DATA ASSIMILATION
319
state to the observations by the observation operator H. The lower part
describes the model’s temporal evolution by operators A and G that embody
the model physics and dynamics. The right hand side of the model equations
(lower part of Eq 1) is identically zero as all terms of the model are generally
uncertain and are placed on the left hand side. (Sources of model error are
included in u.)
For simplicity, we assume a linear model for most of this discussion.
The problem above and the solutions discussed below can be extended to
non-linear models with suitable linearization. Bold upper and lower case
characters represent matrices and column vectors, respectively. The time
increment from t to t+1 above denotes an arbitrary increment, as opposed to
a single model time-step, and corresponds to instances at which observations
are available.
As in most geophysical inverse problems
1 , Eq (1) is rank deficient. In
particular, there are generally more unknowns than the number of constraints.
For example, the dimension of x, excluding the temporal dimension, is of
order several million for typical general circulation models, whereas there
are only about 20,000 hydrographic profiles during the entire World Ocean
Circulation Experiment. Consequently, there are an infinite number of
solutions that could satisfy Eq (1). Different criteria are used to derive
particular solutions. One such criterion is least-squares, and is reviewed
below.
3. Kalman filter and Rauch-Tung-Striebel smoother as
least squares inversions
The least squares solution (cf. Chapter 10) provides a general solution to
(^
inverse problem,
ˆ
a
ˆ ˆ
aa
(2)
Ea b
when the right hand side b is given (known), are,
1
0
0
ˆ
T
T
aa
aa
bb
a a R E ER E R
b Ea
(3)
1
ˆ ˆ
T
T
aa
aa
aa
aa
bb
aa
R
R
R E ER E R
ER
(4)
1
inverse problems such as Eq (1). Namely, the least squares solution
denotes an estimate) and its error covariance matrix R for a general linear
See, for instance, Wunsch (1996) for a general discussion of inverse methods that includes a brief summary
Basic matrix algebra is fundamental to mathematical discussions below and data assimilation in general.
of matrix and vector algeba relevant to the subject.
319
state to the observations by the observation operator H. The lower part
describes the model’s temporal evolution by operators A and G that embody
the model physics and dynamics. The right hand side of the model equations
(lower part of Eq 1) is identically zero as all terms of the model are generally
uncertain and are placed on the left hand side. (Sources of model error are
included in u.)
For simplicity, we assume a linear model for most of this discussion.
The problem above and the solutions discussed below can be extended to
non-linear models with suitable linearization. Bold upper and lower case
characters represent matrices and column vectors, respectively. The time
increment from t to t+1 above denotes an arbitrary increment, as opposed to
a single model time-step, and corresponds to instances at which observations
are available.
As in most geophysical inverse problems
1 , Eq (1) is rank deficient. In
particular, there are generally more unknowns than the number of constraints.
For example, the dimension of x, excluding the temporal dimension, is of
order several million for typical general circulation models, whereas there
are only about 20,000 hydrographic profiles during the entire World Ocean
Circulation Experiment. Consequently, there are an infinite number of
solutions that could satisfy Eq (1). Different criteria are used to derive
particular solutions. One such criterion is least-squares, and is reviewed
below.
3. Kalman filter and Rauch-Tung-Striebel smoother as
least squares inversions
The least squares solution (cf. Chapter 10) provides a general solution to
(^
inverse problem,
ˆ
a
ˆ ˆ
aa
(2)
Ea b
when the right hand side b is given (known), are,
1
0
0
ˆ
T
T
aa
aa
bb
a a R E ER E R
b Ea
(3)
1
ˆ ˆ
T
T
aa
aa
aa
aa
bb
aa
R
R
R E ER E R
ER
(4)
1
inverse problems such as Eq (1). Namely, the least squares solution
denotes an estimate) and its error covariance matrix R for a general linear
See, for instance, Wunsch (1996) for a general discussion of inverse methods that includes a brief summary
Basic matrix algebra is fundamental to mathematical discussions below and data assimilation in general.
of matrix and vector algeba relevant to the subject.