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where M
t
;
to
is the transpose of the tangent-linear operator matrix obtained by
linearizing the forcing terms of the full nonlinear forecast model equations, and
@
x
t
is the differentiated response function R with respect to the atmospheric
state at forecast time. The response function R can be any differentiable function
of the forecast state variables, and is typically chosen to diagnose a specific aspect
of the atmospheric state, such as low-level wind speed or localized precipitation
amount. The term
R
=@
x
o
exists at every model grid point and represents the
adjoint sensitivity of R with respect to the initial-time atmospheric state. For large
sensitivity values, small perturbations to the initial-time atmospheric state will result
in large perturbations to the forecast response function R. On the other hand, very
large initial-time perturbations hardly influence R where sensitivity values are very
small. In turn, adjoint sensitivity reveals regions where analysis errors would grow
rapidly to cause large errors in the forecast response function R, revealing areas
where it would be undesirable to have initial condition error.
Singular vectors (SVs) are similar to adjoint sensitivity in that they also
utilize the tangent linear propagator matrix M
t
;
to
.
Gelaro et al.
(
1999
) provide
an overview of how SVs can be obtained by calculating the eigenvectors of the
eigenvalue/eigenvector problem:
M
t
;
to
M
t
;
to
u
i
D
i
@
R
=@
u
i
(23.2)
where u
i
are the orthogonal initial-time SVs of M
t
;
to
(or eigenvectors of M
t
;
to
M
t
;
to
/
with growth rates
i
. The SVs with largest growth rates are the fastest growing
perturbations with respect to the Euclidean norm
u
i
u
i
1=2
.
Gelaro et al.
(
1999
)
show how the fastest growing perturbations with respect to more sophisticated
norms, such as the dry total energy norm can be found, which adds additional
weighting terms to equation (
23.2
) and presents a new eigenvalue/eigenvector
problem that must be solved. In any case, the leading SVs reveal where errors
would grow most rapidly with regard to a specified norm, and like adjoint sensitivity
reveal areas where analysis error is undesirable with regard to the predictability
of a specified aspect of the forecast state. For both adjoint sensitivity and SV
applications, perturbation growth is measured about a previously run forecast. Both
methods possess errors associated with the assumption of linear perturbation growth
and the lack of a tangent-linear propagator containing the linearization of certain
complex physics that exist in the full nonlinear model.
Perhaps motivated by studies that supported the notion of key analysis errors
in regions of large adjoint sensitivity and leading SVs being most detrimental
to forecasts (
Rabier et al. 1996
;
Klinker et al. 1998
), early observation targeting
techniques were based on these locations. The basic idea was that by reducing
errors where they would grow rapidly is the most effective way to improve forecasts.
Buizza and Montani
(
1999
),
Gelaro et al.
(
1999
),
Langland et al.
(
1999
), and
Liu
and Zou
(
2001
) all found that by ingesting targeted observations in areas of leading
SVs or large adjoint sensitivity, significant forecast error reductions (from 10 % to
50 %) were produced. These studies revealed the usefulness and value of SV and
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