The transit time of water is a fundamental property of catchments, revealing
information about the flow pathways, source of water and storage in a single
integrated measure. While several studies have investigated the relationship
between catchment topography and transit times, few studies expanded the analysis to a wide range of catchment properties and assessed
the influence of the selected transfer function (TF) model. We used stable water
isotopes from mostly baseflow samples with lumped convolution models of time-invariant TFs to estimate the transit time distributions of 24
meso-scale catchments covering different geomorphic and geologic regions in
Switzerland. The sparse network of 13 precipitation isotope sampling sites
required the development of a new spatial interpolation method for the
monthly isotopic composition of precipitation. A point-energy-balance based
snow model was adapted to account for the seasonal water isotope storage in
snow dominated catchments. Transit time distributions were estimated with
three established TFs (exponential, gamma distribution and two
parallel linear reservoirs). While the exponential TF proved
to be less suitable to simulate the isotopic signal in most of the
catchments, the gamma distribution and the two parallel linear reservoirs
transfer function reached similarly good model fits to the fortnightly
observed isotopic compositions in discharge, although in many catchments the
transit time distributions implied by equally well fitted models differed
markedly from each other and in extreme cases, the resulting mean transit time (MTT)
differed by orders of magnitude. A more thorough comparison showed that
equally suited models corresponded to agreeing values of cumulated transit
time distributions only between 3 and 6 months. The short-term
(

Stable water isotopes or other natural constituents, such as chloride, in
precipitation act as environmental tracers whose signals are altered by
hydrological processes, storage and mixing inside a catchment. Measurements
of those environmental tracers in discharge can be used to infer transit (or
travel) time distributions (TTDs) and mean transit times (MTTs) on the
catchment scale

Transit time estimations based on lumped convolution modelling approaches
have been carried out in various studies, reviewed by

Some more recent studies

While these more recent approaches seem to be more suited to reproduce a
natural catchment's TTDs, they all come at an increased cost. In order to
keep the computational cost of the optimisation manageable,

So even though the lumped convolution modelling approach with time-invariant
TTDs has several shortcomings and is likely to be superseded by more
sophisticated modelling approaches in the future, to date the only
practical alternatives to consider a greater number of catchments without
additional assumptions to reduce the number of parameters using commonly
available computing resources are time-invariant TTDs. Neither the fitting of
sine waves

Several studies were dedicated to the investigation of the potential
relationship between catchment properties and MTTs.

The objective of this study was to determine TTDs of 24 catchments in Switzerland and to assess the relationship of MTT and other proxies to catchments' topographical indices, with the final aim of finding a topography driven regionalisation method. Another focus of this study was on a comparison of the MTT estimates from different TFs to assess the suitability of different TF types. Furthermore, the influence of seasonal snow storage in alpine catchments necessitated the development of a snow module, which accounts for the isotopic composition of snow storage and melt water. The sparse network of precipitation isotope sampling sites required the development of a new spatial interpolation method for the monthly isotopic composition of precipitation.

Map of the study area with elevation and catchment borders. The catchment that is not shown, Oberer Rietholzbach, is a subcatchment of the Rietholzbach catchment. The symbols indicate positions of isotope measurement sites of various sources.

This study focused on 24 catchments distributed across the Swiss Plateau and
the Swiss Alps (see Fig.

Mean annual catchment precipitation sums range from 1012 to 2600 mm. The seasonal distribution of precipitation is fairly balanced, with 54 to 61 % of annual precipitation occurring during the summer half year. Primarily elevation dependent temperature differences cause a range of discharge regimes from pluvial for catchments of the Foothill zone and the Submontane zone to nival for catchments of the Alpine zone. Different underlying geologies, from crystalline and limestone in the Alps to flysch and molasse in the Swiss Plateau, in connection with varying topographical conditions led to a variety of soils and further differences in discharge behaviour among the catchments.

The Swiss Federal Office for the Environment (FOEN) provided the daily discharge data for most of the catchments. Discharge data for the catchments Luempenenbach, Erlenbach and Vogelbach were obtained from the Swiss Federal Institute for Forest, Snow and Landscape Research (WSL). Additional discharge data for the catchments Roethebach and Emme were provided by the Amt für Abwasser und Umwelt (AWA) of the Swiss Canton Berne.

Areas, elevations and mean annual precipitation sums of the 24 studied catchments.

The climate data, including average catchment precipitation, temperature,
relative air humidity, wind speed and global radiation for 100 m
elevation bands in each catchment based on interpolated site data from the
national meteorological service of Switzerland (MeteoSwiss) were provided by
the PREVAH working group

All isotopic compositions in this study are expressed in the

In our study region, the ratio of

The National Network for the Observation of Isotopes in the Water Cycle
(NAQUA-ISOT) of the Federal Office for the Environment (FOEN) of
Switzerland measures stable water isotopes (

In order to derive topography based indices for the 24 catchments,
a topographic terrain analysis based on a digital elevation model (DEM) with
a resolution of 25 m was carried out with the free open source
software SAGA-GIS

The SAGA module “Overland Flow Distance to Channel network” was used to
calculate the flow path lengths

The isotopic composition of precipitation was not directly measured within
the catchments. Instead, the following procedure to interpolate the available
site data was applied:

Based on the

Monthly and average monthly

where

The average monthly elevation corrected

To derive the

First, the measurement site closest to the location

To compare the results obtained by the lumped convolution approach using
time-invariant TFs with a more simplistic approach, we also adopted the
transit time proxy approach described by

The model framework in this study is based on the TRANSEP-framework

Overview of transfer functions with specification of the parameters and analytical MTT.

Since several of the selected catchments are notably influenced by snow
accumulation and snow melt processes, the implementation of a snow model was
crucial. Due to a lack of suitable snow data for the calibration of a simple
parameterised snow model and the availability of the appropriate climatic
input data, a point-energy-balance based approach was chosen. This study uses
a modified implementation of ESCIMO (Energy Balance
Snow Cover Integrated Model by

change of time step length from hourly to daily (significant snowfall
rate of 0.5 mm h

calculation of incoming long-wave radiation with available input data and
an empirical relationship given in

Discharge and its isotopic compositions were simulated with two similar
lumped convolution models. Both of these models require effective
precipitation as their input. The effective precipitation was computed with
a rainfall-loss module. While the proposed modelling framework is not bound
to any particular method for computing the effective precipitation, we used
the approach described by

Discharge

In this time-invariant modelling approach, the optimised tracer TF can be considered to represent the respective catchments' TTD.

Table

Overview scheme of the model modules. Grey boxes represent input data, blue boxes represent data computed by model modules (white boxes).

The discharge convolution module was mainly needed as an auxiliary means to
constrain the parameters of the rainfall-loss module. As initial testing
revealed, the TPLR was clearly outperforming the GM and the EM as hydraulic
TF and was therefore a priori selected as the sole hydraulic TF

Regardless of whether or not previous transit time studies mentioned different tracer
TFs, for catchment comparisons most of them focused on one of them:

Due to the large number of optimisations (three models with seven to nine
parameters for 24 catchments), Monte Carlo sampling was deemed impracticable
for this study. Instead, a multi-objective optimisation approach using the
NSGA-II algorithm after

Three objective functions were applied to evaluate the model: KGE

When dealing with multiple separate objective functions, there is no clear
best solution, as the improvement of one objective function value can impair
another. All combinations of objective
function values where this is the case are Pareto optimal. The
multi-objective NSGA-II optimisation algorithm

Monthly maps of interpolated sea level precipitation

Not all of the Pareto optimal parameter sets lead to sensible solutions, as
at a certain point minimal improvements in respect to the value of one
objective function lead to substantial deterioration of the values of the
other objective functions. Similarly to combining three single objective
functions into one for the Kling–Gupta efficiency

where

The results of the iterative meta-heuristic NSGA-II algorithm are not suited
to be used within the established Generalised Likelihood Uncertainty
Estimation

To compare the characteristics of TTDs resulting from the three TF types
across all catchments, we started by identifying the best TF type for each
catchment, i.e. the TF type with the highest median

Monthly elevation gradients of

Values of the three objective functions for all catchments for the three different TF types.

For some catchments, the intended number of 300 Pareto optimal solutions was
exceeded after the first run and it could easily be increased to 1000; for
other catchments the required number of 300 Pareto optimal solutions demanded
several repetitions of the optimisation algorithm. Consequently, the number
of acceptable solutions and the quality of the obtained Pareto fronts varied
between the catchments and the TF types, so that the final analyses were
based on 30–100 (10 % of 300–1000) parameter sets for each catchment and TF
type. The parameters of the rainfall loss module after

Independent from the three different tracer TF types, the rainfall-runoff
component of the model performed equally satisfactorily for most of the
studied catchments, reaching KGE

Optimisation results for selected catchments. Left panels: observed and predicted isotope concentrations in discharge. Right panels: cumulated TTDs (thicker lines represent the median values of all accepted solutions, thinner lines indicate their range). Centre panels: objective function values for isotopic composition predictions, biases of the predictions and MTTs implied by the optimised TFs; lines indicate the full value range, diamonds indicate the 25, 50 and 75 percentiles of the accepted solutions.

Objective function values for the prediction of isotopic compositions in
discharge for the three different TF types are listed in the lower part of
Fig.

Top panel: cumulated response time distributions (RTDs) from the rainfall-discharge module; lower three panels: cumulated TTDs obtained by the three TF types. The shown curves are the median values of all (i.e. 30–100) accepted solutions.

Regardless of the applied TF types, all predicted

Cumulative distribution of catchment MTT estimates based on the three TF types.

Despite the quite similar performances of the simulations based on TPLR and
GM TFs, clear differences of the TTD shapes were observed
(Fig.

The MTTs for all TFs agreed only for two catchments: Schaechen (MTT
of 1.2 years) and Sitter (MTTs between 0.7 and 0.9 years). For the
other catchments, MTT estimates of different TF types occasionally varied by
orders of magnitude (see Table

Spearman rank correlation coefficients (

Combined matrix of Pearson correlation coefficients (lower left) and
Spearman rank correlation coefficients (upper right) for MTTs of all
catchments derived by the three different TF types and the TTP. All
correlations were significant (

The comparison of the cumulated TTDs of the three model types (examples for
five selected catchments in the right column of
Fig.

Without discussing all topographic indices (see Table

Comparison of cumulated discharge fractions after certain elapsed times. Top panel: correlation coefficients between TTDs of specific TFs and a selection of the best TTDs for each catchment. Bottom panel: mean value ratios between specific and selected best TFs.

We observed a high agreement between the cumulated TTD fractions of the first
3 months (hereafter CF3M) for GM and TPLR (see
Fig.

However, the strongest correlations were found between the transit time
metrics (CF3M and MTTs including TTPs) and the mean discharge of the
catchments (

Results of the topographic analysis.

Pearson correlation coefficients (

Minimum, median and maximum MTT in years for each catchment and TF type and the respective TTP values.

In order to estimate the effective precipitation amounts, the discharge
amounts were considered during the multi-objective optimisation procedure.
The relatively simple TPLR discharge convolution module managed to predict
annual discharge reasonably well for most catchments. As it turned out, the
optimised parameters for the rainfall loss module and the discharge
convolution module did not depend on the chosen isotopic TF. This suggests
that both of them could have been calibrated before and independently from
the isotopic convolution module and only once for all TFs, as done by

Despite the availability of precipitation isotope concentration data being
suboptimal (insufficient precipitation isotope data, directly measured only
within a few of the study catchments and a sparse measurement network in
a region with distinct topography), the interpolation method described in
Sect.

The convolution model could adequately reproduce the seasonal variations of the isotope concentrations in stream water; however, all predictions exhibited a bias. For most of the catchments, the biases were independent from the applied TF, indicating that the systematic bias was not caused by the choice of TFs. Upon closer inspection, three possible reasons for this bias have to be considered.

First, there could be a bias in the precipitation isotopes, caused by incorrect assumptions made during the interpolation of the sparse measurement site data. The resulting biases could be positive or negative and are more likely to occur in regions where the surrounding measurement sites are further apart and the catchment elevations exceed the elevations of the measurement sites.

Another error source for the input isotope concentration of alpine catchments
could be assumptions made for the snow module. Particularly the assumption of
isotopical homogeneous melt from the snow pack without significant enrichment
is debatable, as

The third possible cause of the prediction biases is inherent to the model,
more precisely its rainfall-loss module. Since there is no representation of
a soil storage, where winter and summer precipitation can mix to a certain
extent, the simulated evapotranspiration, occurring predominantly during
summer, consists almost entirely of the isotopically heavier summer
precipitation. On the other hand, nearly all of the isotopically lighter
winter precipitation is routed to discharge. While it is likely that the
largest part of the yearly evapotranspiration stems from summer precipitation
and that a larger fraction of winter precipitation contributes to discharge,
it can be assumed that the missing model representation of a mixing soil
storage necessarily leads to a prediction bias towards lighter discharge
isotope concentrations. This kind of bias might be prevalent at the
non-alpine catchments, where all predictions have a slightly negative bias
between 0 and

As all simulated values can only be compared to the observed values, the
coarse temporal resolution of the isotopic input data (fortnightly data in
streamflow and monthly bulk sampled precipitation isotope data) is not suited
to evaluate the short-term components of the TTDs. At the same time, the
increased dampening of the seasonal variation of the

The inter-model comparison in Fig.

This might help to explain the low identifiability of the TPLR model's
parameter representing the MTT of the slow reservoir

As mentioned in the previous section, a TTD containing longer transit times
cannot be properly assessed solely with a cyclical annually varying
environmental tracer such as

Even though the MTT estimates vary between the different model types (see
Fig.

Given a sufficiently high measurement frequency, stable water isotope data should be suited to characterise the short-term and intermediate part of a catchment's TTD, but it certainly does not contain enough information to determine complete TTDs or actual MTTs of a catchment.

Despite the distinct differences between MTT estimates based on
different TF types, the results in Table

However, in this study most of the observed correlations were only
significant as long as the external climatic forcing was not taken into
account. The correlation between MTTs and mean annual discharge was higher
than for any of the topographical indices. For two hypothetical catchments,
which share identical properties regarding geology, topography, soils and
vegetation, the catchment with the higher effective precipitation would
undoubtedly have higher turnover rates and hence lower MTTs. Consequently, a
catchment's MTT actually always will be determined by two components:
external forcings (precipitation and potential evapotranspiration) and
catchment internal properties. When the aim of a study is the assessment of
the influence of catchment properties (such as topography) on MTTs, it would
appear that to be essential to take external forcings into account. Yet many
studies (e.g.

Together with the aforementioned issue, the uncertainties connected to the determination of MTTs (Which is the most appropriate TF? Is the time-invariant TF approach suited at all? How can the TTDs' tailing be properly assessed?) will lead to high degrees of uncertainty for any approach to regionalise MTTs.

On top of that,

In this study, we used three different TF types in a time-invariant lumped
convolution modelling approach to estimate the TTDs and MTTs of 24 meso-scale
catchments in Switzerland on the basis of

The poorly identifiable tailings of the TTDs greatly influenced MTT estimates, which partially exhibited high uncertainties. For catchments with longer MTTs, different model types' MTT estimates could differ by orders of magnitude while the available data were not suited to determine the most appropriate model type. In many cases the EM proved to be less appropriate than the more flexible GM and the TPLR. Given the fact that the easily computable TTP values showed significant rank correlations to MTT estimates of all TF types, they might serve as a coequal replacement for them, as long as the latter are as underdetermined as in this study and only relative differences among the catchments are the focus.

The results of this study suggest that seemingly good correlations between
MTTs and the ratio of median flow path lengths over median flow path
gradients

Within the frame project of this study, bulk precipitation samples have been taken to determine the isotopic composition of the precipitation at five sites in Central Switzerland. With lengths of not more than 1 year and limited spatial coverage, these time series were of little use as model input data. Three of those sites, Benglen, Schallenberg and Aeschau, have been chosen to validate the interpolated precipitation isotope data.

Further isotope composition data were thankfully obtained from

The Institute for Atmospheric and Climate Science (IAC) of the Swiss Federal
Institute of Technology Zurich maintains the field measurement site
Messtelle Büel within the catchment Rietholzbach for
which fortnightly bulk sample data for

The method described in Sect.

Figure

Comparison of measured

This work has been funded as part of the National Research Programme NRP 61 by the Swiss National Science Foundation. We are grateful to Massimiliano Zappa from the Swiss Federal Institute for Forest, Snow and Landscape Research WSL, who provided the preprocessed PREVAH-climate data, and Manfred Stähli (WSL) for discharge data on the catchments Vogelbach, Erlenbach and Luempenenbach. Furthermore, we would like to thank Matthias H. Mueller (University of Basel) for the provision of supplemental precipitation isotope data. The article processing charge was funded by the German Research Foundation (DFG) and the Albert Ludwigs University Freiburg in the funding programme Open Access Publishing. Edited by: B. Schaefli