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In summary, it can be seen that this heuristic analysis rests firmly on two
assumptions—namely, that the large fraction of the change in the solution is due to
the model and that the evolution of the solution and the analysis from
t D T
can be approximated linearly. A necessary condition for this latter assumption to
hold is that the time horizon
t D 0
to
Œ0;T
must be small. Brill et al. ( 1991 )take
t D 3h
in
their analysis.
By discretizing ( 2.9 ) using an Euler scheme, we get
N X
!
N X
x.N/ x.0/ D
F.x.k//
t C G
f.k/Œx a .k/ x.k/t
(2.20)
k D 0
k D 0
which is a direct analog of ( 2.12 ) in discrete form. By following the above
arguments, we leave it to the reader to arrive at an expression for
G
similar to ( 2.16 ).
2.3
Estimating Optimal Nudging Coefficient: Problems
and Challenges
There are two basic approaches to the problem of estimating the optimal value of the
nudging matrix
. The first approach is due to Stauffer and Seaman ( 1990 ), Stauffer
and Bao ( 1993 )and Zou et al. ( 1992 ). Using the classic four-dimensional variational
(4D-Var) data assimilation method, they independently found the optimal
G
.The
second approach is due to Vidard et al. ( 2003 ) where a combination of Kalman filter
and 4D-Var is used to estimate the optimal
G
. In this section we provide a summary
of these two approaches. As will be seen, these approaches are incomplete in the
sense that they do not account for the inherent serial correlation of forecast errors
that constitute the basis for the estimation algorithm. A direct impact of excluding
the underlying correlation introduces a bias into the problem that directly affects the
value of the so-called optimal estimate.
For definiteness in the following development, we use the observation-based
nudging scheme that easily extends to the analysis-based nudging scheme.
G
2.3.1
Estimation of G Using the Variational Approach
Let the observation and the nudged dynamics be given by ( 2.2 )and( 2.7 ), respec-
tively. Let the forecast error
e.k/ 2 R m be given by ( 2.6 ). Define a vector
e.1 W N/ D e T .1/;e T .2/;:::;e T .N/ T 2 R Nm
(2.21)
consisting of the forecast errors at times
1 k N
.
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