There are two fundamentally different approaches to model calibration and conditioning given observational data. The first is based on treating model error as an additive term to the model prediction:

Y(x,t) = M(θ, x, t) + ε(x, t)

Where Y is an observed values, the function M represents a model variable predicted using parameter set θ, ε is an error, x is space and t is time. Formal statistical assumptions about modelling errors are of this type and may involve additional parameters (such as bias and variance) in the error model ε(). Multiplicative errors can be treated in the same way by using log values of the observations and model predicted variables. The error model is normally evaluated using predictions based on some “optimal” parameter set. This is the basis of the regression and Bayesian methodologies described below. In some studies, the evaluation of uncertainties around an optimal model is carried out after optimisation, rather than as an intrinsic part of the model calibration process.

The second approach rejects the concept of an optimal model in favour of the equifinality concept of allowing for multiple acceptable models. It is a rejectionist approach, in that only those models considered to give acceptable predictions in calibration will be retained for use in prediction. The errors associated with the predictions of a particular parameter set are treated implicitly in that it is assumed that the structure of the errors found in calibration (in all their complexity) will be “similar” when that model is used in prediction. This is the basis of the Generalized Likelihood Uncertainty Estimation (GLUE) approach. In this approach, informal model performance measures can be used to decide whether a model is retained (but can use formal error assumptions as a special case, treating the parameters of the error model as additional parameter dimensions).

The distinction between this chapter and Real-time data assimilation is fuzzy. Some methods from Real-time data assimilation could be also under this heading

Methods

• Nonlinear regression

• Bayesian methods

• Generalized Likelihood Uncertainty Estimation (GLUE) Methods and Extended Generalized Likelihood Uncertainty Estimation (GLUE) (rejectionist) methods