Geoscience Reference
In-Depth Information
If the particular problem is sufficiently nonlinear that the cost function is not well
represented by the linear model, it is desirable to update
x
b
with
z
a
, re-linearize, and
find a new update
z
a
. This iterative procedure is referred to as the “outer-loop.”
In many 4D-Var applications, the problem is sufficiently linear that multiple
outer-loops are not used. When it is truly linear, the increments would not change
the subsequent linearization about the perturbed
x
b
. However, any nonlinearities
may result in differing cost functions between the nonlinear and linear spaces. To
deal with this, one may choose to update the nonlinear trajectory with the increments
to avoid local minima in the nonlinear cost function. During these additional outer-
loops, consideration must be given that the increments remain constrained to the
original background,
x
b
; otherwise, the iterates will overfit the data at the expense
of the background covariance. Solving the problem in model-space (see Eq.
10.6
)
provides an implicit constraint to the initial-state and additional outer-loops are not
an issue (
Tshimanga et al. 2008
). This is not the case in data-space and during
additional outer-loops, the cost-function must be modified to add an additional
constraint.
Upon completing the inner-loops, an estimate of
z
k
is found, where
k
signifies
). The operators
G
k
1
and
G
k
1
are linearized about the prior trajectory
x
k
1
(where
x
0
D
the outer-loop iteration (beginning with
k
D
1
).
If the problem were perfectly linear,
G
k
D
G
k
1
andtherewouldbenoneed
for additional outer-loops. The next iteration of the outer-loop (
x
b
and
z
0
D
0
k
C
1
) requires
z
k
/
linearization about the new prior,
integrated by (
10.1
). The
question becomes how to keep the original background constraint,
x
b
, active when
further iterating in the data-space methods. The typical approach is to simply
reapply (
10.7
) such that
.
x
k
D
x
k
1
C
PG
k
1
G
k
1
PG
k
1
C
R
1
d
k
1
;
z
k
D
(10.9)
where
.
d
k
1
/
i
D
y
i
H
i
x
k
1
.t
i
/
as was used in
Zaron et al.
(
2011
). This constrains
z
k
to be small; however, the total increment,
P
j
D
1
z
j
, is required to be small.
Hence (
10.9
) constrains the increments only against the prior nonlinear circulation
(
x
k
1
), which allows the increments to deviate from the background trajectory.
To rectify, the total constraint must be incorporated to yield a new cost function,
J
k
D
1
G
k
1
z
k
/
T
R
1
.
2
.
d
k
1
d
k
1
G
k
1
z
k
/
0
1
0
1
T
k
X
k
X
C
1
2
@
z
k
C
A
@
z
k
C
A
:
P
1
z
j
z
j
(10.10)
j
D
1
j
D
1
The gradient at its minimum is given by
0
1
k
1
X
@
J
k
@
@
z
k
C
A
D
0:
D
G
k
1
R
1
.
P
1
z
j
d
k
1
G
k
1
z
k
/
C
(10.11)
z
k
j
D
1
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