Geoscience Reference
In-Depth Information
10.2
Background
An abbreviated description of variational assimilation is presented, but detailed
descriptions may be found in
Talagrand and Courtier
(
1987
),
Le Dimet and
Talag r an d
(
1986
),
Courtier et al.
(
1994
),
Courtier
(
1997
),
Chua and Bennett
(
2001
),
and
Moore et al.
(
2011b
) among others. Assuming a linearly-forced, nonlinear
model, the time-step integration is expressed as:
x
.t
i
C
1
/
D
M
.
x
.t
i
//
C
L
.
f
.t
i
//
C
q
.t
i
/;
(10.1)
where
x
.t
i
/
is the model state vector at time
t
i
,
M
.
x
.t
i
//
is the forward nonlinear
integration of state
x
.t
i
/
to state
x
.t
i
C
1
/
,
f
.t
i
/
is the forcing and boundary condition
parameter vector,
transforms the given vector to model forcing and boundary
condition influence, and
q
L
.
/
.t
i
/
represents the model errors. Using a model guess for
the initial conditions,
x
, the model can be integrated to produce
a reference or “background” trajectory
x
b
.t/
.0/
, and forcing,
f
.t/
.
The residuals between the observations and the model are given by the innovation
vector,
d
i
D
y
i
H
i
.
x
b
.t
i
//;
(10.2)
for
maps data from the model to a given observation
y
i
location in time and space. The goal of any assimilation scheme is to reduce these
residuals.
Assuming that perturbations,
N
observations, where
H
i
.
/
to the background are
within the realm of linearity, then these perturbations evolve by the tangent-linear
model linearized around the background trajectory,
x
b
, with a forcing function,
L
,
linearized about
ı
x
.t
i
/
,
ı
f
.t
i
/
,and
ı
q
.t
i
/
.
The residuals between the perturbed model solution and the observations are
given by
L
r
i
D
y
i
.
H
i
.
x
b
.t
i
//
C
H
i
M
i
z
.t
i
//
D
d
i
G
i
z
.t
i
/;
(10.3)
where
H
i
is the linearized sampling matrix,
M
i
represents the integration of the
tangent-linear model over the interval
,
G
i
D
H
i
M
i
,and
z
is a vector that
comprises the perturbations to the initial state, forcing, and model error fields:
z
D
.ı
.t
0
;t
i
/
/
T
.
In 4D-Var, our goal is to solve for
z
that minimizes (
10.3
) via least-squares by
employing a quadratic cost function,
x
;ı
f
;ı
q
N
X
J
D
1
2
r
i
R
1
r
i
C
1
2
z
T
P
1
z
;
(10.4)
i
D
1
where
P
is the covariance of uncertainty in the model initial state, forcing, and model
error
.
x
b
;
f
;
q
/
and
R
is the covariance of uncertainty in the residuals,
r
.
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