1. Better Data Assimilation through Gradient Descent Leonard A. Smith, Kevin Judd and Hailiang Du Centre for the Analysis of Time Series London School of Economics London Mathematical Society - EPSRC Durham Symposium Mathematics of Data Assimilation

2. Outline  Perfect model scenario (PMS) GD method GD is NOT 4DVAR Results compared with Ensemble KF  Imperfect model scenario (IPMS) GD method with stopping criteria GD is NOT WC4DVAR Results compared with Ensemble KF  Conclusion & Further discussion

3. Experiment Design (PMS)

4. Ensemble techniques  Generate ensemble directly, e.g. Particle Filter, Ensemble Kalman Filter  Generate ensemble from perturbations of a reference trajectory, e.g. SVD on 4DVAR Gradient Descent (GD) Method K Judd & LA Smith (2001) Indistinguishable States I: The Perfect Model Scenario, Physica D 151: 125-141.

5. Gradient Descent (Shadowing Filter)

10. GD is NOT 4DVAR Difference in cost function Noise model assumption Observational noise model 4DVAR cost function GD cost function not depend on noise model Assimilation window 4DVAR dilemma:  difficulties of locating the global minima with long assimilation window  losing information of model dynamics and observations without long window

11. Methodology

12. Form ensemble Obs t=0 Reference trajectory GD result

13. Form ensemble t=0 Candidate trajectories Sample the local space Perturb observations and run GD

14. Form ensemble t=0 Ensemble trajectory Draw ensemble members according to likelihood

15. Form ensemble Obs t=0 Ensemble trajectory

16. Ensemble members in the state space Compare ensemble members generated by Gradient Descent method and Ensemble Adjustment Kalman Filter method in the state space. Low dimensional example to visualize, higher dimensional results later.

17. Ikeda Map, Std of observational noise 0.05, 512 ensemble members

18. Evaluate ensemble via Ignorance The Ignorance Score is defined by: where Y is the verification. Ikeda Map and Lorenz96 System, the noise model is N(0, 0.4) and N(0, 0.05) respectively. Lower and Upper are the 90 percent bootstrap resampling bounds of Ignorance score Ensemble->p(.)

19. Imperfect Model Scenario

20. Toy model-system pairs Ikeda system: Imperfect model is obtained by using the truncated polynomial, i.e.

21. Toy model-system pairs Lorenz96 system: Imperfect model:

22. Insight of Gradient Descent Define the implied noise to be and the imperfection error to be

23. Insight of Gradient Descent

26. Statistics of the pseudo-orbit as a function of the number of Gradient Descent iterations for both higher dimension Lorenz96 system-model pair experiment (left) and low dimension Ikeda system-model pair experiment (right). Implied noise Imperfection error Distance from the “truth”

27. GD with stopping criteria GD minimization with “intermediate” runs produces more consistent pseudo-orbits Certain criteria need to be defined in advance to decide when to stop or how to tune the number of iterations. The stopping criteria can be built by testing the consistency between implied noise and the noise model or by minimizing other relevant utility function

28. Imperfection error vs model error Model errorImperfection error Obs Noise level: 0.01 Not accessible!

29. Imperfection error vs model error Imperfection error Obs Noise level: 0.002Obs Noise level: 0.05

30. GD vs WC4DVAR WC4DVAR Model error assumption GD Model error estimates

31. Forming ensemble Apply the GD method on perturbed observations. Apply the GD method on perturbed pseudo-orbit. Apply the GD method on the results of other data assimilation methods. Particle filter?

32. Imperfect model experiment: Ikeda system-model pair, Std of observational noise 0.05, 1024 EnKF ensemble members, 64 GD ensemble members

33. Evaluate ensemble via Ignorance The Ignorance Score is defined by: where Y is the verification. Ikeda system-model pair and Lorenz96 system-model pair, the noise model is N(0, 0.5) and N(0, 0.05) respectively. Lower and Upper are the 90 percent bootstrap resampling bounds of Ignorance score SystemsIgnoranceLowerUpper EnKFGDEnKFGDEnKFGD Ikeda-2.67-3.62-2.77-3.70-2.52-3.55 Lorenz9 6 -3.52-4.13-3.60-4.18-3.39-4.08

34. Conclusion Methodology of applying GD for data assimilation in PMS is demonstrated outperforms the 4DVAR and Ensemble Kalman filter methods Outside PMS, mmethodology of applying GD for data assimilation with a stopping criteria is introduced and shown to outperform the WC4DVAR and Ensemble Kalman filter methods. Applying the GD method with a stopping criteria also produces informative estimation of model error. No data assimilation without dynamics.

35. Thank you! H.L.Du@lse.ac.uk Centre for the Analysis of Time Series: http://www2.lse.ac.uk/CATS/home.aspx