Bayesian ensemble forecast of river stages and ensemble size requirements

Summary The problem is to provide a short-term, probabilistic forecast of a river stage time series { H 1 , … , H N } based on a probabilistic quantitative precipitation forecast. The Bayesian forecasting system (BFS) for this problem is implemented as a Monte-Carlo algorithm that generates an ensemble of realizations of the river stage time series. This article (i) shows how the analytic-numerical BFS can be used as a generator of the Bayesian ensemble forecast (BEF), (ii) demonstrates the properties of the BEF, and (iii) investigates the sample size requirements for ensemble forecasts (produced by the BFS or by any other system). The investigation of the ensemble size requirements exploits the unique advantage of the BFS, which outputs the exact, analytic, predictive distribution function of the stochastic process { H 1 , … , H N } , as well as can generate an ensemble of realizations of this process from which a sample estimate of the predictive distribution function can be constructed. By comparing the analytic distribution with its sample estimates from ensembles of different sizes, the smallest ensemble size M ∗ required to ensure a specified expected accuracy can be inferred. Numerical experiments in four river basins demonstrate that M ∗ depends upon the kind of probabilistic forecast that is constructed from the ensemble. Three kinds of forecasts are constructed: (i) a probabilistic river stage forecast (PRSF), which for each time n ( n = 1 , … , N ) specifies a predictive distribution function of H n ; (ii) a probabilistic stage transition forecast (PSTF), which for each time n specifies a family (for all h n - 1 ) of predictive one-step transition distribution functions from H n - 1 = h n - 1 to H n ; and (iii) a probabilistic flood forecast (PFF), which for each time n specifies a predictive distribution function of max { H 1 , … , H n } . Overall, the experimental results demonstrate that the smallest ensemble size M ∗ required for accurate estimation (or numerical representation) of these predictive distribution functions is (i) insensitive to experimental factors and on the order of several hundreds for the PRSF and the PFF and (ii) sensitive to experimental factors and on the order of several thousands for the PSTF. The general conclusions for system developers are that the ensemble size is an important design variable, and that the optimal ensemble size M ∗ depends upon the purpose of the forecast: for dynamic control problems (which require a PSTF), M ∗ is likely to be larger by a factor of 3–20 than it is for static decision problems (which require a PRSF or a PFF).