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Data Assimilation Methods Data Assimilation Methods for Characterizing for Characterizing

Radiation Belt DynamicsRadiation Belt Dynamics

E.J. Rigler1, D.N. Baker1, D. Vassiliadis2, R.S. Weigel1

(1) Laboratory for Atmospheric and Space PhysicsUniversity of Colorado at Boulder

(2) Universities Space Research AssociationNASA / Goddard Space Flight Center

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• Using Data Assimilation (DA) algorithms for identification of empirical dynamical systems

• Finite Impulse Response (FIR) linear prediction filters– Intuitive model structure– Robust and proven predictive capabilities

• Adaptive System Identification (RLS vs. EKF)– Weighted least squares estimates of model parameters

– Tracking non-linear systems with adaptive linear models

• Better Model Structures:– Multiple input, multiple output (MIMO) models– Dynamic feedback and noise models (ARMAX, Box-Jenkins)– Combining RB state with dynamical model parameters

Introduction and OutlineIntroduction and Outline

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Dynamic Model IdentificationDynamic Model Identification

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SISO Impulse Response

Operational Forecasts(NOAA REFM)

Why Linear Prediction Filters?Why Linear Prediction Filters?

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Recursive System IdentificationRecursive System Identification

• RLS minimizes least-squares criterion recursively.– Forgetting factor (λ) allows tracking of non-time-stationary

dynamic processes.– Weighting factor (q) (de)emphasizes certain observations.

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• Model parameters can be incorporated into a state-space configuration.

• Process noise (vt) describes time-varying parameters as a random walk.

• Observation error noise (et) measures confidence in the measurements.

• Provides a more flexible and robust identification algorithm than RLS.

Extended Kalman Filter (EKF)Extended Kalman Filter (EKF)

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Adaptive Single-Input, Adaptive Single-Input, Single-Output (SISO) Linear FiltersSingle-Output (SISO) Linear Filters

EKF-Derived Model Coefficients (w/o Process Noise)

EKF-Derived Model Coefficients

(with Process Noise)

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SISO Model ResidualsSISO Model Residuals

EKF-FIR Residuals (with Process Noise)

EKF-FIR Residuals (w/o Process Noise)

FIR Residuals

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Multiple Input / Output (MIMO)Multiple Input / Output (MIMO)

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Average Prediction EfficienciesAverage Prediction Efficiencies

MIMO PE EKF-MIMO PE (w/o process noise)

EKF-MIMO PE (with process noise)

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Alternative Model StructuresAlternative Model Structures• ARMAX, Box-Jenkins, etc.

– Adaptive colored noise filters.– True dynamic feedback. Better separation between driven and recurrent dynamics.

Combining the State and Model ParametersCombining the State and Model Parameters• True data assimilation:

– Ideal for on-line, real-time RB specification and forecasting.– Framework is easily adapted to incorporate semi-empirical or

physics-based dynamics modules.

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AcknowledgementsAcknowledgements

• Special thanks are extended to Drs. Scot Elkington and Alex Klimas for their valuable time and feedback.

• The data used for this study was generously provided by the National Space Science Data Center (NSSDC) OmniWeb project and the SAMPEX data team.

• This work was supported by the NSF Space Weather Program (grant ATM-0208341), and the NASA Graduate Student Research Program (GSRP, grant NGT5-132).