CSDA Conference, Limassol, 2005

1
CSDA Conference, Limassol, 2005
Implementation issues related to data
assimilation using Kalman filtering
Gabriel Dimitriu University of Medicine and
Pharmacy of Iasi Department of Mathematics and
Informatics 700115 Iasi, Romania, email
dimitriu_at_umfiasi.ro
University of Medicine and Pharmacy Gr. T. Popa
Iasi Department of Mathematics and Informatics
2
CSDA Conference, Limassol, 2005
Contents

• Introduction
• Kalman filter with full covariance matrix
• Factorization of the covariance matrix
• Kalman filter in square root form
• Examples of factorized filters (RRSQRT filter,
  Ensemble filter and POEnK filter)
• Implementation issues
• Conclusions

3
Introduction
CSDA Conference, Limassol, 2005

• The purpose of data assimilation is to
  incorporate measured observations into a
  dynamical system model in order to produce
  accurate estimates of all the current (and
  future) state variables of the system.
• In the atmospheric chemistry field, for example
  the knowledge of initial conditions (and
  sometimes boundary conditions) of chemical
  concentrations is a very challenging task.

4
Introduction
CSDA Conference, Limassol, 2005

• Both the Kalman filter and the variational
  methods are suitable to be used in online
  forecast applications.
• In the variational data assimilation,
  information provided by the observations is used
  to find an optimal set of model parameters
  through a minimization process.
• There is a tendency to extend the assimilation
  procedure with Kalman filter techniques, for
  example to obtain useful background covariances
  for the cost function.

5
Introduction
CSDA Conference, Limassol, 2005

• For this study, the Kalman filter was chosen as
  a data assimilation tool. Not needing to build an
  adjoint model is seen as a major advantage here,
  since an adjoint of the chemistry model is
  complicated.
• Other advantages of the Kalman filter which are
  considered are the availability of an analyzed
  covariance, describing the quality of the result,
  and the simple introduction of uncertainties in
  model parameters.
• Numerical results from some investigations
  (based mainly on running simple tests using
  Matlab routines) are presented.

6
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
7
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
8
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
9
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
10
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
11
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
12
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
13
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
14
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
15
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
16
Kalman filter with full covariance matrix
CSDA Conference, Limassol, 2005
17
Factorization of the covariance matrix
CSDA Conference, Limassol, 2005
18
Factorization of the covariance matrix
CSDA Conference, Limassol, 2005
19
Factorization of the covariance matrix
CSDA Conference, Limassol, 2005
20
Factorization of the covariance matrix
CSDA Conference, Limassol, 2005
21
Factorization of the covariance matrix
CSDA Conference, Limassol, 2005
22
Factorization of the covariance matrix
CSDA Conference, Limassol, 2005
23
Factorization of the covariance matrix
CSDA Conference, Limassol, 2005
24
Kalman filter in square root form
CSDA Conference, Limassol, 2005
25
Kalman filter in square root form
CSDA Conference, Limassol, 2005
26
Kalman filter in square root form
CSDA Conference, Limassol, 2005
27
Kalman filter in square root form
CSDA Conference, Limassol, 2005
28
Examples of factorized filters
CSDA Conference, Limassol, 2005

• The filters are based on the concept that,
  although the degree of freedom in the state is
  very large, the errors in the state are described
  very well by a limited number of directions,
  typically less than 100.
• Whether these directions are called singular
  vectors, modes, the basic equations in all filter
  implementations remain the same.

29
RRSQRT filter
CSDA Conference, Limassol, 2005

• The Reduced Rank Square RooT (RRSQRT) filter was
  developed for asimilation of water level
  measurements in a shallow water models.
• In the RRSQRT formulation of the Kalman filter,
  the covariance matrix is expressed in a limited
  number of (orthogonal) modes, which are
  re-orthogonalized and truncated to a fixed number
  during each time step.

30
Ensemble filter
CSDA Conference, Limassol, 2005

• In comparison with RRSQRT approach, the ENsemble
  Kalman Filter (ENKF) is based on convergence of
  large numbers.
• Both approaches lead to a low-rank approximation
  of the covariance matrix. The ensemble filter was
  introduced by Evensen (1994) for assimilation of
  data in oceanographic models.
• The basic idea behind the ensemble filter is to
  express the probability function of the state in
  an ensemble of possible states.

31
POEnK filter
CSDA Conference, Limassol, 2005

• A new direction in implementation of low-rank
  filters is the use of two filters next to each
  other. The combination should compensate for
  errors in one or both of the individual filters.
• The Partially Orthogonal Ensemble Kalman Filter
  (POENKF) proposed in (Heemink et al., 2001) runs
  a RRSQRT filter next to an ENKF.
• The basic idea is to let the RRSQRT part compute
  the bulk of the covariance structure, described
  in the first modes.

32
POEnK filter
CSDA Conference, Limassol, 2005

• The ENKF part should account for the truncation
  error, by introducing directions in the
  covariance matrix that have been lost during
  reduction.
• This procedure incorporates the advantages of
  both filter types, and accounts for their major
  disadvantages. Ensemble filters suffer from a
  lack of convergence many ensembles are required
  before sample mean and correlations are stable.
  An ensemble filter is able to estimate and
  maintain any correlation introduced by the
  stochastic model, however.
• The reverse holds for the RRSQRT filter a few
  modes are sufficient to describe the main part of
  the covariance structure, but some of the
  correlation structure is lost during the
  reduction.

33
Implementation issues
CSDA Conference, Limassol, 2005
34
Implementation issues
CSDA Conference, Limassol, 2005
35
Implementation issues
CSDA Conference, Limassol, 2005
36
Implementation issues
CSDA Conference, Limassol, 2005
37
Conclusions
CSDA Conference, Limassol, 2005

• In this study, the background, implementation and
  some numerical results in data assimilation of
  some common low-rank filters have been discussed.
• Low-rank filters are either based on
  factorizations of the covariance matrix (e.g.
  RRSQRT filter), or approximation of statistics
  from a finite ensemble (ENKF).
• A new direction in filter implementation is to
  use of two filters next to each other of the same
  form or hybrid (POENKF).

38
Conclusions
CSDA Conference, Limassol, 2005

• The factorization approch is often based on the
  linear Kalman filter which has been extended
  towards nonlinear models the ensemble technique
  is a reformulation of the filter problem in a
  statistic approach.
• In spite of the different philosophies, all
  low-rank filters turn out to have a similar
  implementation. Evolution of mean and covariance
  is in each of the filters performed by
  propagation of an ensemble of state vectors by
  the model.
• The propagation of the forecast ensemble is the
  most expensive part of the filter.

39
Conclusions
CSDA Conference, Limassol, 2005

• There are several approaches for the analysis of
  measurements, based on whether the gain will lead
  to a minimal variance or not, and whether the
  filter is based on the factorization or the
  ensemble approach.
• The forms with a minimum variance gain are in
  practice most often used, and differ hardly from
  each other in computational costs.
• The main data structure in all filters is the
  covariance square root a large low-rank matrix,
  with state vectors stored in the columns.

40
Conclusions
CSDA Conference, Limassol, 2005

• The covariance square root needs to be
  transfomed at least one time during each time
  step, which is an expensive operation.
• In addition to forecast and analysis, the filters
  based on factorization require a singular value
  decomposition or re-orthogonalization of the
  covariance square root.
• For comparable costs, the RRSQRT filter produces
  stable and more accurate results than POENKF.