Challenges for including error updating in real-time hydrological error models

Abstract

Accurate and reliable quantification of uncertainty in hydrological models promises to make real-time streamflow predictions both more accurate and more useful. One simple but effective approach to quantifying hydrological uncertainty is to apply an error model to streamflow simulations. An error model collates errors from all sources into prediction errors and builds up a statistical model of the error time series. Streamflows are often highly auto-correlated, and updating real-time hydrological error models with information from recent observations is an obvious means to improving the accuracy of the predictions. We attempt to include an error updating component in a two-stage error model. In Stage 1, we use a logarithmic hyperbolic sine transform to normalise both simulated and observed time series and stabilize variance. A bias-correction component is introduced to correct the bias of transformed simulations. In Stage 2, we apply an error updating procedure to the simulations obtained from Stage 1 by using the information from the previous time step. The error updating is based on the auto-correlation of hydrological errors. All model parameters in both stages are assumed to be seasonally dependent, because hydrological models often perform differently for different seasons. The residual term from both stages are assumed to be Gaussian. Stage 1 only uses the information from the present time step and can be applied without Stage 2, while Stage 2 relies on Stage 1 and requires information from the previous time step. We test the error model on hydrological simulations of four catchments generated with a conceptual daily rainfall-runoff model. In isolation, Stage 1 leads to similar or marginally more accurate predictions than the original hydrological simulations. The Stage 1 uncertainty estimation is generally reliable. Applying the error updating model (Stage 2) markedly improves the accuracy of Stage 1 simulations. However, we show that the uncertainty in the Stage 2 simulations is no longer reliably quantified after the error updating is applied. This is associated with our assumption that the residual term is Gaussian. After applying the error updating, most residuals in the error model are significantly reduced, while a few residuals remain. This causes the distribution of residuals to have a longer tail than a Gaussian distribution. The two-stage estimation of hydrological prediction uncertainty is shown to be simple but effective. The prediction accuracy is progressively improved after bias correction in the transform domain (Stage 1) and error updating in original domain (Stage 2). However, it remains challenging to offer reliable uncertainty estimation after including error updating. We recommend replacing the Gaussian distribution with more sophisticated distributions in the error model.