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The elicitation of the background error covariance matrix (becm) is a major difficulty of Bayesian assimilation used in atmospheric and oceanic sciences based on Kalman filter in the form of 3Dvar or 4Dvar. In a preliminary companion paper, the definition of a becm and link to earlier information were seen poorly practicable. Here, more decisive arguments are obtained by reviewing mathematical studies, many in optimal control, a field dual to filtering. When the system is discretized, and in the simplest steady and linear case, controlla-bility and detectability conditions are generally fulfilled implying convergence in theory, not necessarily in practice as the limit may be very large. The non steady or nonlinear case may be stabilized by covariance inflation. However, mathematical works about inflation disprove the common belief that the converged result would be the becm of the system. Considering the complete infinite dimensional system is even more disappointing. The becm is replaced by a trace class operator solution, in the steady linear case, to an infinite-dimensional Riccati equation. This equation has never been studied corresponding to the specific features of geophysical assimilation of data. These correspond to limit cases where the existence of solutions is no longer guaranteed. There is no ground, accordingly, for presuming filter convergence in terms of the true infinite-dimensional system. Counterexamples are proposed. This conclusion jeopardizes the physical meaning of any finite-dimensional becm that could be computed based on discretization.

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.