Prof. Pierre Lermusiaux was an invited lecturer at the International Workshop on “Probabilistic Approaches to Data Assimilation for Earth Systems” which took place in Banff, Canada from 17-22 February, 2013 […]

The properties and capabilities of the GMM-DO filter are assessed and exemplified by applications to two dynamical systems: (1) the Double Well Diffusion and (2) Sudden Expansion flows; both of which admit far-from-Gaussian statistics. The former test case, or twin experiment, validates the use of the EM algorithm and Bayesian Information Criterion with Gaussian Mixture Models in a filtering context; the latter further exemplifies its ability to efficiently handle state vectors of non-trivial dimensionality and dynamics with jets and eddies. For each test case, qualitative and quantitative comparisons are made with contemporary filters. The sensitivity to input parameters is illustrated and discussed. Properties of the filter are examined and its estimates are described, including: the equation-based and adaptive prediction of the probability densities; the evolution of the mean field, stochastic subspace modes and stochastic coefficients; the fitting of Gaussian Mixture Models; and, the efficient and analytical Bayesian updates at assimilation times and the corresponding data impacts. The advantages of respecting nonlinear dynamics and preserving non-Gaussian statistics are brought to light. For realistic test cases admitting complex distributions and with sparse or noisy measurements, the GMM-DO filter is shown to fundamentally improve the filtering skill, outperforming simpler schemes invoking the Gaussian parametric distribution.

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.