Geoscience Reference
In-Depth Information
6.1.3
Adjoint Observation Impact
A robust procedure for quantitatively evaluating the impact of an observation's
assimilation on short-range forecast error utilizing the adjoint observation sensitivity
framework was developed by
Langland and Baker
(
2004
). This procedure is
responsible for most of the results presented in this chapter so the details found
in
Langland and Baker
(
2004
) are summarized here for clarity.
The error of two forecasts of lengths
f
and
g
can be measured against an analysis
x
t
available at verification time
t
in an inner product h
;
i using the following two
equations,
e
f
D
˝
.
x
f
x
t
/
˛
;
x
f
x
t
/;
C
.
(6.11)
and,
e
g
D
˝
.
x
g
x
t
/
˛
:
x
g
x
t
/;
C
.
(6.12)
The coefficients in
C
weight the model fields so that the error is measured in an
energy norm. The forecast for
g
begins at an earlier time than
f
, and a short-term
field (usually 6 or 12 h) from the
forecast serves as the background field
x
b
in
the analysis procedure to produce
x
a
for the
g
f
forecast. For global atmospheric
NWP models, the value of
e
g
due to the assimilation of
observations
y
to update
x
b
. If no observations are assimilated to produce
x
a
for the
f
e
f
is generally less than
e
g
. For limited area
models, the lateral boundaries are also updated during the analysis procedure, which
can lead to a change in
forecast, then
x
a
will be the same field as
x
b
and
e
f
will equal
e
f
even if no observations are assimilated (Sect.
6.3.1
).
To quantify the value of observations in reducing forecast error, an equation for
the difference in
e
f
and
e
g
is defined,
e
f
D
e
f
e
g
:
(6.13)
e
g
f
Using the adjoint NWP model,
can be mapped backward in time to analysis
space. To do this, two cost functions are defined along with their corresponding first
derivatives, which will serve as input for two adjoint model integrations along the
f
and
g
forecast trajectories,
J
f
D
1
2
e
f
;
(6.14)
J
g
D
1
2
e
g
;
(6.15)
@J
f
@
D
C
.
x
f
x
t
/;
(6.16)
x
f
@J
g
@
x
g
D
C
.
x
g
x
t
/:
(6.17)
Eqs.
6.11
-
6.12
and
6.16
-
6.17
can be used to rewrite Eq.
6.13
as,
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