unbalanced pressure pu and total water mixing ratio qt. The
aim of the three-dimensional variational approach is to produce
the best compromise between an a priori estimation of
the analysis field and observations through the iterative solution
that minimizes a cost function J . Most leading assimilation
schemes do not perform the minimization process in
the model space, but they use a transformed or control variable
space that is the space allowed for the corrections to the
background. This new space is chosen to have a special and
desirable property when the background field is represented
in this space: