This study explores how catchment heterogeneity and variability can be summarized in simplified models, representing the dominant hydrological processes. It focuses on Mediterranean catchments, characterized by heterogeneous geology, pedology and land use, as well as steep topography and a rainfall regime in which summer droughts contrast with high-rainfall periods in autumn. The Ardèche catchment (Southeast France), typical of this environment, is chosen to explore the following questions: (1) can such a Mediterranean catchment be adequately characterized by a simple dynamical systems approach and what are the limits of the method under such conditions? (2) what information about dominant predictors of hydrological variability can be retrieved from this analysis in such catchments?
In this work we apply the data-driven approach of Kirchner (2009) to estimate discharge sensitivity functions that summarize the behaviour of four sub-catchments of the Ardèche, using low-vegetation periods (November–March) from 9 years of measurements (2000–2008) from operational networks. The relevance of the inferred sensitivity function is assessed through hydrograph simulations, and through estimating precipitation rates from discharge fluctuations. We find that the discharge sensitivity function is downward-curving in double-logarithmic space, thus allowing further simulation of discharge and non-divergence of the model, only during low-vegetation periods. The analysis is complemented by a Monte Carlo sensitivity analysis showing how the parameters summarizing the discharge sensitivity function impact the simulated hydrographs. The resulting discharge simulation results are good for granite catchments, which are likely to be characterized by shallow subsurface flow at the interface between soil and bedrock. The simple dynamical system hypothesis works especially well in wet conditions (peaks and recessions are well modelled). On the other hand, poor model performance is associated with summer and dry periods when evapotranspiration is high and low-flow discharge observations are inaccurate. In the Ardèche catchment, inferred precipitation rates agree well in timing and amount with observed gauging stations and SAFRAN climatic data reanalysis during the low-vegetation periods. The model should further be improved to include a more accurate representation of actual evapotranspiration, but provides a satisfying summary of the catchment functioning during wet and winter periods.
Catchments show a high degree of heterogeneity and variability, both in space and time (McDonnell et al., 2007) raising questions about the degree of complexity that must be used to model their behaviour (Sivapalan, 2003a). Many hydrological models are based on the bottom-up or reductionist approach (Sivapalan, 2003b; Zehe et al., 2006), following the blueprint proposed by Freeze and Harlan (1969). Governing equations such as the Darcy or Richards' equation, which are inherent in many hydrological models, are suitable for point-scale processes (Bloschl and Sivapalan, 1995; Kirchner, 2006). Their use to describe processes at larger scales leads to the calibration of “effective parameters” which are sometimes difficult to link with measurable quantities (Sivapalan, 2003b), although recent methods combining the use of small-scale variability and regionalization techniques were shown to be efficient in preserving spatial patterns of variability (Samaniego et al., 2010). Such “effective” large-scale equations might not, however, describe hydrologic processes realistically, even if they can be calibrated to reproduce observed catchment behaviour (Kirchner, 2006). Klemeš (1983) was one of the first hydrologists proposing the use of alternative modelling concepts. He defines the top-down or downward approach as the “route that starts with trying to find a distinct conceptual node directly at the level of interest (or higher) and then looks for the steps that could have led to it from a lower level”. To go in this direction, Sivapalan (2003b) and Kirchner (2006) promote a combination of data analysis and process conceptualization (the top-down approach). This allows understanding of the main drivers of the system functioning (the perceptual model; Beven, 2002) and inferring the system's “emergent properties” (Sivapalan, 2003b) or “functional traits” (McDonnell et al., 2007). Thus, models obtained through this approach are simple, with a limited number of parameters that can be estimated from the available data.
Kirchner (2009) represents a catchment with a simple bucket model in which
system parameters are derived directly from measured streamflow fluctuations
during recession periods. He based his analysis on storage–discharge
relationships with one essential assumption: discharge depends only on the
total water stored in the catchment. This approach allows the derivation of a
first-order nonlinear differential equation for simulating rainfall–runoff
behaviour. Until now, this approach has mostly been applied in small, humid
catchments. Kirchner (2009) obtained good results for the Severn
(8.70 km
Krier et al. (2012) applied the concept of doing hydrology backwards to infer
spatially distributed rainfall rates in the Alzette catchment (1092 km
To our knowledge, the simple dynamical system approach (SDSA) proposed by
Kirchner (2009) has not been evaluated in a Mediterranean context, where the
rainfall regime exhibits strong contrasts between dry conditions in summer
and intense rainfall events, often related to stationary Mesoscale Convective
Systems (Hernández et al., 1998), during autumns. Wittenberg and
Sivapalan (1999), for instance, used recession analyses to estimate
groundwater recharge in a Mediterranean type of climate in Australia, but
they did not consider the storage–discharge relationship in its implicit
differential form, the sensitivity function
Mediterranean catchments are also characterized by heterogeneous topography,
vegetation and geology. The study of the water cycle in such Mediterranean
conditions, as well as a better understanding and modelling of processes
triggering flash floods, are central research topics addressed in the
HyMeX (Hydrological Cycle in the Mediterranean
Experiment; Drobinski et al., 2013) program
Map of the Ardèche catchment with gauging and rainfall stations, dam locations, and catchments that were examined: #1. Ardèche at Meyras; #2. Borne at Nicolaud Bridge; #3. Thines at Gournier Bridge; #4. Altier at Goulette.
Our study area is the Ardèche catchment (2388 km
As a complementary approach to the modelling studies mentioned above, we adopt in this study the data-based approach proposed by Kirchner (2009) to estimate the hydrological water balance of the Ardèche catchment and to gain insight into the dominant associated processes. Like in the work of Melsen et al. (2014), we divided our examined period into vegetation period (April–October) and low-vegetation period (November–March) where evapotranspiration can be considered as a low.
The idea is to use this insight to propose simple models with very few parameters to learn more about hydrological functioning at the catchment scale.
In the present paper, we focus on the following questions: (1) what is the applicability of the simple dynamical systems approach (SDSA) and what are its limitations in a Mediterranean type catchment like the Ardèche with its particular conditions (size, climate, geological and pedological heterogeneity), and when data from operational networks are used? (2) what can we learn about dominant hydrological processes using this methodology?
To answer those questions, we first estimate the discharge sensitivity function using the available discharge data. Then we assess the relevance of the obtained function by testing how well the simple model based on it can simulate observed discharge, and can retrieve rainfall. The study is complemented by examining the sensitivity of the results to the parameters of the discharge sensitivity function.
The Ardèche catchment is located in southern France (Fig. 1). The
catchment has an area of 2388 km
The main lithologies found in the Ardèche are schist, granite, and limestone (Fig. 2). Upstream, the Ardèche flows from west to east in a deep granite valley, then flows through basalt formations and schist in a north–south direction. Downstream, it flows through bedded and massive limestone before flowing into the Rhone River (see for example the description provided by Naulet et al., 2005).
Geological map of the Ardèche catchment (extracted and
processed from geological map of France 1
Average hourly discharge
Among the land use types found in the Ardèche, forest dominates
throughout the basin (Corine Land Cover database
In the Ardèche Basin, there is a strong influence of the Mediterranean climate with seasonally heavy rainfall events during autumn. Historical data show that these events usually lead to flash floods: Lang et al. (2002) mention seven rainfall events locally exceeding 400 mm during the 1961–1996 period. They also comment on the relatively quick flow response (a couple of hours) to precipitation due to the steepness of the upstream part of the catchment and presence of granitic and basaltic rocks.
Figure 3 shows the average hourly regime of the main terms of the water
balance equation for all months between 2000 and 2008, differently coloured with
respect to vegetation (red) and low-vegetation periods (blue). Under the
main terms of the water balance we consider discharge, evapotranspiration
and precipitation. As we consider interannual values, change in water
storage is assumed to be zero. This hypothesis is consistent with the lack
of a regional aquifer in the Ardèche catchment. The hydrological year
consists mainly of two periods. There is a rainy season (September–February)
with maximum precipitation intensity in autumn, characterized by rainfall
amounts greatly exceeding reference evapotranspiration ET
Physiographic characteristics of the four examined Ardèche sub-catchments. Strahler stream order, channel length and drainage density are calculated from the 25 m IGN DTM using TauDEM tools (Tarboton et al., 2009).
In the Ardèche catchment, measurements of the hydrological state
variables were mainly started in the 1960s for the purpose of flood
forecasting. In our study, we use hourly data of precipitation (
The analysis is mostly constrained by the availability of discharge data,
which were obtained from the national Banque Hydro website (
As the stations were not designed and managed for low-flow measurements, the low-flow time series were investigated by contacting the operational services in charge of the stations. Consequently, two stations had to be removed from further analysis due to unreliable measurements and agriculture water withdrawals in summer periods. Ultimately, four sub-catchments could be examined: the Ardèche at Meyras (#1), the Borne at Nicolaud Bridge (#2), the Thines at Gournier Bridge (#3), and the Altier at Goulette (#4); see locations in Fig. 1. These four sub-catchments are characterized by steep slopes (> 15 %), average altitude of around 1000 m and igneous and metamorphic bedrock. We have also computed Strahler stream order and channel length using TauDEM tools (Tarboton et al., 2009) in order to classify and measure the size of the river network. The analysis was conducted using the 25 m resolution IGN DTM and the D8 flow direction algorithm, so the resulting network statistics may only loosely resemble those that would be obtained from more accurate procedures such as field mapping. Main physiographic catchment characteristics are summarized in Table 1.
The discharge data were available at varying time intervals, and were
aggregated to hourly sums. Two types of precipitation data have been
examined and are used throughout the analysis. Local rain gauges at the
hourly time step provided by the OHM-CV database (Boudevillain et al.,
2011) are used as the primary source of rainfall data for the catchment
Ardèche at Meyras (#1). For the catchments Borne at Nicolaud Bridge
(#2), Thines at Gournier Bridge (#3) and Altier at Goulette (#4)
we use the SAFRAN reanalysis of Météo-France, based on 8 by 8 km
To further assess data quality, we evaluated the consistency of the local rainfall station with SAFRAN data for the Ardèche at Meyras (#1) catchment at the hourly time step. The resulting coefficient of determination was 0.99. For the rest of the sub-catchments, we first assumed that SAFRAN rainfall is representative of the catchment average. However, by looking at the mean annual water fluxes (Table 3) and estimated runoff coefficients, we infer that the mass balances for catchments #2, #3 and # 4 are implausible.
Weighted average crop coefficient for each examined catchment per growing stage.
Hydro-climatic characteristics of the four examined Ardèche sub-catchments (2000–2008).
Scaling hydro-climatic characteristics of the four examined Ardèche sub-catchments (2000–2008).
For these reasons, two actual evapotranspiration (AET) estimates and runoff
coefficients are provided to gain useful insight about data uncertainty. In
Table 3 the first evapotranspiration estimate comes from the water balance
AET
In Table 4 the second AET estimate corresponds to Turc (1951) annual actual
evapotranspiration, which is calculated using the following formula:
The values of the water balance components differ from catchment to
catchment as illustrated in Table 3. By comparing Tables 3 and 4, we note
that the mass balance AET
Discharge data uncertainty has been addressed in many works and sometimes it can be quite large, especially in catchments where high flows are seldom gauged due to safety reasons (Le Coz et al., 2010) or where low flows may be difficult to measure accurately. Nevertheless, here we decided to go ahead with the available operational discharge data, to assess whether the SDSA can provide useful information about catchment hydrological functioning in a Mediterranean context, even in the presence of some uncertainty in the discharge data.
However, in order to apply the SDSA with data where water balance closure is
more representative, we rescaled precipitation and
The first step in the rescaling analysis was to obtain a robust estimate of actual evapotranspiration.
We used the following equation of Fu (1981) to draw Budyko (1974) type curves for
the Ardèche catchments:
The parameter
In our study, we drew Fu curves with parameter
We then make the following assumptions. We assume that the long-term average
Description of different empirical formulas for estimating mean
annual actual evapotranspiration: AET is actual evapotranspiration (mm
yr
The hourly precipitation values are then rescaled by multiplying them by the
ratio found in the previous step between the average
The new precipitation and new AET values for catchments #2, #3 and #4 are then used in further analysis, whereas original data were conserved only for catchment #1.
In this part, we first present the estimation of the discharge sensitivity
function,
Kirchner (2009) proposed a method for determining nonlinear reservoir
parameters for a simple bucket model with the assumption that discharge
It is known that precipitation measurements are spatially variable. Rain gauges reflect precipitation on areas much smaller than the catchment itself. The same comment is valid for evapotranspiration estimates, which are typically representative of much smaller areas than the catchment.
In Eq. (5), only discharge can be considered as a state variable that
characterizes the entire catchment. This observation led Kirchner (2009) to
make the fundamental assumption that discharge is uniquely dependent on
total water storage
Combining Eqs. (7) and (8) we can express
Following the approach of Kirchner (2009), we consider periods when
precipitation and actual evapotranspiration are relatively small compared to
discharge, obtaining the following equation, which shows that under these
conditions the discharge sensitivity function can be estimated from
discharge data alone:
The sensitivity function
We avoid the vegetation period for the estimation of the
These rainless nighttime hours are further used to determine the sensitivity
function
A quadratic function (Kirchner, 2009) is then fitted to the binned means
leading to the following empirical equation in log space:
Discharge sensitivity functions can be used to
simulate discharge (Kirchner, 2009) by combining Eqs. (9) and (10),
resulting in the following expression, where the quadratic function of Eq. (12) is used to describe
Equation (13) indicates that dQ/dt depends on the balance between precipitation,
actual evapotranspiration and discharge. However, variations in
In order to minimize numerical instabilities, Eq. (13) is solved using its
log transform (Kirchner, 2009):
To estimate the AET term in Eq. (14), Kirchner (2009) originally used
Penman–Monteith reference evapotranspiration and a rescaling effective
parameter (
In our study, we assumed that actual evapotranspiration is equal to potential
evapotranspiration (PET) throughout the year, being defined as reference
evapotranspiration ET
To show how data inconsistency problems may affect the performance of discharge simulation, we also ran the model with non-rescaled values of precipitation and evapotranspiration. The resulting model performance is reported in Sect. 4.5.
Mean annual evapotranspiration ratio AET
Until recently, it was considered infeasible to infer precipitation from streamflow fluctuations. Spatial variability of precipitation is high and conventional rain gauges can only measure precipitation over an area that is many orders of magnitude smaller than a catchment itself. We assess the relevance of the inferred storage–discharge relationship for the examined catchments in the Ardèche using the rainfall retrieval scheme (“doing hydrology backward”) as proposed by Kirchner (2009) and further tested by Krier et al. (2012).
Assuming that the assumptions of the SDSA are valid, we can infer temporal
patterns of precipitation rates from streamflow fluctuations using the
following inversion of Eq. (13), as outlined by Kirchner (2009):
The time lag is optimized for each sub-catchment by calculating the correlation coefficient between estimated and measured rainfall using the lag times of 1, 2, 3, 4, 5, 6, 12, 24 and 48 h. The lag time that shows the best correlation is used. This approach is similar to the one used by Krier et al. (2012).
To make this concept of “doing hydrology backward” feasible, we identify
periods when the contribution of evapotranspiration in the water balance
equation can be neglected. This includes rainy periods when relative
humidity should be relatively high, resulting in low evapotranspiration
fluxes and thus P-AET
To measure the agreement between the reference values and the retrieved
values we use the coefficient of determination
To assess model efficiency, we use Nash–Sutcliffe efficiency and percent
bias as model evaluation criteria for discharge simulations, and coefficient
of determination for rainfall retrieval. Nash–Sutcliffe efficiency, NSE (Nash
and Sutcliffe, 1970) is used as a dimensionless model evaluation statistic
indicating how well the simulated discharges fit the observations. We
compute the NSE to emphasize the high flows as shown in the following equation:
NSE values range between
In addition, percent bias (PBIAS) was also calculated as a part of the model
evaluation statistics. It measures total volume difference between two time
series, as Eq. (19) indicates:
The optimal value of PBIAS is 0.0 where positive values indicate model overestimation bias, and negative values indicate model underestimation bias (e.g. Gupta et al., 1999).
In rainfall retrieval, model performance is assessed by using the
coefficient of determination (
In this part, we performed a Monte Carlo analysis to sample the parameter
space defined by the three parameters
A representative set of 10 000 (
The number of simulations (10 000) was assumed to be adequate in view of the relative simplicity of the parametric model, and because the best-fit NSE did not change significantly beyond 10 000 simulations. For comparison, Zhang et al. (2008) and Tekleab et al. (2011) used 20 000 simulations for a four-parameter dynamic water balance model, and Uhlenbrook et al. (1999) used more than 400 000 model runs for the much more complex HBV model with 12 parameters.
The results section is divided into five parts. In the first part, results
concerning estimation of
Figure 5 shows an example of a recession plot for the Ardèche at Meyras (#1) catchment for the all low-vegetation periods between 2000 and 2008. We observe that the recession plot exhibits large scatter at low discharge. This result is consistent with the findings of Kirchner (2009) and Teuling et al. (2010). They argue that this is possibly due to measurement errors or differences between the modelling concept and reality.
Recession plots for the Ardèche at Meyras (#1) catchment
for all low-vegetation periods between 2000 and 2008: left, flow recession
rates (
Table 6 provides values of the recession plot parameters for all four
catchments during low-vegetation periods between 2000 and 2008. It shows one
parameter set for each catchment. We observe that our choice of the
low-vegetation period for estimation of
We have also tested
Parameter values for the examined catchments for all low-vegetation periods (2000–2008).
Summary statistics of computed NSE, NSE log and PBIAS for each examined catchment in the Ardèche Basin.
Continuous discharge simulations were performed for 2000–2008. Figure 6 presents a simulation extract (year 2004) for the Ardèche at Meyras (#1) catchment. Table 7 presents a model performance summary (NSE, NSE of the logarithm of discharge, and PBIAS) for each catchment and each year.
Looking at Fig. 6, we can see that discharge simulations reproduce the observed hydrograph behaviour better in winter and low-vegetation periods. The low-flow (summer) periods are less well reproduced, even if the overall performance of the simulation is good. The influence of evapotranspiration in summer periods can be one of the explaining factors for that. It should be noted that high evapotranspiration influence is visible only when discharge is evaluated in log space. In linear space, evapotranspiration has a negligible influence on (already quite small) discharge, and the model runs well under dry conditions.
Series of simulated hourly hydrographs (red) for the Ardèche at Meyras (#1) catchment for the year 2004, compared with observed discharge (blue).
Inferred versus measured daily precipitation for the study catchments: #1. Ardèche at Meyras; #2. Borne at Nicolaud Bridge; #3. Thines at Gournier Bridge; #4. Altier at Goulette. Blue dots correspond to the inferred daily totals from low-vegetation periods; red points correspond to the inferred daily totals from vegetation periods; blue line is correlation for low-vegetation periods, red line for vegetation periods and green line for total examined periods.
We note in Table 7 that the Ardèche at Meyras (#1) catchment shows
satisfactory performance with NSE
Furthermore, Gupta et al. (1999) show that PBIAS values for streamflow tend
to vary more than other performance criteria between dry and wet years. This
could be another possible explanation of the overall poor model performance
in 2005 for the Ardèche at Meyras catchment. The Borne at Nicolaud Bridge
(#2) and Thines at Gournier Bridge (#3) catchments show good overall
performance for the 9-year period with NSE
Model performance of inferred versus measured daily rainfall in four sub-catchments for all low-vegetation periods 2000–2008.
NSE values of log discharge for the Ardèche at Meyras (#1)
catchment, illustrating sensitivity to changes in the
Following SDSA we retrieved precipitation from discharge
fluctuations. We used the same
The coefficient of determination, mean bias, and slope of the relationship
between inferred and measured rainfall for examined catchments and
low-vegetation periods, as well as information about lag time, can be found
in Table 8. Other lag times (
Figure 7 shows daily precipitation retrieval for the four studied
sub-catchments of the Ardèche during low-vegetation periods, vegetation
periods and for the entire study period 2000–2008 using the same
Good correlation between retrieved precipitation and observed precipitation
can be observed for low-vegetation periods where the slope of the regression
line shows a modest degree of overestimation. Figure 7 illustrates that the
inferred precipitation daily totals from low-vegetation periods (blue line)
agree quite well with the precipitation measurements in the Altier at
Goulette (#4) catchment, yielding
The optimized time lags are generally very small (less than 2 h), which
confirms the very short response time in the Ardèche catchment. In order
to see whether the retrieved daily rainfalls were sensitive to the lag time,
we compared the results obtained with different lag times for two catchments:
the Ardèche at Meyras (#1) and Altier at Goulette (#4). The
Ardèche at Meyras (#1, 98 km
As a first approach, a manual sensitivity analysis was done by successively
varying the values of each parameter and plotting the corresponding simulated
hydrographs (grey areas in Figs. 8 and 9). The results for the Ardèche at
Meyras (#1) catchment (year 2004) are presented; see Figs. 8 and 9 for the
We can see that
Observed versus simulated hydrograph (
Observed versus simulated hydrograph (
Dotty plots for the Ardèche at Meyras (#1) catchment
(left: plots with NSE efficiencies; right: plots with NSE efficiencies calculated
on log
Series of simulated hourly hydrographs (red) for Altier at
Goulette (#4) catchment for the year 2000 and its comparison with
observed discharge (blue), using original non-scaled data (top), with
rescaled
From Table 9 we can also observe that the model efficiency for the parameter values that were obtained from the recession plots is close to optimal (at least for this year at this site), and cannot be substantially improved by manual parameter adjustments.
In order to complement the manual sensitivity analysis presented above, to
explore the range of these parameters and to assess whether the parameters of
the
Comparison of the chosen parameter range and parameters obtained from low-vegetation periods for the Ardèche at Meyras (#1) catchment.
Model performance for three examined catchments over the whole examined period (2000–2008), comparing the original operational data and rescaled precipitation and evapotranspiration data.
The results show that when the parameters are calibrated to discharge
simulations, their ranges are quite large. The maximum model performance
appears to be around 0.8 for all three parameters and both indicators.
Low-flow performance (NSE log) is not very sensitive to the variations of
the parameters. Giving peak flow more weight (NSE) allows the
identification of clear optima and a narrower range for the
In Sect. 2.3 we introduced a rescaling technique to obtain more representative water balances for catchments #2, #3 and #4. Here, we show the consequences of foregoing this rescaling for those three catchments that showed unrealistic mass balances (Table 3). Figure 11 shows observed discharge and simulated hourly hydrographs for the Altier at Goulette (#4) catchment for the year 2000, obtained with non-scaled data, rescaling of precipitation alone, and rescaling of both precipitation and evapotranspiration.
The lack of water balance closure may contribute substantially to poor model performance, as can be seen from Fig. 11. The simple dynamical systems approach, like many modelling approaches, is based on conservation of mass; it is therefore unsurprising that it may perform poorly when tested against data sets that violate mass conservation. We observe that when the original non-scaled data are used, discharge is generally underestimated. By introducing the rescaled precipitation, flow peaks can be better reproduced, but model performance is still poor during the vegetation period. If both the rescaled evapotranspiration and rescaled precipitation are used, significantly better results are obtained in both vegetation and low-vegetation periods.
As a complement to assessing modelling performance with non-scaled data, we re-ran the SDSA model for these catchments to see how this affects the hydrograph simulation and performance indicators. Table 11 compares model performance with the original operational data and the rescaled data, using NSE, NSE on log of discharge and PBIAS as performance metrics. We observe that model performance is markedly improved by using the rescaled precipitation as forcing (runoff coefficients are more representative as shown in Table 4). In addition, model performance is improved by also introducing rescaled evapotranspiration (better NSE and lower PBIAS values are obtained).
In this study, the SDSA method was applied to four sub-catchments in the Ardèche catchment (France), representative of Western Mediterranean catchments. We first discuss the advantages and limits of the method for this type of catchment. Then we discuss how the application of this approach was useful in deriving information about the catchment functioning and possible dominant processes.
The application of this method to the Ardèche catchment was at first quite challenging. In particular, the basins are larger and less humid than those of the original case studies; in addition, data availability is more limited and data quality is distinctly lower.
The drainage area does not seem to be a limiting factor at the scale of our
catchments. The catchments where this theory has been applied so far in order
to reproduce the hydrograph were typically smaller than
Our study demonstrates that data quality is particularly important for the
application of this method. Concerning discharge data, the method is based on
the discharge-sensitivity function
In the present study, we used discharge data from operational networks. We
have shown in Sect. 2.2 that there are known issues with the quality of these
data for our purposes. Nevertheless, when data consistency is sufficient
(e.g. Ardèche at Meyras (#1) station), a robust estimation of the
The quality of the rainfall data was questioned early in our work, and
rescaling of precipitation was needed to obtain realistic results. As
mentioned in Sect. 2.2, the gridded SAFRAN product underestimates
precipitation especially in mountainous areas and underestimates the
occurrence of strong precipitation (
Assuming that the discharge data are reliable, it was shown that when input
rainfall and ET
One assumption behind the rescaling approach proposed in Sect. 2.2.2 is that discharge data are reliable enough to provide an accurate estimate of annual runoff. This is of course questionable, because stage–discharge relationships are known to be highly extrapolated in this region due to the difficulty of gauging high discharges (e.g. Le Coz et al., 2010). As also mentioned in Sect. 2.2.1, low discharges are also highly uncertain, because these stations were often designed for flood warning purposes. Work is currently in progress in order to quantify the runoff data accuracy. This work is based on the BaRatin method (Le Coz et al., 2014) which provides an uncertainty range on the estimated discharge. The uncertainty can be propagated to the whole discharge time series (Branger et al., 2015) and the next step will be the propagation to the hydrological water balance and the quantification of uncertainty for the annual and monthly values. This work will help quantify which of the data (rainfall, discharge or both) need to be improved.
In addition, the operational discharge measurement network has recently been
complemented by research instrumentation covering nested scales (see Braud
et al., 2014 for details). In particular, small catchments ranging from 0.5
to 100 km
Regarding discharge uncertainty, if data have to be rescaled, an approach like the one proposed by Yan et al. (2012) should be preferred, as it allows a consolidation of the water balance at the scale of the whole Ardèche catchment, taking into account data uncertainties on all the components, and constraining the results with the water balance equation along the river network.
The simulation results show that additional effort must be put into quantifying data uncertainty in both discharge and rainfall. The derivation of more accurate rainfall fields combining various data sources (such as radar data and in situ gauges (see, for instance, Delrieu et al., 2014) should also be encouraged. It could also be interesting to use actual evapotranspiration estimates derived from remote sensing techniques adapted to complex topography (e.g. Gao et al., 2011; Seiler and Moene, 2010) to obtain independent estimates of AET and better constrain this component in hydrological modelling.
The sampling strategy of deriving the
Independently from the data quality issues, we also showed that the SDSA
model performs better during the wet, winter periods than the dry, summer
periods and dry years (see Sect. 4.2). We interpret these results as an
indication that the current model is not fully adapted to the high
evapotranspiration conditions of our Mediterranean catchments. We must also
point out that, when assessing the relevance of the estimated
In addition, the recent study of Brauer et al. (2013) showed that the two-parameter model they used cannot deal with complexity of hydrological processes in their catchment (only 39 % of the hydrographs had NSE over 0.5). In the Ardèche catchments, the three-parameter model succeeds in capturing the catchment behaviour, with quite good response of discharge to rainfall in low-vegetation periods (peaks and recession were nicely reproduced).
Recession analysis has been used to build hydrological models for many years (e.g. Brutsaert and Nieber, 1977). What is new in the SDSA is not the reservoir itself, but the manner to derive its structure and parameters from the data analysis: in particular, here the functional form of the storage–discharge relationship is not specified a priori, but determined directly from data without calibration (Kirchner, 2009). This is the very definition of the top-down or data-driven modelling approach, that is acknowledged to be a major paradigm shift in modelling by the hydrological community (and which was a major emphasis of the PUB decade; see, for instance, Sivapalan, 2003b and Hrachowitz et al., 2013). We argue that testing this kind of approach on new data sets, for various climatic conditions, contributes to the advance of hydrological science in itself. We have also compared the model results with other models that are based on similar data-driven methodology (e.g. Brauer et al., 2013 and Melsen et al., 2014) and obtained similar results.
The major limitation of the SDSA is of course the availability of good quality discharge data with a short time step, in catchments representative of the spatial variability of hydro-climatic conditions. Discharge must also be representative of natural conditions, which could also limit its applicability in catchments impacted by human activity.
The most important output from our application of the simple dynamical systems approach is the validation of underlying hypotheses and information about the dominant processes that can be derived from the model parametrization.
The SDSA model is based on an underlying hypothesis that regards a catchment as a single nonlinear bucket model. In our study we note the good performance of the model in each sub-catchment which suggests that SDSA, although it was developed for humid regions, remains valid for these Mediterranean sub-catchments as well. We can thus interpret that these sub-catchments do follow the model's functioning hypotheses, especially in winter and low-vegetation periods. These results are consistent with the findings of Brauer et al. (2013) for the Hupsel Brook catchment, Kirchner (2009) for Plynlimon and Teuling et al. (2010) for the Rietholzbach catchment. In contrast, during the vegetation period the model seems to be less adapted to our Mediterranean setting. The catchments seem to behave differently when they are dry. This is probably due to the strong influence of evapotranspiration. In our hydroclimatic context (see details in next section), and taking into account that no regional groundwater exists in the Ardèche catchment, discharge provided by the SDSA can be associated with subsurface flow (generally assumed to occur via lateral flow along perched water tables in shallow soils), which is less active in summer and when evapotranspiration is high. It could be necessary to consider another storage, probably more superficial than the “SDSA” storage, which could be used to supply evapotranspiration over shorter timescales, and which may be largely decoupled from subsurface lateral flow that sustains base flows.
The model works better in the Ardèche at Meyras (#1) and Thines at Gournier Bridge (#3) catchments, which both are granitic (see Fig. 2). The hypothesis of shallow subsurface flow caused by saturation of at interface between soil and bedrock makes particular sense in this geology (e.g. Cosandey and Didon-Lescot, 1989; Tramblay et al., 2010).
In the forested granitic catchments of this region, infiltration capacity is generally very high and runoff occurs due to soil saturation (e.g. Tramblay et al., 2010). However, this saturation mostly occurs at the interface between the very thin soil and the large altered bedrock, where contrasts of hydraulic conductivity can be encountered, leading to quick lateral subsurface flow. Experiments are currently being conducted on infiltration plots to quantify the velocity of this lateral flow (see Braud et al., 2014 for their description). Therefore the main mechanism we are speaking about is quick lateral subsurface flow which transits through the reservoir considered in the Simple Dynamical Systems Approach. On agricultural areas, in the intermediate part of the Ardèche catchment, infiltration excess surface runoff is likely to occur (and has been observed in the field). Its contribution is also under investigation using detailed experiments (see Braud et al., 2014).
In addition, unaltered bedrock tends to be impermeable, but flow pathways are created in the many fractures, joints and fissures of the altered horizons. During extended rainfall those flow pathways might become connected, generating rapid subsurface flow (Krier et al., 2012). Moreover, the parameter values of the granite catchments are quite similar (see Table 6).
To quantify the relative influence of several predictors of the catchment
response (and values of
This is also consistent with the contemporary literature, as geology has been invoked in numerous recent studies as a controlling factor of flood response (Gaál et al., 2012; Garambois et al., 2013; Krier et al., 2012; Vannier et al., 2014). As also discussed by Kirchner (2009), the theory is challenged by catchments with heterogeneous geology and thus with many disconnected subsurface storage reservoirs. This might explain the good modelling performance in granite catchments (see also Vannier (2013) for similar conclusions using a reductionist modelling approach).
Our study describes in detail the application of SDSA methodology to
four catchments of the Ardèche Basin ranging from 16 to 103 km
To have more representative water balance fluxes, we rescaled precipitation and evapotranspiration for three sub-catchments (#2, #3 and #4). In our work we used average annual scaling coefficients for the whole time series (for both precipitation and evapotranspiration). In the future, varying the scaling coefficients according to different seasons could possibly lead to a better approximation of hourly precipitation and evapotranspiration fluxes.
We calculated the discharge sensitivity functions from low-vegetation periods and performed continuous discharge simulations with an hourly time step for the period 2000–2008. We also inferred precipitation and performed sensitivity analyses of the three parameters of the discharge sensitivity function.
Our results show that good results for discharge simulation can be obtained, especially under winter humid conditions and for catchments characterized by predominantly granitic lithology. Under dry conditions, poor model performance is mainly related to the disturbed water balance terms, high influence of AET and imprecise discharge measurements. Improving AET estimation is recommended for better model performance in summer periods when evapotranspiration is high and when the unsaturated zone has a significant role in attenuating the precipitation input. Working on the quantification of data accuracy and error reduction is also recommended in order to get more robust and reliable results.
As a perspective to this study, dominant predictors of runoff variability other than geology (such as land use, soil properties, drainage density, topographic steepness etc.) still need to be explored and linked to catchment hydrological behaviour. Relating the obtained parameters of the discharge sensitivity function to the catchment characteristics using different statistical classification techniques (e.g. principal component analysis (PCA) and factor analysis of mixed data (FAMD) or self-organized maps) could allow us to apply the method also to ungauged basins, thus contributing to the PUB initiative (Hrachowitz et al., 2013). Another step would be then to create a distributed “Kirchner type” hydrological model where a parameter set would be attributed to “regions” discretized on the basis of their physiographic characteristics. This would allow us to determine the rainfall–runoff behaviour in large scale river basins by taking into account the precipitation spatial distribution and flood flow routing through the channel network. We would then be able to broaden our understanding of nonlinear catchment response and travel time lags as suggested by Kirchner (2009).
The study is conducted within the FloodScale project, funded by the French National Research Agency (ANR) under contract no. ANR 2011 BS56 027, which contributes to the HyMeX program. The HyMeX database teams (ESPRI/IPSL and SEDOO/Observatoire Midi-Pyrénées) helped in accessing the data. The authors acknowledge Brice Boudevillain for providing the OHM-CV rainfall data, Météo-France for their rainfall and SAFRAN climate data. EdF-DTG provided discharge data from three of the gauges used in this study. We thank the Region Rhône-Alpes for its funding of the PhD grant of the first author. We thank R. Krier for providing us the codes used to perform the recession analysis and E. Leblois for constructive comments on the paper.Edited by: M. Mikos