Theoretical Issues in the Application of Bayesian Data Assimilation in Context of Meteorology, Part 1: Bayesian assimilation as a Tikhonov regularization

For improving or updating the knowledge of atmospheric or oceanic state, observational data are assimilated into numerical evolution models. This is performed, most of the time, following a Bayesian framework to compromise, like Kalman filter, between observation errors and model errors. This article is a preliminary discussion about the structure of the theory, in particular the pivotal definition of a background error covariance matrix. The well known difficulties associated with the practical implementation and the so called catastrophic filter divergence are addressed in a companion paper. Geophysical Bayesian assimilation may be seen as a Tikhonov regularization, the background error covariance matrix corresponding to the regularizing term. Tikhonov regular-ization was originally developped to stabilize overdetermined inverse problems against measurement errors. However, Geophysical assimilation is always strongly underdeter-mined. This changes totally the role of the regularizing term: instead of stabilizing the estimation, it determines its main features. The derivation of the matrix becomes, accordingly , essential. It is shown here that the traditional definition of the background error covariance matrix and its presumed dependence on earlier information raises practical difficulties. These difficulties are not compatible with the simplifications generally assumed. They are also not compatible with a practical implementation of Bayesian assimilation.