# 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.