General Introduction
Data assimilation is the name given to the process of combining observed data with a numerical model to produce the best estimate of the state of a system. It is used for example in numerical weather prediction, but not only, to obtain the initial conditions from which to run a weather forecasting model.
Modern techniques of data assimilation include four-dimensional variational data assimilation (4D-Var), which uses a sequence of observations over a given time window to estimate the best model trajectory within the window.
This method considers the data assimilation problem as a non-linear least squares problem, which is solved using an iterative process. However the problem is very large and so approximations must be made to ensure an efficient solution process. In this course we introduce the concept of four-dimensional variational data assimilation and explore its connections with linear algebra, statistics and numerical optimization theory.
We will study in depth the Gauss-Newton method for solving a nonlinear least squares problem, which is equivalent to the incremental approached that is commonly used in 4D-Var, and involve important algorithmic features such as truncation, preconditioning, and multilevel approximations. Numerical results with simple models will be proposed to illustrate and better understand the numerical properties of the presented algorithms
