球面上のベクトル場の推定とその応用
中野 慎也
モデリング研究系 准教授
2020年10月27日 統計数理研究所 オープンハウス
Localized basis functions for spherical vector field
A divergence-free vector field can be represented by a stream function as follows:
A curl-free vector field can be represented by a potential function as follows:
According to the Helmholtz theorem, an arbitrary vector field on a sphere can be written as a sum of a divergence-free field and a curl-free field. We can thus represent any vector field in the
following form:
( ) .
df df
= − ×∇Ψ r
V r e
( ) .
cf = −∇Ψ cf
V r
( ) = df ( ) + cf ( ) = − ×∇Ψ − ∇Ψ r df cf .
V r V r V r e
Ψ df
Ψ cf
We expand the stream function and the potential function by using localized basis functions and :
Defining the following vector-valued basis functions:
an arbitrary spherical field can be represented using these basis functions :
Ψ df Ψ cf
( ) ( , ), ( ) ( , ).
df df df cf cf cf
i i i i
i i
w ψ w ψ
Ψ r = ∑ r r Ψ r = ∑ r r
( ) ( )
( ) ( , ) ( , )
( , ) ( , ).
df df cf cf
i r i i i
i i
df df cf cf
i i i i
i i
w w
w w
ψ ψ
= − ×∇ + −∇
= +
∑ ∑
∑ ∑
V r e r r r r
v r r v r r
ψ df ψ cf
( , ) ( , ), ( , ) ( , ),
df df cf cf
i = − ×∇ r ψ i i = −∇ ψ i
v r r e r r v r r r r
Modeling of divergence-free vector field
We employ spherical Gaussian functions for obtaining a set of basis functions
We then obtain
2 , 2
( , ) ( )exp 1 , ( , ) ( sin )exp 1 .
df i cf i
i
= η
i× η R ⋅ −
i=
θ iη ∆ θ η R ⋅ −
r r r r
v r r r r v r r e
( )
( , ) ( , ) exp
21 exp cos 1 .
df cf i
i i
R
Iψ = ψ = η ⋅ − = η θ ′ −
r r r r r r
When we use the expansion of our basis functions
the node points r i can be placed arbitrarily.
We placed 2500 node points randomly and uniformly distributed in the region above 40 degree in latitude, which approximates the following Monte Carlo convolution:
/2 2 2
0 0
( ) ( ) ( , ) ( ) ( , ) sin .
S w ′ d π π w r ′ R θ θ φ d d
= ∫ = ∫ ∫
V r r v r r r v r r
( ) i ( , ), i
i
w
= ∑
V r v r r
Now we consider a divergence-free vector field (no source, no sink) and expand the field by using the divergence-free basis functions:
( ) i df df ( , ). i
i
w
= ∑
V r v r r
>> Next page
Application to ionospheric physics Design of covariance matrices
Now we apply this method for estimating the ionospheric plasma velocity distribution which can be assumed to be divergence-free.
We fit the model to the data of SuperDARN, which is a radar network observing the ionospheric plasma velocity.
The gaps of the data coverage of SuperDARN are filled with the empirical model (Weimber 2001).
We assume the weight w can be decomposed into the model- based value ζ and the residual β :
and the residual β is estimated with the Kalman filter.
, w = + ζ β
We assume the temporal evolution of the weights β
The residual component can then be estimated with the following Kalman filter algorithm:
| 1 1| 1
| 1 2 1| 1
,
.
k k k k
k k k k
α α
− − −
− − −
=
= +
β β
P
P Q
Prediction:
Filtering:
1 1 1 1
| | 1 | 1
1 1 1
| | 1
( )
. (
( ,)
T T
k k k k k k k k k k k k k k
k k k k T k k k
− − − −
− −
− − −
−
+ +
+
= −
=
β β P H R H y H β
R H
H R
P P H )
1 1
( k | k ) ( k , ) . p β β − = α β − Q
Results >> See our paper for detail ( https://doi.org/10.1186/s40623-020-01168-4 ).
In order to ensure spatial smoothness, the covariance matrix Q k is set as follows:
where is the following correlation function:
where
1 1 1
2
1
( , ) ( , )
,
( , ) ( , )
Q Q n
Q
Q n Q n n
C C
C C
σ
=
r r r r
r r r r
Q
, )
( i
Q j
C r r
( , i j ) ( , exp i j ) 2 i 1 , C Q
ρ κ R ⋅ −
= r r
r r r r
2
(sin sin 40 )(sin sin 40 )
( , .
(1 sin 40
) )
i j λ i λ j
ρ
° °
− °
= − −
r r
The matrix R k is set as follows:
where b i and g i denote the beam number and the range gate of the i-th element of the observation y and
1 1
1
1 1
1 1 1
