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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