Biomedical Engineering Reference
In-Depth Information
due to the columns of U b summing to 0. To form the analysis we first add w a to each
column vector, w a ( i ) ,of W a . Denote the resulting vectors w a ( i ) and matrix W a .Then
the i th analysis ensemble member is defined by u a ( i ) =
U a w a ( i ) . The updated
u b +
analysis mean is
k
i = 1 ( u b + U a w a ( i ) )
k 1
u a =
k
i = 1 w a ( i )
k 1 U b
=
u b +
U b w a +
k 1 U b W a v
=
u b +
U b w a +
=
u b +
U b w a ,
as desired.
4.3
Computational Implementation of the LETKF
Computation of the analysis mean, covariance and ensemble as derived above
is accomplished through the following steps. The LETKF procedure begins with
several preliminary calculations carried out over the entire model grid. First the
observation operator is applied to the m -dimensional background ensemble vectors
u b ( i ) to form the background observation ensemble y b ( i ) . Next both ensembles are
averaged and the vectors y b ( i )
y b and u b ( i )
u b are computed. These vectors are
then used to form the perturbation matrices U b and Y b . The remaining steps below
are performed for each local region.
1. Select the components of u b ( i ) , y b ( i ) , U b , Y b ,and R that correspond to the local
region. We denote the resulting local background ensemble and background
ensemble perturbation matrix x b ( i ) and X b , respectively.
2. Compute the k
Y b R 1 (If the observations are not independent
and R is not diagonal, it is computationally more efficient to solve the system
RC T
×
matrix C
=
Y b instead of inverting R .).
3. Compute the k
=
k symmetric matrix P a =[(
CY b ] 1 (See below for
×
k
1
)
I
/ ρ +
more discussion of
ρ
.).
P a ]
2 . This choice ensures that W a
depends continuously on the elements of P a (Otherwise, small changes in P a at
neighboring grid points can lead to very different analysis ensembles [ 13 , 34 ].).
5. Compute the k -vector w a =
1
/
4. Compute the k
×
k matrix W a =[(
k
1
)
P a C
y o
(
y b )
and add it to each column of W a to
k analysis weight matrix W a .
6. Compute the analysis perturbation matrix X a
form the k
×
X b W a .
7. The analysis ensemble, x a ( i ) , is formed by adding x b to the i th column of X a ,
i
=
=
1
,
2
,...,
k .
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