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of the forecast error ranging from 1 (0 % reduction) to 0 (100 % reduction)
for each analysis variable at all model grid points. The analysis error solution
is a local approximation performed within the grid decomposition blocks that is
improved upon though the use of halo regions to bring in the influence of additional
observations. The analysis error estimation uses the same data inputs as the 3DVAR
other than the innovations. In this way the analysis error calculation can be done
at the same time as the analysis, albeit on a different set of processors, to improve
throughput of the entire data assimilation system. The primary application of the
analysis error covariance program is as a constraint in the Ensemble Transform
technique (Sect. 13.5.3 ).
13.5.2
Adjoint
Adjoint-based observation sensitivity provides a feasible all at once approach to
estimating observation impact. Observation impact is calculated in a two-step
process that involves the adjoint of the forecast model and the adjoint of the
assimilation system. First, a cost function ( J ) is defined that is a scalar measure of
some aspect of the forecast error. The forecast model adjoint is used to calculate
the gradient of the cost function with respect to the forecast initial conditions
(
. The second step is to extend the initial condition sensitivity gradient from
model space to observation space using the adjoint of the assimilation procedure
(
@
J
=@
x a /
y D K T @
,where K D P b H T
HP b H T
1 is the Kalman gain matrix
@
J
=@
J
=@
x a /
Œ
C R
of ( 13.1 ) and the adjoint of K is given by K T
1 HP b . The only
difference between the forward and adjoint of the analysis system is in the post-
multiplication of going from the solution in observation space to grid space. The
pre-conditioned, conjugate gradient solver
HP b H T
D Œ
C R
HP b H T
is symmetric or self-adjoint
and operates the same way in the forward and adjoint directions. The NCODA
3DVAR adjoint was coded directly from the forward 3DVAR by transposition of the
post-multiplier to a pre-multiplier that is invoked first to convert adjoint sensitivities
from grid space to observation space prior to execution of the solver for calculation
of observation sensitivities and data impacts.
Œ
C R
13.5.3
Ensemble Transformation
The ensemble transform (ET) ensemble generation technique ( Bishop and
Toth 1999 ) transforms an ensemble of forecast perturbations into an ensemble
of analysis perturbations. The method ensures that the analysis perturbations are
consistent with the analysis error covariance matrix
, computed using ( 13.9 ). To
compute the required transform matrix an eigenvector decomposition is performed,
.
P a /
.X f P a X f /=n D CC T
(13.10)
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