The art of Stokes inversions
Luis R. Bellot Rubio
Instituto de Astrofísica de Andalucía (CSIC), Spain
National Astronomical Observatory, Japan

NAOJ, 17 July 2007

Luis R. Bellot Rubio

Outline• What is an inversion technique?
• How do they work?
• ME inversions• Accounting for asymmetric Stokes profiles• Be careful with the choice of model atmosphere!
• Available codes• Tips and tricks• Stray-light considerations• Running SIR– Input files– Visualization of results

NAOJ, 17 July 2007

Luis R. Bellot Rubio

What is an inversion technique?
• Any method used to infer the physical conditions of theatmosphere from the interpretation of Stokes profiles– Center-of-gravity method, bisector analyses, …
– Forward modeling– PCA, artificial neural networks– Least-squares fitting• What to expect: a model atmosphere capable of reproducing
the observations…. nothing else!

NAOJ, 17 July 2007

Luis R. Bellot Rubio

Radiative transfer• The Stokes parameters obey the RTE⎛ I ⎞ ⎛ ηI⎜ ⎟ ⎜d ⎜ Q ⎟ ⎜ηQ=⎜η U⎜⎟dτ U⎜ ⎟ ⎜⎝ V ⎠ ⎝η VηQ
ηI− ρVρUηUηV ⎞ ⎛ I − S ⎞⎟ ⎜⎟ρ V − ρU ⎟ ⎜ Q ⎟ηIρQ ⎟ ⎜ U ⎟⎟ ⎜⎟− ρ Q ηI ⎠ ⎝ V ⎠(Unno 1956;Rachkovsky 1962)• ηI,Q,U,V and ρQ,U,V depend on a ≡ (B, γ, χ, vLOS, T, Pe, vmic)• This means that
– Four Stokes parameters needed to understand just one of them
– Proper interpretations of the Stokes vector require a good knowledgeof the atmosphere (a)NAOJ, 17 July 2007

Luis R. Bellot Rubio

Least-square inversions• The complete line transfer problem has to be solved• Self-consistent inferences → Inversion techniques
INITIAL MODELATMOSPHERE

SPECTRALSYNTHESIS
Nonlinear, least-squares fit

OBSERVED AND
SYNTHETIC PROFILESPERTURBEDMODELATMOSPHERE––––No simplifying assumptions
Full Stokes vector fitted
Complex model atmospheres
All atmospheric parametersinferred at the same time

FINAL MODELATMOSPHERE

NAOJ, 17 July 2007

Luis R. Bellot Rubio

How do they work?
• Inversion driven by χ2-minimization:

χ 2 (a) = ∑ [I obs (λi ) − I syn (λi , a)]2
• Linearization: Levenberg-Marquardt algorithm∇χ 2 (a) + A( χ 2 ) ⋅ δ a = 0• Keeping the number of free parameters small:– Atmospheric parameters perturbed in coarse grid
– Full stratifications in finer grid by cubic spline interpolation• Regularization techniques (when in doubt, smooth)– Penalty function, or– Modified SVD method (Ruiz Cobo & del Toro Iniesta 1992)
NAOJ, 17 July 2007

Luis R. Bellot Rubio

Inversions based on ME atmospheres• ME atmosphere:– Source function is linear with optical depth– Absorption matrix does not vary with optical depth• Analytical Stokes profiles• Fast inversion• Smooth maps of physical quantities• Results are relatively accurate and easy to interpret

NAOJ, 17 July 2007

Luis R. Bellot Rubio

ME inversions of high spatial resolution profiles
MHD simulations (Vögler et al. 2005)+• Fe I 630.1 and 630.2 nm
profiles degraded toHINODE/SP pixel size

NAOJ, 17 July 2007

Luis R. Bellot Rubio

ME inversions of high spatial resolution profiles
MHD simulations (Vögler et al. 2005)+• Fe I 630.1 and 630.2 nm
profiles degraded toHINODE/SP pixel size
• Maps of inferred B and vLOS
very similar to real ones!
NAOJ, 17 July 2007

Orozco Suárez et al. 2007, ApJL
Luis R. Bellot Rubio

Inversions based on ME atmospheres• ME atmosphere:– Source function is linear with optical depth– Absorption matrix does not vary with optical depth• Analytical Stokes profiles• Fast inversion• Smooth maps of physical quantities• Results are easy to interpret• Simplistic treatment of radiation transfer• Little thermal information. No height variations• Cannot account for asymmetric Stokes profiles

NAOJ, 17 July 2007

Luis R. Bellot Rubio

Asymmetric Stokes profiles
0.020.01
0.00
-0.01
-0.02
-0.03
-0.04
00.04

Stokes V/IQS

204060QS80

204060QS

8020406080

0.02++++0.00
-0.02
-0.04

++-0.06
00.02
0.00• KIS/IAA Visible Imaging Polarimeter + TESOS + KAOS
• VTT, Observatorio del Teide• Spatial resolution: ∼0.4"• Pore near disk center, Fe I 630.15 and 630.25 nm
NAOJ, 17 July 2007

-0.02
-0.04
0

Bellot Rubio et al. (2007)
Luis R. Bellot Rubio

ME inversions of asymmetric profiles
Stokes V/IQS
MHD simulations (Vögler et al. 2005)

ME inv+ME inv+
• Stokes profiles not well fitted• ME results are some kind of“average” of physicalparameters along the LOS

NAOJ, 17 July 2007

Luis R. Bellot Rubio

The origin of asymmetries
Amplitude asymmetry/
Multi-lobed Stokes profiles

Different magnetic atmospheres
coexisting in resolution element

Area asymmetry

Gradients/discontinuities ofphysical parameters along LOS
Auer & Heasley (1978)

RF of Stokes V to B

Cabrera Solana et al. (2005)

NAOJ, 17 July 2007

The area asymmetry gives
information on the height variationof atmospheric parametersLuis R. Bellot Rubio

Accounting for asymmetries• Inversion codes capable of dealing with asymmetries– Are based on numerical solution of RTE– Provide reliable thermal information– Use less free parameters than ME codes– Infer stratifications of physical parameters with depth

NAOJ, 17 July 2007

Luis R. Bellot Rubio

Accounting for asymmetries
1.0

Stokes I/IQS

0.8
0.6
0.4
0.2
020

40

60

80

0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.04
0Temperature [kK]7.5
7.0

NAOJ, 17 July 2007

80

Field strength [kG]0.4
0.3-3-2-10LOS velocity [km/s]

0.2-414521400135-2130-3-2-1
log tau0-3-2-10Field inclination [deg]4-4-460

0.5

5.5• VIP + TESOS + KAOS
• SIR with 10 free parameters• Bellot Rubio et al. (2007)

40

0.6

6.0
5.0
4.5-4

20

0.7

6.5+Stokes V/IQS

125-4-3-2-1
log tau

0Luis R. Bellot Rubio

Inversions with gradients
LOS velocity at log τ500 = 0 [km s-1]

Temperature at log τ500 = 0 [K]
60006200

6400-2-1

0Center-side penumbra

1I/Ic_QS

5800

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1-40

4400

4600

4800

-200∆λ [pm]2040-40-20020

40

LOS velocity at log τ500 = -2 [km s-1]

Temperature at log τ500 = -2 [K]4200

Limb-side penumbra

0.8-0.5

0.0

0.5

1.0

• Spatial resolution ∼ 0.5"
• Intensity profiles of Fe I 557.6 nm• SIR with 7 free parameters• Thermal/kinematic structure of AR 0019
at different heights in the photosphere
Bellot Rubio, Schlichenmaier, & Tritschler 2006, A&A 453, 1117
NAOJ, 17 July 2007

Luis R. Bellot RubioBe careful with the atmospheric model!
Inversion results change if the physical model is changed
Models are often simplistic and do not describe the real atmosphereB [kG]

2.5
2.0

Flux tube model

1.5
1.0
0.5
0.0-5 -4 -3 -2 -1 0log tau

12Fe I 630.25 nm

Magnetic flux tubes in facular regions

NAOJ, 17 July 2007

0.10
0.05

0.8

V/IQS

I/IQS

1.0
0.9
0.7
0.6
0.5-400 -200
0200
Lambda [mA]0.00-0.05

400-0.10-400 -200
0200
Lambda [mA]

400Luis R. Bellot RubioBe careful with the atmospheric model!
Inversion results change if the physical model is changed
Models are often simplistic and do not describe the real atmosphereB [kG]

2.5
2.0

Flux tube model
Best-fit 1C model

1.5
1.0
0.5
0.0-5 -4 -3 -2 -1 0log tau

12Fe I 630.25 nm

Magnetic flux tubes in facular regions

NAOJ, 17 July 2007

0.10
0.05

0.8

V/IQS

I/IQS

1.0
0.9
0.7
0.6
0.5-400 -200
0200
Lambda [mA]

0.00-0.05

400-0.10-400 -200
0200
Lambda [mA]

400Luis R. Bellot RubioBe careful with the atmospheric model!
Inversion results change if the physical model is changed
Models are often simplistic and do not describe the real atmosphereB [kG]

2.5
2.0

Flux tube model
Best-fit 1C model

1.5
1.0
0.5
0.0-5 -4 -3 -2 -1 0log tau

12

Fe I 630.25 nm

Magnetic flux tubes in facular regions

NAOJ, 17 July 2007

0.10
0.05

0.8

V/IQS

I/IQS

1.00.90.7
0.6
0.5-400 -200
0200Lambda [mA]

0.00
-0.05

400-0.10-400 -200
0200
Lambda [mA]

400

Luis R. Bellot Rubio

Available codes for inversions with gradients
SIRRuiz Cobo &del Toro Iniesta (1992)

1C & 2C atmospheres, arbitrary
stratifications, any photospheric line

SIR/FT

Bellot Rubio et al. (1996)

Flux tube model, arbitrary stratifications,any photospheric line

SIR/NLTE

Socas-Navarro et al. (1998)

NLTE line transfer, arbitrary stratifications

SIR/GAUS

Bellot Rubio (2003)

Uncombed penumbral model, arbitrary
stratifications

SIR/JUMP

Bellot Rubio (2007)

Canopy-like atmospheres

SPINOR

Frutiger & Solanki (2001)

1C & 2C atmospheres, arbitrary
stratifications, any photospheric line,
molecular lines, flux tube model,
uncombed model

LILIA

Socas-Navarro (2001)

1C atmospheres, arbitrary stratifications

MISMA IC

Sánchez Almeida (1997)

MISMA model, arbitrary stratifications,any photospheric line

NAOJ, 17 July 2007

Luis R. Bellot Rubio

Tips and tricks• First of all, look at the profiles• Try a ME inversion, it usually works– If the V profiles are very asymmetric, fit only I, Q, and U• Examine the fits: are they reasonably good?
• Identify– Pixels with bad fits and/or large asymmetries– Regions where interesting physical processes occur• Run SIR inversions on these pixels– Which model are you going to use?
1C model, 2C model, flux tube model, uncombed model?
– Use ME results as initialization– Give more weight to the strangest Stokes parameter– Do it simple! See if linear stratifications (2 nodes) are sufficient– If not, increase the number of nodes gradually• Ask yourself if the retrieved model atmosphere makes sense!!
NAOJ, 17 July 2007

Luis R. Bellot Rubio

1C SIR inversion of Hinode/SP data

NAOJ, 17 July 2007

Luis R. Bellot Rubio

1C SIR inversion of Hinode/SP data++NAOJ, 17 July 2007

Luis R. Bellot Rubio

Tips and tricks• First of all, look at the profiles• Try a ME inversion, it usually works– If the V profiles are very asymmetric, fit only I, Q, and U• Examine the fits: are they reasonably good?
• Identify– Pixels with bad fits and/or large asymmetries– Regions where interesting physical processes occur• Run SIR inversions on these pixels– Which model are you going to use?
1C model, 2C model, flux tube model, uncombed model?
– Use ME results as initialization (with f=1?)– Give more weight to the strangest Stokes parameter– Do it simple! See if linear stratifications (2 nodes) are sufficient– If not, increase the number of nodes gradually• Ask yourself if the retrieved model atmosphere makes sense!!
NAOJ, 17 July 2007

Luis R. Bellot Rubio

Uncombed inversion of penumbral profiles
SIR/GAUS inversion of simultaneous visible and IR observations
Fe I 630.25 nm

Fe I 1564.8 nm

Fe I 1565.2 nm

Stokes V/Iqs

Fe I 630.15 nm

Beck et al. (in prep)
NAOJ, 17 July 2007

Luis R. Bellot Rubio

Uncombed inversion of penumbral profiles
Temp

Best-fit uncombed modelγ

Visible and IR profiles observed at
1" are compatible with flux tubes
embedded in a more vertical andstronger background field

Stokes V/Iqs

BvLOS

Beck et al. (in prep)
NAOJ, 17 July 2007

Luis R. Bellot Rubio

Tips and tricks• First of all, look at the profiles• Try a ME inversion, it usually works– If the V profiles are very asymmetric, fit only I, Q, and U• Examine the fits: are they reasonably good?
• Identify– Pixels with bad fits and/or large asymmetries– Regions where interesting physical processes occur• Run SIR inversions on these pixels– Which model are you going to use?
1C model, 2C model, flux tube model, uncombed model?
– Use ME results as initialization– Give more weight to the strangest Stokes parameter– Keep it simple! See if linear stratifications (2 nodes) are sufficient• Ask yourself if the retrieved model atmosphere makes sense!!
• You have experts in-house: ask them for advice (Jan-san)NAOJ, 17 July 2007

Luis R. Bellot Rubio

Running SIR: input files• Input files

PROFILE FILE

Lineindex∆λ[mA] I/Iqs Q/Iqs U/Iqs V/Iqs

lperfiles.pro
wperfiles.pro
NAOJ, 17 July 2007

Luis R. Bellot Rubio

Running SIR: input files
WAVELENGTH GRID FILE

NAOJ, 17 July 2007

Luis R. Bellot Rubio

Running SIR: input files
ATOMIC PARAMETER FILE

Line
index Atom λNAOJ, 17 July 2007

E χlog gf transition

Luis R. Bellot Rubio

Running SIR: input files

MODEL FILE

leemod.pro
escribemod.pro
modelador2.x
equilibrium.x
geometrical.x
NAOJ, 17 July 2007log τTPevmic

BvLOSγϕz [km]Pg

ρLuis R. Bellot Rubio

Running SIR/GAUSS: input files
GAUSSIAN FILE

NAOJ, 17 July 2007

Luis R. Bellot Rubio

Concept of nodes• Keeping the number of free parameters small:

Temperature– Atmospheric parameters perturbed in coarse grid (nodes)– Full stratifications in finer grid by cubic spline interpolation

●

●

●

★

●

●

●

●

★

★

▲

-5●

▲

-3-1Location of 8 nodes
Location of 3 nodes
Location of 2 nodes

1log τ
NAOJ, 17 July 2007

Luis R. Bellot Rubio

Stray-light considerations• Stray-light in ME inversions:– Equivalent to a magnetic filling factor f = 1- α

• Stray-light in SIR inversions:– It is NOT equivalent to a magnetic filling factor– In fact, SIR has two free parameters: α and f• Global vs local stray-light profile– Classical treatment: global stray-light profile (over FOV)
– Orozco Suárez et al. (2007): local stray-light profile accounts
for telescope diffraction

NAOJ, 17 July 2007

Luis R. Bellot RubioLocal stray light• Telescope diffraction reduces the polarization signal• Important for weak fields and noisy signals!

Unpolarized stray light
Average over a box 1″wide centered on the
pixel of interest
NAOJ, 17 July 2007

Luis R. Bellot RubioExecuting the inversion

echo sir.trol | sir.x

NAOJ, 17 July 2007

Luis R. Bellot Rubio

Visualizing SIR results: graphics.pro

Model atmospheres

Stokes profiles

NAOJ, 17 July 2007

Luis R. Bellot Rubio