The theory, which derives from Bayesian principles of rationality, offers a methodological framework and mathematical concepts for modeling warning systems in communities exposed to flash floods or rapid riverine floods. Models may be used (1) to develop optimal decision rules for issuing warnings, (2) to evaluate system performance statistically, and (3) to compute the expected economic benefits from a system [ Krzysztofowicz, 1993a].
A flood warning system is decomposed into a monitor, a forecaster, and a decider. The monitor is characterized by the diagnosticity and reliability parameters. Floods are characterized by a prior distribution of the crest height and time to crest; the forecaster is characterized by a family of likelihood functions of the actual crest, conditional on the forecasted crest. The decider is characterized by a disutility function which quantifies the relative undesirability of outcomes (such as response cost, property damage, lives lost) and whose expectation provides a criterion for decision making. Given these inputs, the theory provides three principal outputs: (1) an optimal warning rule, which prescribes whether or not to issue a warning based on an imperfect forecast of the flood crest, (2) a relative operating characteristic, which shows feasible trade-offs that a given system offers between the probability of detection and the probability of false warning, and (3) utilitarian measures of performance: the annual value of a given warning system, and the potential annual value of a warning system for a given floodplain.
The models of the monitor and the forecaster quantify all uncertainties associated with operation of the warning system, from the viewpoint of a decision maker. At the heart of this quantification is the Bayesian processor of forecasts (BPF) which outputs a posterior description of uncertainty about flood occurrence and crest height, conditional on a flood crest forecast.
One of the formidable challenges in applying the BPF in this and other contexts has been modeling of likelihood functions and derivation, or computation, of the posterior distribution when the prior distribution is not a member of the conjugate family for a specified likelihood model. Aside from the convenient normal-linear BPF, there are virtually no suitable analytic models. This has been a major hurdle in applying Bayesian methods to forecasts of flood peaks whose distributions are anything but Gaussian. A general analytic solution for the BPF has been found: the posterior distribution can be obtained for any prior distribution, parametric or nonparametric [ Kelly and Krzysztofowicz, 1994a]. This result paves the way for applications.