The GR4H model is a catchment water balance model that relates runoff to rainfall and evapotranspiration using hourly data. The model contains two stores and has four parameters.
Scale
GR4H operates at a catchment scale with an hourly timestep.
Principal developer
The GR4H model is a version of the daily time step, GR4J model, which was modified to be suitable for running at hourly time step. Further details on the development of GR4J are provided in the scientific reference guide for GR4J Add link to GR4J here. Details of the GR4H model are provided in Perrin et al. (2003).
Scientific provenance
The successive versions of GR4H were widely tested on large sets of catchments in France but also in other countries, using demanding testing frameworks (Andréassian et al., 2009). The GR4H model has also been compared with other hydrological models and has provided comparatively good results (see e.g. Perrin et al., 2001; 2003).
Dependencies
None.
Structure and processes
The mathematical details provided below follow the presentation of the model made by Perrin et al. (2003). Figure 1 shows a schematic diagram of the model.
Figure 1. Schematic diagram of the GR4J model
In the following, for calculations at a given timestep, we note P the rainfall depth and E the potential evapotranspiration estimate that are inputs to the model. P is an estimate of the areal catchment rainfall that can be computed by any interpolation method from available rain gauges. E can be based on longterm average monthly or daily values, which means the same potential evapotranspiration series could be repeated every year, although a recorded time series of E would be expected to give a better result.
All water quantities (input, output, internal variables) are expressed in mm, by dividing water volumes by catchment area, when necessary. All the operations described below are relative to a given timestep and correspond to a discrete model formulation (obtained after integration of the continuous formulation over the timestep).
Determination of net rainfall and PE
The first operation is the subtraction of E from P to determine either a net rainfall P_{n} or a net evapotranspiration capacity E_{n}. In GR4J, this operation is computed as if there were an interception storage of zero capacity. P_{n} and E_{n} are computed with the following equations:
Equation 1 
otherwise:
Equation 2 
Production store
This store can be considered as a soil moisture accounting (SMA) store. In case P_{n} is not zero, a part P_{s} of P_{n} fills the production store. It is determined as a function of the level S in the store by:

where the terms are defined in Table 1.
Table1. Model parameter definitions
Parameter  Definition 

E  Potential areal evapotranspiration 
E_{n}  Net evapotranspiration capacity 
E_{s}  Actual evaporation rate 
F(x_{2})  Groundwater exchange term 
P  Areal catchment rainfall 
Perc  Percolation leakage 
P_{n}  Net rainfall 
P_{r}  Total quantity of water to reach routing functions 
P_{n}P_{s}  Amount of net rainfall that goes directly to the routing functions 
P_{s}  Amount of net rainfall that goes directly to the production store 
Q  Total stream flow 
Q1  Output of UH2 
Q9  Output of UH1 
Q_{d}  Direct flow component 
Q_{r}  Routed flow component 
R  Water content in the routing store 
S  Water content in the production store 
UH1, UH2  Unit hydrographs 
x_{1}  Capacity of the production soil (SMA) store (mm) 
x_{2}  Water exchange coefficient (mm) 
x_{3}  Capacity of the routing store (mm) 
x_{4}  Time parameter (days) for unit hydrographs 
Equation 3 and Equation 4 result from the integration over the timestep of the differential equations that have a parabolic form with terms in (S/x_{1})², as detailed by (Edijatno and Michel, 1989).
In the other case, when E_{n} is not zero, an actual evaporation rate is determined as a function of the level in the production store to calculate the quantity Es of water that will evaporate from the store. It is obtained by:

The water content in the production store is then updated with:
Equation 5 
Note that S can never exceed x_{1}. A representation of the rating curves obtained with Equation 3 and Equation 4 is shown in Figure 2.
Figure 2. Behaviour of the production functions (E_{s}/E_{n}: solid line; P_{s}/P_{n}: dashed line) as a function of storage rate S/x_{1} for different values of E_{n}/x_{1} or P_{n}/x_{1}
A percolation leakage Perc from the production store is then calculated as a power function of the reservoir content:
 Perc=S11+14Sx141/4 
Perc is always lower than S. The reservoir content becomes:
Equation 7 
The percolation function in Equation 6 occurs as if it originated from a store with a maximum capacity of 4•x_{1}. Given the power law of the mathematical formulation, this means that the percolation does not contribute much to the stream flow and is interesting mainly for low flow simulation.
Linear routing with unit hygrographs
The total quantity P_{r} of water that reaches the routing functions is given by:
Equation 8 
P_{r} is divided into two flow components according to a fixed split: 90 % of P_{r} is routed by a unit hydrograph UH1 and then a non linear routing store, and the remaining 10% of P_{r} is routed by a single unit hydrograph UH2. With UH1 and UH2, one can simulate the time lag between the rainfall event and the resulting stream flow peak. Their ordinates are used in the model to spread effective rainfall over several successive timesteps. Both unit hydrographs depend on the same time parameter x_{4} expressed in hours. However, UH1 has a time base of x_{4} hours whereas UH2 has a time base of 2•x_{4} hours. x_{4} can take real values and is greater than 0.5 hours.
In their discrete form, unit hydrographs UH1 and UH2 have n and m ordinates respectively, where n and m are the smallest integers exceeding x_{4} and 2•x_{4} respectively. This means that the water is staggered into n unit hydrograph inputs for UH1 and m inputs for UH2. The ordinates of both unit hydrographs are derived from the corresponding Scurves (cumulative proportion of the input with time) denoted by SH1and SH2 respectively. SH1 is defined along time t by:
Equation 9  
Equation 10  For 0<t<x4, SH1t=tx454 
Equation 11 

SH2 is similarly defined by:
Equation 12  

 For 0<t≤x4, SH2t=12tx454 
 For 0<t<2x4, SH2t=1122tx454 

Equation 15 
UH1 and UH2 ordinates are then calculated by:
Equation 16  

Equation 17 
where:
j is an integer.
If 0.5 ≤ x_{4} ≤ 1, UH1 has a single ordinate equal to one and UH2 has only two ordinates.
At each timestep, the outputs Q9 and Q1 of the two unit hydrographs correspond to the discrete convolution products and are given by:
 


where:
Equation 20 

Inter catchment groundwater exchange
A groundwater exchange term F that acts on both flow components, is then calculated as:

Where R is the level in the routing store, x_{3} its "reference" capacity and x_{2} the water exchange coefficient. x_{2} can be either positive in case of water imports, negative for water exports or zero when there is no water exchange. The higher the level in the routing store, the larger the exchange. In absolute value, F cannot be greater than x_{2}: x_{2} represents the maximum quantity of water that can be added (or released) to (from) each model flow component when the routing store level equals x_{3}. Note that Le Moine (2008) proposed an improved formulation of this function, with an additional parameter.
Non linear routing store
The level in the routing store is updated by adding the output Q9 of UH1 and F as follows:
Equation 22 

The outflow Q_{r }of the reservoir is then calculated as:


Q_{r} is always lower than R, as shown in Figure 4. The formulation of the output of the store is the same as the percolation from the SMA store. The level in the reservoir becomes:
Equation 24 

Note that, although the reservoir can receive a water input greater than the saturation deficit x_{3}R at the beginning of a timestep, the level in the reservoir can never exceed the capacity x_{3} at the end of a timestep, as shown in Figure 4. Therefore, the capacity x_{3} could be called the "one hour ahead maximum capacity". This routing store is able to simulate long stream flow recessions, when necessary.
Figure 4. Illustration of the outflow Q_{r} from the routing reservoir as a function of the level in the store after the introduction of input Q9
Total stream flow
Like the content of the routing store, the output Q1 of UH2 is subject to the same water exchange F to give the flow component Q_{d} as follows:
Equation 25 

Total stream flow Q is finally obtained by:
Equation 26 

Input data
The model requires daily rainfall and potential evapotranspiration data. The rainfall and evaporation data sets need to be continuous and overlapping.
Parameters or settings
Information on parameters is provided in Table 2. All four parameters are real numbers. x_{1} and x_{3} are positive, x_{4} is greater than 0.5 and x_{2} can be either positive zero or negative.
Guidance on expected median values and ranges for parameters were obtained from van Esse (2012, p. 83)
Table 2: Parameters in GR4H and their default values
Parameter  Description  Units  Default  Range 
x_{1}  Capacity of the production soil (SMA) store  mm  350  11200 
x_{2}  Water exchange coefficient  none  0  5.03.0 
x_{3}  Capacity of the routing store  mm  199  44663 
x_{4}  Time parameter for unit hydrographs  hours  5.0  0.596 
Most optimisation algorithms used to calibrate the model parameter values require knowledge of an initial parameter set. This initial set may consist of median values obtained on a large variety of catchments (for example, see Table 3). Given the small number of model parameters, simple optimisation algorithms are generally capable of identifying parameter values yielding satisfactory results. The choice of an objective function depends on the objectives of model user. Note that care should be taken to set appropriate initial conditions of the internal state variables in the model to avoid discrepancies at the beginning of the simulation periods. One year can be used for model warmup at the beginning of each simulation.
Table 3: Values of median model parameters and approximate 80% confidence intervals
Parameter  Median Value  80% Confidence Interval 

x_{1}  350  1001200 
x_{2}  0  5 to 3 
x_{3}  90  20300 
x_{4}  1.7  1.12.9 
Output data
The model outputs daily surface flow and intercatchment groundwater exchange flow, expressed in mm/day.
References
Perrin, C., C. Michel, and V. Andréassian (2003), Improvement of a parsimonious model for streamflow simulation, J. Hydrol., 279, 275289.
van Esse, W.R. (2012) Demystifying hydrological monsters, Can flexibility in model structure explain monster catchments?, M.Sc. Thesis, University of Twente, Enshede, Netherlands.
Bibliography
Le Moine, N. (2008), Le bassin versant de surface vu par le souterrain : une voie d'amélioration des performances et du réalisme des modèles pluiedébit ?, PhD thesis (french), UPMC, Paris, France.