Biomedical Engineering Reference
In-Depth Information
k
background ensemble perturbation matrix,
U
b
,is
where the
i
th column of the
m
×
given by
u
b
(
i
)
−
u
b
. The problem now is to determine a suitable analysis ensemble,
u
a
(
i
)
:
i
{
=
1
...
k
}
, that has the appropriate mean and covariance matrix
k
i
=
1
u
a
(
i
)
,
k
−
1
u
a
=
k
i
=
1
(
u
a
(
i
)
−
u
a
)(
u
a
(
i
)
−
u
a
)
)
−
1
T
P
a
=(
k
−
1
(45)
)
−
1
U
a
U
a
T
=(
k
−
1
(
)
.
(46)
Cost Function
Formally the LETKF computes the analysis by approximately minimizing the
analogue of Eq. (
16
) adapted to the nonlinear observation operator:
T
P
−
1
b
y
o
T
R
−
1
y
o
J
(
u
)=[
u
−
u
b
]
[
u
−
u
b
]+[
−
H
(
u
)]
[
−
H
(
u
)]
.
(47)
Difficulties ensue in adapting the Kalman filter Eqs. (
40
)-(
42
) because the columns
of
U
b
sum to zero, which implies
rank
k
. It follows then that
P
−
1
b
is not generally defined for all model vectors. However,
P
b
is symmetric, thus one-
to-one of its column space,
S
. This space is the same as the column space of
U
b
,i.e.,
the span of the background ensemble perturbations vectors. Now if we let
S
denote a
general
k
-dimensional space, we can treat
U
b
as a linear transformation from
S
onto
S
. Then our strategy is to find the appropriate linear combination of background
ensemble perturbations,
w
a
∈
(
P
b
)=
rank
(
U
b
)
<
S
,sothat
u
a
=
U
b
w
a
minimizes Eq. (
47
).
To justify this approach rigorously, suppose
w
has a Gaussian distribution with
mean
0
and covariance matrix,
u
b
+
)
−
1
I
. From properties of Gaussian random
(
k
−
1
vectors we know
u
U
b
w
is Gaussian with mean
u
b
and covariance given
by Eq. (
44
). Then the analogous cost function for
w
defined on the space
S
is
=
u
b
+
J
w
T
w
y
o
T
R
−
1
y
o
(
w
)=(
k
−
1
)
+[
−
H
(
u
b
+
U
b
w
)]
[
−
H
(
u
b
+
U
b
w
)]
.
(48)
J
,then
u
a
=
U
b
w
a
minimizes
J
.Tosee
this let
P
be the orthogonal projection matrix from
S
onto the subspace spanned
by the columns of
U
b
.Then
P
We prove that if
w
a
minimizes
u
b
+
U
b
U
b
]
−
1
U
b
. Next decompose the vector
w
as
=
U
b
[
w
=
Pw
+(
I
−
P
)
w
. Substituting this into Eq. (
48
)for
w
only in the first term we get
w
T
w
w
T
(
k
−
1
)
=(
k
−
1
)
[
Pw
+(
I
−
P
)
w
]
(49)
w
T
Pw
)
−
1
w
T
=(
k
−
1
)
+(
k
−
1
(
I
−
P
)
w
.
(50)
U
−
1
b
Utilizing
w
=
(
u
−
u
b
)
and the formula for
P
,wehave
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