Sequential State and Variance Estimation within the Ensemble Kalman Filter

Jonathan R. Stroud Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania

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Thomas Bengtsson Statistics and Data Mining Group, Bell Laboratories, Murray Hill, New Jersey

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Abstract

Kalman filter methods for real-time assimilation of observations and dynamical systems typically assume knowledge of the system parameters. However, relatively little work has been done on extending state estimation procedures to include parameter estimation. Here, in the context of the ensemble Kalman filter, a Monte Carlo–based algorithm is proposed for sequential estimation of the states and an unknown scalar observation variance. A Bayesian approach is adopted that yields analytical updating of the parameter distribution and provides samples from the posterior distribution of the states and parameters. The proposed assimilation algorithm extends standard ensemble methods, including perturbed observations, and serial and square root assimilation schemes. The method is illustrated on the Lorenz 40-variable system and is shown to be robust with system nonlinearities, sparse observation networks, and the choice of the initial parameter distribution.

Corresponding author address: Jonathan R. Stroud, Department of Statistics, The Wharton School, University of Pennsylvania, 400 Jon M. Huntsman Hall, 3730 Walnut St., Philadelphia, PA 19104-6340. Email: stroud@wharton.upenn.edu

Abstract

Kalman filter methods for real-time assimilation of observations and dynamical systems typically assume knowledge of the system parameters. However, relatively little work has been done on extending state estimation procedures to include parameter estimation. Here, in the context of the ensemble Kalman filter, a Monte Carlo–based algorithm is proposed for sequential estimation of the states and an unknown scalar observation variance. A Bayesian approach is adopted that yields analytical updating of the parameter distribution and provides samples from the posterior distribution of the states and parameters. The proposed assimilation algorithm extends standard ensemble methods, including perturbed observations, and serial and square root assimilation schemes. The method is illustrated on the Lorenz 40-variable system and is shown to be robust with system nonlinearities, sparse observation networks, and the choice of the initial parameter distribution.

Corresponding author address: Jonathan R. Stroud, Department of Statistics, The Wharton School, University of Pennsylvania, 400 Jon M. Huntsman Hall, 3730 Walnut St., Philadelphia, PA 19104-6340. Email: stroud@wharton.upenn.edu

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