Representing uncertainty in objective functions: extension to include the influence of serial correlation

Abstract

The role of performance indicators is to give an accurate indication of the fit between a model and the system being modelled. As all measurements have an associated uncertainty (determining the significance that should be given to the measurement), performance indicators should take into account uncertainties in the observed quantities being modelled as well as in the model predictions (due to uncertainties in inputs, both data and model parameters). In the presence of significant heteroscedasticity in the uncertainty in observed and modelled output of a system, failure to adequately account for variations in the uncertainties means that the objective function only gives a measure of how well the model fits the observations, not how well the model fits the system being modelled. Since in most cases, the interest lies in fitting the system response, it is vital that the objective function(s) be designed to account for any variations in the uncertainties. Most objective functions (e.g. those based on the sum of squared residuals) assume homoscedastic uncertainties. If model contribution to the variations in residuals can be ignored, then transformations (e.g. Box-Cox) can be used to remove (or at least significantly reduce) heteroscedasticity. An alternative which is more generally applicable is to explicitly represent the uncertainties in the observed and modelled values in the objective function. Previous work on this topic addressed the modifications to standard objective functions (Nash-Sutcliffe efficiency - NSE, Root Mean Square Error - RMSE, chi-squared, coefficient of determination) using the optimal weighted averaging approach. A Monte Carlo trial using synthetic data with known uncertainty in the rating curve shows that the modified NSE gives significantly lower uncertainty in the estimated parameter values compared to those derived using the standard NSE. In addition to the heteroscedasticity in the uncertainties, there may also be significant serial correlation in the uncertainties for different time steps as a result of the use of a rating curve in estimating the observed flows, and for modelled flows, the influence of system memory (propagation of input uncertainties through the model). This includes not only first-order serial correlation (correlation between the uncertainties of the values for neighbouring time steps), but also longer-term serial correlation induced through the uncertainty in the rating curve. The extent of the longer-term serial correlation will depend on the stability of the rating curve. Thus, the requirement for exploiting the long-term serial correlation is information on the uncertainty and stability in the rating curve, information which is not necessarily available at the current time. This requires a change in the way that streamflow databases are constructed. A modified form of the NSE has been proposed, and the effect of including first-order serial correlation tested against synthetic data. The result of a Monte Carlo trial shows significant reduction in parameter uncertainty when first-order serial correlation is included in the objective function.