Geoscience Reference
In-Depth Information
x f x g /; @J f
@
x f C @J g
g
f D
e
.
:
(6.18)
@
x g
f
g
The difference between forecast trajectories
and
at the analysis time is the
. The adjoint model maps @J f
@ x f
to @J f
@ x a
and @J f
@ x g
to @J f
increment
@ x b . Assuming
that the analysis increment evolves approximately tangent linearly, then an estimate
of
.
x a x b /
e f
in analysis space can be written as,
x a x b /; @J f
@
x a C @J g
ıe f D
.
:
(6.19)
@
x b
e f
Equation 6.19 is not an exact match to
because the adjoint model does not
capture all of the processes of the nonlinear model used to calculate the error. To
determine the impacts in observation space, the analysis increment is replaced in
Eq. 6.19 in the following manner,
K
y Hx b /; @J f
@
x a C @J g
ıe f D
.
:
(6.20)
@
x b
Using the properties of an adjoint operator in an inner product, the following
expression in observation space,
K T @J f
@
x a C @J g
ıe f D
.
y Hx b /;
;
(6.21)
@
x b
is obtained. The observation impacts are a product of the innovation vector
components and the vector obtained from the adjoint DA process. The inner product
in Eq. 6.21 gives a total estimate for all observations, but the inner product can
be partitioned into any particular subset of interest. For example, the impact of a
particular instrument, observation type (temperature, winds, etc.), or even a single
measurement at a specific location can be determined using the method outlined
above.
In practice, to calculate the observation impacts for a cycling NWP system the
following steps are involved:
Save appropriate forecast trajectories during the nonlinear NWP model run.
When a verifying analysis x t becomes available, calculate cost functions
(Eqs. 6.14 and 6.15 ) and forcings for the adjoint NWP runs (Eqs. 6.16 and
6.17 ).
Perform two adjoint NWP integrations along trajectories
f
and
g
back to the
analysis time of
f
.
Add the two resulting model space vectors together to create the input vector for
the adjoint DA operator.
Run the adjoint DA scheme to obtain the gradient in observation space, the inner
product of this vector with the innovation will provide observation impacts.
 
Search WWH ::




Custom Search