Entropy as a Model Diagnostic in Hydrological Modelling


“From a philosophical point of view, it is worth noting that measures such as Nash-Sutcliffe Efficiency attempt to match data of low statistical significance (e.g. high flow values) - and in many applications are highly biased towards those ranges. We raise the question: is this a good or bad thing in terms of data information extraction?” (Pechlivanidis et al., 2014a) 


Research Team

Dr Ilias G. Pechlivanidis

Dr Bethanna Jackson (Victoria University of Wellington, New Zealand)

Dr Hilary McMillan (National Institute for Water & Atmospheric Research, New Zealand)

Prof. Hoshin Gupta (University of Arizona, USA)



Calibration of rainfall-runoff models is made complicated by uncertainties in data, and by the arbitrary emphasis placed on various magnitudes of the model residuals by most traditional measures of fit. Different measures emphasise different systematic and/or dynamic behaviours within the hydrological system; hence a robust assessment of model performance using single measures is difficult (Schaefli and Gupta, 2007). Current research highlights the importance of moving from model calibration to diagnostic model evaluation, which aims to: 1) determine the information contained in the data and in the model, 2) examine the extent to which a model can be reconciled with observations, and 3) point towards the aspects of the model (or data limitations) that need improvement (Gupta et al., 2008).

Recent work has proposed the concept of a diagnostic evaluation approach rooted in information theory (Weijs et al., 2010). Information theoretic entropy-based measures provide a promising avenue to allow us to better identify where information is present and/or conflicting. If placed within a hydrologically relevant context, these measures may assist in the generation of more robust modelling frameworks allowing us to better diagnose model/data/hypotheses inconsistencies (Pechlivanidis et al., 2010). Information theoretic computations ultimately rely on quantities such as entropy, which has drawn the scientific community’s attention in a range of problems in hydrology and water resources; however the potential of information entropy measures to serve as objective functions, and the uses of entropy in conjunction with other measures as diagnostics in hydrological modelling are still unexplored.

We have recently developed an entropy measure, named Conditioned Entropy Difference (CED) metric, suited to capturing the static (non-dynamical) information contained in streamflow signals as described by the probability distribution, and hence of the flow duration curve (Pechlivandis et al., 2012). When combining CED with timing sensitive metrics in a multi-objective framework both parameter identifiability and model fit (both to the time series and the flow duration curve) are improved compared to more traditional multi-objective approaches (Pechlivanidis et al., 2014b).



Gupta, H., Wagener, T. and Liu, Y., 2008. Reconciling theory with observations: elements of a diagnostic approach to model evaluation, Hydrological Processes, 22(18): 3802-3813.

Pechlivanidis, I.G., Jackson, B. and McMillan, H., 2010. The use of entropy as a model diagnostic in rainfall-runoff modelling, iEMSs 2010: International Congress on Environmental Modelling and Software, 5-8 July, Ottawa, Canada, 2, 1780-1787.

Pechlivanidis I.G., Jackson B., McMillan H. and Gupta H., 2012. Using an informational entropy-based metric as a diagnostic of flow duration to drive model parameter identification, Global NEST Journal, 14(3), 325-334.

Pechlivanidis I.G., Jackson B., McMillan H. and Gupta H., 2014a. Robust informational entropy-based descriptors of flow in catchment hydrology, Hydrological Sciences Journal, doi: 10.1080/02626667.2014.983516 

Pechlivanidis I.G., Jackson B., McMillan H. and Gupta H., 2014b. Use of an entropy-based metric in multiobjective calibration to improve model performance, Water Resour. Res., 50, 8066–8083, doi: 10.1002/2013WR014537

Schaefli, B., and Gupta, H.V., 2007. Do Nash values have value?, Hydrological Processes, 21: 2075-2080.

Weijs, S. V., Schoups, G. and van de Giesen, N., 2010. Why hydrological predictions should be evaluated using information theory, Hydrology and Earth System Sciences, 14(12): 2545-2558.

Selected Research Outcomes