Geoscience Reference
In-Depth Information
ˇ
ˇ
ˇ
x
b
/
ˇ
ˇ
ˇ
x
a
x
b
/
M.
.
x
a
/
M.
M
ˇ
ˇ
ˇM.
ˇ
ˇ
ˇ
r
er
D
(11.8)
x
a
/
M.
x
b
/
D
"
exp
"
ref
"
ref
(11.9)
Validity tests of the tangent-linear approximations are mostly performed over the
time period and at the resolution at which adjoint models will be applied in practice:
resolution and time length of an inner-loop integration of 4D-Var system (e.g. 12 h,
T255 and 91 vertical levels at ECMWF) or longer time periods for singular vectors
and sensitivity applications (e.g. 24 h at ECMWF). An example of the results from
such TL approximation assessment will be given in Sect.
11.6.1
.
11.4.4
Adjoint Version
The adjoint of a linearized operator,
M
, is the linear operator,
M
, such that:
M
:
8
x
;
8
y
<
M
:
x
;
y
>
D
<
x
;
y
>
(11.10)
where
denotes the inner product and
x
and
y
are input vectors.
Besides, the adjoint model
M
can provide the gradient of any objective function,
J
, with respect to
x
<;>
.t
i
/
from the gradient of the objective function with respect to
x
.t
i
C
1
/
D
M
@
J
@
J
(11.11)
@
x
.t
i
/
@
x
.t
i
C
1
/
The integration of the AD forecast model works backward in time. One should
remember that,
being non-linear,
M
as well as
M
depend on the particular
state of the atmosphere,
x
, about which the linearization is performed. The adjoint
operator, for the simplest canonical scalar product
M
(
11.10
), is actually the
transpose of the tangent linear operator,
M
T
(not its inverse).
For the practical verification of the adjoint code, one must test the identity
described in (
11.10
). It should be emphasized that it is absolutely essential to ensure
that the TL and AD codes verify (
11.10
) to the level of machine precision, even
when vectors
x
and
y
are global 3D atmospheric states and even for time integrations
up to 12 or 24 h. Note that a correct adjoint test does not imply the correctness of
tangent-linear code.
<;>
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