Leveraging sparsity in variational data assimilation

Abstract

Nowadays data assimilation is an essential component of any effective environmental prediction system. Environmental prediction models are, indeed, initial value problems and their forecast skills highly depend on the quality of their initialization. Data assimilation (DA) seeks the best estimate of the initial condition of a (numerical) model, given observations and physical constraints coming from the underlying dynamics. This important problem is typically addressed by two major classes of methodologies, namely sequential and variational methods. The sequential methods are typically built on the theory of mathematical filtering and recursive weighted least-squares, while the variational methods are mainly rooted in the theories of mathematical optimization and batch mode weighted least-squares. The former methods, typically use observations in sequential mode to obtain the best estimate of the geophysical state of interest at present time. In this thesis, we briefly review the mathematical and statistical aspects of classic data assimilation methodologies with particular emphasis on the family of variational methods. We explore the use of regularization in variational data assimilation problem and focus on sparsity-promoting approaches in a pre-selected basis. Central results suggest that in the presence of sparsity, the $\ell_{1}$-norm regularization in an appropriately chosen basis produces more accurate and stable solutions than the classic data assimilation methods. To motivate further developments of the proposed methodology, assimilation experiments are conducted in the wavelet and spectral domain using the linear advection-diffusion equation.