Geoscience Reference
In-Depth Information
n
X
R
i
.
y
i
y
i
/
r
y
i
J
o
D
(11.1)
i
D
0
is the model observed equivalent,
y
i
where
is the vector of available
observations and
R
i
is the observation error covariance matrix. Using the adjoint of
the observation operator,
H
i
y
i
D
H.
x
i
/
, one can then calculate the gradient of
J
o
with respect
to the model state at observation time,
x
.t
i
/
,
X
n
H
i
R
i
.H
i
Œ
.t
i
/
y
i
/
r
x
i
J
o
D
x
(11.2)
i
D
0
Finally, the gradient of
J
o
with respect to the model state at time 0 can be obtained
by applying the adjoint (AD) of the forecast model,
M
T
.t
i
;t
0
/
,
n
X
M
T
.t
i
;t
0
/
H
i
R
i
.H
i
Œ
.t
i
/
y
i
/
r
x
.t
0
/
J
o
D
x
(11.3)
i
D
0
Again, since the adjoint version of the forecast model can be seen as the transpose
of its Jacobian matrix, the forecast model first needs to be differentiated with respect
to its inputs, yielding the so-called tangent-linear (TL) model,
M
.
In contrast with the full non-linear model, the tangent-linear model works on
perturbations of the input variables rather than on full model fields and is fully linear
by construction. The adjoint is therefore a fully linear operator as well and, in the
case of 4D-Var, its inputs are the components of r
x
.t
i
/
J
. As a consequence, solving
the 4D-Var minimization requires the linearization of the forecast model's physical
parameterizations (e.g. vertical diffusion, radiation, convection, large-scale moist
processes) so that their TL and AD versions can be used to describe the (forward,
respectively backward) time evolution of the model state during the minimization
as seen from (
11.3
).
11.3
Implication of the Linearity Constraint
The minimization of the 4D-Var cost function is solved with an iterative algorithm
and is therefore computationally rather demanding. Even though the minimization
is usually performed at a much lower resolution (T159/T255
1
in current ECMWF's
operations) than in the standard forecast model (T1279
2
at ECMWF), the several
tens of iterations required to obtain the optimal model state means that the linearized
1
T159/T255 corresponding approximately to 130/80 km
2
T1279 corresponding approximately to 16 km
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