Geoscience Reference
In-Depth Information
24.2
Methodologies
24.2.1
CNOP
In this part, we briefly introduce the method of conditional nonlinear optimal
perturbation. Suppose we have the following model
@
X
@t
C
F.
/
D
0
X
j
t
D
0
D
X
0
X
(24.1)
where
X
is the state vector of the model with initial value
X
0
.
is a nonlinear partial
differential operator. The solution of (
24.1
) can be expressed in discrete form:
F
X
t
D
M
.
X
0
/
(24.2)
where
M
is a nonlinear propagator, and
X
t
is the value of
X
at time
.
To measure the development of
X
0
, appropriate norms must be chosen. In
discrete form, this is equivalent to choosing symmetric positive definite matrices
C
1
and
C
2
. An initial perturbation
t
X
0
•
of vector
X
0
is called CNOP if and only if
X
0
/
D
J.
•
•
X
0
C
1
ı
X
0
ˇ
J.
ı
max
X
0
/
(24.3)
Where
T
C
2
Œ
J.
•
X
0
/
D
Œ
.
X
0
/
P
M
.
X
0
/
.
X
0
/
P
M
.
X
0
/
P
M
X
0
C •
P
M
X
0
C •
(24.4)
and
is a constraint condition of initial perturbations with the
presumed positive constant
ı
X
0
C
1
ı
X
0
ˇ
representing the magnitude of the initial uncertainty.
The first guess of the initial perturbation
ˇ
X
0
, which is usually taken as the
difference between the model outputs at two times, should be adjusted to satisfy the
constraint condition
ı
.
P
is a local projection operator and takes value
1(0) within (without) the targeted region. The superscript “
T
” denotes the transpose
of vectors or matrices. Note that the norms used in the cost function and the initial
constraint condition may be the same, depending on the physical problem. It is clear
that the CNOPs depend on the nonlinear model
M
, the initial state vector
X
0
,and
the parameters
ı
X
0
C
1
ı
X
0
ˇ
ˇ
,
P
,
C
1
,and
C
2
.
24.2.2
SV
Suppose that the initial perturbation
X
0
is sufficiently small and the integration
time interval is of moderate length, then the development of
ı
ı
X
0
in discrete form
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