Data Assimilation Methods
for Characterizing
Radiation Belt Dynamics
E.J. Rigler1, D.N. Baker1, D. Vassiliadis2, R.S. Weigel1
(1) Laboratory for Atmospheric and Space Physics
University of Colorado at Boulder
(2) Universities Space Research Association
NASA / Goddard Space Flight Center
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Introduction and Outline
• 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
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Dynamic Model Identification
n
yˆ t   ut 
p
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
θˆ  U T U

1
UT Y
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Electron Flux Response
Why Linear Prediction Filters?
SISO Impulse Response
Days Since Solar Wind Impulse
Operational Forecasts
(NOAA REFM)
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Recursive 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.
θˆ t  θˆ t 1  γ t et , et  yt  uTt θˆ t 1
Pt 1u t
γt  T
u t Pt 1u t   / qt
Pt 
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
I  γ u P

1
t
T
t
t 1
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Extended Kalman Filter (EKF)
• Model parameters can be
incorporated into a statespace 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.
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θt 1  θt  vt ,
E{vvT }  R1
yt  uTt θt  et , E{eeT }  R 2

θˆ t  θˆ t 1  γ t yt  uTt θˆ t 1

 P
P  I  γ u  P  R
γ t  u Pt 1u t  R 2
T
t
t
t
T
t
t 1
1

u
t 1 t
1
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Adaptive Single-Input,
Single-Output (SISO) Linear Filters
EKF-Derived
Model Coefficients
(w/o Process Noise)
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EKF-Derived
Model Coefficients
(with Process Noise)
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SISO Model Residuals
FIR Residuals
Auto-covariance
EKF-FIR Residuals (w/o Process Noise)
EKF-FIR Residuals (with Process Noise)
Lagged Days
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Multiple Input / Output (MIMO)
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Average Prediction Efficiencies
MIMO PE
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EKF-MIMO PE
(w/o process noise)
EKF-MIMO PE
(with process noise)
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Alternative 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 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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Acknowledgements
• 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).
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