Propagating reliable estimates of hydrological forecast uncertainty to many lead times

We propose a revised version of the ERRIS (error reduction and representation in stages) error model capable of reliably propagating hydrological uncertainty to long lead times (>150 time steps). ERRIS employs four stages: a transformation to handle heteroscedasticity, a moving average bias-correction, an autoregressive model and two mixture Gaussian distributions. To propagate uncertainty through multiple lead times, ERRIS makes use of a technique termed ‘stochastic updating’. Ensemble spread at long lead times is partly controlled by the interplay of the autoregression coefficient p and the width of the error distribution. When p approaches 1 and the error distribution is wide, this causes over-wide ensemble distributions at long lead times. We control this interplay with the moving average bias-correction, which reduces the value of p and the width of the error distribution, resulting in reliable ensembles at long lead times. An additional control on the width of the ensemble at longer lead times is a restriction applied to the autoregressive model. This restriction guards against large overcorrections, which can lead to very poor forecasts. Applying the restriction when parameters are inferred can result in over-wide residual distributions. We propose the simple expedient of applying the restriction only when forecasts are generated, not when parameters are inferred. We show through a comparison with an earlier version of ERRIS that this produces more reliable ensemble distributions at long lead times, whilst still guarding against overcorrection. The resulting error model is simple and computationally efficient, and thus suitable for deployment in operational streamflow forecasting systems.