Data Assimilation for the Lorenz (1963) Model using Ensemble and Extended Kalman Filter
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Presentation prepared for the Biogeodynamics and Earth System Sciences Summer School (BESS), 23rd June 2011. http://www.istitutoveneto.org/bess/
Data Assimilation for the Lorenz (1963) Model using Ensemble and Extended Kalman Filter
- 1. Biogeodynamics and Earth System Sciences Summer School (BESS)Sciences Summer School (BESS) Data Assimilation for the Lorenz (1963) Model using Ensemble d E t d d K l Filtand Extended Kalman Filter D. Pasetto (a) and C. Vitolo (b) Advisor M.Ghil, ENS & UCLA (a) Universita' di Padova (Italy) (b) Imperial College London (UK)
- 2. Data Assimilation for the Lorenz model using Ensemble and Extended Kalman FilterEnsemble and Extended Kalman Filter OUTLINE Data AssimilationData Assimilation Kalman Filter (EnKF and EKF) Lorenz ModelLorenz Model Results of Sensitivity Tests Future Challenges
- 3. DATA ASSIMILATIONSS O Data Assimilation is usually defined as Estimation and prediction (analysis) of an unknown trueEstimation and prediction (analysis) of an unknown true state by combining observations and system dynamics (model output)( p ) It is needed in order to: - Reduce uncertainties and biases - Improve forecasting - Estimate initial state of a system (e g hydrologic system)Estimate initial state of a system (e.g. hydrologic system) from multiple sources of information - Permit forecast adjustmentsPermit forecast adjustments
- 4. EXAMPLE Lorenz's Model a simplified model of thermal convectiona simplified model of thermal convection in the atmosphere. Outcome: - No predictable pattern - Butterfly effect
- 5. A BIT OF MATHS...O S The Lorenz model Bistability and chaotic behaviour Where: For the bistable behaviour:Matlab code to simulate For the bistable behaviour: = 8/3, =1.01, = 10 For the Lorenz attractor: Matlab code to simulate the model dynamics Perturbation of a true run = 8/3, =28, = 10 Perturbation of a true run with a random noise to get pseudo-observationsp
- 6. A BIT MORE MATHSO S Kalman FilterKalman Filter Data assimilation in non linear models: EKF, EnKF and Particle Filters
- 7. RESULTS
- 8. RESULTS
- 9. RESULTS
- 10. RESULTS
- 11. SENSITIVITY TESTS: N b f li ti f th E KFNumber of realization of the EnKF 10 50 100 7 00E 001 ComparisonbetweenRMSE(normalized) 4,00E001 5,00E001 6,00E001 7,00E001 EnKF 1,00E001 2,00E001 3,00E001 , NRMSE EKF OpenLoop 0 20 40 60 80 100 120 0,00E+000 NumberofRealizations
- 12. SENSITIVITY TESTS: Ob ti Ti StObservation Time Step 0.1 0.5 1 7,00E001 ComparisonbetweenRMSE(normalized) 4,00E001 5,00E001 6,00E001 , EnKF 1,00E001 2,00E001 3,00E001 NRMSE EKF OpenLoop 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,00E+000 ObservationTimeSteps
- 13. CONCLUSIONSCO C US O S Data Assimilation on the Lorenz Model using EnKF and EKF was performed:and EKF was performed: Sensitivity tests on the number of realization on the EnKF show that N=10 is already optimal for this small model Sensitivity tests on the observation time steps sho the error increases as the amo nt ofshow the error increases as the amount of information provided decreases EnKF and EKF perform similarly but EKF is to be prefered because computationally more efficientprefered because computationally more efficient
- 14. FUTURE CHALLENGESU U C G S Further sensitivity tests (e.g. bistable dynamics) Parameter estimationParameter estimation Utilize Data Assimilation for real problems (e.g. Weather predictions)
- 15. Data Assimilation for the Lorenz model using Ensemble and Extended Kalman FilterEnsemble and Extended Kalman Filter Thanks for your attentionThanks for your attention