Geoscience Reference
In-Depth Information
12.1
Introduction
Four-dimensional variational data assimilation (4D-Var) is an estimation technique
which finds a model state x
t 0 , that minimizes a quadratic objective
function, the sum of the distance between the initial state x
.t 0 /
, at initial time
.t 0 / 2 R n and a prior
estimate (the so-called background field) x b 2 R n , and the distance between a real-
valued vector of observations y 2 R m and measurements
H .
x
/
of the trajectory x
.t/
obtained by integration of a dynamical model from x
.t 0 /
. The objective function
J
is written
.t 0 / x b / T B 1 .
.t 0 / x b /
J Œ
.t 0 / D .
x
x
x
(12.1)
/ T R 1 Œ
C Œ
y H .
x
y H .
x
/;
where B and R are estimates of the background and observation error covariance
matrices, respectively, and the observations, y Df y i g i D 1
, are nonlinear functions of
the initial state,
y i D H i ΠM .t i ;t 0 /
x
.t 0 / C ı i :
(12.2)
Here we assume that
M .t i ;t 0 /
propagates the model state from
t 0 to
t i ,
H i is the
i th observation operator, and
ı i is the observation error. Note that if the initial
condition and observation errors are Gaussian distributed with covariances B and
R , if the observation errors are unbiased, and if the background field x b is equal to
the statistical mean of x
.t 0 /
, then the minimizer of
J
is the maximum likelihood
estimate of x
.
In addition to errors in the initial conditions, it is clear that oceanic and
atmospheric models contain other sources of error which must be considered.
Specifically, there are errors in model inhomogeneities such as boundary conditions
and radiative forcing. Weak-constraint four-dimensional variational data assimila-
tion (W4D-Var) is a generalization of 4D-Var which permits one to estimate these
additional inhomogeneities, denoted f . Assuming that prior or background values
of the forcing fields are available, f b , then the above objective function naturally
generalizes to
.t 0 /
f f b / T F 1 .
f f b /
J Œ
x
.t 0 /;
f
D .
.t 0 / x b / T B 1 .
.t 0 / x b /
C .
x
x
(12.3)
/ T R 1 Œ.
C Œ.
y H .
y H .
/;
x
x
where it should be understood that the model propagator
now depends on both
the space-time-dependent inhomogeneities, f , and the initial conditions, x
M
.
In the incremental formulation ( Courtier et al. 1994 ), the d yn amics and measure-
ment operators are linearized around a background traje c tory x , and an incremental
objective function is defined in terms of
.t 0 /
x D x x . Of course, if the model
dynamics and observation operator are linear, the extremum of the incremental
ı
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