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A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Theory and Scheme

Lolla, T. and P.F.J. Lermusiaux, 2017a. A Gaussian--Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Theory and Scheme. Monthly Weather Review. doi:10.1175/MWR-D-16-0064.1.

Retrospective inference through Bayesian smoothing is indispensable in geophysics, with crucial applications in ocean estimation, numerical weather prediction, climate dynamics and Earth system modeling. However, dealing with the high–dimensionality and nonlinearity of geophysical processes remains a major challenge in the development of Bayesian smoothers. Addressing this issue, we obtain a novel smoothing methodology for high– dimensional stochastic fields governed by general nonlinear dynamics. Building on recent Bayesian filters and classic Kalman smoothers, the equations and forward–backward algorithm of the new smoother are derived. The smoother uses the stochastic Dynamically–Orthogonal (DO) field equations and their time–evolving stochastic subspace to predict the prior probabilities. Bayesian inference, both forward and backward in time, is then analytically carried out in the dominant DO subspace, after fitting semi–parametric Gaussian Mixture Models (GMMs) to joint DO realizations. The theoretical properties and computational cost of the new GMM-DO smoother are presented and discussed.