Accounting for Model Error in Ensemble-Based State Estimation and Forecasting

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Abstract
Accurate forecasts require accurate initial conditions. For systems of interest, even given a perfect model and an infinitely long time series of observations, it is impossible to determine a system's exact initial state. This motivates a probabilistic approach to both state estimation and forecasting. Two approaches to probabilistic state estimation, the ensemble Kalman filter, and a probabilistic approach to 4DVAR are compared in the perfect model framework using a two-dimensional chaotic map. Probabilistic forecasts are fed back into the probabilistic state estimation routines in the form of background weighting information. It is found that both approaches are capable of producing correct probabilistic forecasts when a perfect model is in hand, but the probabilistic approach to 4DVAR appears to be the least sensitive to nonlinearities. When only imperfect models are available (i.e., always), one does not have access to the distribution that produces truth, and it is therefore impossible to produce a correct probabilistic forecast. A multimodel approach to ensemble forecasting provides an opportunity to generate ensembles that systematically bound the true system state. Results suggest that a beneficial approach to ensemble construction is to produce multimodel ensemble members that lie on their respective model attractors, and to select model attractors that systematically bound the system attractor. The inclusion of multimodel uncertainty information in ensemble Kalman filter–like approaches allows ensemble members to be drawn off their respective model attractors, while the dynamical constraints intrinsic to probabilistic 4DVAR enables the approach to ignore ensemble spread due to model differences and to produce analyses that remain on their respective model attractors. Rather than implying that probabilistic 4DVAR is the preferred technique for multimodel data assimilation, these results suggest that it is best to ignore multimodel information during data assimilation, either implicitly (e.g., probabilistic 4DVAR) or explicitly (e.g., the “poor man's ensemble”).