a non-parametric regression model for estimation of...
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A non-parametric regression model for estimation of ionospheric plasma velocity distribution from SuperDARN data
S. Nakano (The Institute of Statistical Mathematics)
T. Hori (ISEE, Nagoya University)
K. Seki (University of Tokyo)
N. Nishitani (ISEE, Nagoya University)
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Ionospheric convection pattern
The ionospheric flow pattern is one of fundamental property of the ionospheric science.
There exist several studies which deduced the velocity distribution pattern in the ionosphere.
Ionospheric convection pattern deduced by the spherical harmonics method (http://vt.superdarn.org/ Virginia Tech Website)
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Existing methods
Spherical harmonic fitting Assuming that the divergence-free condition,
the vector field can be represented by a scalar stream function. The stream function is expanded with spherical harmonics funcitons.
Sensitive to local distrubances and noises
Matrix-valued kernel (Narcowich et al., 2007; Fuselier and Wright, 2009) Localized basis function
Computationally demanding
Designed for interpolating vector-valued data
Spherical elementary current system (Amm, 1997) Localized basis function
Diverge at the singular point of a basis function
Ionospheric convection pattern deduced by the spherical harmonics method (http://vt.superdarn.org/ Virginia Tech Website)
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Spherical elementary current system (SECS)
The divergence-free SECS basis function is defined so that the curl is constant except at the pole, and the curl-free SECS basis function is defined so that the divergence is constant except at the pole.
Nodes of the SECS basis functions can be placed arbitrarily. This can be regarded as one of radial basis function (RBF) networks.
They diverge to infinity at the pole.
An example of node distribution
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Generalization of SECS
The ionospheric plasma drift velocity distribution can be assumed to be divergence-free (no source, no sink).
The divergence-free vector field can be represented by a stream function Ψ as follows:
We expand the stream function Ψ by using localized basis functions:
Thus,
( ) .= − ×∇ΨV r r
( ) ( , ).i ii
wψΨ = ∑r r r
( ) ( , ).i ii
w ψ= − ×∇Ψ = − ×∇∑V r r r r r
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Generalization of SECS
Defining vector-valued localized basis function:
we obtain
If where
This is the original divergence-free SECS basis function.
( ) ( , ).i ii
w= ∑V r v r r
( , ) ( , ),i r iψ= − ×∇v r r e r r
2arccos ,i
Rθ ⋅ ′ =
r r
,| |( , ) cot .
2i iφθ∆ ′
=v r r e
( , ) 2 log sin2iθψ ∆ ′
=r r
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Generalization of SECS
We represent the divergence-free velocity field by
We choose the spherical Gaussian function for ψ :
and obtain the following divergence-free basis function
( )( ) ( , ), ( , ) ( , ) .i i i ii
w ψ= = − ×∇∑V r v r r v r r r r r
( )2( , ) exp 1 exp cos 1 ,ii
IRψ η η θ
⋅ = − = ′ −
r rr r
2( , ) ( ) exp 1 .ii i
IRη η
⋅= × −
r rv r r r r
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Kalman filter
We assume the temporal evolution of the weights w
The residual component can then be estimated with the following Kalman filter algorithm:
| 1 1| 1
2| 1 1| 1
,
.k k k k
k k k k
α
α− − −
− − −
=
= +
w w
PP Q
Prediction:
Filtering:1 1 1 1
| | 1 | 1
1 1 1| | 1
( )
.(
( ,)T Tk k k k k k k k k k k k k k
Tk k k k k k k
− − − −− −
− − −−
+ +
+
= −
=
w w y wP H R H H
R H
H R
P P H )
1 1( )| , )( .k k kp α− −=w w w QN
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Estimation
The covariance matrices Q is given so as to satisfy
And R is assumed to be diagonal: The parameters and are determined by
maximizing the marginal likelihood:
1:( | ) ( | )| .) (K k k k kk
pp p d= ∏∫y θ y w w θ w
222
1 22 2
exp( cos )cos .cos cos
1T k kk k Q
i i i
w wξ λσ λλ λ λ λ φ
− ∂ ∂∂ = + ∂ ∂ ∂
∑w wQ
2 ( 500).k R Rσ σ= =R I,
Qσ ξ
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Experiment
We conducted experiments with synthetic radar data generated from a certain velocity distribution model.
The observation sites and observed echoes were assumed to be the same as observed on March 17, 2015.
The nodes of the basis functions were placed at every 5 and 2 degrees in longitude and in latitude, respectively.
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Reconstruction with proposed functions
0800 UT, Mar. 17, 2015
Estimated velocity distribution Original velocity distribution
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Reconstruction with SECS functions
0800 UT, Mar. 17, 2015
Estimated velocity distribution Original velocity distribution
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Stream function with proposed
0800 UT, Mar. 17, 2015
The estimate with proposed basis funcitons.
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Stream function with proposed
0810 UT, Mar. 17, 2015
The estimate with proposed basis funcitons.
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Stream function with proposed
0820 UT, Mar. 17, 2015
The estimate with proposed basis funcitons.
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Stream function with proposed
0830 UT, Mar. 17, 2015
The estimate with proposed basis funcitons.
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Stream function with proposed
0840 UT, Mar. 17, 2015
The estimate with proposed basis funcitons.
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Stream function with proposed
0850 UT, Mar. 17, 2015
The estimate with proposed basis funcitons.
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Stream function with SECS
0800 UT, Mar. 17, 2015
The estimate with SECS basis funcitons.
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Stream function with SECS
0810 UT, Mar. 17, 2015
The estimate with SECS basis funcitons.
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Stream function with SECS
0820 UT, Mar. 17, 2015
The estimate with SECS basis funcitons.
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Stream function with SECS
0830 UT, Mar. 17, 2015
The estimate with SECS basis funcitons.
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Stream function with SECS
0840 UT, Mar. 17, 2015
The estimate with SECS basis funcitons.
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Stream function with SECS
0850 UT, Mar. 17, 2015
The estimate with SECS basis funcitons.
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Use of empirical model
An empirical model is referred to in estimating the velocity distribution.
We assume the weight w can be decomposed into the model-based value ζ and the residual β :
The model-based value is determined so as to fit an empirical model by Weimber 2001.
The residual β is estimated with the Kalman filter.
.= +w ζ β
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Kalman filter
We assume the temporal evolution of the resudial component obeys
The residual component can then be estimated with the following Kalman filter algorithm:
| 1 1| 1
2| 1 1| 1
,
.k k k k
k k k k
α
α− − −
− − −
=
= +
β β
PP Q
Prediction:
Filtering:1
| | 1 | 1 | 1
1| | 1 | 1 | 1 | 1
( [ ] ,
.
( ) )
( )
T Tk k k k k k k k k k k k k k k k
T Tk k k k k k k k k k k k k k k
−− − −
−− − − −
= + −+ +
−= +
β β y ζ βP H P H
P
H R H
P HH PHH RP P
1 1( )| , )( .k k kp α− −=β β β QN
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17 March 2015
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Result
0800 UT, Mar. 17, 2015
The estimate with actual data
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Result
0810 UT, Mar. 17, 2015
The estimate with actual data
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Result
0820 UT, Mar. 17, 2015
The estimate with actual data
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Result
0830 UT, Mar. 17, 2015
The estimate with actual data
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Result
0840 UT, Mar. 17, 2015
The estimate with actual data
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Result
0850 UT, Mar. 17, 2015
The estimate with actual data
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27 March 2017
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Result
1000 UT, Mar. 27, 2017
The estimate with actual data
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Result
1010 UT, Mar. 27, 2017
The estimate with actual data
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Result
1020 UT, Mar. 27, 2017
The estimate with actual data
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Result
1030 UT, Mar. 27, 2017
The estimate with actual data
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Result
1040 UT, Mar. 27, 2017
The estimate with actual data
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Result
1050 UT, Mar. 27, 2017
The estimate with actual data
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Summary
We have proposed a framework for obtaining global flow vector distribution in the ionosphere.
In our framework, the vector field is represented by weighted sum of localized basis functions derived from a spherical Gaussian function. The basis functions consist of two types of functions: curl-free functions and divergence-free functions.
Our framework also allows us to combine the SuperDARN data with existing statistical pattern of the velocity distribution.