for the time-varying corona and measurements. State estimation .... l=1 such that the sample mean (Ëxi|j. ) and co- variance (ËPi|j. ) approximate the KF Ìxi|jand ...
DYNAMIC TOMOGRAPHIC IMAGING OF THE SOLAR CORONA Mark D. Butala1, Richard A. Frazin2, Yuguo Chen3, and Farzad Kamalabadi1 Department of Electrical and Computer Engineering1, Department of Statistics3, University of Illinois, USA Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, USA
3 Dynamic Reconstruction
Electron Density: The measured polarized light yi(·) at time index i is related to the unknown electron density xi(·) by Z yi(θi, s) = C dl H(s, s + lθ i) xi(s + lθ i) + vi(θi, s) (1)
General Hidden Markov p(x1) hi(y i|xi) fi(xi+1|xi)
Under the above signal models, dynamic tomographic reconstructions may be computed as minimum mean R b ?i|j , RN xi p(xi|y 1:i) dxi. Each MMSE estimate x b ?i|j is the optimal square error (MMSE) state estimates: x estimate of the N dimensional physical state xi at time index i given the set of data y 1:j , {y 1, . . . , y j }. It is in b ?i|j when the state dimension N is large. general computationally intractable to compute x
y i = Σi Λ xi + v i = H i xi + v i
The PF processes the set of particles {(wil , xli|j )}Ll=1 such that their empirical distribution converges to the posterior p(xi|y 1:j ) as the # of particles L → ∞.
Measurement Update
yi
i.i.d. xl1|0 ∼
px1 (x1 ), 1≤l≤L
w0l
=
If
1 L
L X
L X
wil , xi|i =
l=1
(µli )2
L X
xi|i
l=1
Mk4 2/18/08
b i|i xi|i ≈ x
(xli|i → xli+1|i )
˚ EIT 171 A
˚ EIT 195 A
3.2 Kalman Filter (KF) [5] b i|i ) Measurement Update (b xi|i−1 → x
yi i=2
i=1 b 1|0 = µx1 x
K i = P i|i−1 H Ti H i P i|i−1 H Ti + Ri b i|i = x b i|i−1 b i|i−1 + K i y i − H i x x P i|i = P i|i−1 − K i H i P i|i−1
P 1|0 = Π1
b i+1|i ) Time Update (b xi|i → x
i=i+1
b i|i x
−1
P i|i
b i+1|i = F i x b i|i x P i+1|i = F i P i|i F Ti + Qi
b i+1|i , P i+1|i x
y 1:I = H 1:I x + v 1:I
(5)
where, e.g., y 1:I , (y T1 , . . . , y TI )T . Estimates may then be computed through a constrained, regularized optimization: b = arg min ky 1:I − H 1:I xk22 + λkD xk22. x
The LEnsKF processes an ensemble {xli|j }Ll=1 such that the sample mean (e xi|j ) and coe i|j ) approximate variance (P b i|j and P i|j . the KF x
5
0.8
10
0.6
15
0.4
20
0.2
25
0
1
2.5e8
0.5
solar y (R⊙ )
Solar y
1
Row index
Carington longitude: 17.8 (◦ )
0
5 −0.5
0.5e8 −1
−1 −1
−0.5
0 Solar x
0.5
1
−1
b Density x
−0.5
0 0.5 solar x (R⊙ )
1
b “Smearing” in static x
˚ EIT 171 A
b Emissivity x
Projection index
Projection index
224
i = 256
256
128
256
384
512 Time index i
i = 512
640
768
896
1024
i = 1024
yi i=2
i=1 i.i.d. e l1|0 ∼ x
N (µ1 , Π1 )
1≤l≤L
e li|i ) Measurement Update (e xli|i−1 → x P e li|i−1 , z li|i−1 = x e li|i−1 − x e i|i−1 e i|i−1 = L1 L x l=1 x P L T T l l e i|i−1 )H Ti = [C i ◦ 1 B i , (C i ◦ P l=1 z i|i−1 (z i|i−1 ) ]H i L−1
e i|i x
fi = B i [H i B i + Ri ]−1 K
Time Update
e i+1|i ≈ P i+1|i e i+1|i ≈ x b i+1|i , P x
• The LEnsKF converges to the localized KF (LKF) as the ensemble size L → ∞ and shows the bias introduced by covariance tapering. • The (L)KF is much more computationally demanding and scales more poorly than the LEnsKF.
i.i.d.
l fi (y li − H i xl xli|i = xli|i−1 + K i|i−1 ), y i ∼ N (y i , Ri )
i = i+1
KF, LKF LEnsKF Time Mem. Time Mem. L 40m 2 MB 26s 0.5 MB 128 576d 512 MB 6h 16 MB 256
e li+1|i ) (e xli|i → x
e i|i ≈ x b i|i x e i|i ≈ P i|i P
e li+1|i = F i x e li|i + uli x i.i.d.
uli ∼ N (0, Qi )
6 Conclusions The LEnsKF shows great promise for dynamic tomography applications as demonstrated in the numerical experiment. Scaling the method up for 4D solar tomography will require further developments. Fortunately, the method lends itself to parallelization, an endeavor that shows great preliminary promise.
4 Covariance Tapering
(6)
x≥0
1.0e8
192
3.3 Localized Ensemble Kalman Filter (LEnsKF) [6]
Prior work has assumed the unknown is fixed over I = 14 days
−0.5
160
b i|i x
2 Static Reconstruction
1.5e8
128 Time index i
−: Requires the storage and operation on a massive matrix, i.e., P i|j requires 2 TB of storage when N = 106.
e i|j is the localized estiCi ◦ P mate of P i|j . (◦: the elementby-element matrix product)
˚ EIT 284 A
100
+: Operates under the general hidden Markov model, i.e., nonlinear problems. −: Computationally limited to state dimensions no greater than N ≈ 100, e.g., [3], [4].
+: Computationally tractable for huge state dimension N applications when tapering is applicable.
2.0e8
96
b∞ KF: x i|i
(4)
with T given by a plasma emission model such as CHIANTI [2].
0
64
0
y i = (T ⊗ Σi) xi = H i xi + v i.
50
150
40
i = 128
(i.e., simulate the evolution of xli to xli+1 according to fi (·|xi ))
(2)
Discretizing (3) results in the following: (⊗: Kronecker product)
0.5
30
µli xli
xli+1|i ∼ fi (xli+1 |xli )
b i+1|i xi+1|i ≈ x
Electron Temperature: The observed light at extreme ultraviolet wavelength λ is related to the density of electrons at a given temperature T by the following, where ψ(·) is the emission process: Z Z ∞ yi(θi, s, λ) = C dl dT ψ(λ, T ) xi(s + lθ i, T ) + v(θ i, s, λ). (3) R
High Resolution: N = 1282, M = 184, I = 2048
l=1
i =i+1
The KF is a deterministic algorithm while the PF and LEnsKF are Monte-Carlo algorithms.
3.0e8
20
32
exceeds a threshold then resample.
Time Update
The PF estimates xi|j converge b ?i|j . to the MMSE estimates x
l (xli|i−1 , wi−1 → xli|i , wil )
l wil = wi−1 hi (y i |xli ), µli = wil /
i=2
i=1
The KF computes linear MMSE (LMMSE) state esb i|j which are MMSE optimal under the timates x constraint that they must be an affine function of the data y 1:j .
1
Low resolution: N = 322, M = 46, I = 256 Data: y i
3.1 Particle Filter (PF) [3]
where y i ∈ RM , Σi corresponds to line integrals, the diagonal matrix Λ is the spatially-dependent scattering potential, and xi ∈ RN . Note that N may be large, e.g., dividing the 3D solar atmosphere into 100 × 100 × 100 voxels results in N 6!
Mk4 2/11/08
The performance of the LEnsKF is demonstrated in the following tomography simulation. The N dimensional unknown xi is a highly dynamic random process with complicated motion, where the time index 1 ≤ i ≤ I. Both a low and high sampling of the process are considered. The measurements y i are a set of M parallel line integrals of xi at some angle θi with 1% additive white Gaussian noise. The state transition model is F i = I, i.e., the state evolution is modeled as a random walk. More sophisticated models may be easily incorporated.
10
R
where θ i is the orientation of the observer relative to the Sun, s is the coordinate in the measurement plane, l is the distance along the line of sight, H(·) is a known scattering function [1] that encapsulates the Thomson scattering physics, and vi(·) is instrumentdependent noise. Discretizing (1) results in the linear system
Linear Additive Noise µ , E[x1], Π1 = Var(x1) y i = H ixi + v i, Var(v i) = Ri xi+1 = F ixi + ui, Var(ui) = Qi
Truth: xi
1 Forward Problem
Dynamic Signal Model: Initial prior: Measurement model: State-transition model:
e i|i LEnsKF: x
New developments in data assimilation and statistical estimation theory are necessary to characterize the transient phenomena responsible for space weather. We present a state-space model for the time-varying corona and measurements. State estimation methods may then be used to solve the resultant time-dependent inverse problem. However, such a formulation demands new state-estimation methods that scale well with problem size. We describe a Monte Carlo state estimation algorithm, the localized ensemble Kalman filter (LEnsKF), that results in dramatic reductions in computation for tomographic inverse problems, thus enabling data-assimilative models of the stellar atmosphere.
5 Numerical Experiment
e∞ LKF: x i|i
Overview
10 15 Column index
20
25
• The LEnsKF is computationally tractable only if the the ensemble size L is small. e i|i−1 is a provably poor estimator of P i|i−1 when L is • The sample covariance P e i|i−1 as the estimator of P i|i−1 [7]. small. The LEnsKF instead uses C i ◦ P • For typical problems, error correlations are strongest amongst spatial neighbors. The example taper matrix C i (left) enforces this intuition where the (m, n)th element is close to 1 near the diagonal and 0 when far away.
• Covariance tapering introduces a bias but can greatly reduce the estimator variance. Ultimately, the proper choice of taper matrix enables a reduction in ensemble size L, many additional computational simplifications as a result of the sparsity of C i, and preserves a positive definite covariance matrix estimate.
References [1] H. C. van de Hulst, “The electron density of the solar corona,” Bull. Astron. Inst. Netherlands, vol. 11, no. 410, pp. 135–150, 1950. [2] E. Landi, G. Del Zanna, P. R. Young, K. P. Dere, H. E. Mason, and M. Landini, “CHIANTI - an atomic database for emission lines - VII. New data for X-rays and other improvements,” Astrophys. J. Suppl., vol. 162, pp. 261–280, 2006. [3] A. Doucet, N. de Freitas, and N. Gordon, Eds., Sequential Monte Carlo Methods in Practice.
New York, NY: Springer-Verlag, 2001.
[4] F. Daum and J. Huang, “Curse of dimensionality and particle filters,” in Proc. IEEE Aerospace Conf., vol. 4, Big Sky, Montana, 2003, pp. 1979–1993. [5] B. D. O. Anderson and J. B. Moore, Optimal Filtering.
Englewood Cliffs, New Jersey: Prentice-Hall, 1979.
[6] G. Evensen, “Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics,” J. Geophys. Res., vol. 99, pp. 10 143–10 162, 1994. [7] P. L. Houtekamer and H. L. Mitchell, “A sequential ensemble Kalman filter for atmospheric data assimilation,” Mon. Weather Rev., vol. 129, pp. 123–137, 2001.
