Hydroinformatics combines topics of modelling and decision-making, both of which have attracted a great deal of attention outside hydroinformatics from the point of view of uncertainty. Epistemic uncertainties are due to the inevitably incomplete evidence about the dependability of a model or set of competing models. Inherent uncertainties are due to the varying information content inherent in measurements or model predictions, be they probabilistic or fuzzy. Decision-making in management of the aquatic environment is, more often than not, a complex, discursive, multi-player process. The requirement for hydroinformatics systems is to support rather than replace human judgment in this process, a requirement that has significant bearing on the treatment of uncertainty. Furthermore, a formal language is required to encode uncertainty in computer systems. We therefore review the modern mathematics of uncertainty, starting first with probability theory and then extending to fuzzy set theory and possibility theory, the theory of evidence (and its random set counterpart), which generalises probability and possibility theory, and higher-order generalisations. A simple example from coastal hydraulics illustrates how a range of types of uncertain information (including probability distributions, interval measurements and fuzzy sets) can be handled in the types of algebraic or numerical functions that form the kernel of most hydroinformatic systems.

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