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Lagrangian Data Assimilation and Overcoming the Saddle Effect

Lagrangian Data Assimilation and Overcoming the Saddle Effect. Liyan Liu, Christopher Jones, Hayder Salman UNC-Chapel Hill Kayo Ide, UCLA. Supported by NSF and ONR. Augmented System Strategy. Kalman Filter. Challenge of LaDA. Lagrangian trajectories can be “chaotic”

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Lagrangian Data Assimilation and Overcoming the Saddle Effect

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  1. Lagrangian Data Assimilation and Overcoming the Saddle Effect • Liyan Liu, Christopher Jones, Hayder Salman • UNC-Chapel Hill • Kayo Ide, UCLA Supported by NSF and ONR

  2. Augmented System Strategy

  3. Kalman Filter

  4. Challenge of LaDA • Lagrangian trajectories can be “chaotic” • What information do they contain if sensitivity to initial conditions is present? • Hyperbolic (saddle) points are the “engines” of chaotic dynamics • Passage near saddle point is critical challenge

  5. Saddle Effect in 2-Point Vortex Flow

  6. pdf after pdf before Passage near Saddle forecast true

  7. Saddle Effect in 2x Gyre • Jump in drifter position estimate can be large near saddle • PDF is bimodal • Resolve using Ensemble Kalman Filter Updating the mean Ensemble spread

  8. Layer Model Represent layered features by 2-layer point vortex model • λ is the deformation between layers • Г is the vortex strength (circulation) • ψ represents the streamfunction in each layer

  9. Assimilation of top layer tracer EKF in a two-layer vortex system, one tracer is observed in top layer. System noise=0.02, observation error=0.02. Actual error in vortex positions in the model assimilating tracer positions, and without assimilation.

  10. Saddle Effect • Exponential separation of trajectories near the saddle causes divergence of filter.

  11. Tracer Control Make correction if EKF-reinitialize state and covariances Tracer-use analysis Vortex-do not update

  12. Top layer tracer controlled

  13. Extended Kalman Filter-no tracer control Extended Kalman Filter with ΔT=1.5

  14. Extended Kalman Filter-w/tracer control Extended Kalman Filter with ΔT=1.5

  15. EnKF Resettings no update update update update do not use use use use

  16. Ensemble Kalman Filter-no tracer control Ensemble Kalman Filter with ΔT=1.5

  17. Ensemble Kalman Filter-w/ tracer control Ensemble Kalman Filter with ΔT=1.5

  18. Ensemble Kalman Filter with TC • Apply tracer control technique to each ensemble member • The update state vector is the mean of the ensembles which represent the true evolution • More accurate error covariance matrix Error: vortex position error averaged over time and noise realizations. Failure: error exceeds successful tracking threshold(=1)

  19. Conclusions • The saddle effect is a serious impediment to the implementation of Lagrangian data assimilation • Tracer control works very well in resolving the issue • With EKF it works well • With EnKF, its success is striking

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