A non-parametric regression model for estimation of ionospheric plasma velocity distribution from SuperDARN data

Size: px

Start display at page:

Download "A non-parametric regression model for estimation of ionospheric plasma velocity distribution from SuperDARN data"

Elwin Murphy
5 years ago
Views:

1 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)

2 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 ( Virginia Tech Website)

Existing methods Spherical harmonic fitting Assuming that the divergence-free condition, the vector field can be represented by a scalar stream function.

3 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 ( Virginia Tech Website)

defined so that the divergence is constant except at the pole.

4 Spherical elementary current system (SECS) An example of node distribution 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.

5 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: V() r = r Ψ. We expand the stream function Ψ by using localized basis functions: Thus, Ψ ( r) = wψ ( rr, ). i i Vr ( ) = r Ψ = w r ψ ( rr, ). i i i i

6 Generalization of SECS Defining vector-valued localized basis function: we obtain vrr (, ) = e ψ (, rr), i r i Vr ( ) = w vrr (, ). θ If ψ ( rr, i ) = 2log sin where 2 θ vrr (, i) = eφ, icot. 2 i i i θ r r arccos, R i = 2 This is the original divergence-free SECS basis function.

7 Generalization of SECS We represent the divergence-free velocity field by ( ψ ) Vr () = wivrr (, i), vrr (, i) = r (, rri). i We choose the spherical Gaussian function for ψ : r r i ψ( rr, i ) = exp η 1 exp 2 = η( cosθ 1 ), R I and obtain the following divergence-free basis function i (, i) ( η )exp r r vrr = ri r η 1. 2 R I

8 Kalman filter We assume the temporal evolution of the weights w p( w w ) = N ( αw, Q). k k 1 k 1 The residual component can then be estimated with the following Kalman filter algorithm: Prediction: w P Filtering: = αw kk 1 k 1 k 1 = α P + Q. 2 kk 1 k 1 k 1, w = w + ( P + H R H ) HR ( y H w ), 1 T 1 1 T 1 kk kk 1 kk 1 k k k k k k k k P = ( P + H R H ) 1 T 1 1 kk kk 1 k k k.

9 Estimation w The covariance matrices Q is given so as to satisfy 1 w exp( ξcos λ) w cos. 2 2 T 1 2 k k kq wk = σq λ 2 2 i cos λ λ + λ cos λ φ i i And R is assumed to be diagonal: The parameters σ Q and ξ are determined by maximizing the marginal likelihood: p( y θ) p( y w ) p( w θ) dw. = 2 k R R 1: K k k k k k R = σ I, ( σ = 500). 2

10 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, The nodes of the basis functions were placed at every 5 and 2 degrees in longitude and in latitude, respectively.

11 Reconstruction with proposed functions Estimated velocity distribution Original velocity distribution 0800 UT, Mar. 17, 2015

12 Reconstruction with SECS functions Estimated velocity distribution Original velocity distribution 0800 UT, Mar. 17, 2015

13 Stream function with proposed The estimate with proposed basis funcitons UT, Mar. 17, 2015

14 Stream function with proposed The estimate with proposed basis funcitons UT, Mar. 17, 2015

15 Stream function with proposed The estimate with proposed basis funcitons UT, Mar. 17, 2015

16 Stream function with proposed The estimate with proposed basis funcitons UT, Mar. 17, 2015

17 Stream function with proposed The estimate with proposed basis funcitons UT, Mar. 17, 2015

18 Stream function with proposed The estimate with proposed basis funcitons UT, Mar. 17, 2015

19 Stream function with SECS The estimate with SECS basis funcitons UT, Mar. 17, 2015

20 Stream function with SECS The estimate with SECS basis funcitons UT, Mar. 17, 2015

21 Stream function with SECS The estimate with SECS basis funcitons UT, Mar. 17, 2015

22 Stream function with SECS The estimate with SECS basis funcitons UT, Mar. 17, 2015

23 Stream function with SECS The estimate with SECS basis funcitons UT, Mar. 17, 2015

24 Stream function with SECS The estimate with SECS basis funcitons UT, Mar. 17, 2015

25 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 : w = ζ + β. The model-based value is determined so as to fit an empirical model by Weimber The residual β is estimated with the Kalman filter.

26 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: Prediction: β P p( β β ) = N ( α β, Q). k k 1 k 1 = α β kk 1 k 1 k 1 = α P + Q. 2 kk 1 k 1 k 1, Filtering: β β ( y [ ζ β ], P T T 1 kk = kk 1 + Pkk 1 Hk( HkPkk 1Hk + Rk) k Hk k + k ) = P P H ( H P H + R ) T T 1 kk kk 1 kk 1 k k kk 1 k k k kk 1 H P.

27 17 March 2015

28 Result The estimate with actual data 0800 UT, Mar. 17, 2015

29 Result The estimate with actual data 0810 UT, Mar. 17, 2015

30 Result The estimate with actual data 0820 UT, Mar. 17, 2015

31 Result The estimate with actual data 0830 UT, Mar. 17, 2015

32 Result The estimate with actual data 0840 UT, Mar. 17, 2015

33 Result The estimate with actual data 0850 UT, Mar. 17, 2015

34 27 March 2017

35 Result The estimate with actual data 1000 UT, Mar. 27, 2017

36 Result The estimate with actual data 1010 UT, Mar. 27, 2017

37 Result The estimate with actual data 1020 UT, Mar. 27, 2017

38 Result The estimate with actual data 1030 UT, Mar. 27, 2017

39 Result The estimate with actual data 1040 UT, Mar. 27, 2017

40 Result The estimate with actual data 1050 UT, Mar. 27, 2017

41 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.

Lecture 2: Reconstruction and decomposition of vector fields on the sphere with applications

Lecture 2: Reconstruction and decomposition of vector fields on the sphere with applications 2013 Dolomites Research Week on Approximation : Reconstruction and decomposition of vector fields on the sphere with applications Grady B. Wright Boise State University What's the problem with vector fields