Data Assimilation Theory CTCD Data Assimilation Workshop Nov 2005

Sarah Dance

(apologies to Rube Goldberg)

Data assimilation is often treated as a black box algorithm

OUTAnalysis

INObservations

anda priori

information

BUT, understanding and developing what goes on inside the box is crucial !!

Some DARC DA Theory Group Projects

Nonlinear assimilation techniques

Convergence of 4D-Var

Reduced order modelling

Stochastic Processes in DA

Treatment of observation error correlations

Background error modelling and balance

Multiscale DA Phase errors and rearrangement theory

Model errors and bias correction

EnKF and bias

Information content Observation targeting

Formulations of the Ensemble Kalman Filter and Bias

MSc thesis by David Livings, supervised by Sarah Dance and Nancy

Nichols

Outline

• Bayesian state estimation and the Kalman Filter

• The EnKF• Bias and the EnKF• Conclusions

e.g. Suppose xk = M xk-1+

M is linear, the prior and model noise are Gaussian

P(xk-1) ~ N(xb, P) ~ N(0, Q)

Then P(xk |xk-1) ~N(Mxb, MPMT+Q)

Prediction (between observations)

At an observation we use Bayes rule

)(xp

)|()()|( xyxyx ppp

Prior Background error distribution

)|( xypLikelihood of observationsObservation error pdf

Bayes rule

Bayes rule illustrated

)(xp)|( xyp

Bayes rule illustrated (cont)

)|( xyp )(xp

)|()()|( xyxyx ppp

The Kalman Filter

• Use prediction equation and Bayes rule • Assume linear models (forecast and observation)• Assume Gaussian statistics

Kalman filter BUT• Models are nonlinear• Evolving large covariance matrices is

expensive (106 x 106 in meteorology)• So use an ensemble (Monte Carlo idea)

=

=

=

N=10, Perfect observations

Red ensemble mean

Blue ensemble std.

Error bars indicate obs std.

Results with ETKF (old formulation) and Peter Lynch’s swinging spring model

Ensemble statistics not consistent with the truth!

Bias and the EnKF

• Many EnKF algorithms, can be put into a “square root” framework.

• Define an ensemble perturbation matrix:

x1x

2x

3x4x

5x

So, by definition of the ensemble mean

01XX

1

1

The mean of the ensemble is updated separately.

Ensemble perturbations are updated as

where T is a (non-unique) square root of an update equation.

Thus, for consistency,

David discovered that not all implementations preserve this property.

We have now found nec. and suff. conditions for consistency.

Square-root ensemble updates

0T1X1X fa

Consequences

• The ensemble will be biased

• The size of the ensemble spread will be too small

• Filter divergence is more likely to occur !

• Care must be taken in algorithm choice and implementation