process includes identifying and representing key sources of uncertainty; identifying physical observations; experiments, or other information sources for the assessment; assessing prediction uncertainty; assessing the reliability or quality of the prediction; supplying information on how to improve the assessment; and communicating results.

Identifying and representing uncertainties typically involves sensitivity analysis to determine which features or inputs of the model affect key model outputs. Once they are identified, one must determine how best to represent these important contributors to uncertainty—parametric representations of input conditions, forcings, or physical modeling schemes (e.g., turbulent mixing of fluids). In addition to parametric forms, some analyses might assess the impact of alternative physical representations/schemes within the model. If solution errors or other sources of model discrepancy are likely to be important contributors to prediction uncertainty, their impact must also be captured in some way.

The available physical observations are key to any validation assessment. In some cases these data are observational, provided by nature (e.g., meteorological measurements, supernova luminosities); in other cases, data come from a carefully planned hierarchy of controlled experiments—e.g., the Predictive Engineering and Computational Sciences (PECOS) case study in Section 5.9. In addition to physical observations, information may come from the literature or expert judgment that may incorporate historical data or known physical behavior.

Estimating prediction uncertainty requires the combination of computational models, physical observations, and possibly other information sources. Exactly how this estimation is carried out can range from very direct, as in the weather forecasting example in Figure 5.1, to quite complicated, as described in the case studies in this chapter. In these examples, some physical observations are used to refine or constrain uncertainties that contribute to prediction uncertainty. Estimating prediction uncertainty is a vibrant research topic whose methods vary depending on the features of the problem at hand.

For any prediction, assessing the quality, or reliability, of the prediction is crucial. This concept of prediction reliability is more qualitative than is prediction uncertainty. It includes verifying the assumptions on which an estimate is based, examining the available physical measurements and the features of the computational model, and applying expert judgment. For example, well-designed sets of experiments can lead to stronger statements regarding the quality and reliability of more extrapolative predictions, as compared to observational data from a single source. Here the concept of “nearness” of the physical observations to the predictions of the intended use of the model becomes relevant, as does the notion of the domain of applicability for the prediction. However,


FIGURE 5.1 Daily maximum temperatures for Norman, Oklahoma (left), and histograms of next-day prediction errors (right) using two prediction models. The top histogram shows residuals from the persistence model, predicting tomorrow’s high temperature with today’s high temperature. The bottom histogram shows residuals from the National Weather Service (NWS) forecast. Ninety percent of the actual temperatures are within ±14°F for the persistence-model forecasts and within ±6°F for the NWS forecasts. The greater accuracy of the NWS forecasts is due to NWS’s use of computational models and additional meteorological information. The assessment of these two forecast methods is relatively straightforward because of the large number of comparisons of model forecast to measurement. SOURCE: Data from Brooks and Doswell (1996).

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