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where
denotes the operators of the non-linear ROMS that advance the
state vector forward in time over the interval
M.t
i
;t
i
1
/
denotes the ocean
surface forcing (i.e. surface fluxes of momentum, heat and freshwater), and
b
Œt
i
1
;t
i
, while
f
.t
i
/
.t
i
/
denotes the lateral open boundary conditions, both over the same interval
.
Data assimilation requires the specification of
prior
or background estimates
of all the control variables for the model, which in the case of ROMS includes
the model initial conditions,
x
b
.t
0
/
Œt
i
1
;t
i
, the surface forcing,
f
b
.t
i
/
, and open boundary
conditions,
b
b
.t
i
/
. For each
prior
, there will also be an associated
prior
or
background error covariance matrix, namely
B
x
,
B
f
,and
B
b
which embody all of the
hypotheses about errors and uncertainties in the
prior
fields. For ease of notation, we
will denote the vector that comprises all control variables as
z
b
T
T
,
where
f
and
b
in the absence of a time argument denote the concatenation in
time of the vectors of surface forcing and open boundary conditions over the
entire time interval of interest,
D
x
T
f
T
.t
0
/;
;
. Similarly we will denote the combined
prior
error covariance matrix of all control variables by the block diagonal matrix
D
Œt
0
;t
N
. In the systems described in later sections it is assumed that
the model is free of errors, so the inclusion of control vector elements to account
for model errors is not considered here. Furthermore, the
prior
or background error
covariance matrix
D
is assumed to be time invariant.
In addition to the
prior
estimate of the circulation
x
b
.t/
D
diag
.
B
x
;
B
f
;
B
b
/
from ROMS, there
will also be available observations during the same interval
. The vector
of observations is traditionally denoted as
y
o
with an associated observation error
covariance matrix
R
. According to Bayes' theorem, the optimal choice of
z
that
yields the most likely
posterior
circulation estimate is that which minimizes the
cost function:
Œt
0
;t
N
T
D
1
.
y
o
T
R
1
.
y
o
J
NL
D
.
z
z
b
/
z
z
b
/
C
.
H.
x
//
H.
x
//
(14.2)
where
H
denotes the observation operator, and
H.
x
/
denotes the circulation esti-
mate
x
evaluated at the appropriate observation times and locations (
Lorenc 1986
;
Wikle and Berliner 2007
). Since
x
.t/
.t/
is the solution of the nonlinear model
(
14.1
), the cost function
J
NL
in (
14.2
) is a non linear function of the state vector,
which in practical terms means that it may not possess a unique global minimum
value. Even in the event that a global minimum does exist for (
14.2
), locating the
global minimum in what may be a complicated topology may be very challenging.
Therefore instead of minimizing (
14.2
) directly,
Courtier et al.
(
1994
) proposed the
incremental approach in which the desired
posterior
circulation estimate
x
a
.t/
can
be considered as a small departure from the
prior
, namely
x
a
.t/
D
x
b
.t/
C
ı
x
a
.t/
.
The increment
is a solution of the tangent linearization of (
14.1
) subject to
the surface forcing increments
ı
x
a
.t/
ı
f
a
.t/
and open boundary increments
ı
b
a
.t/
which
are also assumed to be small compared to the
prior
estimates
f
b
.t
i
/
and
b
b
.t
i
/
respectively. Specifically, we will denote the tangent linearization of (
14.1
), the so
called tangent linear ROMS (hereafter TLROMS), as:
ı
x
.t
i
/
D
M
b
.t
i
;t
0
/ı
x
.t
0
/
(14.3)
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