Generalized martingale model of the uncertainty evolution of streamflow forecasts

Streamflow forecasts are dynamically updated in real-time, thus facilitating a process of forecast uncertainty evolution. Forecast uncertainty generally decreases over time and as more hydrologic information becomes available. The process of forecasting and uncertainty updating can be described by the martingale model of forecast evolution (MMFE), which formulates the total forecast uncertainty of a streamflow in one future period as the sum of forecast improvements in the intermediate periods. This study tests the assumptions, i.e., unbiasedness, Gaussianity, temporal independence, and stationarity, of MMFE using real-world streamflow forecast data. The results show that (1) real-world forecasts can be biased and tend to underestimate the actual streamflow, and (2) real-world forecast uncertainty is non-Gaussian and heavy-tailed. Based on these statistical tests, this study proposes a generalized martingale model GMMFE for the simulation of biased and non-Gaussian forecast uncertainties. The new model combines the normal quantile transform (NQT) with MMFE to formulate the uncertainty evolution of real-world streamflow forecasts. Reservoir operations based on a synthetic forecast by GMMFE illustrates that applications of streamflow forecasting facilitate utility improvements and that special attention should be focused on the statistical distribution of forecast uncertainty.