Sondergaard, T. and P.F.J. Lermusiaux, 2012a. Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal
Field Equations. Part I. Theory and Scheme. Monthly Weather Review, 141, 6, 1737-1760, doi:10.1175/MWR-D-11-00295.1.
This work introduces and derives an efficient, data-driven assimilation scheme, focused on a
time-dependent stochastic subspace, that respects nonlinear dynamics and captures non-Gaussian
statistics as it occurs. The motivation is to obtain a filter that is applicable to realistic geophysical
applications but that also rigorously utilizes the governing dynamical equations with information
theory and learning theory for efficient Bayesian data assimilation. Building on the foundations of
classical filters, the underlying theory and algorithmic implementation of the new filter are developed
and derived. The stochastic Dynamically Orthogonal (DO) field equations and their adaptive
stochastic subspace are employed to predict prior probabilities for the full dynamical state, effectively
approximating the Fokker-Planck equation. At assimilation times, the DO realizations are fit to
semiparametric Gaussian mixture models (GMMs) using the Expectation-Maximization algorithm
and the Bayesian Information Criterion. Bayes’ Law is then efficiently carried out analytically within
the evolving stochastic subspace. The resulting GMM-DO filter is illustrated in a very simple example.
Variations of the GMM-DO filter are also provided along with comparisons with related schemes.