With the formulation of Equation (1), one seeks to obtain the probability distribution of calibration parameters, ș by comparing model predictions ysim(x t, θ), to physical observations y obs(x t) while simultaneously making an independent estimate for the discrepancy bias, į(xt). As seen, the discrepancy bias is fundamentally different than the commonly adapted concept of “goodness-of-fit.” Instead, the discrepancy estimated over the domain of applicability provides a notion of predictive maturity of the model or rather, its lack of maturity. The accurate estimation of discrepancy term depends heavily on the quantity of the available experimental measurements for discrepancy bias. Naturally, the discrepancy function would be more reliable if it is trained with a larger amount of discrete data points. Kennedy and O’Hagan’s [1] estimation of the discrepancy term, is closely associated with the calibration of the model parameters. In a later implementation of this approach, Higdon et al. [2] used fast running Gaussian Process Models (GPM) as surrogates to estimate the discrepancy bias. It must be noted that the mathematical definition of the discrepancy term inherently considers the presence of experimental and numerical uncertainty. 3 USEFULNESS OF A SIMULATION MODEL FOR FORECASTING In a treatment combining parameter calibration and bias correction procedures, experimental measurements improve the numerical model in two distinct ways: (1) by reducing the uncertainty in the model parameters and (2) by quantifying the discrepancy more accurately. As a result, a model calibrated with an increasing amount of experimental information should have increasing fidelity. To formalize this seemingly obvious statement, a series of conceptual assertions must first be made. This section introduces a sequence of three conceptual assertions that ultimately lead to the notion of forecasting metrics. 3.1 Requirements for Forecasting Predictions We postulate that, a numerical model to be useful for forecasting purposes must satisfy the following two requirements: (a) Model predictions must be consistent with the experimental measurements that are relevant to the specific application of the model. The estimated functional form for the discrepancy bias introduced in the previous section is a quantitative and rigorous representation of the model’s consistency with measurements. This very 435
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