Links between Kalman Filtering and Data Assimilation with Generalized Least Squares

Kalman filtering (KF) is a popular form of data assimilation, especially in real-time applications. It combines observations with an equation that describes the dynamic evolution of a system to produce an estimate of its present-time state. Although KF does not use future information in producing an estimate of the state vector, later reanalysis of the archival data set can produce an improved estimate, in which all data, past, present and future, contribute. We examine the case in which the reanalysis is performed using generalized least squares (GLS), and establish the relationship between the real-time Kalman estimate and the GLS reanalysis. We show that the KF solution at a given time is equal to the GLS solution that one would obtain if data excluded future times. Furthermore, we show that the recursive procedure in KF is exactly equivalent to the solution of the GLS problem via Thomas’ algorithm for solving the block-tridiagonal matrix that arises in the reanalysis problem. This connection suggests that GLS reanalysis is better considered the final step of a single process, rather than a “different method” arbitrarily being applied, post factor. The connection also allows the concept of resolution, so important in other areas of inverse theory, to be applied to KF formulations. In an exemplary thermal diffusion problem, model resolution is found to be somewhat localized in both time and space, but with an extremely rough averaging kernel.