Introduction to
Data Assimilation
Quintin Schiller
GEM Student Tutorial
June 17th 2012
Snowmass, CO
Data Assimilation
Data assimilation combines sparse
observations with a physical model to
generate a better estimate of the global
system.
Generally, data assimilation incorporates
the errors associated with the
measurements and model to find the best
estimate.
Introduction
Kalman Filters
A Simple Case
Applications
Weather Prediction
60% Probability Cone
Introduction
Kalman Filters
A Simple Case
Applications
Orbit Determination
Error Ellipse
Introduction
Kalman Filters
A Simple Case
Applications
Electric Potential
AMIE
Ionosphere Prediction
GAIM
TEC
Introduction
Kalman Filters
A Simple Case
Applications
Global Magnetosphere Environment
Introduction
Kalman Filters
A Simple Case
Applications
Complexity
Types of Assimilation
Introduction
Non-linearity
4DVAR
Kalman Filters
3DVAR
Direct Injection
Optimum Interpolation
‘Nudging’
Interpolation of observations
Kalman Filters
A Simple Case
Applications
Complexity
Types of Assimilation
Introduction
Non-linearity
4DVAR
Kalman Filters
3DVAR
Direct Injection
Optimum Interpolation
‘Nudging’
Interpolation of observations
Kalman Filters
A Simple Case
Applications
Kalman Filter
An optimal recursive estimator that
minimizes the mean of the squared error
in observations and model.
Optimized for Gaussian errors.
Consists of two major operations:
1) Analysis Step (Update Step)
2) Forecast Step
Introduction
Kalman Filters
A Simple Case
Applications
Observations
Observation Errors
Update Step
t
• Compare observation
and model errors
• Use obs to update state
• Update state covariance
Introduction
Kalman Filters
A Simple Case
Applications
Observations
Observation Errors
Update Step
t
• Compare observation
and model errors
• Use obs to update state
• Update state covariance
Forecast Step
t+1
• Use model to propagate
state vector
• Use model to propagte
state covariance
Analysis State
• Analysis state vector
• Analysis covariance
Observations
Observation Errors
Update Step
t
• Compare observation
and model errors
• Use obs to update state
• Update state covariance
Forecast Step
t+1
• Use model to propagate
state vector
• Use model to propagte
state covariance
Analysis State
• Analysis state vector
• Analysis covariance
Forecast State
• Forecast state vector
• Forecast covariance
Simple Case
Introduction
Kalman Filters
A Simple Case
Applications
Simple Case
Introduction
Kalman Filters
A Simple Case
Applications
Simple Case
Introduction
Kalman Filters
A Simple Case
Applications
Simple Case
Introduction
Kalman Filters
A Simple Case
Applications
Simple Case
You
Introduction
Kalman Filters
A Simple Case
Applications
Observations
h 
10%
y    y  

v 
10%

Introduction
Kalman Filters
A Simple Case
Applications
Observations
h 
10%
y    y  

v 
10%
Physical Model
1 2
h

h

v
t

at
0
0

2
 h  10   y 
  

 x   v  10   y 

 a 
 
 0.001 

h
v
Introduction
Kalman Filters
A Simple Case
Applications
Observations
h 
10%
y    y  

v 
10%
Physical Model
1 2
h

h

v
t

at
0
0

2
 h  10   y 
  

 x   v  10   y 

 a 
 
 0.001 

h
State Vector
(Estimateables)
h 
 
x  v 

a 

v
Introduction
Kalman Filters
A Simple Case
Applications
1 2
h  h0  v 0 t  at
2
Introduction
Kalman Filters
A Simple Case
Applications
Height
Assimilated State
Time
Introduction
Kalman Filters
A Simple Case
Applications
Model State
Assimilated State
Height
Forecast Step
Time
Introduction
Kalman Filters
A Simple Case
Applications
Model State
Observation
Assimilated State
Height
Update Step
Update
Time
Introduction
Kalman Filters
A Simple Case
Applications
Model State
Observation
Assimilated State
Height
Forecast Step
Time
Introduction
Kalman Filters
A Simple Case
Applications
Model State
Observation
Assimilated State
Height
Update Step
Update
Time
Introduction
Kalman Filters
A Simple Case
Applications
Model State
Observation
Assimilated State
Height
Forecast Step
Time
Introduction
Kalman Filters
A Simple Case
Applications
Model State
Observation
Assimilated State
Height
Repeat!
Time
Introduction
Kalman Filters
A Simple Case
Applications
1 2
h  h0  v 0 t  at
2
2

1
v Cd A  2
h  h0  v 0 t  a 
t
2 
m 
h 
 
x  v 

a 

Introduction
Kalman Filters
A Simple Case
Applications
Introduction
Kalman Filters
A Simple Case
Applications
h 
 
x  v 

a 

Introduction
Kalman Filters
A Simple Case
Applications
h 
 
x  v 

a 

2

1
v Cd A  2
h  h0  v 0 t  a 
t
2 
m 
Introduction
Kalman Filters
A Simple Case
Applications
Application to the Magnetosphere
Introduction
Kalman Filters
A Simple Case
Applications
Observations
30%(?) 
PSDSat1 
y  

 y  
70%(?) 
PSDSat2 
(For constant 1st and 2nd adiabatic invariants)

Introduction
Kalman Filters
A Simple Case
Applications
Observations
30%(?) 
PSDSat1 
y  

 y  
70%(?) 
PSDSat2 
Physical Model
DLL f  f
f
2 
L
 2
  S
t  L  L L  
  10   (?) 
f L 10
y

 

 x   M   M 
  10   (?) 

y
 f L 2  
Introduction
Kalman Filters
A Simple Case
Applications
Observations
30%(?) 
PSDSat1 
y  

 y  
70%(?) 
PSDSat2 
Physical Model
DLL f  f
f
2 
L
 2
  S
t  L  L L  
State Vector
(Estimateables)
 PSDL 10 


PSD
L 9.97 

x   M 


PSD
L 2.25 


 PSDL 2 

  10   (?) 
f L 10
y

 

 x   M   M 
  10   (?) 

y
 f L 2  
Introduction
Kalman Filters
A Simple Case
Applications
State Vector
Observations
Error
Covariance
Backups
Obs. Information
yt Rt
Update Step
t=ti
Kt=PftHTt[HtPftHTt+Rt]-1
xat=xft+Kt [yt-Htxft]
Pat=(I-KtHt)Pft
Prediction Step
tt+1
Xft+1=Φt+1,txat
P=Φt+1,tPatΦTt+1,t+Qt
Analysis State
xat
Pat
Forecast State
xft
Pft