analysis of sst images by weighted ensemble transform kalman filter1.pdf
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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 assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
Particle filter
Monte-Carlo implementation
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
Particle filter
Monte-Carlo implementation
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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 )
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
Results
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
Results
RMSE on a simulated oceanic image sequence
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
Results
RMSE on a simulated oceanic image sequence
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
Thank you!
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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).
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF
Data assimilationFilters
Vorticity image assimilationResults
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
Sai Subrahmanyam Gorthi, Sebastien Beyou, Etienne Memin Analysis of SST images by WETKF