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Ensemble methods for meteorological predictions
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    Since the atmospheric system is a nonlinear chaotic system, its numerical prediction is bound by a predictability limit due to imperfect initial conditions and models. Ensemble forecasting is a dynamical approach to quantify the predictability of weather, climate and water forecasts. This chapter introduces various methods to create an ensemble of forecasts based on three aspects: perturbing initial conditions (IC), perturbing a model and building a virtual ensemble. For generating IC perturbations, methods include 1) random, 2) time-lagged, 3) bred vector, 4) ensemble transform (ET), 5) singular vector (SV), 6) conditional nonlinear optimal perturbation (CNOP), 7) ensemble transform Kalman filter (ETKF), 8) ensemble Kalman filter (EnKF), and 9) perturbations in boundaries including land surface and topography. For generating model perturbations, methods include 1) multi-model and multi-physics, 2) stochastically perturbed parametrization tendency (SPPT), 3) stochastically kinetic energy backscatter (SKEB), 4) convection triggering, 5) stochastic boundary-layer humidity (SHUM), 6) stochastic total tendency perturbation (STTP), and 7) vorticity confinement. A method to create a spatially correlated random pattern (mask) needed by SPPT and SKEB, etc., is introduced based on the Markov process; a factor-separation method is introduced to estimate the relative impact of various physics schemes and their interactions. A method of perturbing a dynamic core to create an ensemble is also mentioned. Quantitative forecast uncertainty information and ensemble products can also be generated from "virtual ensembles" based on existing deterministic forecasts through at least five different approaches including 1) time-lagged, 2) poor-man's, 3) hybrid, 4) neighborhood, and 5) analog ensembles. Generally speaking, the selection of perturbation methods in constructing an EPS is more important for smaller scale and shorter range forecasts and less critical for larger scale and longer range forecasts. Finally, the frequently asked question about the trade-off between ensemble size and model resolution is discussed. By introducing these methods, we hope to help readers who are interested in ensemble forecasting but not familiar with these approaches to build their own EPS or produce ensemble products as well as for students to learn the subject of ensemble forecasting.

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