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Evapotranspiration observations diagnostic
Another diagnostic for the evaluation of the validity of the datasets is performed using the actual evapotranspiration (AETwb) inferred using water balance Eq.(2). Considering that at the annual time scale the variation of stored water is negligible, then, AETwb is equal to the difference between precipitation and runoff. 𝐴𝐸𝑇𝑤𝑏=𝑃−𝑄 (2).
ET0 corresponds to evapotranspiration of an ideal canopy (well watered grass) receiving as much as water it needs. Potential evapotranspiration (PET) for another canopy or crop corresponds to evapotranspiration that would occur if the canopy of crop was well-watered. It can be deduced from ET0 using crop coefficients which are tabulated in FAO (1998) for instance. AET depends on PET and soil moisture. By definition, AET is lower or equal to PET. AETwb is also compared with (Turc, 1961) annual actual evapotranspiration (AETTurc) which is calculated using the following formula: 𝐴𝐸𝑇𝑇𝑢𝑟𝑐=𝑃√0.9+𝑃²𝐿² (3).
Where 𝐿=300+25𝑇+0.05𝑇3, with T the mean annual temperature in °C.
Influence of hydroelectric complexes and watershed inconsistencies
Unrealistic flow regimes are also tracked using flow data and knowledge of the watersheds. All the drainage basins of the Ardèche River presented Table 1, apart the Ardèche at Meyras, are influenced by the hydroelectric complex of Montpezat. Runoff coefficient is influenced by water from the Loire catchment that is drained into the Ardèche River and overestimated what would be the natural runoff coefficient. The Chassezac at Gravière is also influenced by several hydroelectric complexes. The station is located between the dam of Malarce and the hydro plant of Salelles. Thus, the station does not record the flow passing through the penstock between the factory and the plant. Although the Chassezac River is highly dammed, this is not visible considering an annual time scale. The runoff coefficient is calculated for the station of Chambonas, approximately 2.5km downstream the station of Gravières. Regarding the influenced hydrologic stations of the Ardèche River, Jean-François Noël (2014) performed a naturalisation of river flow using observed discharge and knowledge of input daily discharge from the Loire catchment at the Montpezat dam. These values were provided by EDF (Electricité De France), and J.F Noël calculated the daily naturalized for the Ardèche River. These data are used to compare influenced and naturalised flow regimes for the 5 stations downstream of the complex.
Rescaling of the meteorological data
The last diagnostics highlight implausible water balance for several catchments. However, Kirchner (2009) approach can only be applied for catchments on which water balance is closed. Flow measurements can also include large uncertainty. These uncertainties derive from a combination of sources, including errors in stage and velocity measurements during gaugings, assumption of a particular form of stage–discharge relationship, extrapolation of the stage–discharge relationship beyond the maximum gauging and cross-section change due to vegetation growth or bed movement (McMillan, et al., 2010). On the other hand, precipitation and evapotranspiration are spatially variable and are representative of much smaller areas than the catchment. In this study, flow measurements are assumed to be the most reliable source of data. Krichner method is applied directly on raw discharge data to assess whether useful information can still be extracted for Mediterranean catchments.
Hence, rescaling of the meteorological data (P and PET) is performed in order to apply Krichner method with data where water balance closure is more representative. According to the analysis of Adamovic, et al. (2014), AETTurc is assumed to be a robust estimation of actual evapotranspiration for the Mediterranean region while estimated using SAFRAN annual precipitation and temperature. The mean annual AETTurc formula (Turc, 1954) is solved iteratively to find AETTurc and PTurc consistent with one another and consistent with the mean annual Q. Rescaling coefficients are defined such as 𝑃𝑇𝑢𝑟𝑐/𝑃 and 𝐴𝐸𝑇𝑇𝑢𝑟𝑐/𝐾𝑐×𝐸𝑇0. The hourly precipitation and actual evapotranspiration get rescaled after being multiplied by the rescaling coefficients. The Kc used to calculate PET are provided in Table 2.
Sensitivity of g(Q) to the data used for the extraction
Several methods for extracting recession events are tested. (1) Events were extracted according to Kirchner conditions (Solar flux < 1W/m² and no rain for the previous 6h and next 2h) for the whole record period. (2) A snow filter was also applied in order to exclude recession events that may be caused by snow melt. Possible snowmelt periods are considered when at least 5 mm of snow occurred within the last 15 days (Vannier, et al., 2013). (3) Kirchner’s conditions were applied only for low-vegetated period (November to March). Finally, (4) the snow filter was applied to recession events extracted using Kirchner’s condition for low-vegetated period.
Evaluation of g(Q) relevance using discharge simulation
Discharge simulation is performed in order to evaluate/validate the estimated sensitivity functions. Simulations are performed only for watersheds that are non-influenced. Sensitivity function𝑔(𝑄), described by Eq.(10) is used in the following combination of Eq.(8) and Eq.(9) to simulate discharge: 𝑑𝑄𝑑𝑡= 𝑑𝑄𝑑𝑆𝑑𝑆𝑑𝑡=𝑔(𝑄)(𝑃−𝐴𝐸𝑇−𝑄) 11).
Streamflow hydrograph can be directly simulated by integrating Eq.(11) through time. A hypothesis is required for the specification of AET. It is assumed that AET=Kc*ET0, where the scaled Kc*ET0 defined in section 2.1.4 is used .In the simulation the scaled rainfall was also used. This implementation requires attention onto two practical details (Kirchner, 2009). (a) Changes in measured discharge at the catchment outlet may lag behind the changes in catchment storage and potentially introduce error in the synthetic hydrograph. Kirchner (2009) and Adamovic, et al. (2014) pointed out that such lags can be retrieved using cross-correlation between P and Q records, although, they are not very important since discharge time series are highly auto-correlated. However, they suggest to take into account these lags to enhance model performance. (b) Numerical instabilities can be triggered solving Eq.(11). In order to minimize numerical instabilities the log transform of Eq.(11) is preferred: 𝑑(ln(𝑄))𝑑𝑡= 1𝑄𝑑𝑄𝑑𝑡=𝑔(𝑄)𝑄(𝑃−A𝐸𝑇−𝑄)=𝑔(𝑄)(𝑃−A𝐸𝑇𝑄−1) (12).
This ordinary differential equation can be solved using Runge-Kutta 4th order scheme (see details in Appendix I) and is computed iterating at hourly time steps. Simulations are performed for a selection of catchments for the reference years 2000 – 2008. Goodness-of-fit measures are computed. Their definition is presented in Appendix II.
Parameter regionalisation using Factorial Analysis of Mixed Data (FAMD)
In order to set up the SIMPLEFLOOD model in a large catchment, it is necessary to assign a parameter set to all the sub-catchments of the domain even if they are not gauged. This is the model parameter regionalisation step. Adamovic (2014) used factorial analysis of mixed data (FAMD) for this purpose and found that geology and the period over which the discharge sensitivity function is estimated (low-vegetation versus vegetation period) were the main predictors of the discharge sensitivity functions parameters. However, his analysis was based on only five catchments. The objective here is to apply the same analysis to our larger catchment sample in order to assess the robustness of Adamovic (2014) results. We present a brief introduction on the theory of factorial analysis of mixed data (FAMD), the dataset used, and the principles of classification performed further.
Principle of FAMD
FAMD was introduced by Pagès (2004) and a nice description can be found in Wikipedia2. A FAMD deals with I individuals that present measurements/information (quantitative and quantitative) for K variables. In our case, individuals are represented by the different catchments and the variables correspond to the 3 model parameters, topographic measurements, geology, land use, and meteorological and hydrological data. We are interested to learn whether there are relations between the variables and especially involving the models parameters. We are also aiming to find similarities catchment to catchment. A classification is then performed to reveal patterns that could be helpful in the understanding of hydrological cycle in the Mediterranean area. FAMD tends to synthetize information of a dataset by representing it geometrically. Quantitative data are normalized allowing the comparison and in order to give the same importance between the different variables. Thus, a dataset composed by I individuals described by 3 variables can be visualized in a 3D scatterplot; each points corresponding to one individuals plotted using its variable values. Similarly, variables can be plotted in a domain of individuals. If the number of individuals or variables is superior to 3 it is not possible to plot the data but mathematically, I individuals will be represented in a domain of K dimension, and K variables will be represented in a domain of I dimensions. As regard to the qualitative variables, the intraclass correlation ², is calculated. This is a measure of the relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample.
Presentation of the SIMPLEFLLOD model
The SIMPLEFLOOD model (Adamovic, 2014) is a distributed model implemented in JAMS (Jean Adaptable Modelling System) modelling framework. JAMS modelling framework permits modular model creation using different environmental and hydrological processes at discrete point in time and/or space. Each process has its own algorithm-common feature that can be found in many distributed model nowadays.
The SIMPLEFLOOD model allows discharge simulation following Kirchner’s approach (Section 2.2). The catchment discretization is composed of Hydrological Response Units (HRUs), which are sub-catchments, in order to be consistent with the principles of the simple dynamical system approach. The size of the sub-catchments is chosen is agreement with the mean size of rainfall events. Meteorological inputs are spatialized over each sub-catchment. At each time step, the discharge is simulated thanks to the runoff generator (Figure 3) that applies Kirchner’s method considering each sub-catchment as a simple dynamical system. The runoff is directly transferred to the river network where it can be routed to the outlet. The model forcing is composed of hourly rainfall and reference evapotranspiration, as well as a monthly time series of crop coefficient per modelling unit. We present in the next section the model set up for the Gardon catchment. For our application, only interflow is considered and the bypassing mechanism is not implemented (Figure 3).
Table of contents :
1. Introduction
1.1. Context and objectives of the study
1.2 Study area, previous modelling studies and available data presentation
1.2.1 Hydrological modelling in the Cevennes-Vivarais region
1.2.2 Meteorological and hydrological observations
2. Materials and methods
2.1 Data consistency
2.1.1 Water balance analysis
2.1.2 Evapotranspiration estimation diagnostic
2.1.3 Influence of hydroelectric complexes and watershed inconsistencies
2.1.4 Rescaling of the meteorological data
2.2 Application of the simple dynamical system approach to the Cévennes-Vivarais region
2.2.1 Estimation of the catchment sensitivity function g(Q)
2.2.2 Sensitivity of g(Q) to the data used for the extraction
2.2.3 Evaluation of g(Q) relevance using discharge simulation
2.3 Parameter regionalization of the SIMPLEFLOOD model using FAMD
2.3.1 Principle of FAMD
2.3.2 Dataset used
2.3.3 Hierarchical Classification in Principal Component (HCPC)
2.4 Distributed modelling of the Gard catchment using the SIMPLEFLOOD model (Adamovic, 2014)
2.4.1 Presentation of the SIMPLEFLLOD model
2.4.2 Spatial discretization in Hydraulic Response Unit (HRU)
2.4.3 Discharge simulation of HRUs
2.4.4 River routing parameters
3. Results
3.1. Data consistency analysis
3.1.1 Water balance analysis and evapotranspiration observations diagnostic
3.1.2 Influence of hydroelectric complexes and watershed inconsistencies
3.1.3 Rescaling of the meteorological data
3.2. Estimation and evaluation of the discharge sensitivity function g(Q)
3.2.1 Estimated catchment sensitivity functions over the Cévennes region
3.2.2 Sensitivity analysis of g(Q) to the data selected for recession extraction
3.2.3 Evaluation of the relevance of the estimated discharge sensitivity function using discharge simulation
3.3. Parameter regionalization of SIMPLEFLOOD model using FAMD
3.4. SIMPLEFLOOD model applied to the Gardon catchment
4. Discussion
4.1 Comparison of g(Q) with Adamovic, et al. (2014)
4.3 About the discharge sensitivity function estimation
4.4 In terms of SIMPLEFLOOD model performance
5. Conclusions and perspectives
6. References
