In this thesis, several important topics in the area of particle filtering for applications in Data Assimilation are covered. The main focus of my research
has been to explore ways to overcome the weight degeneracy problem that Importance Sampling (IS) faces in high dimensional problems or for very
informative observations. It is also considered the case where it is necessary to determine what sensor took the observations before formulating the filtering
problem. Here are the main findings of the research:
• In the filtering problem where we have a continuous time signal following a stochastic partial differential equation (SPDE) and discrete time observations, I have found that the combination of tempering the likelihood to bridge the sequence of posterior distributions, Markov Chain
Monte Carlo (MCMC) steps to reintroduce diversity of the particles and the use of IS to guide the particles to regions of interest allows us to use particle filters to tackle harder problems that are intractable
otherwise.
• When both the signal and the observation are continuous in time, the methods proposed in discrete time cannot be applied straightforwardly. I study an IS method based on modifying the drift of the signal, focus-
ing on how to approximate the drift. I have found that this term could be approximated via the solution of a Partial Differential Equation (PDE) or via approximating the Smoothing density and concluded nu-
merically that the second option was more promising computationally. I also show how this importance sampling improves the performance of particle filtering when compared to a basic implementation. • Finally, I show that the use of machine learning algorithms has the potential to considerably improve the accuracy of previous algorithms based on a deterministic classification tree for the problem of classifying historical sea temperature and salinity data according to which sensor was used to collect this data.