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estimate the impact on a response function R due to innovations (
y
o
/
with the
calculation:
R D
@
R
=@
y
o
y
o
(23.4)
The only drawback of this method is that innovations associated with hypothetical
observations are not known prior to obtaining the observations, although the
technique is still very useful for understanding relative forecast impacts from a fixed
innovation anywhere in the model domain.
Langland and Baker
(
2004
) derived the observation impact methodology directly
from (
23.3
)and(
23.4
), noting that
R is composed of a sum of terms, each term
containing a coefficient representing a different innovation (and thus a different
observation). In this way, the contribution from each observation or any subset
of observations to
R can be easily calculated, and the impact of different,
assimilated observations or subsets of observations can easily be produced. Con-
ceptually, this is equivalent to the analysis increment produced from a specific set
of assimilated observations projected onto the adjoint sensitivity field, yielding an
estimate of
R. This technique is the foundation for a number of observation impact
studies (
Langland and Baker 2004
;
Tremolet 2008
;
Gelaro and Zhu 2009
;
Gelaro
et al. 2010
), although these studies expand the observation impact method to account
for nonlinear terms in the definition of the response function.
Errico
(
2007
)and
Gelaro et al.
(
2007
) discuss the accuracy of the expanded higher-order methodology,
and also offer a more in depth interpretation of equation (
23.4
) noting that cross-
correlations appear in each observational term that sum to produce
R. The
important issue these studies address through the observation impact technique is to
understand which types of observations, such as those at different heights or those
associated with different observational platforms, contribute to reducing forecast
error and which do not. These results are crucial toward designing the most effective
routine observational networks for operational assimilation/forecasting systems.
Significant observation impact and targeting developments were also made
using ensemble data assimilation systems.
Bishop et al.
(
2001
) developed an
ensemble transform Kalman filter (ETKF) observation targeting method based on
the ensemble transform technique of
Bishop and Toth
(
1999
). The ETKF method is
able to estimate the reduction in forecast variance due to hypothetical observations.
A similar method was provided by
Ancell and Hakim
(
2007a
) within an EnKF
assimilation system that also estimates the reduction in forecast variance of a chosen
response function R due to hypothetical observations. This method is based on
ensemble sensitivity which can be calculated in the following way (
Ancell and
Hakim 2007a
):
y
o
/
D
1
@
R
=@
y
o
D Cov
.
R
;
(23.5)
where
y
o
is a row vector representing the ensemble sensitivity of R with
respect to each analysis variable, Cov
@
R
=@
is a row vector representing the
covariance between the response function R and each analysis variable, and D
is a diagonal matrix containing the variance of each analysis variable.
Ancell
and Hakim
(
2007a
) explain that ensemble sensitivity allows one to estimate the
perturbation to the response function R resulting from the temporal evolution of
.
R
;
y
o
/
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