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