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 time-step.

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 time-step, 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 long-term 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 time-step and correspond to a discrete model formulation (obtained after integration of the continuous formulation over the time-step).

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:

Equation 3

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(x₂)	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
x₁	Capacity of the production soil (SMA) store (mm)
x₂	Water exchange coefficient (mm)
x₃	Capacity of the routing store (mm)
x₄	Time parameter (days) for unit hydrographs

Equation 3 and Equation 4 result from the integration over the time-step of the differential equations that have a parabolic form with terms in (S/x₁)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:

Equation 4

The water content in the production store is then updated with:

Equation 5

Note that S can never exceed x₁. 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/x₁ for different values of E_n/x₁ or P_n/x₁

A percolation leakage Perc from the production store is then calculated as a power function of the reservoir content:

Equation 6

Perc=S1-1+14Sx14-1/4

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•x₁. 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 time-steps. Both unit hydrographs depend on the same time parameter x₄ expressed in hours. However, UH1 has a time base of x₄ hours whereas UH2 has a time base of 2•x₄ hours. x₄ 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 x₄ and 2•x₄ 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 S-curves (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
Equation 13	For 0<t≤x4, SH2t=12tx454

Equation 14	For 0<t<2x4, SH2t=1-122-tx454
Equation 15

UH1 and UH2 ordinates are then calculated by:

Equation 16
Equation 17

where:
j is an integer.
If 0.5 ≤ x₄ ≤ 1, UH1 has a single ordinate equal to one and UH2 has only two ordinates.
At each time-step, the outputs Q9 and Q1 of the two unit hydrographs correspond to the discrete convolution products and are given by:

Equation 19
Equation 18

where:

Equation 20

Inter catchment groundwater exchange

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

Equation 21

Where R is the level in the routing store, x₃ its "reference" capacity and x₂ the water exchange coefficient. x₂ 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 x₂: x₂ represents the maximum quantity of water that can be added (or released) to (from) each model flow component when the routing store level equals x₃. 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_rof the reservoir is then calculated as:

Equation 23

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 x₃-R at the beginning of a time-step, the level in the reservoir can never exceed the capacity x₃ at the end of a time-step, as shown in Figure 4. Therefore, the capacity x₃ 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. x₁ and x₃ are positive, x₄ is greater than 0.5 and x₂ 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
x₁	Capacity of the production soil (SMA) store	mm	350	1-1200
x₂	Water exchange coefficient	none	0	-5.0-3.0
x₃	Capacity of the routing store	mm	199	44-663
x₄	Time parameter for unit hydrographs	hours	5.0	0.5-96

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 warm-up at the beginning of each simulation.

Table 3: Values of median model parameters and approximate 80% confidence intervals

Parameter	Median Value	80% Confidence Interval
x₁	350	100-1200
x₂	0	-5 to 3
x₃	90	20-300
x₄	1.7	1.1-2.9

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, 275-289.
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 pluie-débit ?, PhD thesis (french), UPMC, Paris, France.

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