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R
n
.
Also, the expansion in terms of representer functions is valid even in the continuum
limit of the discretized dynamics, in which case (
12.5
) become the Euler-Lagrange
equations for the extremum of the objective functional. The columns of the
BH
T
matrix, which are approximations to the representer functions in the continuum
limit, span the space of observable increments; i.e., they are exactly the
R
m
, whereas
x
lies in
x
lies in
In this dual formulation the unknown vector
m
degrees
of freedom which are determined by the measurements (
Bennett 1992
).
The dual formulation and representer expansion have by now been utilized in
many data assimilative modeling studies of the ocean and atmosphere. Because the
dimension of the vector of unknowns is
m
in either case of 4D-Var or W4D-Var,
there is no intrinsic limitation of the method in the latter case. In order to fix the
notation so that a single system describes both 4D-Var and W4D-Var, consider the
following augmented vectors and covariance matrices:
x
B
H
0
.t
0
/
f
0
x
0
D
B
0
D
H
0
D
R
0
D
R
y
0
D
y
;
;
;
;
:
(12.7)
0
F
Henceforth, we drop primes and simply write the objective function as
x
x
b
/
T
B
1
.
x
x
b
/
J
Œ
D
.
x
(12.8)
/
T
R
1
.
C
.
y
Hx
y
Hx
/;
noting that the extremal conditions (
12.5
) and dual formulation (
12.6
) are formally
unchanged.
Recent advances for representer-based variational assimilation have been con-
nected with technologies for solving (
12.6
), e.g., preconditioners and iterative
solvers, and with developing justifiable error models for the background and model
forcing errors,
B
and
F
.
In the next section, recent technological developments for solving (
12.6
)are
discussed, and we share our experience concerning the primal and dual forms of
the variational data assimilation algorithms, as has been the focus of recent papers
(
El Akkraoui and Gauthier 2010
;
El Akkraoui et al. 2008
;
Gratton and Tshimanga
2009
). Following that, recent work on covariance modeling is described. The latter
developments are not unique to representer-based approaches.
12.2
Solver Improvements
Several considerations have led to improvements in representer-based solvers for
variational data assimilation.
First, it has been noted that iterative solvers for (
12.6
) may yield a non-monotonic
sequence of
of
the iterative solver (
El Akkraoui et al. 2008
). This phenomenon has been observed
J
.
x
p
/
values, where
x
p
represents the approximate solution at step
p
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