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 entropybased
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 km2 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.