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A consistent Bayesian multiscale framework for data assimilation.

Task 3c, principal investigator: CEREA/CLIME.

In geophysical data assimilation, the control space is by definition the set of parameters that are estimated through the assimilation of observations. It has recently been proposed to design the discretisation of control space in order to optimally assimilate observations (cf. Bocquet 2009, in the MSDAG reference list). These designs are adaptive and multiscale (grid-cells can be of different sizes in space and in time). This multiscale formalism can be made consistent with the Bayesian paradigm, and hence standard data assimialtion schemes such as BLUE. To do so, one define a restriction operator to a project a control vector at fine resolution onto a coarser one, and a prolongation that refines such control vector from the coarse grid to the fine one. However, this is ambiguous since there are many possible prolongation operators. We advocate to choose the prolongation operator that makes an optimal use of knowledge from both the coarse control vector and from the background information available in the data assimilation scheme. This leads to a prolongation operator which is affine. Using this algebra, one can consistently define all the necessary operators for data assimilation through the scales.

Schematic representation of the restriction and prolongation operator acting on a finest grid and on an arbitrary coarser discretisation of control space.