Mathias et al. (2025) A theoretical appraisal of the GR4J rainfall-runoff modelling framework

Identification

• Journal: Journal of Hydrology
• Year: 2025
• Date: 2025-10-16
• Authors: Simon A. Mathias, Cyril Thébault, Andrew Ireson
• DOI: 10.1016/j.jhydrol.2025.134393

Research Groups

• Department of Engineering, Durham University, Durham, UK
• Schulich School of Engineering, University of Calgary, Canada
• Global Institute for Water Security, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Short Summary

This study theoretically appraises the GR4J rainfall-runoff model, linking its heuristic components to physically-based processes and proposing modifications to eliminate operator splitting. The revised model structure maintains calibration and validation performance across 671 UK catchments while offering improved physical interpretability and computational efficiency.

Objective

• To establish clearer links between heuristic aspects within GR4J and more physically-based counterparts.
• To identify practical changes to minimise the need for operator splitting within GR4J’s original analytical solution framework.

Study Configuration

• Spatial Scale: 671 catchments across Great Britain (UK CAMELS-GB database), with catchment areas ranging from 1.63 km² to 9930 km².
• Temporal Scale: Daily time-step for hydrometeorological data. Warm-up period: 1970–2005; Calibration period: 2005–2010; Validation period: 2010–2015.

Methodology and Data

• Models used:
  • GR4J (Génie Rural à 4 paramètres Journalier)
  • GR4Ja (GR4J with no percolation and linear inter-catchment groundwater exchange)
  • GR4Jb (GR4Ja incorporating a moment-matched diffusion wave convolution kernel and the Exponential Storage Function Analytical Solution (ESFAS) for non-linear flow routing)
  • Probability Distributed Model (PDM)
  • Diffusion wave model (moment-matched with GR4J unit hydrograph)
  • Exponential Storage Function Analytical Solution (ESFAS)
  • Numerical solvers: Euler explicit, Euler implicit, Heun explicit, Heun implicit
  • Reference solver: MATLAB's ODE45 (explicit Runge–Kutta (4,5) Dormand–Prince pair)
• Data sources: CAMELS-GB dataset, providing catchment-averaged daily potential evapotranspiration (Penman–Monteith), daily precipitation, and daily mean river flow data for 671 catchments in Great Britain.

Main Results

• GR4J’s soil water model is mathematically identical to the Probability Distributed Model (PDM), employing a fixed probability density function that resembles a log-normal distribution.
• The GR4J empirical unit hydrograph can be moment-matched with a diffusion wave model, allowing its parameters (x4 and η) to be related to open-channel flow hydraulic parameters (wave speed, mean velocity, Peclet number, friction slope).
• Operator splitting in GR4J can be entirely avoided by omitting percolation (κ=0), assuming inter-catchment groundwater exchange is a linear function of discharge rate (bg=b), and adopting the Exponential Storage Function Analytical Solution (ESFAS) for the non-linear flow routing store.
• Testing the modified GR4J formulations (GR4Ja and GR4Jb) across 671 UK catchments revealed minimal impacts on model calibration and validation performance (Nash–Sutcliffe Efficiency and absolute percentage bias) compared to the original GR4J.
• For catchments where GR4J performed reliably (NSE > 0.8), GR4Ja and GR4Jb showed strong correlations (≥0.884) in both calibrated parameters and performance indices.
• Numerical analysis on four diverse catchments showed that ESFAS outperformed the operator splitting scheme in catchments with more dynamic flow regimes (NSE > 0.977 for ESFAS vs. > 0.933 for operator splitting, compared to ODE45 reference).
• The ESFAS was found to be at least 200 times faster than the ODE45 solution and consistently slightly faster than the operator splitting scheme. The Heun explicit scheme was identified as the most accurate and computationally efficient among the standard numerical solvers tested.

Contributions

• Provides a deeper theoretical basis for GR4J's heuristic components by establishing mathematical links to physically-based hydrological models (PDM, diffusion wave equation derived from Saint-Venant equations).
• Introduces a revised GR4J model structure (GR4Jb) that is entirely free of operator splitting, leading to more numerically reliable and computationally efficient simulations.
• Demonstrates that these theoretical and structural modifications maintain the model's predictive performance across a large and diverse set of catchments.
• Offers a more physically interpretable framework for GR4J, enhancing understanding of its internal processes.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Citation

@article{Mathias2025theoretical,
  author = {Mathias, Simon A. and Thébault, Cyril and Ireson, Andrew},
  title = {A theoretical appraisal of the GR4J rainfall-runoff modelling framework},
  journal = {Journal of Hydrology},
  year = {2025},
  doi = {10.1016/j.jhydrol.2025.134393},
  url = {https://doi.org/10.1016/j.jhydrol.2025.134393}
}