Analysis of SST images by Weighted Ensemble Transform Kalman Filter1.pdf

Data assimilationFilters

Vorticity image assimilationResults

Analysis of SST images by Weighted EnsembleTransform Kalman Filter

Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin

INRIA Rennes – Bretagne Atlantique

July 29, 2011

Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF



Data assimilation

Retrieve the state of a dynamical system given some observations.

System only partially known: dx = M(x , t)dt + dBt

Noisy observations: zk = h(xk ) + εk

In our case:

system = vorticity on the surface of the ocean

observations = satellite images of Sea-Surface Temperature




Prototype: Kalman filter

System represented by a Gaussian distribution: mean and covariance

Forecast:xk|k−1 = Mk xk−1|k−1

Pk|k−1 = MkPk−1|k−1MTk + Qk

Update/analysis:

xk|k = xk|k−1 + Kk (zk −Hk xk|k−1)Pk|k = (I −KkHk )Pk|k−1

Kk = Pk|k−1HTk (HkPk|k−1H

Tk + Rk )−1

Drawback: needs to store and compute covariance matrices




Ensemble Kalman filter (EnKF)

Monte-Carlo implementation

Distributions represented by an ensemble of particles[X1 X2 . . . XN ] = X

Sample means and covariances: 1N X 1N and 1

N−1 XX T

Same equations as the Kalman Filter

Never compute neither store covariances matrices

Ensemble Transform Kalman Filter (ETKF)

”Same” as EnKF

Analytical analysis







Sample means and covariances:

1N X 1N and 1

N−1 XX T




”Same” as EnKF

Analytical analysis








N−1 XX T




”Same” as EnKF

Analytical analysis








N−1 XX T




”Same” as EnKF

Analytical analysis








N−1 XX T




”Same” as EnKF

Analytical analysis




Particle filter


Very general: nonlinearities, non-gaussianity

Set of particles

2 steps:1 Proposal distribution (a priori)2 Weight the particles according to the likelihood (a posteriori)

Convergence to the Bayesian filter when the number of particlestends to infinity

Generally needs a lot of particles




Particle filter


Very general: nonlinearities, non-gaussianity

Set of particles

2 steps:1 Proposal distribution (a priori)2 Weight the particles according to the likelihood (a posteriori)

Convergence to the Bayesian filter when the number of particlestends to infinity

Generally needs a lot of particles




WETKF

2 steps:

1 Proposal distribution with ETKF

2 Weight the particles according to the likelihood

w(i)k ∝ w

(i)k−1

p(zk |x(i)k )p(x

(i)k |x

(i)k−1)

N (x(i)k ;µ

(i)k ,Σk )




Vorticity image assimilation

Dynamical model

dξ = −∇ξ · vdt + ν∆ξdt + ηdBt

Observation given by displaced image difference

Ik−1(x) = Ik (x + d(x)) + γ(x)εk where d(x) =

∫ k−δt

k−1v(x , t)dt

Dealing with missing data: uncertainty proportional to thenumber of missing data pixels around




Results




Results

RMSE on a simulated oceanic image sequence




Results

RMSE on a simulated oceanic image sequence




Conclusion

Assimilation of oceanic images to retrieve currents

Use of a particle filter embedding an ETKF as proposaldistribution

Future:

Use of a more realistic dynamics and noise

Other strategies of filtering




Conclusion

Assimilation of oceanic images to retrieve currents

Use of a particle filter embedding an ETKF as proposaldistribution

Future:

Use of a more realistic dynamics and noise

Other strategies of filtering




Thank you!




References

Papadakis, N., Memin, E., Cuzol, A. Gengembre, N. Data assimilationwith the weighted ensemble Kalman filter. Tellus A, 62 (2010).

Tippett, M. K., Anderson, J. L., Bishop, C. H., Hamill, T. M.Whitaker, J.S. Ensemble Square Root Filters. Monthly Weather Review,131 (2003).

Evensen, G. The Ensemble Kalman Filter: theoretical formulation andpractical implementation. Ocean Dynamics, 53 (2003).




Appendix: ETKF

Xk|k = Xk|k−1A and Pk|k = Xk|k−1DXTk|k−1

D = AAT = I− XTk|k−1H

Tk (HkXk|k−1X

Tk|k−1H

Tk + Rk )−1HkXk|k−1

= (I + XTk|k−1H

Tk R−1

k HkXk|k−1)−1

D = UΛUT

A = UΛ1/2UT


Technology

Analysis of SST images by Weighted Ensemble Transform Kalman Filter1.pdf