Preprocessing of Vector Magnetograms for Extrapolation of

Coronal Magnetic Fields

General Discussion

• Assumption: force-free:

with

• Force-free condition holds above 400km– Measurements might be affected by non-magnetic

effects– Extrapolation should take into account of plasma

conditions

0BJ

)()()( xBxxJ

BJ

Test case & Reality Low&Lou Bastille Event

Preprocessing

• Problems on real magnetograms– Force-free ?– Noise (lack of smootheness)– Hidden problems?

• Philosophy: – Different mgms, close to the original one– Staying within given errorbars

Preprocessing: Minimization

• Force-freeness [Molodenskii (1969)]

– Magnetic-field Stress Tensor T with ext. forces F

– Force-free condition

0

dsTx ik

s i

ijjiij BBBT

2

2

1

4

1

BB

0 B

FT

ˆ

0F

Preprocessing: Minimization

• Force-freeness– Transforming surface integral

into an integral along contour L

– Since the magnetic field weakens as the diameter of contour L increases, surface integral will approach to zero

– Magnetic field decreases with height

0321

S

k

L

kk dsTz

dxTdyT

S

k constdsT3

S

kdsT 03

Preprocessing: Minimization

• Torque-Freeness– Same procedure, but

multiply vectorially with r

– Then integrate over surface S, the z-component of the resulting equation

– Seperating out the divergence part

– Similar as before

rFT

ˆ

0

S

yzyyyxxzxyxx

z

T

y

T

x

Tx

z

T

y

T

x

Tyds

0

yzxzyyxyyxxx xTyTz

xTyTy

xTyTx

ds

0yzxy xTyTds

How to Preprocess: Minimization

• Force-free

• Torque-free

0 dxdyBBdxdyBB z

surface

yz

surface

x

surface

yxz dxdyBBB 0222

surface

yxz dxdyBBBx 0222

surface

yxz dxdyBBBy 0222

0 dxdyByBBxBsurface

zxzy

Preprocessing: Minimization

• Smootheness

– Current Method: Laplacian 4th order

– Pro: smoothes the Magnetogram, because of averaging the discontinuities

– Contra: flattens the whole Magnetogram

Numerical Expressions

– L* : treshold to control smoothing» L4=0 no curvature

2

222

22

1

mgmyxz

mgmzy

mgmzx BBBBBBBL

24 *),( LyxBL

22

222

2

2222

mgmzxzy

mgmyxz

mgmyxz ByBBxBBBByBBBxL

2

2

/)2,(121)2,(12

1),(25),(3

4),(34

/),2(121),2(12

1),(25),(3

4),(34

),(

yyyyy

xxxxx

hhyxBhyxByxBhyxBhyxB

hyhxByhxByxByhxByhxB

yxB

Normalizations

2

222222

2

2221

)(

)(

mgmzyx

mgmzyx

BBByxLN

BBBLN

2

2

2

4

/)2,(121)2,(12

1),(25),(3

4),(34

/),2(121),2(12

1),(25),(3

4),(34

)(

yyyyy

xxxxx

hhyxBhyxByxBhyxBhyxB

hyhxByhxByxByhxByhxB

LN

Minimizing functional

)()()( 4

44

2

22

1

11 LN

L

LN

L

LN

LL

]1.0...005.0[

1

4

21

)()( 2

22

1

111212 LN

L

LN

LL

Method of Minimization: Simulated Annealing

• Usual Minimizing(Newton-Raphson, etc.) of a function leads to nearest local minimum, depends on starting point

• Current Problem: – functional with several constraints– Several local minima

• Simulated annealing finds also local minima, but leaves a chance to get out of it, in order to find better local minima

How does Simulated Annealing Work?

• Choose randomly new value of Bx,y,z – But within a small interval (~± 0.5%)

• Calculate L, check if LNEW ≤ LOLD– TRUE: Take Bnew

x,y,z & forget Boldx,y,z,

– FALSE: Take with probability ~ exp{-(LNEW - LOLD)}

• Repeat – To scan the whole domain– Domain within given errorbars of the measurement (depends if

transversal or normal component of Magnetogram)

Results: preprocessing just force- & torque-freeness

Quality of force- & torque-freeness

Results: preprocessing just force- & torque-freeness

Magnetogram

Results: preprocessing both constraintsMagnetogram

Results: preprocessing both constraints Quality of force-freeness

Compared Results

 σJ

Low & Lou semi-analytical solution 0

extrapolation of Low & Lou Magnetogram 0.018

Noise introduced 0.214

preprocessed: force- & torquefreeness 0.196

preprocessed: smootheness 0.108

preprocessed: both constraints 0.094

Conclusions

• Extrapolation successful, but on real cases lack of force-freeness (σJ ~ 10-1)

• Preprocessing: – successful to make magnetogram force-free

better agreement with initial assumption

slightly better results, but not enough– Difficulties to smooth without flattening

have to find a better smoothing term!