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The GR4J model is a catchment water balance model that relates runoff to rainfall and evapotranspiration using daily data. The model contains two stores and has 4 parameters.

GR4J operates at a catchment scale with a daily time-step.

The development of the GR4J model was initiated by Claude Michel at the beginning of the 1980s at Cemagref, a public research institute in France. The first version of the model only had a single parameter. Further development of the GR4J model was undertaken using a modelling approach where large numbers of catchments were used to evaluate and improve the model.

The GR4J modelling approach is mainly empirical (Michel et al., 2006), and consists of searching data for the most efficient model structures, with the objective of getting a general, efficient and robust model. The result being a parsimonious hydrological model, with successive improved versions. The main stages of the GR4J model development were:

• 3-parameter version proposed by Edijatno and Michel (1989) and Edijatno (1991). This provided the groundwork for further model development through testing and refinement;
• 4-parameter version proposed by Nascimento (1995) and detailed by Edijatno et al. (1999) (with one fixed parameter);
• 4-parameter version proposed by Perrin (2000) and detailed by Perrin (2002) and Perrin et al. (2003);
• 5-parameter version proposed by Le Moine (2008).

The successive versions of GR4J were widely tested on large sets of catchments in France but also in other countries, using demanding testing frameworks (Andréassian et al., 2009). The GR4J model has also been compared with other hydrological models and has provided comparatively good results (see eg Perrin et al., 2001; 2003; Vaze et al., 2011).

Source v2.10

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The mathematical details provided below follow the presentation of the model made by Perrin et al. (2003). Figure 11 shows a schematic diagram of the 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 is repeated every year.

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).

The first operation is the subtraction of E from P to determine either a net rainfall Pn or a net evapotranspiration capacity En. In GR4J, this operation is computed as if there were an interception storage of zero capacity. Pn and En are computed with the following equations:

Figure 3

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otherwise:

Equation

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This store can be considered as a soil moisture accounting (SMA) store. In case Pn is not zero, a part Ps of Pn fills the production store. It is determined as a function of the level S in the store by:

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where the terms are defined in Table 3.

Equation 5 and Equation 6 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 En 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:

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

Equation

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Note that S can never exceed x1. A representation of the rating curves obtained with Equation 5 and Equation 6 is shown in Figure 12. Appendix A details the calculations yielding to Equation 5 and Equation 6.

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

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Perc is always lower than S. This formulation is similar to the one of the routing store detailed in Appendix B. The reservoir content becomes:

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The percolation function in Equation 8 occurs as if it originated from a store with a maximum capacity of 9÷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.

The total quantity Pr of water that reaches the routing functions is given by:

Equation

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Pr is divided into two flow components according to a fixed split: 90 % of Pr is routed by a unit hydrograph UH1 and then a non linear routing store, and the remaining 10 % of Pr 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 days. However, UH1 has a time base of x4 days whereas UH2 has a time base of 2•x4 days. x4 can take real values and is greater than 0.5 days.

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 SH1 and SH2 respectively. SH1 is defined along time t by:

Equation

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Equation

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SH2 is similarly defined by:

Equation

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Figure 5

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Figure 6

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Equation

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UH1 and UH2 ordinates are then calculated by:

Equation

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where:

j is an integer.

If 0.5 ≤ x4 ≤ 1, UH1 has a single ordinate equal to one and UH2 has only two ordinates. Figure 13 shows an example of unit hydrograph ordinates for x4 = 3.8 days.

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

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Equation

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where:

l = int(x4)+1 et m = int(2•x4)+1, with int() the integer part.

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

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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.

The level in the routing store is updated by adding the output Q9 of UH1 and F as follows:

Equation

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The outflow Qr of the reservoir is then calculated as:

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Qr 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 (see Appendix B). The level in the reservoir becomes:

Equation

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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 14. Therefore, the capacity x3 could be called the "one day ahead maximum capacity". This routing store is able to simulate long stream flow recessions, when necessary.

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 Qd as follows:

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Total stream flow Q is finally obtained by:

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The model requires daily rainfall and potential evapotranspiration data. The rainfall and evaporation data sets need to be continuous and overlapping.

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.

Note that some figures in the model equations may appear as fixed parameter values, eg a power 4 in Equation 8 and Equation 22, a fixed split 10 % - 90 % of effective rainfall, a 2.5 exponent in the computation of the unit hydrographs, a 2.25 coefficient related to the percolation function in Equation 8. These values were chosen as those yielding the best model results in many different test conditions. They were fixed because leaving them free did not significantly improve (or even degraded) the model results while adding unhelpful complexity to the model structure.

Most optimisation algorithms used to calibrate the model parameters require knowledge of an initial parameters set. This initial set may consist of median values obtained on a large variety of catchments (see Table 5). Approximate 80 % confidence intervals for the four parameters are provided in Table 5. They were derived from the 0.1 and 0.9 percentiles of the distributions of model parameters obtained over a large sample of catchments. 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.

The model outputs daily surface flow and intercatchment groundwater exchange flow, expressed in mm/day.

Perrin, C., C. Michel, and V. Andréassian (2003), Improvement of a parsimonious model for streamflow simulation, J. Hydrol., 279, 275-289.

Andréassian, V., C. Perrin, L. Berthet, N. Le Moine, J. Lerat, C. Loumagne, L. Oudin, T. Mathevet, M. H. Ramos, and A. Valéry (2009), Crash tests for a standardized evaluation of hydrological models, Hydrol. Earth. Syst. Sci., 13, 1757-1764.

Edijatno (1991), Mise au point d’un modèle élémentaire pluie-débit au pas de temps journalier, Thèse de Doctorat thesis, 242 pp, Université Louis Pasteur/ENGEES, Strasbourg.

Edijatno, and C. Michel (1989), Un modèle pluie-débit journalier à trois paramètres, La Houille Blanche, 113-121.

Edijatno, N. O. Nascimento, X. Yang, Z. Makhlouf, and C. Michel (1999), GR3J: a daily watershed model with three free parameters, Hydrol. Sci. J., 44, 263-277.

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 ?, Thèse de Doctorat thesis, 324 pp, Université Pierre et Marie Curie, Paris.

Michel, C., Perrin, C., Andréassian, V., Oudin, L. and Mathevet, T.(2006) Has basin-scale modelling advanced beyond empiricism? In: Andréassian, V., Hall, A., Chahinian, N. and Schaake, J. (eds) Large sample basin experiments for hydrological model parameterization: results of the model parameter experiment (MOPEX) 2006 pp. 108-116. IAHS Publication 307.

Nascimento, N. O. (1995), Appréciation à l’aide d’un modèle empirique des effets d’action anthropiques sur la relation pluie-débit à l’échelle du bassin versant, Thèse de Doctorat thesis, 550 pp, CERGRENE/ENPC, Paris.

Perrin, C. (2000), Vers une amélioration d’un modèle global pluie-débit au travers d’une approche comparative, Thèse de Doctorat thesis, 530 pp, INPG (Grenoble) / Cemagref (Antony).

Perrin, C. (2002), Vers une amélioration d’un modèle global pluie-débit au travers d’une approche comparative, La Houille Blanche, 84-91.

Perrin, C., C. Michel, and V. Andréassian (2001), Does a large number of parameters enhance model performance? Comparative assessment of common catchment model structures on 429 catchments, J. Hydrol., 242(3-4), 275-301.

Vaze, J., Chiew, F. H. S., Perraud, JM., Viney, N., Post, D. A., Teng, J., Wang, B., Lerat, J., Goswami, M., 2011. Rainfall-runoff modelling across southeast Australia: datasets, models and results. Australian Journal of Water Resources, Vol 14, No 2, pp. 101-116.