| Summary: | Weather forecasts produced with numerical weather prediction (NWP) models of the atmosphere possess intrinsic uncertainty. This uncertainty is caused through both errors in the specification of the initial state of the model, as well as errors in the model formulation itself. In the process of NWP, the consideration of both error sources is important, because the nature of atmospheric dynamics is such that it acts to increase errors originating from either error source. In addition to this overall error-growth effect, forecast error possesses considerable day-to-day variability depending, among other things, on the flow regime. The quantitative and reliable assessment (i.e., prediction) of the uncertainty of weather forecasts is important, both for scientific and economic reasons. Scientifically, quantification of atmospheric predictability asks for the rate at which two initially dose trajectories diverge (on average) for given atmospheric dynamics. Such estimates place upper bounds on time horizons over which useful forecasts may be expected. Economically, a reliable estimate of the uncertainty of a particular forecast will lead to increased credibility and utility of weather forecasts. The description and discussion of strategies and methods to predict the uncertainty of weather forecasts produced with NWP models are the subject of this artide. The limited predictability of atmospheric flows, considered here on time scales of days, as it results essentially from the intrinsic error growth in the atmosphere is briefly discussed. The Liouville equation as the theoretical concept for dealing with the prediction problem of forecast uncertainty is described, as it governs the time-evolution of the probability density function (pdf) of the NWP model state. Related concepts more readily applicable in operational contexts are reviewed. Among these concepts are stochastic-dynamic prediction, the lagged-average forecasting technique, and the Monte Carlo approach. Particular attention is given to the description of methodology and results from currently operational efforts at major forecasting centers directed towards the prediction of forecast uncertainty through multiple (i.e., an ensemble of) NWP model integrations started from different initial states. Such time-evolved ensembles provide partial information about the time-evolved pdf. These ensemble prediction systems at the European Centre for Medium-Range Weather Forecasts, as well as the National Centers for Environmental Prediction are discussed in some detail (e.g., with respect to the selection of the initial states of individual ensemble members). Results presently obtained with ensemble prediction systems are highly promising, although various questions related to, for example, the modeling error source, the validation of products from ensemble prediction systems, as well as to the methodology for the selection of initial states remain to be answered. This review is conduded by mentioning briefly similar efforts at other operational NWP centers, as well as applications of the methodology used in ensemble prediction in related contexts such as stability analysis and data assimilation.
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