Using an informational entropy-based metric as a diagnostic of flow duration to drive model parameter identification

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. Current research highlights the importance of driving model identification by assimilating information from the data. In this paper, we evaluate the potential use of an entropy based measure as an objective function or as a model diagnostic in hydrological modelling, with particular interest in providing an appropriate quantitative measure of fit to the flow duration curve (FDC). The proposed Conditioned Entropy Difference (CED) metric is capable of characterising the information in the flow frequency distribution and thereby constrain the model calibration to respect this distributional information. Four years of hourly data from the 46.6 km ² Mahurangi catchment, NZ, are used to calibrate the 6-parameter Probability Distributed Moisture model. Results are analysed using three measures: the proposed entropy-based measure, the Nash-Sutcliffe (NSE), and the recently proposed Kling-Gupta efficiency (KGE). We also examine a conditioned entropy metric that trades-off and reweights different segments of the FDC to drive model calibration in a way that is based on modelling objectives. Overall, the entropy-based measure results in good performance in terms of NSE but poor performance in terms of KGE. This entropy measure is strongly sensitive to the shape of the flow distribution and is, from some viewpoints, the single best descriptor of the FDC. By conditioning entropy to respect multiple segments of the FDC, we can reweight entropy to respect those parts of the flow distribution of most interest to the modelling application. This approach constrains the behavioural parameter space so as to better identify parameters that represent both the "fast" and "slow" runoff processes. Use of this importance-weighted, conditioned entropy metric can constrain high flow predictions equally well as the NSE and KGE, while simultaneously providing wellconstrained low flow predictions that the NSE or KGE are unable to achieve.