Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 240

w T Pw U − 1

b

T U b [

U b U b ] − 1 U b U − 1 ( k − 1 ) =( k − 1 )( [ u − u b ]) [ u − u b ] (51)

b

U − 1
b

T U − 1
b

=( k − 1 )( [ u − u b ]) [ u − u b ] (52)

T

U − 1
b

T U − 1
b

=( − )[ − u b ] ( ) [ − u b ]

k 1 u u (53)

T T

] − 1 =( k − 1 )[ u − u b ] ([ U b ( U b ) )[ u − u b ] (54)

T

P b ) − 1 =[ u − u b ] ( [ u − u b ] . (55)

Combining Eqs. ( 50 )and( 55 )inEq.( 48 ) we deduce that

J

w T ( w )=( k − 1 ) ( I − P ) w + J ( u b + U b w ) . (56)

The first term on the right is the orthogonal projection of w onto the null space, N ,
of U b ; thus, it depends only on the components of w in the null space. Similarly
the second term only depends on the components in the column space, S ,of U b .It
follows that w a minimizes J if and only if it is orthogonal to N and u a minimizes J .

Analysis Mean and Covariance

We now proceed to derive the updated analysis and covariance matrix by computing
an approximate minimizer to J based on the Kalman filter equations. The results are
the analogue of Eqs. ( 40 )and( 41 ). To do so we first obtain the linear approximation
of the observation operator. This is accomplished by first applying H to each
background trajectory, u b ( i ) . This produces the

≤

-dimensional ( m ) vectors that

comprise the background observation ensemble

y b ( i ) = u b ( i ) ) .

(

H (57)

Here denotes the spatial dimension of the OBSERVATIONS. Denote y b as the

mean background observation and Y b the k background observation ensemble

perturbation matrix whose i th column is y b ( i ) −

×

y b . We then take the linear

approximation

H ( u b + U b w ) ≈ y b + Y b w . (58)

Replacing H with the above and using the assumption that the background mean
w b =

0, the analysis mean satisfies the analogue of Eq. ( 25 ):

y o w a = K [ − y b − Y b w ] . (59)

Applying Eqs. ( 40 )and( 41 ) with Y b playing the role of H produces

P a Y b R − 1

y o w a = ( − y b ) , (60)

P a =[(

Y b R − 1 Y b ] − 1 k − 1 ) I + . (61)

In model variables the analysis mean and covariance matrix are determined by