Observability, a Problem in Data Assimilation

Transcription

1 Observability, Data Assimilation with the Extended Kalman Filter 1 Observability, a Problem in Data Assimilation Chris Danforth Department of Applied Mathematics and Scientific Computation, UMD March 10, 2004 Advisors Joaquim Ballabrera, UMD/ESSIC James Yorke, UMD/IPST Eugenia Kalnay, UMD/METO D.J. Patil, UMD/IPST Bob Cahalan, NASA/GSFC

2 Observability, Data Assimilation with the Extended Kalman Filter 2 Sources of Numerical Forecast Error Displacement error (standard chaos) initial conditions are approximate indistinguishable conditions of the atmosphere diverge Model error improper physical parameterizations sub-grid scale phenomena

3 Observability, Data Assimilation with the Extended Kalman Filter 3 Ensemble Forecasts and Shadowing

4 Observability, Data Assimilation with the Extended Kalman Filter 4 Ensemble Forecasts and Shadowing

5 Observability, Data Assimilation with the Extended Kalman Filter 5 Ensemble Forecasts and Shadowing

6 Observability, Data Assimilation with the Extended Kalman Filter 6 Ensemble Forecasts and Shadowing

7 Observability, Data Assimilation with the Extended Kalman Filter 7 Model Error and Nudging Conservation law q t = F(q)

8 Observability, Data Assimilation with the Extended Kalman Filter 8 Conservation law Model Error and Nudging q t = F(q) Nudge model forecast to truth through relaxation q t = F(q) + q obs q τ

9 Observability, Data Assimilation with the Extended Kalman Filter 9 Conservation law Model Error and Nudging q t = F(q) Nudge model forecast to truth through relaxation q t = F(q) + q obs q τ Hourly nudging terms correct state-dependent tendency error Time-averaged nudging terms represent systematic model error

10 Observability, Data Assimilation with the Extended Kalman Filter 10 Data Assimilation Data Assimilation Cycle: Start with best guess of initial conditions, background Integrate model to generate prediction, forecast Make measurements of truth, observations Combine model prediction with observations, analysis

11 Observability, Data Assimilation with the Extended Kalman Filter 11 Data Assimilation Data Assimilation Cycle: Start with best guess of initial conditions, background Integrate model to generate prediction, forecast Make measurements of truth, observations Combine model prediction with observations, analysis Sources of difficulty: Model grid vs observational grid Model variables vs observations Observability : Does the model respond to measurements?

12 Observability, Data Assimilation with the Extended Kalman Filter 12 Kalman Filter Analysis cycle: Combine forecast with observations K [ o H f ] + f = a Operator H transforms model forecast state f into the space of observation o Kalman Gain matrix K weights the observational increment with knowledge of confidence in measurements and forecast Analysis state a is our new best guess

13 Observability, Data Assimilation with the Extended Kalman Filter 13 Lorenz Model dx dt = σy(t) σx(t) dy dt = ρx(t) x(t)z(t) y(t) dz dt = x(t)y(t) βz(t) Solutions represent simplified convection in the atmosphere Chaotic for certain parameter values Suitable for testing data assimilation techniques

14 Observability, Data Assimilation with the Extended Kalman Filter 14 Twin Experiments Generate reference state (truth) from model integration of an arbitrary initial condition Start forecast from a different arbitrary initial state Observe truth at relevant time steps, combine with forecast Generate analysis (best estimate) of current state Does the forecast stay close to the truth?

15 Observability, Data Assimilation with the Extended Kalman Filter 15 Plot of y(t), assimilating x every two time steps

16 Observability, Data Assimilation with the Extended Kalman Filter 16 Relative Error remains small observing x,y and combinations

17 Observability, Data Assimilation with the Extended Kalman Filter 17 Plot of x(t), assimilating z every time step

18 Observability, Data Assimilation with the Extended Kalman Filter 18 Observability Conclusions The EKF fails to push forecasts to truth in the classic toy weather model of Lorenz, when measuring the variable z Nonlinear systems do not necessarily respond to assimilation of all state variables, not all measurements are the same! Operational weather models need to be tested for observability

19 Observability, Data Assimilation with the Extended Kalman Filter 19 Current Work Researching 40-d Lorenz model to develop techniques of ensemble variance inflation Model error experiments with Marshall and Molteni global 3-level QG model show nudging terms correct model bias Displacement error and model error cooperate to destroy weather forecasts......to keep the truth contained within our ensemble ellipse, and to evaluate/ correct model error, we MUST effectively assimilate accurate and representative observations!

20 Observability, Data Assimilation with the Extended Kalman Filter 20 References [1] Robert Miller, Michael Ghil, Francois Gauthiez, Advanced Data Assimilation in Strongly Nonlinear Dynamical Systems, Journal of the Atmospheric Sciences, Vol. 51, No. 8, April [2] Eugenia Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2002 [3] Edward Lorenz, Predictability - a problem partly solved, in Predictability, edited by T. Palmer, European Centre for Medium- Range Weather Forecasting, Shinfield Park, Reading, UK, [4] D.J. Patil, E. Ott, B.R. Hunt, E.Kalnay, J.A. Yorke, Local low dimensionality of atmospheric dynamics, Physical Review Letters, Vol. 86, No 26, 2001, The End Thank you Contact: danforth@math.umd.edu