Chloé Guennou1, F. Auchère2, K. Bocchialini2, J.A. Klimchuk3

1Instituto de Astrofisica de Canarias, Spain2Institut d'Astrophysique Spatiale, France3Nasa Goddard Space Flight Center, MD, USA

On the accuracy of the DEM inversion problem

ISSI meeting   

Chloé Guennou11/05/2015The DEM inversion  problem 2/17

Corona Optically thin plasma. Intensity observed in a band→ b

Temperature response function

Contribution function for each emission lines

Population equilibrium + Ionization Equilibrium + Abundance + Abundance H/e-

Einstein coefficient spontaneous emission

Sb : Instrument sensibility

UV Coronal observations

Chloé Guennou11/05/2015The DEM inversion  problem 3/17

The DEM: a powerful but complex tool

  → : mean density weighted by the inverse of the temperature gradient

Te

Te + dTe

dp(Te)dp(Te)

(Craig & Brown, 1978)

n02

Be careful with DEM interpretation!e.g. regions with strong temperature gradient do not contribute a lot

Chloé Guennou11/05/2015The DEM inversion  problem 4/17

Simple scheme : 1. → Simulations of observations + errors by Monte-Carlo2. Inversion process→

3. Comparison → inversion results/Observations

Uncertainties

P  sol

(Te)

sol

P

Te

Simulation of the inversion process

How to test DEM inversion robustness

Chloé Guennou11/05/2015The DEM inversion  problem 5/17

Temperature responsefunction Rb(Te)

DEMs models(k1, k2, …, kM)

« True » Plasma DEM P

Random errors nb Photon and detection noise

Systematic errors sb Atomic + calibration

Intensities space (k1, k2, …, kM)I bth « Observed » Intensities (P)

Results → I = argmin (C())

I b0 (P)

I bobs

(k1, k2, …, kM) I b0I b0spaceI b

0

C (ξ)=∑b=1

N b

( I bobs

−I bth(k1, k 2, ... , kM )σunc

)2

NN

Robustness Analysis

NN

N

Chloé Guennou11/05/2015The DEM inversion  problem 6/17

: Likelihood function

: Probability a posteriori (Bayes' theorem)

Probability of all the solutions = {

11 

2

sol, ...}

consistent with the data + errors, knowing the initial DEM P

→ Can't be used on « real data »

Probability of all the DEMs P consistent with a

given inversion result

→ Can be used on « real data » Guennou et al. (2012a) and (2012b)

A probabilistic approach

N Results → I = argmin (C())

(Te)

sol

P

Te

Chloé Guennou11/05/2015The DEM inversion  problem 7/17

A very simple example

AIA: EUV imager with 6 coronal narrow band channels : 94, 131, 171, 193, 211, 335 Å

Simplest case = Plasmas isothermes

→

104 105 106 107 108

10-25

10-24

10-26

10-27AIA

tem

pera

ture

resp

onse

func

tions

[D

N.c

m5 .s

-1]

Temperature [K]

Random nb Systematics sbReadnoise 21e- RMS Calibration 25 %

Shotnoise Atomic physics 25%

Chloé Guennou11/05/2015The DEM inversion  problem 8/17

AIA – 3 bandes – 171/193/211

Probability of the inversion

result, knowing the “True” one

“Measured” temperature

“True” temperature

Chloé Guennou11/05/2015The DEM inversion  problem 9/17

AIA – 6 bandes

Probability of the inversion

result, knowing the “True” one

“Measured” temperature

“True” temperature

Chloé Guennou11/05/2015The DEM inversion  problem 10/17

Application: DEM & Coronal heating

Can the DEM constrain the timescale of the energy deposition in the solar corona ? 

Active Regions DEMs DEM → T (e.g. Jordan 1969, Warren et al. 2011, Schmelz 2012, Winebarger et al. 2012, )

Tripathi, Klimchuk & Mason, (2011)

DEM Slope

→ Indication of the cold/warm material ratio

→ Timescale of the energy deposition in the solar corona

What is the confidence level on the reconstructed slope ?

Chloé Guennou11/05/2015The DEM inversion  problem 11/17

➢ Abundance: 10% à 20 % depending on elements ➢ Ionization equilibrium : 10% ➢ Atomic parameters : 10%

Hinode/EIS set of 30 lines of elements Fe, Si, Mg, S, Ca, Ar → large temperature coverage [105.2: 106.9] K

~ 22-27% Total uncertainties

Spectrometer Isolated emission lines Detailed treatment of uncertainties : → →

Parametric models of ARs DEMs =Power law + half a gaussian

p

Framework : Hinode/EIS

Chloé Guennou11/05/2015The DEM inversion  problem 12/17

log TcP =6.8 

2

3

4

5

6

1

α I

Probability of the true slope knowing the

measured one

“True” slope

“Mea

sure

d” s

lope

Probability distribution of the solutions

αP = 3.2 ± 0.6

αP = 4.9 ± 0.7

Slope probability map

Chloé Guennou11/05/2015The DEM inversion  problem 13/17

2

3

4

5

6

1

α I

log TcP =6.5 

… Slope probability map

αP = 3.42 ± 1.17

αP = 4.95 ± 1.07

Probability of the true slope knowing the

measured one

“True” slope

“Mea

sure

d” s

lope

Chloé Guennou11/05/2015The DEM inversion  problem 14/17

Tp = 106.8 K

Tp = 106.5 K

EIS lines

Chloé Guennou11/05/2015The DEM inversion  problem 15/17

Slope standard deviation

Summary of the results

Chloé Guennou11/05/2015The DEM inversion  problem 16/17

EIS DEM capabilities : Robust reconstruction of isothermal plasma→

→ Difficulty to constrain the timescale of heating events

0.81 ≤ 2.56≤

Schmelz & Pathak (2012)Tripathi, Klimchuk, & Mason (2011)Warren, Brooks, & Winebarger (2011)Warren, Winebarger, & Brooks (2012)Winebarger et al. (2011)

Model of low frequency nanofares →

36% consistent

77% consistent

0% consistent

Δ = ± 1.0

12

22 Active Region Cores (inter-moss regions)

Bradshaw et al. (2012)

≤ 2.0 2.0 < ≤ 2.5 2.5 < ≤ 3.0 3.0 < ≤ 3.5 3.5

3 5 3 6 5Δ 11 6 2 2 1Δ 3 5 14

Comparison Observations/Modèles

Chloé Guennou11/05/2015The DEM inversion  problem 17/17

Going further ...

- Adopt more detailed uncertainties, i.e. different from each line

→ Atomic physics experts?

- Include correlations between each uncertainties sources (e.g. ionization equilibrium, …)

- Test this robustness measurement technique with the new spectrometer Solar Orbiter/Spice

→ Determine the optimal set of lines for various type of coronal structures → Determine the optimal observational parameters

Thanks!