low KS. (c) Catchment shape – lighter
shades of a color indicate top-heavy, darker shades bottom-heavy catchments.
(d) Full saturation and extreme precipitation – black lines indicate fully
saturated initial conditions, pink lines fully saturated initial conditions
and very large event precipitation (+), red lines fully saturated initial
conditions and extreme event precipitation (+++). The horizontal lines
in the box above the diagram indicate periods where actual overland flow was
recorded during the respective runs. The insets show the cumulative TTDs.
Catchment shape
We observe unexpectedly little variation between the TTDs of the differently
shaped catchments (Fig. 13c). While Q1, Q2 and the mTT are all
more or less similar, Q3 increases slightly for catchments with a lower
center of gravity and on average shorter flow paths (Table S11). The influence of the catchment shape is fractionally larger for
dry θant. Still, apparently the differences in catchment shape
need to be a lot more pronounced than those we explored in order to significantly
affect the TTD shape.
Full saturation and extreme precipitation
Starting runs with fully saturated soils increased the fractions of overland
flow for both the high and the low KS scenario (THSB and TLSB). For THSB
the fraction of outflow during the first 10 d that was overland outflow
(SOF10) increased from 1 % to 9 %. For TLSB the increase was even
higher, from 76 % to 91 %. The increase had clear effects on the resulting
transit times. In particular the very short transit times increased in
importance while the longer transit times were less affected. That means the
changes we observed in the shape of the TTDs followed the pattern of
increasing θant (i.e., a higher percentage of increase in the
young fraction of the TTD, smaller impact at later times, and shape
metrics). Increasing the precipitation amount and intensity of the input
event by a factor of 100 (+; from 0.1 to 10 mm h-1) affected only the
low-KS scenario (TLSB+), further increasing the fraction of short
transit times while the high-KS scenario was unaffected (THSB+). We
had to increase the precipitation intensity of the input event by a factor
of 1000 (to 100 mm h-1) to eventually create substantial amounts of
initial overland flow for both scenarios. Once this was triggered, the shape
of the TTDs changed considerably. For these scenarios (THSB+++ and
TLSB+++), all quartiles of the TTDs shortened to less than one day and
the whole distribution became extremely leptokurtic (Fig. 13d and Table S12).
DiscussionUse of theoretical distributions
The fact that TTDs under dry θant are better represented by the
(humped) log-normal distributions can be explained by the circumstance that
the (rather empty) catchment storage has to be filled at least a little bit
before faster flow paths are activated and substantial flow out of the
system can occur. This means that the early response is much better captured
by a distribution that starts with an initial value of close to 0.
Furthermore, log-normal distributions also work better in highly conductive
soils that produce TTD modes that are higher and narrower than the ones of
gamma distributions. Contrary to that, low KS values and wet θant favor gamma distributions because initial outflow values are
generally higher when the soil is closer to saturation while the TTD modes
are lower and wider in soils that are less conductive (Fig. 14).
Modeled TTDs for low KS with high θant (blue) and
high KS with low θant (yellow). Best-fit theoretical
distributions (dotted lines) for the individual scenarios for the log-normal
model (a, c) and the gamma model (b, d). Breaks in the tails
of the modeled TTDs are marked by the solid black lines. Small panels show
cumulative TTDs.
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 c) which
increases equifinality problems (that we set out to eliminate). The same
logic applies to the truncated log-normal distribution. It performs best in
almost all regards (see Table 3) but is more difficult to parameterize (e.g., we found no good relationships between the parameters σ, λ
and F), and no straightforward mathematical expressions exist that define its
moments. Therefore we recommend utilizing the two-parameter log-normal
distribution for high KS and the gamma distribution for low KS
scenarios. When doing that, we have to be careful though and consider the
distribution median as a more reliable transit time estimate than the mean
(see Table 3).
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 F, resulting from different antecedent moisture conditions θant (from blue – wet – on the left to yellow – dry – on the right) and
subsequent precipitation amounts Psub. TTDs for low KS are gamma
distributions (b); for high KS they are log-normal
distributions (c). Individual TTDs start with time shifts so that
they do not overlap (individual start times correspond to the Psub
markers in a).
Connection between the shape of TTDs and the flow path number F
We can pretty accurately predict the general shape of a TTD within the
parameter range of our model scenarios using F alone (Fig. 11). Instead of
using TTDs with constant shapes for determining variable transit times with
transfer function-convolution models, one can use these relationships to
pre-define the TTD shapes – reducing the problem of equifinality that stems
from the simultaneous determination of shape and scale parameters (Fig. 15).
Linked to that, some interesting conclusions can be drawn from the
identified relationships between F and the shape parameters α and
σ:
A flow path number between -1 and +1 characterizes catchments where
the available storage is currently larger than the change in storage caused
by the incoming and outgoing flows – over the characteristic timescale of
the combined average interevent and event duration tIe+tEv
(∼5 d).
If the system receives more water than it can remove during
tIe+tEv, it is inflow-dominated, F is positive and the shape of
TTDs is generally better represented by gamma distributions.
With increasing F, α decreases to values below 1. This decrease in
the shape parameter α is mainly caused by the initial peaks of the
TTDs becoming higher. Our simulation results suggest that the tails of the
TTDs become lighter with increasing positive F values. Therefore α
should increase with increasing positive F values. The circumstance that we find a better relationship between increasing positive F and decreasing
α values is due to the fact that – with regard to mass balance – the change in the initial response
(higher initial values and peaks) outweighs the change in the tails (that are becoming lighter). Therefore we can conclude that the early response
dominates TTD shapes (at least from a mass balance perspective).
If the system has the capacity to remove more water in the subsurface
than it receives during tIe+tEv, it is outflow-dominated, F becomes
negative and the shape of the TTDs is generally better represented by
log-normal distributions.
When F becomes more negative, σ increases from values around 0.5
to values above 1.0 (although the tails of the modeled TTDs become lighter),
indicating higher peaks.
F converges towards 0 for systems with increasing available storage
(because the denominator keeps increasing) or if inflows and outflow
capacity are evenly balanced. For these cases both gamma and log-normal
distributions become more and more dominated by smaller initial and early
values as well as the later arrival of the peak concentration, which is
illustrated by α becoming larger and by σ becoming smaller.
This should not be interpreted as growing dominance of advective over
dispersive transport because the TTD tails still become heavier in these
situations.
The theoretical framework around the flow path number F can also be used to
assess the impact that other catchment and climate properties have on TTD
shapes. For example catchment size would only have an impact on TTD shape if
the cross-sectional area of the outflow boundary Aout changed
disproportionately. If, for example, the catchment area Ain increased but the
cross-sectional area Aout remained the same, then the subsurface outflow
capacity Krem would decrease and hence F would change.
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
KS or Dsoil). That, in turn, could also mean that L-shaped (or
initially slightly humped) TTDs with heavier tails and gamma shape
parameters α around 0.5 are the natural endpoint of catchment
evolution.
Replacing transit time with cumulative outflow
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 KS scenarios (where differences are reduced) we only
generate external flow variability while for the low KS scenarios (where
differences remain) we also cause internal flow variability (Kim et al,
2016). That means that in the high KS scenarios an increase in
Psub increases the flow in all of the available flow paths
proportionally (without changing the flow path partitioning or activating
previously unused flow paths) while for the low KS scenarios an increase
in Psub causes pronounced shifts in the flow path partitioning where the
additional amount of precipitation can bypass the subsurface by
predominantly utilizing overland flow paths (leading to the observation that
a larger amount of Psub is necessary to flush out an equal amount of
tracer). This can serve as direct proof that replacing transit time with
cumulative outflow does not erase all differences between TTDs; however, it
also shows that it may be adequate for many applications where large shifts
in flow path partitioning are not expected.
Limitations and outlook
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 α values that are larger
than 1 – although such high α values were not found in previous
studies (Hrachowitz et al., 2009; Godsey et al., 2010; Berghuijs and
Kirchner, 2017; Birkel et al., 2016). Therefore, in terms of expanding the
modeling effort, it would be very beneficial to include both
evapotranspiration and macropore flow into the simulations. An inclusion of
these processes will shift the flow path number F towards more negative
values. On the one hand, evapotranspiration will provide an additional way
to remove water from the subsurface (representing another sink term similar
to Krem) and macropore flow will enhance the subsurface outflow capacity
of the catchment resulting in a shift towards TTDs with higher initial
peaks. On the other hand, evapotranspiration also has the potential to
reduce θant below moisture levels obtainable with free
drainage alone. This more extreme dryness could lead to even more humped
TTDs with initial values closer to 0. The inclusion of additional
heterogeneity in soil properties (layering, small-scale variations) would
also be a worthwhile exercise that is, however, outside of the scope of our
study. Therefore, since some of the potential shape-controlling parameters
are still excluded from the analysis (like, for example, KBr or the
precipitation event amount PEv), this study is not meant to represent
the full and complete truth about TTD shapes. It is rather an attempt to
find some structure in the way TTD shapes change with certain parameters and
boundary conditions, an attempt to illuminate essential dynamics and to
explore overarching principles in catchment hydrology. Therefore, the next
important step is to verify the generality of these model findings and the
resulting theory on catchment response with field observations. In particular
since under many circumstances, e.g., in areas where soils are characterized
by macropores and preferential flow pathways, traditional hydrological
modeling (i.e., the applicability of the Richards equation) may not be
suitable.
Gamma distributions (solid lines) capture the middle part of the
modeled TTDs (dashed lines; thickness corresponds to Psub amount) quite
well but do not correctly represent the initial parts, breaks in the tails
and heavier tails. Inset: gamma distributions (thick and thin black solid
lines) combine either high initial values with heavier tails or zero initial
values with lighter tails while modeled TTDs often are best described by
high initial values and lighter tails (blue dashed line) or low (albeit
nonzero) initial values with heavier tails (yellow dashed line).
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.
Conclusion
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 (Dsoil, KS, θant and
Psub) so that it is possible to predict the change using the
dimensionless flow path number F. The results can be summarized in three main
conclusions (see also Fig. 11):
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 Psub is
relatively small. Lighter tails appear when the system is in a more
“stressed” state, where the shallow and faster flow paths are
disproportionally used for transport. This can be associated with larger
Psub values. In addition, we observe a distinct break in the TTD tails
if there is a sufficiently large difference in hydraulic conductivity
between the bedrock KBr and the soil KS.
Gamma functions are able to capture the time variance of TTDs in an
appropriate way, especially for low KS and wet θant
scenarios, while log-normal distributions work well for high KS and dry
θant scenarios.
However, neither gamma nor log-normal distributions are able to correctly
represent the early part of the simulated distributions with nonzero
initial values combined with a mode shortly after (i.e., the humped form)
that we observe in most cases. Moreover, we noticed the general pattern that
TTDs with high initial values tend to have lighter tails than TTDs with low
initial values. Gamma distributions, unfortunately, exhibit the opposite
behavior (with high initial values being associated with heavier tails than
low initial values; see Fig. 16). Based on the results from our modeling
efforts, we therefore encourage the exploration of better-fitting
theoretical distributions. These distributions should be able to (a) represent high initial values paired with lighter tails as well as low
initial values paired with heavier tails and (b) take on a “humped” form
with nonzero initial values. We found that truncated distributions fulfil
these requirements a lot better but have more degrees of freedom and are
harder to parameterize.
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.
Data availability
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: https://doi.org/10.5194/hess-24-2895-2020-supplement.
Author contributions
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.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
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.
Financial support
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.
Review statement
This paper was edited by Nunzio Romano and reviewed by three anonymous referees.
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