Indices for assessing the prediction bounds of hydrological models and application by generalised likelihood uncertainty estimation

To reflect the uncertainties of a hydrological model in simulating and forecasting observed discharges according to rainfall inputs, the estimated result for each time step should not be just a point estimate (a single numerical value), but should be expressed as a prediction interval, i.e. a band defined by the prediction bounds of a particular confidence level a. How best to assess the quality of the prediction bounds thus becomes very important for understanding the modelling uncertainty in a comprehensive and objective way. This paper focuses on seven indices for characterizing the prediction bounds from different perspectives. For the three case-study catchments presented, these indices are calculated for the prediction bounds generated by the generalized likelihood uncertainty estimation (GLUE) method for various threshold values. In addition, the relationships among these indices are investigated, particularly that of the containing ratio (CR) to the other indices. In this context, three main findings are obtained for the prediction bounds estimated by GLUE. Firstly, both the average band-width and the average relative band-width are seen to have very strong linear correlations with the CR index. Secondly, a high CR value, a narrow band-width, and a high degree of symmetry with respect to the observed hydrograph, all of which are clearly desirable properties of the prediction bounds estimated by the uncertainty assessment methods, cannot all be achieved simultaneously. Thirdly, for the prediction bounds considered, the higher CR values and the higher degrees of symmetry with respect to the observed hydrograph are found to be associated with both the larger band-widths and the larger deviation amplitudes. It is recommended that a set of different indices, such as those considered in this study, be employed for assessing and comparing the prediction bounds in a more comprehensive and objective way.