Nonlinear data assimilation using synchronisation in a particle filter - Pinheiro, Flávia Rodrigues (2018)
Current data assimilation methodologies still face problems in strongly nonlinear systems. Particle filters are a promising solution, providing a representation of the model probability density function (pdf) by a discrete set of particles. To allow a particle filter to work in high-dimensional systems, the proposal density freedom is a useful tool to be explored. A potential choice of proposal density might come from the synchronisation theory, in which one tries to synchronise the model with the true evolution of a system using one-way coupling, via the observations, by adding an extra term to the model equations that will control the growth of instabilities transversal to the synchronisation manifold. Efficient synchronisation is possible in low-dimensional systems, but these schemes are not well suited for high-dimensional settings. The first part of this thesis introduces a new scheme: the ensemble-based synchronisation, that can handle high-dimensional systems. A detailed description of the formulation is presented and extensive experiments in the nonlinear Lorenz96 model are performed. Successful results are obtained and an analysis of the usefulness of the scheme is made, bringing inspiration for a powerful combination with a particle filter. In the second part, the ensemble synhronisation scheme is used as a proposal density in two different particle filters: the Implicit Equal-Weights Particle Filter and the Equivalent-Weights Particle Filter. Both methodologies avoid filter degeneracy by construction. The formulation proposed and its implementation are described in detail. Tests using the Lorenz96 model for a 1000-dimensional system show qualitatively reasonable results, where particles follow the truth, both for observed and unobserved variables. Further tests in the 2-D barotropic vorticity model were also performed for a grid of up to 16,384 variables, also showing successful results, where the estimated errors are consistent with the true errors. The behavior of the two schemes is described and their advantages and issues exposed, as this is the first comparison ever made between both filters. The overall message is that results suggest that the combination of the ensemble synchronisation with a particle filter is a promising solution for high-dimensional nonlinear problems in the geosciences, connecting the synchronisation field to data assimilation in a very direct way.