Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 237

To ensure this minimizes Eq. ( 26 ) we show that the Hessian is positive semi-definite.
A second calculation using Eqs. ( 35 )and( 36 )showsthat

2 J

∂

2 N T

K 2 =

, (39)

∂

which is positive semi-definite because N is the sum of two covariance matrices
which are positive definite by definition. Note there are many equivalent forms for
K and P a . Here we state the most computationally efficient forms

P a H T R − 1 y o u a = u b + [ − H u b ] , (40)

P b H T R − 1 H ) − 1 P b , P a =( I + (41)

P a H T R − 1

K = . (42)

In the next section we derive the LETKF which is an extension of the basic
Kalman filter.

4.2 Local Ensemble Transform Kalman Filter

Extension of the Kalman filter to the nonlinear scenario entails many difficulties.
First, the propagation of the analysis covariance under the model is no longer
traceable by Eq. ( 14 ). Second, the equations derived for the analysis mean and
covariance matrix must be adapted because of the assumed nonlinearity of H and
F . One approach to this problem that has proven useful in operational meterology
is ensemble Kalman filtering [ 8 ]. The main technique is to select an ensemble of k
trajectories whose covariance is used to approximate P a n − 1 . Each ensemble is then
advanced under the model to time t n , and the resulting background ensemble sample
covariance is used to estimate P b n .
Challenges arise in this approach because the size of k is limited (usually less
than a few hundred) due to computational resources and is typically smaller than
the model resolution, m . If the spread of the ensemble sufficiently approximates
P b n , one can generate an accurate analysis. On the other hand, if the spread poorly
estimates the background covariance, as is the case when the forecast model has
more than k Lyapunov exponents, then analysis fails to correct errors in the forecast
model.
The LETKF [ 13 ] makes use of localization to overcome the challenges related
to the size of k . The strategy is to perform the analysis at each point individually
by forming a local ensemble over a subset of the model domain. The hope is that
the dynamics at a given point are captured over the local region and relatively
low-dimensional. If so the local ensemble will sufficiently estimate the background
uncertainty and subsequently correct the background forecast at each point, thereby