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can be approximated by
•
X
t
D
L
.
•
X
0
/
(24.5)
where
L
is the forward tangent propagator.
•
X
t
is the linear development of
ı
X
0
at time
t
. According to
Barkmeijer et al.
(
2003
), the first singular value
1
of
L
satisfies (with respect to the norms
C
1
and
C
2
/
:
Œ
L
.
•
X
0
/
T
C
2
Œ
L
.
•
X
0
/
1
D
max
(24.6)
.
•
X
0
/
T
C
1
.
•
X
0
/
T
C
1
.
•
X
0
/
¤
0
.
•
X
0
/
Additionally, if
v
i
is the singular vector of
L
,then
C
1
/
1
.
L
T
C
2
L
v
i
D
i
.
/
v
i
(24.7)
where superscript “
1
i
is the singular value
corresponding to
v
i
. The first SV (FSV) maximizes the linear development of the
initial perturbations, the second SV maximizes the development under the constraint
of being orthogonal to the first SV, and the third SV maximizes the development
under the constraint of being orthogonal to the first two SVs, and so on (
Peng
and Reynolds 2006
). A local projection operator P (same meaning as in (
24.4
)) is
employed to localize the development of the perturbation in the verification region.
Actually, the FSV can be obtained by solving the following linear optimization
problem (
Ehrendorfer and Errico 1995
):
” denotes the inverse of the matrix.
X
0
/
D
J.
•
max
X
0
ˇ
J.
•
X
0
/
(24.8)
X
0
C
1
•
•
where
T
C
2
Œ
J.
•
X
0
/
D
Œ
PL
.
•
X
0
/
PL
.
•
X
0
/
(24.9)
According to the linear characteristics of SV, the FSV defined by (
24.8
)and(
24.9
)
equals to the FSV defined by (
24.7
) when it has been normalized to a unit.
24.2.3
The Model and Optimization Algorithm
The CNOP and SVs are calculated with fifth generation Pennsylvania State
University-National Center for Atmospheric Research (PSU-NCAR) Mesoscale
Model (MM5;
Dudhia 1993
) and its corresponding adjoint system (
Zou et al. 1997
).
The following physical parameterizations are used: dry convective adjustment,
grid-resolved large-scale precipitation, the high-resolution PBL scheme, and the
Kuo cumulus parameterization scheme. The simulations initialized by the National
Centers for Environment Predictions (NCEP) FNL (Final) Operational Global
Analysis (
1
ı
1
ı
) interpolated into the MM5 grids serves as the basic states for
most of the calculations of the CNOPs or the SVs. The Weather Research and
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