Entropy in the assessment of uncertainty in hydrologic systems and models

In the representation of natural catchment systems with mathematical models, a relatively wide range of choice exists among models of various degrees of completeness and sophistication. When the cost, as well as the appropriateness, of a model is considered for a particular purpose, it is desirable to have an objective criterion to make this selection. The concept of entropy as used in information theory provides one such criterion. Entropy is a measure of the degree of uncertainty of a particular outcome in a process; therefore, in dealing with the prediction of a hydrologie variable such as streamflow, one can compute the entropy of this variable from historical data and thus characterize the unexpectedness or variability inherent in the process. This represents a property of the system and is called marginal entropy. It is likewise possible to evaluate the uncertainty of the predictions made by a given mathematical model of the catchment by comparing these predictions with the measured flow. This is done through the conditional entropy. By combination of these two entropy functions a criterion called ‘transinformation,’ providing an objective evaluation of the goodness of the model, is obtained. The important point to note is that this assessment depends not only on the model but on the characteristics of the output series resulting from the natural process. The above criteria, which have been discussed in other contexts, were applied to a basin for which a number of years of historical flow record and concurrent model simulations were available. The results show the value of the entropy criterion in judging model performance as well as the difficulties and limitations of the method.