Observability, Data Assimilation with the Extended Kalman Filter 1
Observability,a Problem in Data Assimilation
Chris DanforthDepartment of Applied Mathematicsand Scientific Computation, UMD
March 10, 2004
AdvisorsJoaquim Ballabrera, UMD/ESSIC
James Yorke, UMD/IPSTEugenia Kalnay, UMD/METO
D.J. Patil, UMD/IPSTBob Cahalan, NASA/GSFC
Observability, Data Assimilation with the Extended Kalman Filter 2
Sources of Numerical Forecast Error
● Displacementerror (standard chaos)• initial conditions are approximate• indistinguishable conditions of the atmosphere diverge
● Model error• improper physical parameterizations• sub-grid scale phenomena
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 3
Ensemble Forecasts and Shadowing
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 4
Ensemble Forecasts and Shadowing
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 5
Ensemble Forecasts and Shadowing
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 6
Ensemble Forecasts and Shadowing
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 7
Model Error and Nudging
● Conservation law∂q∂t
= F(q)
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 8
Model Error and Nudging
● Conservation law∂q∂t
= F(q)
● Nudge model forecast to truth through relaxation
∂q∂t
= F(q)+qobs−q
τ
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 9
Model Error and Nudging
● Conservation law∂q∂t
= F(q)
● Nudge model forecast to truth through relaxation
∂q∂t
= F(q)+qobs−q
τ● Hourly nudging terms correct state-dependent tendency error● Time-averaged nudging terms represent systematic model error
Department of Applied Mathematics and Scientific Computing/University of Maryland
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
Department of Applied Mathematics and Scientific Computing/University of Maryland
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 gridvs observational grid•Model variablesvs observations•Observability: Does the model respond to measurements?
Department of Applied Mathematics and Scientific Computing/University of Maryland
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Kalman Filter
● Analysis cycle: Combine forecast with observations
K[o−H f
]+ f = a
● OperatorH transforms model forecast statefinto the space of observationo
● Kalman Gain matrixK weights the observational incrementwith knowledge of confidence in measurements and forecast
● Analysis statea is our new best guess
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 13
Lorenz Model
dxdt
= σy(t)−σx(t)
dydt
= ρx(t)−x(t)z(t)−y(t)
dzdt
= x(t)y(t)−βz(t)
● Solutions represent simplified convection in the atmosphere● Chaotic for certain parameter values● Suitable for testing data assimilation techniques
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 14
Twin Experiments
● Generate reference state (truth) from model integrationof an arbitrary initial condition
● Start forecast from adifferent 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?
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 15
Plot of y(t) , assimilatingx every two time steps
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 16
Relative Error remains small observingx,y and combinations
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 17
Plot of x(t) , assimilatingz every time step
Department of Applied Mathematics and Scientific Computing/University of Maryland
Observability, Data Assimilation with the Extended Kalman Filter 18
Observability Conclusions
● The EKF fails to push forecasts to truth in the classic toy weathermodel of Lorenz, when measuring the variablez
● Nonlinear systems do not necessarily respond to assimilation ofall state variables, not all measurements are the same!
● Operational weather models need to be tested for observability
Department of Applied Mathematics and Scientific Computing/University of Maryland
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Current Work
● Researching 40-d Lorenz model to develop techniques of ensem-ble variance inflation
● Model error experiments with Marshall and Molteni global 3-levelQG model show nudging terms correct model bias
● Displacement error and model error cooperate to destroy weatherforecasts......to keep the truth contained within our ensemble ellipse, and toevaluate/ correct model error, we MUST effectively assimilate ac-curate and representative observations!
Department of Applied Mathematics and Scientific Computing/University of Maryland
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References
[1] Robert Miller, Michael Ghil, Francois Gauthiez,Advanced DataAssimilation in Strongly Nonlinear Dynamical Systems, Journal ofthe Atmospheric Sciences, Vol. 51, No. 8, April 1994.[2] Eugenia Kalnay,Atmospheric Modeling, Data Assimilation andPredictability, Cambridge University Press, 2002[3] Edward Lorenz, ”Predictability - a problem partly solved”, inPredictability, edited by T. Palmer, European Centre for Medium-Range Weather Forecasting, Shinfield Park, Reading, UK, 1996.[4] D.J. Patil, E. Ott, B.R. Hunt, E.Kalnay, J.A. Yorke,Local lowdimensionality of atmospheric dynamics, Physical Review Letters,Vol. 86, No 26, 2001, 5878–5881.
The End — Thank you Contact: [email protected]
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