**Technical note**
06 Mar 2020

**Technical note** | 06 Mar 2020

# Technical note: An improved discharge sensitivity metric for young water fractions

Francesc Gallart Jana von Freyberg María Valiente James W. Kirchner Pilar Llorens and Jérôme Latron

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**Francesc Gallart et al.**Francesc Gallart Jana von Freyberg María Valiente James W. Kirchner Pilar Llorens and Jérôme Latron

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^{1}Surface Hydrology and Erosion group, Department of Geosciences, IDAEA, CSIC, Barcelona, Spain^{2}Department of Environmental Systems Science, ETH Zurich, Zurich, Switzerland^{3}Swiss Federal Institute for Forest, Snow and Landscape Research (WSL), Birmensdorf, Switzerland^{4}Geodynamics Department, University of the Basque Country, Leioa, Spain

^{1}Surface Hydrology and Erosion group, Department of Geosciences, IDAEA, CSIC, Barcelona, Spain^{2}Department of Environmental Systems Science, ETH Zurich, Zurich, Switzerland^{3}Swiss Federal Institute for Forest, Snow and Landscape Research (WSL), Birmensdorf, Switzerland^{4}Geodynamics Department, University of the Basque Country, Leioa, Spain

**Correspondence**: Francesc Gallart (francesc.gallart@idaea.csic.es)

**Correspondence**: Francesc Gallart (francesc.gallart@idaea.csic.es)

Received: 28 Aug 2019 – Discussion started: 12 Sep 2019 – Revised: 24 Jan 2020 – Accepted: 06 Feb 2020 – Published: 06 Mar 2020

Recent virtual and experimental investigations have shown
that the young water fraction *F*_{yw} (i.e. the proportion of catchment
outflow younger than circa 2–3 months) increases with discharge in most
catchments. The discharge sensitivity of *F*_{yw} has been defined as
the rate of increase in *F*_{yw} with increasing discharge (*Q*) and has been
estimated by the linear regression slope between *F*_{yw} and *Q*, hereafter
called DS(*Q*). The combined use of both metrics, *F*_{yw} and DS(*Q*), provides a
promising method for catchment inter-comparison studies that seek to
understand streamflow generation processes. Here we explore the discharge
sensitivity of *F*_{yw} in the intensively sampled small Mediterranean
research catchment Can Vila. Intensive sampling of high flows at Can Vila
allows young water fractions to be estimated for the far upper tail of the
flow frequency distribution. These young water fractions converge toward 1
at the highest flows, illustrating a conceptual limitation in the linear
regression method for estimating DS(*Q*) as a metric of discharge sensitivity:
*F*_{yw} cannot grow with discharge indefinitely, since the fraction of young
water in discharge can never be larger than 1. Here we propose to quantify
discharge sensitivity by the parameter of an exponential-type equation that
expresses how *F*_{yw} varies with discharge. The exponential parameter
(*S*_{d}) approximates DS(*Q*) at moderate discharges where *F*_{yw} is well below
1; however, the exponential equation and its discharge sensitivity metric
better capture the non-linear relationship between *F*_{yw} and *Q* and are
robust with respect to changes in the range of sampled discharges, allowing
comparisons between catchments with strongly contrasting flow regimes.

The seasonal cycles of stable isotopes in precipitation are damped and
phase-shifted as they are transmitted through catchments and thus can be
used to infer properties of catchment travel-time distributions (e.g.
DeWalle et al., 1997; McGuire and McDonnell, 2006). The young water fraction
(*F*_{yw}), or the proportion of catchment outflow younger than circa 2–3 months,
can be estimated as the ratio between the seasonal cycle amplitudes of
stable water isotopes in precipitation and stream water. This ratio
consistently predicts *F*_{yw} across a wide range of transit time
distributions, whereas the same range of distributions yields widely varying
mean transit times (Kirchner, 2016a).

The young water fraction usually increases with stream discharge (Kirchner,
2016b). To account for this flow dependency in their study of 22 Swiss
catchments, von Freyberg et al. (2018) distinguished between time-weighted
(*F*_{yw}) and flow-weighted ($F{{}^{*}}_{\mathrm{yw}}$) young water fractions and introduced
the “discharge sensitivity of the young water fraction” (which we term
DS(*Q*)) as a metric of the progressive increase of *F*_{yw} with
increasing catchment discharge (*Q*). Thus, by combining the mean $F{{}^{*}}_{\mathrm{yw}}$ and
its sensitivity to discharge, catchment young water response can be
classified in two dimensions: catchments with low or high $F{{}^{*}}_{\mathrm{yw}}$ and with
low or high DS(*Q*) (Fig. 10 in von Freyberg et al., 2018). Because these two variables
did not correlate with each other and correlated with different catchment
characteristics, von Freyberg et al. (2018) suggested that $F{{}^{*}}_{\mathrm{yw}}$ and DS(*Q*) are two
independent metrics that can be informative in catchment inter-comparison
studies.

These authors used the linear slope between *F*_{yw} (–) and discharge rate
*Q* (mm d^{−1}) for calculating DS(*Q*) (d mm^{−1}). The use of discharge rate
instead of volume rate (m^{3} d^{−1}) is sensible because of its
independence from catchment area. Von Freyberg et al. (2018) justified the choice
of using *Q* as forcing variable instead of log(*Q*), which is more sensitive to
low flows, by the main focus of the study being storm runoff generation.

Von Freyberg et al. (2018) determined DS(*Q*) through a non-linear fitting algorithm.
They assumed that the seasonal cycle amplitude (*A*_{S}) of the stable water
isotope signal in stream water varies with *Q*, but the corresponding cycle
amplitude in precipitation (*A*_{P}) does not, such that *F*_{yw} varies with
*Q* as

and the isotopic signal of stream water *c*_{S}(*t*) (‰)
follows a sinusoid function

where *φ*_{S} is the phase of the seasonal cycle (rad), *t* is the time
(fractional years), *f* is the frequency (yr^{−1}, equal to 1 for a full
annual cycle) and *k*_{S} (‰) is a constant describing
the vertical offset of the isotope signal.

Then if *A*_{S} is approximated as a linear function of *Q*,

Eq. (2) can be rewritten as

and the slope (*m*_{S}) and the intercept (*n*_{S}) of Eq. (3) can be obtained
from time series of *c*_{S} and *Q* by fitting the four parameters *m*_{S},
*n*_{S}, *φ*_{S}, and *k*_{S} in Eq. (4) using non-linear fitting
methods.

Combining Eqs. (1) and (3) yields

Thus DS(*Q*), the linear slope of the dependence of *F*_{yw} on *Q*, can be approximated
as *m*_{S}∕*A*_{P}, which has units of *Q*^{−1}.

We applied the approach outlined above to the small Mediterranean Can Vila
catchment (Vallcebre Research Catchments, Llorens et al., 2018). The objectives
were to better understand the Can Vila catchment's hydrology and to test the
*F*_{yw} and discharge sensitivity concepts in an environment that was
different, in terms of climate, catchment characteristics and sampling
strategy, from the Swiss catchments studied by von Freyberg et al. (2018). This
technical note focuses only on the aspects of this research that are
relevant to the estimation of *F*_{yw} and its discharge sensitivity, as other
aspects of the Can Vila catchment study will be presented in a separate
publication (Gallart et al., 2020).

The Can Vila catchment (Table 1) is a 0.56 km^{2}, semi-humid
Mediterranean mid-elevation (1115–1458 m a.s.l.) catchment with a
rainfall-dominated flow regime. Stream discharge varies greatly, from zero
flows during some summer periods to several-day-long floods associated with
saturation generation mechanisms during wet periods (Latron and Gallart,
2008; Latron et al., 2009). In addition to long-term hydrometric monitoring since
the early 1990s, precipitation and stream water stable isotopes were
sampled from May 2011 to September 2013 and from May 2015 to May 2016.
During the isotope sampling period, 5 min discharges ranged from zero to
2.621 m^{3} s^{−1} (equivalent to 4.68 m^{3} s^{−1} km^{−2} or 404 mm d^{−1}), with a highly skewed flow duration curve (i.e. 30 % of
total stream discharge flowed through the gauging station during 1 % of
the time). A “smart sampling strategy” was used to obtain
flow-representative water samples, consisting of the combination of two
automatic water samplers, one triggered by time and the other by flow. The
sampling frequency was higher during the rising limb of the hydrograph than
during the falling limb, in order to compensate for the rising limb's
shorter duration. The resulting sampling intervals varied between 30 min
and 26 d with a maximum sampled discharge equivalent to 226 mm d^{−1}.
We investigated the young water fraction and its discharge sensitivity for
the Can Vila catchment using this 40-month-long isotope time series
containing 464 precipitation and 858 streamflow samples. Given the drier
climate, the smaller catchment area and the much finer timescale for
sampling, this data set extends the range of catchments investigated by von
Freyberg et al. (2018).

For the Can Vila catchment, the flow-weighted young water fraction ($F{{}^{*}}_{\mathrm{yw}}=\mathrm{0.226}\pm \mathrm{0.028}$) was much larger than the time-weighted young water fraction (${F}_{\mathrm{yw}}=\mathrm{0.061}\pm \mathrm{0.008}$). Both values fell within the range of those reported by von Freyberg et al. (2018), but the ratio between them was larger than at the Swiss catchments, suggesting that young water fractions are more sensitive to discharge at Can Vila than at most of the Swiss sites.

To further explore the discharge sensitivity DS(*Q*) at Can Vila, we estimated young
water fractions for different quantiles of the flow regime (similar to Fig. 7 in von Freyberg et al., 2018), extending the range to portray the highest flows
(up to the top 0.25 %; Fig. 1). Our flow-dependent sampling strategy
intensively sampled these high flows, which conventional sampling at regular
time intervals would miss. Figure 1 shows that *F*_{yw} increases with
increasing discharge, from nearly 0 at the lowest discharge to nearly 1 for
*Q*≥24 mm d^{−1}. This behaviour partly corresponds to a high-DS(*Q*) type 2
catchment in Fig. 10 in von Freyberg et al. (2018). However, the non-linear
behaviour of *F*_{yw} with increasing flow shown in Fig. 1 is
inconsistent with a linear model of discharge sensitivity. Very small
*F*_{yw} values (*<*0.1) during baseflow are consistent with the long (7.7 years) mean transit time of base flows obtained in this catchment
(Gallart et al., 2016), whereas the high sensitivity of *F*_{yw} to discharge
reflects the varying pre-event water contributions (30 %–90 %) observed for
different flow events (Llorens et al., 2018).

Equations (4) and (5) (numbered 9 and 10 in von Freyberg et al., 2018) yield a
discharge sensitivity DS(*Q*) value of 0.0128±0.0017 d mm^{−1} for the
Can Vila catchment (dashed grey line in Fig. 1), which is among the smallest
discharge sensitivities obtained for the 22 Swiss catchments, in contrast
with the visibly high discharge sensitivity of Can Vila over the range of
its flow regime. Figure 1 shows that the linear design of DS(*Q*) is clearly
inadequate to capture the asymptotic convergence of the young water fraction
toward *F*_{yw}≈1 at the far upper tail of the flow distribution.
Highly dynamic catchments such as Can Vila, and flow sampling strategies
like those employed here, demonstrate that a non-linear discharge
sensitivity function is needed.

An alternative, non-linear model can be derived by noting that the sum of
old and young water fractions is always 1, and by assuming that the old
water fraction decreases with increasing discharge and asymptotically
approaches 0 (and thus the young water fraction asymptotically approaches 1)
as *Q* approaches infinity. We propose the following equation, where the old
water fraction decreases exponentially with increasing *Q*, and the young water
fraction grows accordingly:

where *F*_{0}(–) is the virtual *F*_{yw} for *Q*=0 and *S*_{d} (unit of
*Q*^{−1}) is the new discharge sensitivity metric. The red curve in Fig. 1
shows the application of this equation to the Can Vila data.

On combining Eqs. (1) with (6) and re-arranging the formula so that only
*A*_{S}(*Q*) remains on the left side of the equation, we obtain

Finally, by inserting Eq. (7) into Eq. (2), the *F*_{0} and *S*_{d} parameters can be obtained by fitting a sinusoid function to the seasonal
variation of the isotopic signal of stream water *c*_{S}(*t*):

We obtained the *F*_{0} and *S*_{d} parameters with a non-linear analytic
Gauss–Newton algorithm in which we used streamflow rates as weights.

Taking the derivative of Eq. (6) with respect to *Q* directly yields the result
that the local discharge sensitivity $\frac{\mathrm{d}{F}_{\mathrm{yw}}\left(Q\right)}{\mathrm{d}Q}$ at low discharges will be directly related to (and
in many cases nearly equal to) *S*_{d}:

When *F*_{0} is small, *S*_{d} will be a good approximation to the slope of
the relationship between *F*_{yw} and *Q* at discharges that are low enough to
keep *F*_{yw} still far from 1.

We used the Can Vila data set to test the robustness of the *S*_{d} metric, in
comparison with the original DS(*Q*) metric defined by von Freyberg et al. (2018) and
with several alternative metrics designed to reduce or avoid some of the
DS(*Q*) metric's limitations. We investigated how these metrics changed when we
excluded the discharge and water samples for the highest flows from the Can
Vila data set (Fig. 2). This allowed us to test how these discharge
sensitivity metrics were affected by the availability (or, conversely, the
lack) of tracer data encompassing extreme flows.

For this purpose, we compare the new *S*_{d} metric, the original DS(Q) metric and
several dimensionless options that used log(*Q*), *Q*∕*Q*_{max}, and *Q*∕*Q*_{mean}
instead of *Q* in the calculations (*Q*_{max} and *Q*_{mean} correspond to the
maximum and mean values of the discharge rates *Q*(*t*) associated with stream
water sampling). We call the resulting discharge sensitivity metrics
DS(log*Q*), DS(*Q*_{max}) and DS(*Q*_{mean}), respectively. Note that DS(*Q*_{max}) and DS(*Q*_{mean}) may be obtained by multiplying any
previously calculated DS(*Q*) value by *Q*_{max} or *Q*_{mean}.

The new exponential *S*_{d} metric values (Fig. 2a) show some scatter but are
robust to changes in the underlying data, exhibiting no systematic trend as
the high-flow observations were progressively discarded. In contrast, DS(*Q*) is
highly sensitive to changes in the analysed range of discharges (Fig. 2b),
rapidly increasing (by a factor of 5) on exclusion of the highest flows from
the calculations and reaching its maximum value on exclusion of the upper 5 % of flows (*Q**>*4.82 mm d^{−1}), corresponding to
everything above the green dot (Top 5 %) in Fig. 1. Note that, as
suggested by Eq. (9), DS(*Q*) takes values similar to *S*_{d} when the highest flows
are excluded. DS(log*Q*) declines promptly on omission of the highest flows (Fig. 2c)
but remains stable afterwards. DS(*Q*_{mean}) behaves similarly to DS(*Q*); i.e. it is smallest
when the complete data set is used and is largest on exclusion of the
highest 5 % of flows from the analysis (Fig. 2d). Finally, Fig. 2e shows
that DS(*Q*_{max}) becomes largest with the complete data set and sharply decreases to
much smaller values on exclusion of the highest 1 % of flows from the
calculations, but it undergoes just a little progressive decrease when more
data of the flow distribution are excluded.

In summary, *S*_{d} is clearly more robust than the other discharge
sensitivity metrics to changes in the sampled range of flows. It also has
the distinct advantage that Eqs. (6)–(8), unlike Eqs. (3)–(4), can never
yield *F*_{yw} values larger than 1. One can see from Eqs. (6)–(8) that
*S*_{d} functions as both a shape parameter, controlling how non-linear
*F*_{yw} is as it approaches 1, and a scale parameter, controlling the slope
of the relationship between *F*_{yw} and *Q* at low or moderate discharges.

Figure 3 compares the quantile plot of Fig. 1 for the Can Vila catchment and
the quantile plots of Fig. 7 in von Freyberg et al. (2018) for the Swiss
catchments of Langeten, Biber and Ilfis, which exhibit very different young
water fractions and/or discharge sensitivities (Table 1). The *F*_{0} and
*S*_{d} metrics were calculated from Eq. (8) and good fits were obtained
between the individual *F*_{yw} values and the median discharges as shown by
the red curves. For comparison, grey curves correspond to the linear
approach using Eq. (5).

We find that young water fractions in the Can Vila catchment have a
discharge sensitivity (*S*_{d}) similar to that of the Langeten and the Biber
catchments. By contrast, the young water fractions of the Ilfis catchment
have almost no discharge sensitivity. Although Can Vila has a low *F*_{0}
value, which is in line with its baseflow being several years old, its large
discharge sensitivity expresses well the highly dynamic streamflow regime in
this Mediterranean mountain environment.

Although the linear expression of discharge sensitivity (DS(*Q*), Eq. 5) provides a
reasonable fit for the low-to-medium flow regimes of the Swiss sites, it
fails to capture the highly non-linear dependence of *F*_{yw} on *Q* at Can
Vila, evidenced by the high flows sampled there (Fig. 3a). In addition, Fig. 3 shows a major drawback of the linear approach, namely that it predicts
*F*_{yw} values larger than 1 for high-flow conditions.

The four catchments compared here differ considerably in catchment area and
median discharge (Table 1), which often challenges a robust inter-comparison
analysis. However, Fig. 3 shows that Eq. (6) efficiently estimates the
sensitivities (*S*_{d}) of *F*_{yw} on *Q* across these catchments.

The comparison of the *S*_{d} and DS(*Q*) metrics for Can Vila and the 22 Swiss
catchments studied by von Freyberg et al. (2018) demonstrates that the DS(*Q*) linear
approach approximates small discharge sensitivities reasonably well (Fig. 4). However, for relatively high discharge sensitivities, the linear model
tends to predict smaller and more variable DS(*Q*) values. This behaviour may be
attributed to the fact that, as shown in Fig. 2b, when *S*_{d} is high, the
value of DS(*Q*) decreases if there are high-flow samples that reduce the linear
slope between *F*_{yw} and *Q* (as occurs in Fig. 1).

In order to compare the frequencies of occurrence of *Q* and *F*_{yw} in the
diverse catchments, the same points shown in Fig. 3 are plotted on a single log-probabilistic graph in Fig. 5. It presents the
information as flow duration curves, using the corresponding quantile
frequencies, the log-normal distributions fitted to the flow quantiles and
the *F*_{yw}(*Q*) lines obtained by applying Eq. (6) to the discharges. Figure 5 shows differences in behaviour between Can Vila and the three Swiss
catchments due to the combination of flow distribution and discharge
sensitivity of *F*_{yw} that are only vaguely visible in
Fig. 3. This graph also allows anticipation of the
*F*_{yw} values that might be obtained if more samples would be collected
during high flows (low exceedance frequencies) in the study catchments.

The question arises of where (in what kinds of catchments and in what types of
climates) *F*_{yw} becomes high enough at high flows that it approaches
unity, and thus an exponential model is needed to describe *F*_{yw} at high
flows. This is a question that depends not only on catchment behaviour but
also on the sampling design and the range of investigated discharges. For
example, the results shown in Fig. 3 for the Biber catchment demonstrate
that, if the linear sensitivity were applied, a value of *F*_{yw}=1 would
be reached for a discharge of 26.8 mm d^{−1}. This discharge is exceeded
0.38 % of the time, i.e.1.4 d yr^{−1}, at the Biber catchment (see
the solid green line in Fig. 5). Furthermore, the linear character of DS(*Q*)
makes it sensitive to the sampled discharges (Fig. 2b), so it may be more
vulnerable to insufficient sampling designs and likely to show inconsistent
behaviour in sensitive catchments (Fig. 4).

The discharge sensitivity of the young water fraction is a promising metric
for investigating streamflow generation processes and for catchment
inter-comparison studies. However, the original discharge sensitivity
approach, based on fitting a linear slope between the young water fraction
(*F*_{yw}) and discharge (*Q*), turns out to be inadequate when applied to the
intensively sampled Can Vila catchment; it does not accurately predict
*F*_{yw} during high flows, which consist almost entirely of young water. Can
Vila's young water fractions converge toward 1 at the highest flows,
revealing a conceptual limitation in the linear approach, which can predict
impossible values of *F*_{yw}*>*1. Because *F*_{yw} is confined
between 0 and 1, whereas *Q* may vary by several orders of magnitude, linear
estimates of discharge sensitivity will vary, depending on the highest *Q*
values at which *F*_{yw} estimates are available; this potentially hampers
robust comparisons of discharge sensitivities between catchments with very
different flow regimes and sampling designs.

We propose an alternative, exponential-type approach for estimating
discharge sensitivity (Eq. 6), to overcome the limitations of the linear
approach. The parameters of this exponential equation are *F*_{0}, i.e.
virtual *F*_{yw} for zero discharge, and *S*_{d}, which represents the shape
of the curve for the increase of *F*_{yw} with increasing *Q*. The
exponential *S*_{d} metric outperforms the linear discharge sensitivity
metric in terms of physical soundness and lower sensitivity to changes in
available tracer and discharge information.

As the proposed *S*_{d} metric has dimensions inverse to discharge *Q*, its
value depends on the units of *Q* used in Eqs. (6) and (8). Nevertheless, the
*S*_{d} metric exhibited consistent behaviour across wide ranges of
discharges sampled in the same catchment and between catchments of diverse
sizes and flow regimes.

We hypothesize that, if estimated from tracer samples that adequately
capture the runoff dynamics, the three metrics of *F*_{yw}, *F*_{0} and
*S*_{d} will help in comparing runoff generation behaviour in catchments with
widely varying characteristics. The *F*_{yw} metric, though being sensitive
to catchment wetness, provides an overall measure of the young water
contribution; the *F*_{0} metric characterizes base flows and the *S*_{d}
metric quantifies how much *F*_{yw} changes as catchment wetness increases.

The Swiss isotope data are available online via https://zenodo.org/record/3659679#.Xl6LHi2ZMWp (Staudinger et al., 2020). The Can Vila isotope data are available from Jérôme Latron upon request.

JL and PL designed the isotope sampling strategy at Can Vila and provided measurements. FG and MV analysed the Can Vila data set. FG, JWK and JvF developed the new approach. FG prepared the paper with contributions from JvF, JWK, JL and PL.

The authors declare that they have no conflict of interest.

This research was supported by the projects TransHyMed (CGL2016-75957-R AEI/FEDER, UE) and Drought-CH (National Research Programme NRP 61 by the Swiss National Science Foundation). We are grateful to Carles Cayuela, Gisela Bertràn, Maria Roig-Planasdemunt and Elisenda Sánchez for their support during field work at the Can Vila catchment and to Michael Eaude for his English style improvements.

This research has been supported by the Ministerio de Ciencia, Innovación y Universidades (Spain) (grant no. CGL2016‐75957‐R AEI/FEDER, UE) and the Swiss National Science Foundation (Switzerland) (National Research Programme NRP 61).

This paper was edited by Thom Bogaard and reviewed by two anonymous referees.

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- Abstract
- Recalling the definition of the discharge sensitivity of the young water fraction
- Investigating discharge sensitivity of the young water fraction in a small Mediterranean catchment
- Defining alternative metrics for discharge sensitivity of the young water fraction
- Sensitivity of the discharge sensitivity metrics to changes in data availability at the Can Vila catchment
- Comparing discharge sensitivities at Can Vila and the Swiss catchments
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Recalling the definition of the discharge sensitivity of the young water fraction
- Investigating discharge sensitivity of the young water fraction in a small Mediterranean catchment
- Defining alternative metrics for discharge sensitivity of the young water fraction
- Sensitivity of the discharge sensitivity metrics to changes in data availability at the Can Vila catchment
- Comparing discharge sensitivities at Can Vila and the Swiss catchments
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References